A toy model for the Drinfeld-Lafforgue shtuka construction

arXiv:1908.05420v5 [math.AG] 7 Feb 2022 A TOY MODEL FOR THE DRINFELD-LAFFORGUE SHTUKA CONSTRUCTION D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Abstract. The goal of this paper is to provide a categorical framework that leads to the definition of shtukas à la Drinfeld and of excursion operators à la V. Lafforgue. We take as the point of departure the Hecke action of Rep(Ǧ) on the category Shv(BunG ) of sheaves on BunG , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace. Contents Introduction 0.1. Hecke action 0.2. What is done in this paper? 0.3. Organization of the paper 0.4. Notation and conventions 0.5. Acknowledgements 1. Symmetric monoidal categories integrated over a space 1.1. The integral 1.2. The tensor product A⊗Y as a colimit 1.3. Proof of Theorem 1.2.4 1.4. The category of local systems 1.5. Digression: the stack of G-local systems 1.6. Proof of Theorem 1.5.8 1.7. Functors out of A⊗Y as diagrams parameterized by finite sets 1.8. Objects in and functors out of A⊗Y 2. Excursions 2.1. Description of A⊗Y via the fundamental group 2.2. Proof of Proposition 2.1.5 2.3. Digression: affine functors 2.4. Endomorphisms of the unit object in A⊗Y as a colimit 2.5. Proof of Proposition 2.4.6 2.6. Applying the paradigm 2.7. Endomorphisms of the unit, term-wise 2.8. Action on a module via excursions 2.9. Proof of Theorem 2.8.7 3. Taking the trace 3.1. The usual trace 3.2. Trace in a 2-category 3.3. Properties of the 2-categorical trace 3.4. Trace on DG categories 3.5. Examples 3.6. Trace on DG 2-categories 3.7. The 2-categorical trace and (categorical) Hochschild chains 3.8. The 2-categorical class map Date: February 8, 2022. 1 2 3 3 8 11 12 13 13 14 15 17 20 22 23 25 27 27 28 29 31 32 34 35 38 41 42 42 43 46 47 49 53 56 57 2 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 3.9. A framework for the proof of Theorem 3.8.5 3.10. Proof of Theorem 3.8.5: isomorphism of the underlying objects of Vect 3.11. Proof of Theorem 3.8.5: algebra and module structure 3.12. A more elementary proof of Theorem 3.8.5 4. A few mind-twisters 4.1. The class of a class 4.2. Proof of Proposition 4.1.6 4.3. The “trivial” case and excursions 4.4. Introducing observables 4.5. Cyclicity and observables 5. The “shtuka” construction 5.1. The universal shtuka 5.2. Explicit description of I-legged shtukas 5.3. Partial Frobeniuses 5.4. Description of the action via excursions 5.5. The “S=T” identity, vacuum case 5.6. The “S=T” identity, general case Appendix A. Sheaves and singular support A.1. Sheaf-theoretic contexts A.2. Sheaves on a product A.3. Singular support A.4. Proof of Theorem A.3.9 in case (a) A.5. Proof of Theorem A.3.9 in the ind-constructible contexts Appendix B. Spectral action in the context of Geometric Langlands (after [NY]) B.1. The players B.2. The Hecke action B.3. Digression: naive geometric Satake B.4. Hecke action on the subcategory with nilpotent singular support B.5. Proof of Theorem B.4.2 B.6. Preservation of nilpotence of singular support Appendix C. Integrated actions in the context of D-modules C.1. Definition of integrated action C.2. Lisse actions C.3. A spectral characterization of lisse actions Appendix D. The notion of universal local acyclicity (ULA) D.1. The abstract ULA property D.2. Adding self-duality D.3. The ULA condition in the geometric situation D.4. An aside: the notion of ULA in other sheaf-theoretic contexts D.5. The ULA property for D-modules References 59 61 63 65 68 68 69 72 74 77 79 79 81 82 83 84 85 87 87 90 92 94 97 100 100 101 103 104 105 106 111 111 112 114 116 116 117 119 120 123 127 Introduction Our goal is to provide a categorical framework that leads to the definition of shtukas à la Drinfeld and of excursion operators à la V. Lafforgue. We will capture the main ingredients of V. Lafforgue’s construction, which are: –The action of the algebra of functions on the stack of arithmetic local systems on the space of automorphic functions; –The “S=T” identity. A TOY MODEL FOR SHTUKA 3 However, all of this will be performed in a toy setting: the key technical(?) difficulty in V. Lafforgue’s work is that the sheaf-theoretic context he needs is that of ℓ-adic sheaves on schemes over Fq . By contrast, we will work in the topological context in the spirit of [BN1]. The ℓ-adic context, which gives rise to actual shtukas, will be considered in subsequent work. 0.1. Hecke action. The point of view taken in this paper is that the geometric ingredient that gives rise to the Drinfeld-Lafforgue construction is the categorical Hecke action. In this subsection we will specify what we mean by this. 0.1.1. We will consider the following three geometric/sheaf-theoretic contexts: –ℓ-adic sheaves on schemes over any ground field k; –Sheaves (in the classical topology) with coefficients in a commutative ring e on schemes over C; –D-modules on schemes over a ground field k of characteristic 0. For a scheme/stack Y, let Shv(Y) denote the corresponding category of sheaves; this is a DG category over our field of coefficients (i.e., over Qℓ , e and k, respectively). 0.1.2. Let X be an algebraic curve and G a reductive group (over our ground field). Let BunG denote the stack of principal G-bundles on X. Let Ǧ be the Langlands dual group of G, which is a reductive group over our ring of coefficients. The point of departure is the Hecke action on Shv(BunG ) of the symmetric monoidal category Rep(Ǧ) integrated over X. This is not a completely straightforward notion, and we refer the reader to Sect. 0.2.1 below or Sect. 1.7.3 for a detailed discussion. Here is what this action gives us. 0.1.3. In the context of D-modules, (the rather non-trivial) result of V. Drinfeld and the first-named author (recorded in [Ga1, Corollary 4.5.5]) says that this action gives rise to an action of the category QCoh(LocSysǦ (X)) on Shv(BunG ), where LocSysǦ is the stack of de Rham local systems on X with respect to Ǧ. 0.1.4. In the context of sheaves on the classical topology, Theorem 6.3.5 of [NY] says that this action gives rise to an action of QCoh(LocSysǦ (X)) on ShvNilp (BunG ), where: –LocSysǦ (X) is the stack of Betti local systems on X with respect to Ǧ; –ShvNilp (BunG ) ⊂ Shv(BunG ) is the full subcategory consisting of sheaves with nilpotent singular support. For the reader’s convenience, we will review the construction of this action in Sect. B. We should remark that unlike the D-module context, the action of the Betti QCoh(LocSysǦ (X)) on ShvNilp (BunG ) is obtained relatively easily from what we state in the present paper as Theorem 1.5.5 (which in itself is not a difficult assertion either), combined with the key observation of [NY, Theorem 6.1.1] about the behavior of singular support. 0.1.5. Our main interest, however, is when the sheaf-theoretic context is that of ℓ-adic sheaves. The first conceptual difficulty in this case is that we do not have a direct analog of LocSysǦ (X) as an algebrogeometric object (over Ql ), so we cannot talk about an action of QCoh(LocSysǦ (X)) on Shv(BunG ). Nevertheless, the situation is not as hopeless as it might seem, and we will discuss it in detail in a subsequent publication. 0.2. What is done in this paper? We will now outline the actual mathematical contents of the present paper. Each of the steps we perform is a toy (more precisely, Betti) analog of what one wishes to be able to do in the context of ℓ-adic sheaves. 4 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 0.2.1. Let A be a symmetric monoidal DG category, and let Y be a space, i.e., Y is an object of the category Spc, see Sect. 0.4.1 (we can also think of Y as a homotopy type). To this data one can associate a new symmetric monoidal category A⊗Y , see Sect. 1.2. Sometimes one also uses the notation1 Z A := A⊗Y . Y For example, if Y is a finite set I, we have A⊗Y = A⊗I , i.e., the usual I-fold tensor product of copies of A. For a general Y , the construction can be described by a colimit procedure off the finite set case. For example, we show (see Theorem 1.5.5) that for A = Rep(G) (here G is an algebraic group), under some conditions2, we have a canonical equivalence (0.1) Rep(G)⊗Y ≃ QCoh(LocSysG (Y )), where LocSysG (Y ) is the (derived) stack classifying G-local systems on Y , i.e., its value on a test affine derived scheme S is the space of symmetric monoidal right t-exact functors Rep(G) → QCoh(S), parameterized by Y . Our point of departure is a DG category M, equipped with an action of A⊗Y as a monoidal category. We emphasize that this is not the same as a family of monoidal actions of A on M, parameterized by Y , see Remark 1.7.4. We give an explicit description of tion 1.7.2). Namely, it is equivalent to on M, parameterized by points of Y I . on Shv(BunG ) is given in exactly such what the datum of such an action amounts to (see Proposia compatible family of actions, one for each finite set I, of A⊗I This description is useful, because the Hecke action of Rep(Ǧ) form, see Sect. B.2. Similarly, we show that the datum of a functor from A⊗Y to some DG category C is equivalent to a compatible family of functors, one for each finite set I, SI : Y I × A⊗I → C. This description is useful as it will explain the connection between the universal shtuka and I-legged shtukas, see Sect. 5.2. 0.2.2. Next, we provide the general framework for excursion operators. Let A and Y be as before, and let SY : A⊗Y → C be a functor of DG categories. Consider the object SY (1A⊗Y ) ∈ C. It carries an action of the (commutative) algebra End A⊗Y (1A⊗Y ). We give an explicit description of this action, in the spirit of [Laf, Sect. 9]. First, we give an explicit description of the algebra End A⊗Y (1A⊗Y ) as a colimit, in the case when Y is connected, and A has an affine diagonal, see Corollary 2.4.4. Choose a base point y ∈ Y . The index category in the colimit in question is that of pairs (I, γ I ), where I is a finite set, and γ I is an I-tuple of loops in Y based at y. (Note that this category is sifted, so the colimit in the category of commutative algebras is the same as the colimit of underlying associative algebras and is also the same as the colimit of the underlying vector spaces.) The terms of the colimit are described as follows. The term corresponding to a finite set I is given by (0.2) Hom A (1A , multI+ ◦ multR I+ (1A )), where: 1Yet another name for A⊗Y is “chiral (or factorization) homology of A along Y ”. 2Specifically, if Y has finitely many connected components and the ring of coefficients e contains Q. A TOY MODEL FOR SHTUKA 5 –I+ = I ⊔ {∗}; –For a finite set J, we denote by multJ the tensor product map A⊗J → A; –multR J denotes the right adjoint functor of multJ . The algebra structure on (0.2) comes from the right-lax symmetric monoidal structure on multR J, obtained by adjunction from the symmetric monoidal structure on multJ . 0.2.3. Next, given a functor SY : A⊗Y → C, we show how each term (0.2) acts on SY (1A⊗Y ), see Theorem 2.8.7. Namely, given ξ ∈ Hom A (1A , multI+ ◦ multR I+ (1A )), the corresponding endomorphism of SY (1A⊗Y ) is the excursion operator : ξ −−−−→ S{∗} (y, 1A ) − −−−−→ S{∗} (y, multI+ ◦(multI+ )R (1A )) SY (1A⊗Y ) − ∼  ∼ y SI+ (y I+ , (multI+ )R (1A ))  mon I+ y γ ∼ counit SI+ (y I+ , (multI+ )R (1A ))  ∼ y −−−−− S{∗} (y, 1A ) ← −−−−− S{∗} (y, multI+ ◦(multI+ )R (1A )), SY (1A⊗Y ) ← where –γ I+ = (γ I , γtriv ); –For a finite set J, and a J-tuple γ J of loops in Y based at y, viewed as a loop into Y J based at y J , we denote by monγ J the corresponding automorphism of SJ (y J , −). In the particular case of A = Rep(G), from the colimit expression of Sect. 0.2.2 and (0.1), we obtain an explicit description of the algebra Γ(LocSysG (Y ), OLocSysG (Y ) ) in terms excursion operators. This recovers the analogs of the formulas from [Laf, Sect. 10 and Proposition 11.7]. 0.2.4. We now come to the next main ingredient of this paper, namely the notion of categorical trace. First, we recall that given a symmetric monoidal category O and a dualizable object o ∈ O equipped with an endomorphism F , we can assign to this data a point Tr(F, o) ∈ EndO (1O ), called the trace of F on o, see Sect. 3.1.1. Suppose now that O is actually a symmetric monoidal 2-category (i.e., we have not necessarily invertible 2-morphisms). Let us be given a pair of dualizable objects o1 , o2 ∈ O, each equipped with an endomorphism Fi , i = 1, 2. Assume in addition that we are given a 1-morphism t : o1 → o2 that admits a right adjoint. Finally, suppose that t intertwines F1 and F2 , up to a not necessarily invertible 2-morphism α, i.e., α t ◦ F1 → F2 ◦ t. We show that to this data there corresponds a 2-morphism Tr(t, α) : Tr(F1 , o1 ) → Tr(F2 , o2 ). In fact, the 2-morphism Tr(t, α) can be explicitly described as (0.3) where: Tr(F1 , o1 ) → Tr(tR ◦ t ◦ F1 , o1 ) ≃ Tr(t ◦ F1 ◦ tR , o2 ) → Tr(F2 ◦ t ◦ tR , o2 ) → Tr(F2 , o2 ), 6 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY –tR denotes the right adjoint of t; –The first arrow is induced by the unit of the adjunction ido1 → tR ◦ t; –The second arrow is given by the cyclicity property of trace; –The third arrow is induced by α; –The fourth arrow is induced by the counit of the adjunction t ◦ tR → ido2 . 0.2.5. We apply the above formalism in the following two main examples: O = DGCat and O = Morita(DGCat) (see right below for the definition of the latter). In the case of O = DGCat, to a (dualizable) DG category C equipped with an endofunctor F , we attach Tr(F, C) ∈ Vect, and to a pair of such, equipped with a functor T : C1 → C2 (that admits a continuous right adjoint) and a natural transformation α : T ◦ F1 → F2 ◦ T, we attach a map in Vect (0.4) Tr(T, α) : Tr(F1 , C1 ) → Tr(F2 , C2 ). 0.2.6. The case of O = Morita(DGCat) is obviously richer. By definition, the objects of O = Morita(DGCat) are 2-DG categories of the form R - mod, where R is a monoidal DG category. Further, 1-morphisms in Morita(DGCat) are by definition given by bi-module categories3 . To a 2-DG category C equipped with an endofunctor F we now attach a DG category Tr(F, C). To a pair of such, equipped with a functor T : C1 → C2 (that admits a right adjoint in Morita(DGCat)) and a natural transformation α : T ◦ F1 → F2 ◦ T, we attach a functor between DG categories (0.5) Tr(T, α) : Tr(F1 , C1 ) → Tr(F2 , C2 ). For C = R - mod and F given by Q ∈ (R ⊗ Rrev ) - mod, the resulting DG category Tr(F, C) identifies with HH• (R, Q) := R ⊗ R⊗Rrev Q, this is the category of Hochschild chains on R with coefficients in Q. 0.2.7. In Theorem 3.8.5 we establish a basic compatibility between the categorical and the 2-categorical trace constructions: Namely, let R be a symmetric monoidal DG category; assume that R is rigid. Let M be a module category over R. We can view M as a 1-morphism (0.6) DGCat → R - mod. Assume that M is dualizable as a DG category. This condition is equivalent requiring that the above 1-morphism admit a right adjoint. Let FR be a symmetric monoidal endofunctor of R. Let FM be an endofunctor of M, endowed with a datum of compatibility with FR ; i.e., we have a datum of commutativity of the diagram R⊗M − −−−−→ M    F FR ⊗FM y y M R⊗M − −−−−→ M, 3We note that every object of Morita(DGCat) is dualizable: the dual of R - mod is Rrev - mod, where Rrev is obtained from R by reversing the monoidal structure. A TOY MODEL FOR SHTUKA 7 along with higher compatibilities. We can view this compatibility structure as a data of “α” for the 1-morphism (0.6). Hence, by (0.5), to this data there corresponds a map in DGCat Vect → Tr(FR , R - mod). In other words, we obtain an object cl(M, FM ) ∈ Tr(FR , R - mod) ≃ HH• (R, FR ). Note that since R was symmetric monoidal, and FR was also symmetric monoidal, the category HH• (R, FR ) acquires a symmetric monoidal structure. Let 1HH• (R,FR ) denote the unit object in HH• (R, FR ). Our Theorem 3.8.5 says that there exists a canonical isomorphism (0.7) Hom HH• (R,FR ) (1HH• (R,FR ) , cl(M, FM )) ≃ Tr(FM , M). Isomorphism (0.7) shows that cl(M, FM ) lifts Tr(FM , M) to an object of HH• (R, FR ); this justifies the notation: Trenh R (FM , M) := cl(M, FM ). So Theorem 3.8.5 says that Hom HH• (R,FR ) (1HH• (R,FR ) , Trenh R (FM , M)) ≃ Tr(FM , M). 0.2.8. We now come to the central construction in this paper, which is the prototype of the shtuka construction. Let us be given a symmetric monoidal category A and a space Y . Let now φ be an endomorphism of Y , which induces a symmetric monoidal endofunctor, denoted A⊗φ , of A⊗Y . Let Y /φ be the (homotopy) quotient of Y by φ. First, we note that there exists a canonical equivalence (0.8) HH• (A⊗Y , A⊗φ ) ≃ A⊗Y /φ . Next, let us be given an action of A⊗Y on a (dualizable) DG category M. Let FM be an endofunctor of M compatible with A⊗φ . Let ShtM,FM ,univ ∈ A⊗Y /φ be the object that corresponds under (0.8) to ⊗Y , A⊗φ ). Trenh A⊗Y (FM , M) ∈ HH• (A 0.2.9. In Sect. 5, we show how the object ShtM,FM ,univ encodes (i) the I-legged shtuka construction4, (ii) partial Frobeniuses, (iii) excursion operators, and (iv) the “S=T” relation. In more detail: Using the rigidity of A and hence of A⊗Y /φ , we interpret the datum of ShtM,FM ,univ as a compatible family of functors ShtM,FM ,Y /φ,I : (Y /φ)I × A⊗I → Vect . Let ShtM,FM ,Y,I : Y I × A⊗I → Vect, be the precomposition of ShtM,FM ,Y /φ,I with the tautological projection Y I → (Y /φ)I . (i) In Proposition 5.2.4, we show that the value of the functor ShtM,FM ,Y,I on y ∈ Y I , r ∈ A⊗I , identifies with Tr(Hry ◦ FM , M), where: 4For us shtukas are algebraic objects. What we call “shtukas” is more commonly called the “cohomology of sheaves arising by geometric Satake on (geometric) shtukas”. See Sect. 3.5.11 for an explanation of why this corresponds to cohomology of sheaves on the (geometric) moduli space of shtukas. 8 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY y –ry denotes the object of A⊗Y equal to the image of r under A⊗I −→ A⊗Y ; –For r ′ ∈ A⊗Y , we let Hr′ denotes the endofunctor of M given by the action of r ′ . This should be seen as a direct analog of the I-legged shtuka construction. (ii) The fact that ShtM,FM ,Y,I comes from ShtM,FM ,Y /φ,I means that the former is equivariant under each of the endomorphisms φi of Y I , that acts as φ along the factor of Y I corresponding to i ∈ I and as the identity along the other factors. Following V. Drinfeld, L. Lafforgue and V. Lafforgue, we call these endomorphisms “partial Frobeniuses”. We show (see Proposition 5.3.3) that, in terms of the isomorphism ShtM,FM ,Y,I (y ∈ Y I , r ∈ A⊗I ) ≃ Tr(Hry ◦ FM , M), the formula for the action of partial Frobeniuses translates to the construction from [Laf, Sect. 3]. (We note that a salient feature of this formula is cyclicity property of the trace construction.) (iii) By Theorem 3.8.5 mentioned above, we have a canonical isomorphism Tr(FM , M) ≃ Hom A⊗Y /φ (1A⊗Y /φ , ShtM,FM ,univ ) ≃ ShtM,FM ,Y /φ,∅ . In particular, Tr(FM , M) acquires an action of the algebra End A⊗Y /φ (1A⊗Y /φ ). In Proposition 5.4.3 we describe this action explicitly in terms of the excursion operators, which are direct analogs of those in [Laf, Sect. 9]. (iv) Arguably, the key conceptual and computational place in V. Lafforgue’s paper is the “S=T” relation, which appears as [Laf, Proposition 6.2]. In Theorem 5.5.5 we state and prove an analog of this result in our abstract context. It states the equality of two particular endomorphisms (one called S and the other T ) of Tr(FM , M) corresponding to the data of (y0 , a), where y0 is a φ-fixed point on Y , and a is a compact object in A. The T operator is given by (0.4), for the endofunctor Hay0 ◦ FM of M. The S operator is an explicit excursion operator corresponding to I = {∗}, the tautological loop based at the image ȳ0 of y0 under Y → Y /φ (here we use the fact that y0 is fixed by φ), and a canonical map ξa : 1A → mult ◦ multR (1A ) attached to a, see (4.14). 0.3. Organization of the paper. We will now briefly describe the structure of the paper. 0.3.1. In Sect. 1 we study the operation Y, A 7→ A⊗Y , where Y is a space and A is a symmetric monoidal category. The main results of this section are: –Presentation of A⊗Y as a colimit with terms A⊗I for finite sets I, as a symmetric monoidal category/monoidal category/DG category (Theorem 1.2.4); –Description of functors A⊗Y → C as compatible families of functors A⊗I → C ⊗ LS(Y I ), where LS(−) denotes the category of (topological) local systems on a given space (Proposition 1.7.2); –For a group G and a space Y (with finitely many connected components), an equivalence Rep(G)⊗Y → QCoh(LocSysG (Y )), where LocSysG (Y ) is the derived algebraic stack, classifying (topological) G-local systems on Y (Theorem 1.5.5), and a generalization of this statement when Rep(G) = QCoh(pt /G) is replaced by QCoh of a more general algebraic stack. A TOY MODEL FOR SHTUKA 0.3.2. 9 The main theme of Sect. 2 is excursion operators. The main results of this section are: –In the case when Y is a connected pointed space, a presentation of A⊗Y as a (sifted) colimit with terms A⊗Σ(I+ ) , where I+ = I ⊔∗ is a pointed finite set, and Σ(I+ ) ∈ Spc denotes the suspension of I+ , i.e., Σ(I+ ) ≃ ∨ S 1 I (Corollary 2.1.6); –Assuming that A has an affine diagonal, a description of End A⊗Y (1A⊗Y ) as a colimit with terms End A⊗Σ(I+ ) (1A⊗Σ(I+ ) ) (Corollary 2.4.4); –For A rigid, an identification of End A⊗Σ(J) (1A⊗Σ(J) ) with Hom A (1A , multJ ◦ multR J (1A )) (Corollary 2.7.6); –For a functor SY : A⊗Y → C, an expression for the action of the terms Hom A (1A , multJ ◦ multR J (1A )) → End A⊗Σ(J) (1A⊗Σ(J) ) → End A⊗Y (1A⊗Y ) on SY (1A⊗Y ) in terms of the excursion operators (Theorem 2.8.7). 0.3.3. In Sect. 3 we study the operation of categorical trace. The main results of this section are: –Construction of the categorical trace (Sect. 3.2); –Construction of the 2-categorical trace (Sect. 3.6); –The relationship between the two for rigid symmetric monoidal categories (Theorem 3.8.5). 0.3.4. are: In Sect. 4 we study several generalizations and elaborations of Theorem 3.8.5. The main results –We connect the Chern character of a compact object of a rigid symmetric monoidal category with the excursion operator (Corollary 4.3.6); –We formulate and prove a version of Theorem 3.8.5 “with observables” (Theorem 4.4.4); –We introduce a mechanism that stands behind the Drinfeld-Lafforgue-Lafforgue definition of partial Frobeniuses (Lemma 4.5.4). 0.3.5. In Sect. 5 we combine the material of the previous sections to obtain our toy model for the shtuka construction. The material in this section has been described already in Sect. 0.2.9. 0.3.6. In the main body of this paper we study actions of categories of the form A⊗Y , where A is a symmetric monoidal category, and Y is an object of Spc. A key example is furnished by [NY]: top,sing when working over the ground field C, we have an action of Rep(Ǧ)⊗X on ShvNilp (BunG ), where top,alg X is the object of Spc corresponding to our curve X, and Shv(−) denotes the category of sheaves top,sing in the classical topology. (We recall that according to Theorem 1.5.5, the category Rep(Ǧ)⊗X identifies with QCoh(LocSysǦ,Betti ).) In the Appendix, Sects. A-D we investigate the possibility of extending the construction of [NY] to other sheaf-theoretic contexts. 10 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 0.3.7. In Sect. A we introduce a list of sheaf-theoretic contexts that we will consider. This list includes sheaves in the classical topology (for schemes/stacks over C), D-modules (for schemes/stacks over a ground field of characteristic zero), as well as étale sheaves (over any ground field). We recall the notion of singular support of an object F ∈ Shv(Y), where Y is a scheme or algebraic stack. The main result of Sect. A is Theorem A.3.9, which says that given an algebraic stack Y and a conical subset N ⊂ T ∗ (Y), for a scheme X, the functor of external tensor product defines an equivalence ShvN (Y) ⊗ Shvlisse (X) → ShvN′ (Y × X), N′ := N × {zero-section}, under the assumption that X is proper5. In the above formula Shvlisse (X) ⊂ Shv(X) is the subcategory of lisse objects, see Sect. A.3.7. 0.3.8. In Sect. B we generalize the construction of [NY] about the action of Rep(Ǧ)⊗X ShvNilp (BunG ). top,sing on First, we provide details for the general pattern of Hecke action: we show that in any sheaf-theoretic context, there is a compatible family of actions of Rep(Ǧ)⊗I on Shv(BunG ×X I ) as I ranges over the category of finite sets. Next, we show that the Hecke action gives rise to a compatible family of monoidal functors (0.9) Rep(Ǧ)⊗I → End(ShvNilp (BunG )) ⊗ Shvlisse (X), I ∈ fSet . Finally, if the sheaf-theoretic context is that of sheaves in the classical topology, we show that the functors (0.9) assemble to an action of LocSysǦ (X) on ShvNilp (BunG ). 0.3.9. In Sect. C we specialize to the case when our sheaf-theoretic context is that of D-modules. We formulate several conjectures pertaining to integrated actions in this context. First off, for a symmetric monoidal category A and a scheme X, we define an action of A⊗X on a DG category M, to be the same as a compatible family of monoidal functors A⊗I → End(M) ⊗ D-mod(X I ), I ∈ fSet . This is equivalent to having an action on M of the symmetric monoidal category Fact(A)Ran(X) , defined as in [Ga3, Sect. 2.5]. A key example of such an action is when M = D-mod(BunG ) and A = Rep(Ǧ). Let LocSysǦ (X) be the stack of de Rham Ǧ-local systems on X, defined as in [AG, Sect. 10.1]. According to [Ga1, Sect. 4.3], we have a canonically defined symmetric monoidal functor Rep(Ǧ)⊗X → QCoh(LocSysǦ (X)), which admits a fully faithful right adjoint. Hence, among categories equipped with an action of Rep(Ǧ)⊗X there is a full subcategory formed by those categories on which this action comes from an action of QCoh(LocSysǦ (X)). A result that we mentioned earlier on says that D-mod(BunG ) belongs to the above subcategory. However, the proof of this theorem heavily relied on “non-geometric” constructions, specifically on the interaction between D-mod(BunG ) and representation theory of the affine Kac-Moody Lie algebra corresponding to g at the critical level. Given that, one would have liked to have a more geometric proof of this result. Unfortunately, we do not have a conjecture as to when an action of Rep(Ǧ)⊗X factors via QCoh(LocSysǦ (X)). Next, we introduce what it means for an action of A⊗X on M to be lisse. This means that for any r ∈ A⊗I and m ∈ M, the resulting object of M ⊗ D-mod(X I ) belongs to the full subcategory M ⊗ D-modlisse (X I ) ⊂ M ⊗ D-mod(X I ), 5This assumption is not necessary if Shv(−) is sheaves in the classical topology. A TOY MODEL FOR SHTUKA 11 where D-modlisse (−) ⊂ D-mod(−) is the full subcategory of lisse D-modules (ind-completion of Ocoherent D-modules). For any M, one can consider its maximal full subcategory Mlisse ⊂ M, on which the action is lisse. Take M = D-mod(BunG ). Theorem B.5.2 (due to [NY]) says that we have an inclusion D-modNilp (BunG ) ⊂ (D-mod(BunG ))lisse . In Conjecture C.2.8 we propose that this inclusion is an equality. Finally, we propose Conjecture C.3.7 that describes lisse actions in terms of set-theoretic support. Namely, let M be a category equipped with an action of Rep(Ǧ)⊗X that factors via QCoh(LocSysǦ (X)). Thus, given a compact object of M, one can talk about its set-theoretic support, which is a Zariskiclosed subset in LocSysǦ (X). Conjecture C.3.7 says that a compact object of M belongs to Mlisse if and only if its set-theoretic support lies in the finite union of subsets induced from irreducible local systems for Levi subgroups of G. 0.3.10. In Sect. D we discuss the abstract notion of ULA, which has been used in the definition of ULA actions in Sect. C. First, given a monoidal category C and a module category M, we define what it means for an object m ∈ M to be ULA with respect to C. ! The case of interest for us is C = D-mod(Y ), with the monoidal structure given by the ⊗ operation. We show that for M := D-mod(Z) for a scheme Z over Y we recover the usual notion of what it means for an object in D-mod(Z) to be ULA over Y . We also establish the following criterion for ULA-ness: we recall that the datum of an D-mod(Y )module category is equivalent to that of sheaf of categories over the de Rham prestack YdR of Y . In particular, we can consider the value of this sheaf of categories on Y itself and obtain a QCoh(Y )-module category MY , which can be recovered from M as MY ≃ QCoh(Y ) ⊗ M. D-mod(Y ) In the case of M := D-mod(Z), we have: MY ≃ QCoh(Y × ZdR ); YdR when Z → Y is smooth, this is the (derived) category of modules over the sheaf of vertical differential operators. The categories M and MY are related by an adjoint pair of functors ind : MY ⇄ M : oblv. Our Theorem D.5.8 says that an object m ∈ M is ULA with respect to D-mod(Y ) if and only if its image oblv(m) ∈ MY is compact. 0.4. Notation and conventions. 0.4.1. Higher categories. This paper will substantially use the language of ∞-categories6, as developed in [Lu1]. We let Spc denote the ∞-category of spaces. Given an ∞-category C, and a pair of objects c1 , c2 ∈ C, we let MapsC (c1 , c2 ) ∈ Spc be the mapping space between them. Given a space Y , by a Y -family of maps c1 → c2 we will mean a map Y → MapsC (c1 , c2 ) in Spc. 6We will often omit the adjective “infinity” and refer to ∞-categories simply as “categories” 12 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Recall that given an ∞-category C that contains filtered colimits, an object c ∈ C is said to be compact if the Yoneda functor MapsC (c, −) : C → Spc preserves filtered colimits. We let Cc ⊂ C denote the full subcategory spanned by compact objects. Given a functor F : C1 → C2 between ∞-categories, we will denote by F R (resp., F L ) its right (resp., left) adjoint, provided that it exists. 0.4.2. Higher algebra. Throughout this paper we will be concerned with higher algebra over a commutative ring of coefficients, denoted e. Although e is not necessarily a field, we will denote by Vect the stable ∞-category of chain complexes of e-modules, see, e.g., [GaLu, Example 2.1.4.8]. We will regard Vect as equipped with a symmetric monoidal structure (in the sense on ∞-categories), see, e.g., [GaLu, Sect. 3.1.4]. Thus, we can talk about commutative/associative algebra objects in Vect, see, e.g., [GaLu, Sect. 3.1.3]. Whenever we talk about algebraic geometry over e, we will mean derived algebraic geometry, built off derived affine schemes, the latter being by definition the category opposite to that of commutative algebras in Vect, connective with respect to the natural t-structure. We will denote by DGCat the ∞-category of (presentable) cocomplete stable ∞-categories, equipped with a module structure over Vect with respect to the symmetric monoidal structure on the ∞-category of cocomplete stable ∞-categories given by the Lurie tensor product, see [Lu2, Sect. 4.8.1]. We will refer to objects of DGCat as “DG categories”. We emphasize that as 1-morphisms in DGCat, only colimit-preserving functors are allowed. For a given DG category C, and a pair of objects c1 , c2 ∈ C, we have a well-defined “inner Hom” object Hom C (c1 , c2 ) ∈ Vect, characterized by the requirement that MapsVect (V, Hom C (c1 , c2 )) ≃ MapsC (V ⊗ c1 , c2 ), V ∈ Vect . The category DGCat itself carries a symmetric monoidal structure, given by Lurie tensor product over Vect. In particular, we can talk about the ∞-category of associative/commutative algebras in DGCat, which we denote by DGCatMon (resp., DGCatSymMon ), and refer to as monoidal (resp., symmetric monoidal) DG categories. Unless specified otherwise, all monoidal/symmetric monoidal DG categories will be assumed unital. Given a monoidal/symmetric monoidal DG category A, we will denote by 1A its unit object. 0.4.3. Rigidity. In multiple places in this paper we use the notion of rigidity for a monoidal DG category A. We refer the reader to [GR1, Sect. 9.1.1] for the general definition. That said, in most of the cases of interest, the DG category in question will be compactly generated. In this case, according to [GR1, Lemma 9.1.5], the condition of being rigid is equivalent to the fact that its classes of compact objects and objects that are both left and right dualizable, coincide. 0.4.4. Prestacks. In Appendices A-B we will deal with classical (i.e., non-derived) algebro-geometric objects over a ground field k (which has nothing to do with our ring of coefficients e). We will denote by Schaff the category of affine schemes (of finite type (!)) over k. By a prestack (technically, prestack locally of finite type), we will mean an arbitrary functor (Schaff )op → Spc . 0.5. Acknowledgements. We are grateful to A. Beilinson, V. Drinfeld, V. Lafforgue, J. Lurie and P. Scholze for valuable discussions, which informed our thinking about the subject. The research of D.G. was supported by NSF grant DMS-1707662 and by Gelfand Chair at IHES. The research of Y.V. was supported by the ISF grant 822/17. Part of the project was carried out while D.G, N.R and Y.V were at MSRI and were supported by NSF grant DMS-1440140. The project have received funding from ERC under grant agreement No 669655. A TOY MODEL FOR SHTUKA 13 1. Symmetric monoidal categories integrated over a space Let A be a symmetric monoidal category, and let Y be an object of Spc. The goal of this section is to give an explicit description of the category A⊗Y , (sometimes also denoted R A), which can be thought of as factorization homology of A along Y . Y We will describe A⊗Y as a colimit as a (i) symmetric monoidal category, (ii) just monoidal category, (iii) plain DG category (each time the colimit will be taken within the corresponding ambient category, i.e., inside the category of symmetric monoidal categories, monoidal categories or DG categories, respectively). In particular, we will give an explicit description of what it takes for A⊗Y to act (as a monoidal category) on a DG category M, and what it takes to map out of A⊗Y as a plain DG category. Both descriptions will be formulated in terms of functors out of A⊗I parameterized by points of Y I , for I ranging over the category fSet of finite sets. 1.1. The integral. 1.1.1. Let C be a category with colimits. For an object c ∈ C and Y ∈ Spc define the object Z c := colim c ∈ C. Y Y I.e., we take the colimit along the index category Y of the constant functor Y → C with value c. 1.1.2. (1.1) Tautologically, for c, c′ ∈ C, we have Z MapsC ( c, c′ ) ≃ MapsSpc (Y, MapsC (c, c′ )). Y This shows that the assignment c, Y 7→ (1.2) Z c Y is a functor C × Spc → C. Moreover, the functor (1.2) preserves colimits in each variable. 1.1.3. (1.3) For a fixed c, the functor Y 7→ Z c, Spc → C Y can be characterized as the unique colimit-preserving functor whose value on {∗} ∈ Spc is c. 1.1.4. (1.4) For example for Y a discrete set I, we have Z c ≃ ⊔ c. i∈I I 14 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 1.1.5. Let us recall the notion of left Kan extension. Let i : D → D′ be a functor; let C be a category with colimits, and let F : D → C be a functor. The left Kan extension of F along i is a functor LKEi (F ) : D′ → C with the following universal property MapsFunct(D,C) (F, G ◦ i) ≃ MapsFunct(D′ ,C) (LKEi (F ), G), G ∈ Funct(D′ , C). One can show that the value of LKEi (F ) on an object d′ ∈ D′ is given by (1.5) colim F (d). d∈D/d′ Here and elsewhere, the notation D/d′ is the slice category, i.e., the category of {d ∈ D, i(d) → d′ }. (1.6) 1.1.6. Tautologically, we can rewrite the functor (1.3) as follows. Let {∗} be the point category, and let ∗ denote its single object. By a slight abuse of notation we will also denote by {∗} the corresponding object of Spc, i.e., the point space. Then (1.3) is the left Kan extension along {∗} ֒→ Spc, (1.7) ∗ 7→ {∗} of the functor {∗} → C, 1.1.7. ∗ 7→ c. Let fSet denote the category of finite sets, equipped with the embedding (1.8) fSet ֒→ Spc . By transitivity of the procedure of left Kan extension with respect to the composition {∗} ֒→ fSet ֒→ Spc, we obtain that (1.3) identifies with the left Kan extension along (1.8) of its restriction to fSet, while the latter is given by (1.4). 1.2. The tensor product A⊗Y as a colimit. 1.2.1. We take C to be DGCatSymMon , so an object of C is a symmetric monoidal DG category A. We wish to give an explicit description of the resulting category Z (1.9) A⊗Y := A Y as a colimit, in three different contexts: (i) as a symmetric monoidal category, (ii) as a monoidal category, (iii) as a plain DG category. 1.2.2. For a category D, let TwArr(D) be the corresponding twisted arrows category. I.e., its objects are 1-morphisms in D ds → dt , and morphisms (d0s → d0t ) → (d1s → d1t ) are commutative diagrams d0s − −−−−→   y d0t x   d1s − −−−−→ d1t . In practice we will take D to be the category fSet of finite sets. A TOY MODEL FOR SHTUKA 15 1.2.3. Let A be a symmetric monoidal category, and let C be one of the categories: DGCatSymMon , DGCatMon , DGCat. For a given Y , consider the functor Z (1.10) TwArr(fSet) → C, (I → J) 7→ A⊗I . Maps(J,Y ) ⊗I We emphasize that in the above formula A is understood in the usual sense, i.e., the I-fold tensor product of A with itself, which is again a symmetric monoidal category, but viewed as an object in C using the tautological forgetful functor while R DGCatSymMon → C, is taken within C, i.e., the result of this operation looks different as a category, depending Maps(J,Y ) on which C we choose. We claim: Theorem 1.2.4. In each of the three cases in Sect. 1.2.3, the image of A⊗Y under the forgetful functor DGCatSymMon → C identifies canonically with the colimit in C of the functor (1.10) along TwArr(fSet). Remark 1.2.5. The assertion of Theorem 1.2.4 with the same proof applies more generally, where instead of DGCat we consider a symmetric monoidal category O satisfying the conditions of Remark 1.3.7 below. So instead of DGCatSymMon we will have ComAlg(O) and C can be any of ComAlg(O), AssocAlg(O) or O. 1.3. Proof of Theorem 1.2.4. 1.3.1. Plan of the proof. We will show that both sides, viewed as functors Spc → C, are left Kan extensions of their respective restrictions along (1.8). Then we will show that there restrictions are canonically isomorphic. 1.3.2. The first two steps will use the following general assertion. Recall that a category A is said to be sifted if the diagonal functor A → A × A is cofinal. Lemma 1.3.3. If A has coproducts, then it is sifted. Proof. We need to show that for any a′ , a′′ ∈ A, the category A(a′ ,a′′ )/ := (a ∈ A, a′ → a ← a′′ ) is contractible7. Now, the fact that A has coproducts means that A(a′ ,a′′ )/ has an initial object.  Corollary 1.3.4. Let D have finite coproducts, and let i : D → D′ be a functor that preserves finite coproducts. Then for any d′ ∈ D, the slice category D/d′ (see Equation (1.6)) is sifted. Here is an application of Corollary 1.3.4 that we will use repeatedly: Proposition 1.3.5. Let i : D → D′ be as in Corollary 1.3.4. Let C1 be a category that admits colimits, and let Φ : C1 → C2 be a functor that preserves sifted colimits. Then for F : D → C1 , the natural transformation LKEi (Φ ◦ F ) → Φ ◦ LKEi (F ), obtained by the universal property of LKEi (−), is an isomorphism. Proof. Follows by Corollary 1.3.4 from the formula (1.5) for the values of LKEi (−). We now proceed to the proof of Theorem 1.2.4. 7In our terminology “contractible” is what in [Lu1] is called “weakly contractible”.  16 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 1.3.6. Step 1. Let us show that the functor Spc Y 7→A⊗Y −→ DGCatSymMon → C is the left Kan extension of its restriction along (1.8). We apply Proposition 1.3.5 to the functor fSet → Spc, which obviously satisfies the assumption of Corollary 1.3.4. Since the functor Y 7→ A⊗Y is the left Kan extension of its restriction to fSet, it remains to show that the forgetful functor DGCatSymMon → C preserves sifted colimits. This follows from the fact that both forgetful functors (1.11) DGCatSymMon → DGCat and DGCatMon → DGCat preserve sifted colimits and are conservative. Remark 1.3.7. The fact that the functors (1.11) preserve sifted colimits and are conservative is a consequence of the following: Let O be a symmetric monoidal category which admits sifted colimits and such that the tensor product functor commutes with sifted colimits. Then for an operad P, the forgetful functor oblvP : P -alg(O) → O preserves sifted colimits and is conservative, see [Lu2, Prop. 3.2.3.1 and Lemma 3.3.2.6]. 1.3.8. Step 2. Let us now show that the functor Spc → C that sends Y to the colimit (in C) of the functor (1.10) along TwArr(fSet) is the left Kan extension of its restriction along (1.8). R It suffices to show that for each I and J, the functor that sends Y to A⊗I is the left Kan Maps(J,Y ) extension of its restriction along (1.8). Since the functor (1.2) commutes with sifted colimits in the space variable, it suffices to show that the functor that sends Y to Maps(J, Y ) is the left Kan extension of its restriction along (1.8). Since the functor Maps(J, −) preserves sifted colimits, this follows again from Proposition 1.3.5 applied to C1 = C2 = Spc, using the fact that the identity functor on Spc is the left Kan extension of its restriction along (1.8). 1.3.9. Step 3. It remains to show that the restrictions of both sides in Theorem 1.2.4 to fSet are canonically isomorphic. However, this follows from the next version of the Yoneda lemma (which we prove below for the sake of completeness): Proposition 1.3.10. Let C be a category with colimits and let Φ : D → C be a functor. Then for d ∈ D there is a canonical isomorphism Z Φ(ds ). Φ(d) ≃ colim (ds →dt )∈TwArr(D) Maps(dt ,d) [Theorem 1.2.4] 1.3.11. For the proof of Proposition 1.3.10 we note the following general feature of the twisted arrows category: Let F1 , F2 : D′ → E be a pair of functors. Consider the functor Tw(F1 , F2 ) : TwArr(D′ )op → Spc that sends (d′s → d′t ) 7→ MapsE (F1 (d′s ), F2 (d′t )). We have the following standard fact (see e.g. [GHN, Prop. 5.1]): A TOY MODEL FOR SHTUKA 17 Lemma 1.3.12. There exists a canonical isomorphism lim TwArr(D′ )op 1.3.13. Tw(F1 , F2 ) ≃ MapsFunct(D′ ,E) (F1 , F2 ). To prove Proposition 1.3.10, we apply Lemma 1.3.12 to D′ := Dop , E = Spc and F1 := MapsD (−, d), F2 := MapsC (Φ(−), c), c ∈ C. Note that we have a tautological identification TwArr(D) ≃ TwArr(Dop ). Using Lemma 1.3.12 and (1.1), we obtain that    Z    colim Φ(ds ) , c ≃ MapsC  (ds →dt )∈TwArr(D) Maps(dt ,d) ≃ lim (ds →dt )∈(TwArr(D))op MapsC ( Z Φ(ds ), c) ≃ Maps(dt ,d) ≃ lim (ds →dt )∈(TwArr(D))op MapsSpc (Maps(dt , d), Maps(Φ(ds ), c)) ≃ ≃ MapsFunct(Dop ,Spc) (MapsD (−, d), MapsC (Φ(−), c)) , which by the usual Yoneda lemma identifies with MapsC (Φ(d), c), as desired.  1.4. The category of local systems. In this subsection we will describe the right adjoint to the functor Z C 7→ C, C ∈ C Y for C being DGCatSymMon , DGCatMon , DGCat. 1.4.1. For a category C with limits, an object c ∈ C and Y ∈ Spc, set cY := lim c, Y i.e., the limit of the functor Y → C with constant value c. For Y being a discrete set I, we obtain cI ≃ Π c. i∈I Note that for c, c′ ∈ C, we have MapsC (c′ , cY ) ≃ MapsSpc (Y, MapsC (c′ , c)). The latter expression shows that the assignment c, Y 7→ cY is a functor C × Spcop → C, moreover, this functor preserves limits in each variable. Furthermore, by (1.1), for a fixed Y , the functor c 7→ cY , is the right adjoint of the functor c′ 7→ C→C Z Y c′ . 18 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 1.4.2. Take C = DGCatSymMon and set LS(Y ) := VectY ∈ DGCatSymMon . Since the forgetful functors DGCatSymMon → DGCatMon → DGCat commute with limits (this is valid in the general context of algebras over an operad in Sect. 1.3.6, see [Lu2, Prop. 3.2.2.1]), the image of LS(Y ) under these forgetful functors identifies with VectY , thought of taking values in DGCatMon (resp., DGCat). Remark 1.4.3. Let Y be a topological space8 weakly equivalent to the geometric realization of Y . One can show that LS(Y ) is equivalent to the full subcategory Shvlisse (Y) of the (unbounded) derived category Shv(Y) of sheaves of e-vector spaces on Y consisting of objects with locally constant cohomology sheaves. 1.4.4. We will now describe the category LS(Y ) as a plain DG category more explicitly: Proposition 1.4.5. (a) For a map f : Y1 → Y2 , the restriction map f † : LS(Y2 ) → LS(Y1 ) admits a left adjoint. (a’) The functor Spc → DGCat, (1.12) Y 7→ LS(Y ),   (f † )L LS(Y1 ) −→ LS(Y2 ) f (Y1 → Y2 ) is canonically isomorphic to the functor Y 7→ Z Vect . Y (b) The category LS(Y ) is dualizable. (b’) The functor (1.12) is canonically isomorphic to the functor   (f † )∨ f LS(Y1 )∨ −→ LS(Y2 )∨ (1.13) Spc → DGCat, Y 7→ LS(Y )∨ , (Y1 → Y2 ) The proof of this proposition is given below. R Corollary 1.4.6. The category Vect is dualizable. The functor Y op Spc → DGCat, Y 7→ ( Z ∨ Vect) ,  ( (Y1 → Y2 ) Y is canonically isomorphic to the functor Y 7→ LS(Y ). Z ∨ Vect) → ( Y2 Z Y1 Vect)  ∨ Proof. Combine points (a’) and (b’) of Proposition 1.4.5.  Note that in particular, Proposition 1.4.5 and Corollary 1.4.6 imply that for an individual Y we have the canonical equivalences Z Z Z LS(Y ) ≃ Vect, LS(Y )∨ ≃ LS(Y ), ( Vect)∨ ≃ Vect . Y Y Y 8We assume that Y is sufficiently nice; i.e. a paracompact topological space homotopy equivalent to a CW complex and of finite homotopical dimension (see [Lu1, Sect. 7.1]) A TOY MODEL FOR SHTUKA 19 1.4.7. The proof of Proposition 1.4.5 is based on the next general lemma (see [GR1, Chapter 1, Propositions 2.5.7 and 6.3.4, and Lemma 2.6.4]): Lemma 1.4.8. Let a 7→ Ca , (1.14) A → DGCat be a diagram of DG categories, such that for every arrow a1 → a2 , the corresponding functor Fa1 ,a2 : Ca1 → Ca2 admits a continuous right adjoint. Set C := colim Ca . a∈A Then: (a) The tautological functors insa : Ca → C admit continuous right adjoints. (b) Consider the functor (1.15) Aop → DGCat, a 7→ Ca , (a1 → a2 ) FaR ,a 1 2 Ca1 ), (Ca2 −→ and set e := lim Ca . C op a∈A e given by the system of functors Then the functor C → C, insR a : C → Ca is an equivalence. (c) Let φ : A′ → A be a functor of index categories. Composing with (1.14) and (1.15), respectively, we obtain the functors A′ → DGCat and A′op → DGCat . Set e′ := lim Cφ(a′ ) . C′ := colim Cφ(a′ ) and C ′ ′ ′ ′op a ∈A a ∈A Then under the identifications e and C′ ≃ C e′ , C≃C e′ → C. e the tautological functor C → C identifies with the left adjoint of the restriction functor C (d) Assume that all Ca are dualizable. Consider the functor ′ (1.16) Aop → DGCat, a 7→ C∨ a, Then C is dualizable, and the functor (a1 → a2 ) Fa∨ ,a 1 2 ∨ (C∨ a2 −→ Ca1 ). C∨ → limop C∨ a, a∈A given by the system of functors ∨ ∨ ins∨ a : C → Ca is an equivalence. Proof of Proposition 1.4.5. Recall that Z Vect ≃ colim Vect and LS(Y ) ≃ lim Vect . Y Y Y The identification Z Vect ≃ LS(Y ) Y follows from Lemma 1.4.8(b) applied to the index category Y and the constant family with value Vect. Point (a’) of Proposition 1.4.5 follows from Lemma 1.4.8(c), implying also point (a). Point (b’) of Proposition 1.4.5 follows from Lemma 1.4.8(d), implying also point (b). 20 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY  1.4.9. Let C be one of the three categories from Theorem 1.2.4, and let C be an object of C. I.e., C is either a symmetric monoidal category, or a monoidal category or a plain DG category. Note that for a symmetric monoidal category A, the tensor product C ⊗ A is again naturally an object of C (this is true for DGCat replaced by an arbitrary symmetric monoidal category O and any operad P as in Sect. 1.3.6; this follows, e.g., from [Lu2, Prop. 3.2.4.3]). We claim that we have a natural map (1.17) C ⊗ LS(Y ) → CY . Indeed, by the definition of CY , the datum of such a map is equivalent to that of a family of maps C ⊗ LS(Y ) → C, parameterized by points of Y . For y ∈ Y we take the corresponding map to be C ⊗ LS(Y ) IdC ⊗ evy −→ C ⊗ Vect ≃ C, where evy : LS(Y ) → LS({∗}) ≃ Vect y is the functor of pullback corresponding to {∗} → Y . We claim: Proposition 1.4.10. The map (1.17) is an isomorphism. The proof of Proposition 1.4.10 is given below. Using Sect. 1.4.1, from Proposition 1.4.10, we obtain: Corollary 1.4.11. For a given Y , the right adjoint of the functor Z C′ 7→ C′ Y is given by C 7→ C ⊗ LS(Y ). Proof of Proposition 1.4.10. Since the forgetful functor C → DGCat commutes with the operation − ⊗ LS(Y ), it suffices to prove the assertion for C = DGCat. The map in question is an isomorphism for Y = {∗}. Since Y ≃ colim {∗}, it suffices to show that Y (for a fixed C), the functor C 7→ C ⊗ LS(Y ) takes colimits in Y to limits in DGCat. However, this follows from Corollary 1.4.6, as the functor in question can be rewritten as Z Funct( Vect, C). Y  1.5. Digression: the stack of G-local systems. In this subsection we will assume that the ring e of coefficients is a field of characteristic 0. 1.5.1. Let G be an affine group-scheme over e. We let Rep(G) be the category of G-representations, defined as Rep(G) := QCoh(pt /G). The assumption on e implies that the category Rep(G) is compactly generated; its compact objects are those that are mapped to compact objects of Vecte under the forgetful functor (1.18) oblvG : Rep(G) → Vect = QCoh(pt). A TOY MODEL FOR SHTUKA 21 1.5.2. For Y ∈ Spc, let LocSysG (Y ) denote the (derived) Artin stack that classifies G-local systems on Y . By definition, for a derived affine scheme S, Maps(S, LocSysG (Y )) is the space of right t-exact symmetric monoidal functors Rep(G) → QCoh(S) ⊗ LS(Y ), (where the RHS is equipped with the tensor product t-structure, see [Lu3, Sect. C.4]). Consider the (symmetric monoidal) category QCoh(LocSysG (Y )). We will now construct a (symmetric monoidal) functor Rep(G)⊗Y → QCoh(LocSysG (Y )). (1.19) 1.5.3. First, we note that for Y = {∗}, we have LocSysG ({∗}) ≃ pt /G. Indeed, Maps(S, pt /G) is the groupoid of G-torsors on S, and those can be described as right t-exact symmetric monoidal functors Rep(G) → QCoh(S), see [AG, Sect. 10.2]. Hence, QCoh(LocSysG ({∗})) ≃ QCoh(pt /G) =: Rep(G). For any Y , we have a canonical map of spaces Y ≃ MapsSpc ({∗}, Y ) → MapsPreStk (LocSysG (Y ), LocSysG ({∗})) → → MapsDGCatSymMon (QCoh(LocSysG ({∗})), QCoh(LocSysG (Y ))) ≃ ≃ MapsDGCatSymMon (Rep(G), QCoh(LocSysG (Y ))). Hence, by (1.1), we obtain the desired map Z Rep(G)⊗Y := Rep(G) → QCoh(LocSysG (Y )). Y 1.5.4. We claim: Theorem 1.5.5. Assume that Y has finitely many connected components. Then the functor (1.19) is an equivalence. Remark 1.5.6. The assertion of Theorem 1.5.5 would fail for Y which is the infinite disjoint union of copies of {∗}, and G = Ga . 1.5.7. We will actually prove a generalization of Theorem 1.5.5. Let Z be a prestack satisfying the following conditions: • The diagonal map Z → Z × Z is affine; • The category QCoh(Z) is dualizable; Let us call such prestacks quasi-passable, cf. [GR1, Chapter 3, Sect. 3.5.1]. For Y ∈ Spc consider the prestack Maps(Y, Z) defined as MapsPreStk (S, Maps(Y, Z)) = MapsSpc (Y, MapsPreStk (S, Z)), S ∈ Schaff . As in Sect. 1.5.3, we have a canonically defined symmetric monoidal functor (1.20) QCoh(Z)⊗Y → QCoh(Maps(Y, Z)). We will prove: Theorem 1.5.8. Assume that Y has finitely many connected components and that Z is quasi-passable. Then the prestack Maps(Y, Z) is quasi-passable and the functor (1.20) is an equivalence. 22 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Theorem 1.5.5 is a particular case of Theorem 1.5.8 for Z = pt /G; note that in this case Maps(Y, pt /G) ≃ LocSysG (Y ). 1.6. Proof of Theorem 1.5.8. 1.6.1. We will first establish a few basic facts about quasi-passable prestacks. Lemma 1.6.2. Suppose that f : Z1 → Z2 is an affine morphism, where Z2 is quasi-passable. Then Z1 is quasi-passable. Proof. By the standard argument, Z1 has affine diagonal; namely, by assumption, the maps Z1 → Z2 → Z2 × Z2 and Z1 × Z1 → Z2 × Z2 are affine and by the 2-out-of-3 propery of affine morphisms, we obtain that Z1 → Z1 × Z1 is affine, as desired. It remains to show that QCoh(Z1 ) is dualizable. By [GR1, Chapter 3, Proposition 3.3.3], we have QCoh(Z1 ) ≃ f∗ (OZ1 ) -mod(QCoh(Z2 )). The lemma follows from the following general fact (applied to the monad given by tensoring with  f∗ (OZ1 )). Lemma 1.6.3. Let C be a dualizable DG category and let M : C → C be a monad such that the underlying endo-functor of C preserves colimits. Then the category M -mod(C) is dualizable. Proof. It is easy to see that M ∨ -mod(C∨ ) provides a dual for M -mod(C).  1.6.4. As a consequence, we have following persistence property of quasi-passable prestacks: Corollary 1.6.5. Suppose that we have a pullback diagram Z / Z2   f / Z0 Z1 such that Z0 , Z1 and Z2 are quasi-passable and the map f : Z2 → Z0 is affine. Then Z is quasi-passable and the functor QCoh(Z1 ) ⊗ QCoh(Z2 ) → QCoh(Z) QCoh(Z0 ) is an equivalence. Proof. By base change, the map Z → Z1 is affine and therefore Z is quasi-passable by Lemma 1.6.2. Moreover, by base change and [GR1, Chapter 3, Proposition 3.3.3], we have QCoh(Z) ≃ f∗ (OZ2 ) -mod(QCoh(Z1 )) ≃ f∗ (OZ2 ) -mod(QCoh(Z0 )) ≃ QCoh(Z1 ) ⊗ QCoh(Z0 ) ⊗ QCoh(Z0 ) QCoh(Z1 ) ≃ QCoh(Z2 ), as desired.  1.6.6. We are now ready to prove the Theorem 1.5.8. We first reduce to the case that Y is connected. Indeed, by [GR1, Chapter 3, Theorem 3.1.7] if X1 and X2 are quasi-passable prestacks, so is X1 × X2 and the functor (1.21) QCoh(X1 ) ⊗ QCoh(X2 ) → QCoh(X1 × X2 ) is an equivalence. In particular, if Y = Y1 ⊔ Y2 , we have Maps(Y1 ⊔ Y2 , Z) ≃ Maps(Y1 , Z) × Maps(Y2 , Z). Therefore, if Theorem 1.5.8 is true for Y1 and Y2 , it is true for Y (using the fact that the coproduct in DGCatSymMon is given by tensor product). A TOY MODEL FOR SHTUKA 23 1.6.7. We can thus assume without loss of generality that Y is connected. In this case, we will prove a slightly more precise version of the theorem (which implies Theorem 1.5.8 by Lemma 1.6.2): Theorem 1.6.8. Let (Y, y) be a pointed connected space and Z a quasi-passable prestack. Then the evaluation map evy : Maps(Y, Z) → Z is affine and the functor (1.20) is an equivalence. 1.6.9. By Proposition 2.1.5 below, we can write (Y, y) as a sifted colimit (Y, y) = colim(Yα , y) α∈A in SpcPtd (and therefore Spc) of objects of the form  (Yα , y) = {∗} ≃ {∗} ⊔ {∗} → {∗} {∗} ⊔ I⊔{∗} This gives an isomorphism of prestacks  {∗} ≃ ∨ S 1 , I I ∈ fSet . Maps(Y, Z) ≃ lim Maps(Yα , Z). α∈A Theorem 1.6.8 now follows from the following two assertions: Lemma 1.6.10. Theorem 1.6.8 is true for Y = ∨ S 1 , I ∈ fSet. I Lemma 1.6.11. Let a 7→ Wa be a sifted diagram of prestacks, affine over some base prestack Z; set W := lim Wa . a Then the map W → Z is affine and the induced functor colim QCoh(Wa ) → QCoh(W) a is an equivalence. 1.6.12. Proof of Lemma 1.6.10. Since ∨ S 1 ≃ {∗} I ⊔ I⊔{∗} {∗}, we have a pullback diagram of prestacks Maps(∨ S 1 , Z) /Z   / ZI⊔{∗} I Z Since Z is quasi-passable, so is ZI⊔{∗} and the diagonal map Z → ZI⊔{∗} is affine. The result then follows by Corollary 1.6.5 (using the fact the pushouts in DGCatSymMon are given by tensor products).  1.6.13. Proof of Lemma 1.6.11. The assertion that W → Z is affine follows from the fact a limit of affine schemes is an affine scheme and limits commute with pullbacks. Now, let Aa (resp., A) be the object of ComAlg(QCoh(Z)) equal to the image of the structure sheaf on Wα (resp., W). Then (e.g., by [GR1, Chapter 3, Prop. 3.3.3]), direct image induces an equivalence QCoh(Wa ) ≃ Aa -mod(QCoh(Z)) and QCoh(W) ≃ A-mod(QCoh(Z)). Now the assertion of the lemma follows from Lemma 2.5.2 below, combined with the fact that the forgetful functor ComAlg → AssocAlg commutes with sifted colimits.  1.7. Functors out of A⊗Y as diagrams parameterized by finite sets. In this subsection we will derive some corollaries of Theorem 1.2.4. 24 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 1.7.1. Let C be one of the three categories from Theorem 1.2.4, and let C be an object of C. I.e., C can be a symmetric monoidal DG category, or a monoidal DG category, or a plain DG category. Consider the following two functors fSet → C, I 7→ A⊗I ; (1.22) and I 7→ C ⊗ LS(Y I ). (1.23) We claim: Proposition 1.7.2. The space of maps A⊗Y → C within C (i.e., functors A⊗Y → C that preserve the corresponding structure) is canonically isomorphic to the space of natural transformations from (1.22) to (1.23). Proof. Recall (see Lemma 1.3.12) that given a pair of functors Φ1 , Φ2 : D → C, the space of natural transformations Φ1 → Φ2 identifies with the limit lim (ds →dt )∈TwArr(D)op MapsC (Φ1 (ds ), Φ2 (dt )). Hence, we obtain that the space of natural transformations from (1.22) to (1.23) identifies canonically with lim MapsC (A⊗I , C ⊗ LS(Y J )). op (I→J )∈TwArr(fSet) Applying Corollary 1.4.11, we rewrite this as lim (I→J )∈TwArr(fSet)op MapsC ( Z A⊗I , C). YJ Now the assertion follows from Theorem 1.2.4, which says that Z colim A⊗I ≃ A⊗Y . (I→J )∈TwArr(fSet) YJ  1.7.3. As a first application of Proposition 1.7.2, we will describe what it takes to have an action of A⊗Y on some DG category M. We take C = DGCatMon . Set C := End(M); this is a monoidal category. An action of A⊗Y on M is the same as a map of monoidal categories A⊗Y → End(M). According to Proposition 1.7.2, the latter amounts to a system of monoidal functors (1.24) A⊗I → End(M) ⊗ LS(Y I ), I ∈ fSet, compatible in the sense that they make the following diagrams commute (1.25) A⊗I − −−−−→ End(M) ⊗ LS(Y I )     y y A⊗J − −−−−→ End(M) ⊗ LS(Y J ) for every I → J, along with a system of higher compatibilities. A TOY MODEL FOR SHTUKA 25 Remark 1.7.4. Let usR note the difference between the notion of action of A⊗Y on M, and that of action R of A on M, where − is understood within DGCatMon : Y Y The former contains the data of the homomorphisms (1.24) for all I with the compatibilities expressed by diagrams (1.25). The latter contains just the data of homomorphism A → End(M) ⊗ LS(Y ), i.e., the case I = {∗}. 1.7.5. Example of Y = S 1 . Let us elaborate further on Remark 1.7.4 in the case Y = S 1 . Note that by identifying S 1 = {∗} ⊔ {∗}, {∗}⊔{∗} 1 we obtain that LS(S ) identifies with the category of vector spaces equipped with an automorphism. 1 The monoidal category End(M) ⊗ LS(S an automorR ) is that of endofunctors of M equipped with R A (with understood as taking place inside DGCatMon ) on M is an phism. Hence, an action of assignment S1 a ∈ A 7→ (Ha , mona ), where Ha is an endofunctor of M equipped with an endomorphism mona . These data must be compatible in the sense that for m ∈ M the diagram Ha1 (mona2 |Ha (m) ) 2 −−→ Ha1 ◦ Ha2 (m) −−−−−−−−−−−−   (1.26) ∼y Ha1 ⊗a2 (m) Ha1 ◦ Ha2 (m) mona1 ⊗a2 |H a ⊗a (m) 1 −−−−−−−−−−− −−2−−→ mona1 |Ha (Ha (m)) 2 −−−−−−−−1−−−− −−→ Ha1 ◦ Ha2 (m)   ∼y Ha1 ⊗a2 (m) should commute, along with higher compatibilities. In the case when M and A are derived categories of abelian categories and the action is t-exact, the commutativity of the diagrams (1.26) for objects in the heart is sufficient to ensure the higher compatibilities. 1 By contrast, an action of A⊗S on M requires additional compatibilities. The first of these is that under the identification Ha1 ◦ Ha2 (m) ≃ Ha1 ⊗a2 (m) ≃ Ha2 ⊗a1 (m) ≃ Ha2 ◦ Ha1 (m), the automorphism mona1 |Ha1 (Ha2 (m)) of the LHS should correspond to the automorphism Ha2 (mona1 |Ha1 (m) ) of the RHS. Again, in the situation arising from abelian categories, this condition is sufficient for the higher compatibilities. 1.8. Objects in and functors out of A⊗Y . 1.8.1. Let us now take C = DGCat. For a target DG category C, Proposition 1.7.2 describes what it takes to construct a functor SY : A⊗Y → C. Namely, such a datum is equivalent to a natural transformation between the functors (1.27) I 7→ A⊗I and (1.28) I 7→ C ⊗ LS(Y I ). 26 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY In other words, this is a data of functors SI : A⊗I → C ⊗ LS(Y I ), (1.29) I ∈ fSet, that make the following diagrams commute S I A⊗I − −−− −→ C ⊗ LS(Y I )     y y (1.30) S J A⊗J − −−− −→ C ⊗ LS(Y J ) for every I → J, along with a system of higher compatibilities. 1.8.2. We now make the following observation (which is a special case of Proposition 2.3.9 below applied to the constant diagram Y → DGCatSymMon ). Recall the notion of a rigid monoidal category, see Sect. 0.4.3. Proposition 1.8.3. Let A be a rigid monoidal DG category. Then A⊗Y is also rigid for any Y ∈ Spc. Moreover, if A is compactly generated so is A⊗Y . Applying [GR1, Chapter 1, Sect. 9.2.1], we obtain: Corollary 1.8.4. Assume that A is rigid. Then the functors mult A⊗Y ⊗ A⊗Y −→ A⊗Y and Hom(1A⊗Y ,−) −→ Vect multR 1 ⊗Y A Vect −→ A⊗Y −→ A⊗Y ⊗ A⊗Y , where the first functor in the first line is given by the monoidal operation, and the second functor in the second line is its right adjoint, define an identification (A⊗Y )∨ ≃ A⊗Y . 1.8.5. Taking C = Vect, and applying Corollary 1.8.4, we obtain that the category Funct(A⊗Y , Vect) ≃ (A⊗Y )∨ is canonically equivalent to A⊗Y itself via the map r 7→ Hom A⊗Y (1A⊗Y , r ⊗ −). Note that in the description of Sect. 1.8.1, for Suniv ∈ A⊗Y , the corresponding functor SY : A⊗Y → Vect is given in terms of (1.29) by the family of functors SI : A⊗I → LS(Y I ) (1.31) that are described as follows: The datum of (1.31) is equivalent to the map from Maps(I, Y ) ≃ Y I to space of functors A⊗I → Vect . (1.32) For a point y ∈ Y I ≃ Maps(I, Y ) and r ∈ A⊗I , denote by ry the corresponding object of A⊗Y . By unwinding the definitions, we obtain that (1.32) sends r 7→ Hom A⊗Y (1A⊗Y , ry ⊗ Suniv ). 1.8.6. In particular, under the correspondence of Sect. 1.8.5, the object S∅ ∈ Funct(A⊗∅ , LS(Y ∅ )) ≃ Funct(Vect, Vect) ≃ Vect identifies with Hom(1A⊗Y , Suniv ). A TOY MODEL FOR SHTUKA 27 2. Excursions The goal of this section is to give a (more explicit) description of the category A⊗Y in the case when Y is a pointed connected space, and, more substantially, of the endomorphism algebra of its unit object. The latter will be described in terms of what we will call, following V. Lafforgue, “excursion operators”. 2.1. Description of A⊗Y via the fundamental group. 2.1.1. Let E1 (Spc) denote the category of monoid objects in Spc (a.k.a. E1 -objects in Spc with respect to the Cartesian monoidal structure). I.e., these are objects of Spc equipped with a binary operation and a unit that satisfy the axioms of monoid up to coherent homotopy. We have an adjoint pair (see [Lu2, Example 3.1.3.6, Lemma 3.2.2.6 and Prop. 3.2.3.1]) (2.1) freeE1 : Spc ⇄ E1 (Spc) : oblvE1 , where oblvE1 preserves sifted colimits and is conservative. Explicitly, freeE1 (Y ) = ⊔ Y n . n 2.1.2. Recall also that the category E1 (Spc) is connected by a pair of mutually adjoint functors to the category of pointed spaces SpcPtd := Spc{∗}/ , B : E1 (Spc) ⇄ SpcPtd : Ω. This adjunction factors as E1 (Spc) ⇄ Spccnctd,Ptd ⇄ SpcPtd , where Spccnctd,Ptd ⊂ SpcPtd is the full subcategory of connected pointed spaces. The functor Ω|Spccnctd,Ptd is fully faithful, and its essential image consists of group-like objects in E1 (Spc), i.e., those objects Y , for which π0 (Y ), equipped with a monoid structure induced by that on Y , is actually a group (see [Lu2, Theorem 5.2.6.10] for an ∞-categorical account of this basic fact, which is originally due to Stasheff [St]). 2.1.3. Let FFM denote the full subcategory of E1 (Spc) equal to the essential image of the functor freeE1 applied to the full subcategory fSet ⊂ Spc . Note that the category FFM is discrete (a.k.a. ordinary). Its objects are the usual free finitely generated monoids, and morphisms are morphisms between such. 2.1.4. The following assertion is probably well-known, but we will supply a proof for completeness: Proposition 2.1.5. For Y ∈ Spccnctd,Ptd , the slice category FFM/Ω(Y,y) is sifted, and the natural map (2.2) colim H∈FFM/Ω(Y,y) B(H) → Y is an isomorphism, where the colimit is taken in the category of pointed spaces. As an immediate corollary we obtain: Corollary 2.1.6. For Y ∈ Spccnctd,Ptd , we have a canonical identification A⊗Y ≃ colim H∈FFM/Ω(Y,y) A⊗B(H) . Proof. Since the functor Y 7→ A⊗Y preserves colimits, we only need to show that the isomorphism (2.2) remains valid if the colimit is understood as taking place in Spc rather than SpcPtd . However, this is always the case any time the index category is contractible, which in our case it is: indeed, it is sifted, and any sifted category is contractible (see [Lu1, Prop. 5.5.8.7]).  28 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY In Sect. 2.4 we will use Corollary 2.1.6 to give an explicit description of the algebra EndA⊗Y (1A⊗Y ). 2.2. Proof of Proposition 2.1.5. 2.2.1. We will prove the following: Proposition 2.2.2. For every H0 ∈ E1 (Spc), the slice category FFM/H0 is sifted, and the map (2.3) H → H0 colim H∈FFM/H 0 is an isomorphism. Proposition 2.2.2 implies Proposition 2.1.5: take H0 = Ω(Y, y) for Y ∈ Spccnctd,Ptd , and apply the functor B to (2.3). Remark 2.2.3. Instead of monoids in Proposition 2.2.2, we can just as well consider semi-groups; i.e. E1 spaces which are not required to have a unit element. Indeed, by [Lu2, Corollary 5.4.3.6], the forgetful functor from grouplike E1 -algebras to E1 -semi-groups is fully faithful. A slight adavantage of this is that there are fewer maps between free semi-groups and therefore we obtain a slightly smaller indexing category. 2.2.4. The rest of this subsection is devoted to the proof of Proposition 2.2.2. We claim: Lemma 2.2.5. (a) The category FFM has finite coproducts and the functor FFM → E1 (Spc) preserves finite coproducts. (b) The Yoneda embedding j : E1 (Spc) → PShv(FFM) is conservative and preserves sifted colimits. In point (b) of the lemma, for a small category C (in our case C = FFM), we are using the notation PShv(C) to denote all functors Cop → Spc. 2.2.6. Let us show how Lemma 2.2.5 implies Proposition 2.2.2: By Lemma 2.2.5 and Corollary 1.3.4, the category FFM/H0 is sifted. Next, since j is conservative it suffices to show that the map ! j colim H∈FFM/H H → j(H0 ) 0 is an isomorphism. Since j preserves sifted colimits, we have to show that the map colim H∈FFM/H j(H) → j(H0 ) 0 is an isomorphism. But this follows from the Yoneda Lemma. [Proposition 2.2.2] 2.2.7. We now proceed to the proof of Lemma 2.2.5. First, we state its analog for spaces: Lemma 2.2.8. The Yoneda embedding j : Spc → PShv(fSet) is conservative and preserves sifted colimits. Proof. To prove that j preserves sifted colimits, its suffices to show that for every I ∈ fSet the map MapsSpc (I, −) preserves sifted colimits. But this is standard (and has been used already in Sect. 1.3.8). The functor j is conservative, since it has a left inverse PShv(fSet) → PShv({∗}) = Spc, induced by the inclusion {∗} → fSet.  Let us now prove Lemma 2.2.5: A TOY MODEL FOR SHTUKA 29 Proof. Point (a) follows from the fact that freeE1 is a left adjoint. We now prove point (b). Consider the functor freeE1 : fSet → FFM, and the corresponding pullback functor free∗E1 : PShv(FFM) → PShv(fSet). The functor free∗E1 preserves all colimits (tautologically), and is conservative, since freeE1 is essentially surjective. Hence, it suffices to show that the composition: free∗E1 ◦ j : E1 (Spc) → PShv(fSet) is conservative and preserves sifted colimits. Since freeE1 is the left adjoint of oblvE1 , the above composition factors as j ◦ oblvE1 : E1 (Spc) → Spc → PShv(fSet). Since oblvE1 is conservative and preserves sifted colimits (see [Lu2, Lemma 3.2.2.6 and Prop. 3.2.3.1]), the assertion follows from Lemma 2.2.8.  2.3. Digression: affine functors. 2.3.1. Let F : C0 → C be a map monoidal categories, which admits a continuous right adjoint as a map of plain DG categories. The functor F R has a natural right-lax monoidal structure; hence the object F R (1C ) ∈ C0 has a natural structure of associative algebra in C0 (see [Lu2, Cor. 7.3.2.7]). Moreover, the functor F R naturally upgrades to a functor (F R )enh : C → F R (1C )-mod(C0 ). (2.4) Definition 2.3.2. We shall say that F is affine if (2.4) is an equivalence. For example, if f : Z1 → Z2 is a quasi-affine map between prestacks, the functor f ∗ : QCoh(Z2 ) → QCoh(Z1 ) is affine, see [GR1, Chapter 3, Prop. 3.3.3]. 2.3.3. Here is a handy criterion of affineness (which follows immediately from the Barr-Beck-Lurie theorem): Lemma 2.3.4. A functor F is affine if and only if its right adjoint F R has the following properties: (i) It is continuous; (ii) It is conservative; (iii) The right-lax compatibility of F R with the action of C0 is strict. 2.3.5. Let A be a symmetric monoidal category. Definition 2.3.6. We shall say that A has an affine diagonal if the tensor product functor A⊗A →A is affine. For example (assuming that e contains Q), for an algebraic group G, the category Rep(G) ≃ QCoh(pt /G) has affine diagonal, because the morphism pt /G → pt /G × pt /G is affine (here we are using the equivalence (1.21)). 30 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Remark 2.3.7. It is easy to see that a symmetric monoidal category A is rigid (see Sect. 0.4.3 for what this means) if and only if it has an affine diagonal and its unit object is compact. This follows immediately from Lemma 2.3.4 above. 2.3.8. We can use the characterization of rigid symmetric monoidal categories in Remark 2.3.7 to prove that rigid symmetric monoidal categories are stable under colimits: Proposition 2.3.9. Let I → DGCatSymMon be a diagram of symmetric monoidal DG categories such that for each i ∈ I, the corresponding category Ci is rigid and for each morphism i → j ∈ I, the corresponding functor Ci → Cj admits a continuous right adjoint. Then the colimit colim Ci is rigid. i∈I Moreover, if each Ci is compactly generated, so is colim Ci . i∈I In the proof of Proposition 2.3.9, we will use the following basic fact about colimits: Proposition 2.3.10. Let C be a category that admits all colimits. Suppose that we have a diagram F : I × [1] → C, where I is a contractible category: ... / Xi / Xj / ... ...  / Yi  / Yj / ... Let X = colim Xi . Then the natural map colim Yi → colim (X ⊔ Yi ) i∈I i∈I Xi is an isomorphism. Proof. Since push-outs commute with colimits, we have colim (X ⊔ Yi ) ≃ (colim X) i∈I i∈I Xi ⊔ colim Xi (colim Yi ). i∈I i∈I Now the assertion follows from the fact that colim X ≃ X, i∈I since I is contractible, so (colim X) i∈I ⊔ colim Xi i∈I (colim Yi ) ≃ X ⊔ (colim Yi ) ≃ (colim Yi ). i∈I X i∈I i∈I  Proof of Proposition 2.3.9. Since all colimits decompose as coproducts and sifted colimits, we can treat each case separately. Suppose C0 and C1 are rigid symmetric monoidal DG categories. Clearly, C0 ⊗ C1 (which is the coproduct in DGCatSymMon ) is also rigid. Moreover, by [GR1, Chapter 1, Proposition 7.4.2] it is compactly generated if C0 and C1 are. Now, suppose that I is sifted. In this case, the underlying DG category of C := colim Ci is the i∈I corresponding colimit in DGCat. In particular, for every i ∈ I, the functor insi : Ci → C admits a continuous right adjoint and therefore preserves compact objects. It follows, that 1C ≃ insi (1Ci ) (for any i ∈ I) is compact. Moreover, if each Ci is compactly generated, so is C. It remains to show that C has affine diagonal. Note that since I is sifted, we have colim (Ci ⊗ Ci ) ≃ C ⊗ C. i∈I Therefore, since pushouts in DGCat SymMon are given by tensor products, we have by Proposition 2.3.10, C ≃ colim (C ⊗ C) i∈I ⊗ Ci . Ci ⊗Ci A TOY MODEL FOR SHTUKA 31 For each i ∈ I, the tensor product functor Ci ⊗ Ci → Ci is affine by assumption. Therefore, so is the functor C ⊗ C → (C ⊗ C) ⊗ Ci . Ci ⊗Ci The proposition now follows from the following general assertion.  Proposition 2.3.11. Let C be a symmetric monoidal DG category. The functor ComAlg(C) → DGCatSymMon ≃ C -ComAlg(DGCat) C/ given by A 7→ A -mod(C) is fully faithful and commutes with colimits. Proof. The functor admits a right adjoint given by (C → D) 7→ HomD (1D , 1D ), where HomD (−, −) denotes internal Hom relative to the right action of C on D, i.e., HomC (c, HomD (d1 , d2 )) = HomD (d1 ⊗ c, d2 ). The fully faithfulness of the functor in question follows from the isomorphism HomA -mod(C) (1A -mod(C) , 1A -mod(C) ) ≃ A.  2.4. Endomorphisms of the unit object in A⊗Y as a colimit. 2.4.1. Let us be in the situation of Lemma 1.4.8. Assume that A has an initial object, denoted a0 . Fix two objects c′a0 , c′′a0 ∈ Ca0 with c′a0 compact. Set c′ := insa0 (c′a0 ), c′′ := insa0 (c′′a0 ). We can form an A-family of objects of Vect a 7→ Hom Ca (Fa0 ,a (c′a0 ), Fa0 ,a (c′′a0 )), and we have a natural map (2.5) colim Hom Ca (Fa0 ,a (c′a0 ), Fa0 ,a (c′′a0 )) → Hom C (c′ , c′′ ). a 2.4.2. It is not difficult to show that when the index category A is filtered, then the map (2.5) is an isomorphism (see [Ro]). However, it is also easy to give an example when the map (2.5) is a not an isomorphism. However, we claim: Theorem 2.4.3. Let A have an affine diagonal. Then the map (2.5) is an isomorphism for the diagram of Corollary 2.1.6. A consequence of this theorem that will play a role for us in the sequel is the following: Corollary 2.4.4. Let A have an affine diagonal, and assume that 1A ∈ A is compact. Then for a connected Y ∈ Spc with a marked point y ∈ Y , we have the following expression for End A⊗Y (1A⊗Y ): End A⊗Y (1A⊗Y ) ≃ colim H∈FFM/Ω(Y,y) End A⊗B(H) (1A⊗B(H) ). In Sect. 2.7 we will give an explicit description of the algebras End A⊗B(H) (1A⊗B(H) ) for H ∈ FFM, i.e., for H of the form freeE1 (I) for I ∈ fSet. The next two subsections are devoted to the proof of Theorem 2.4.3. 32 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 2.4.5. We will prove Theorem 2.4.3 in the following general context: Proposition 2.4.6. Assume that in the situation of (2.5) the functor a 7→ Ca Mon comes from a functor A → DGCat . Assume that A is sifted and all the functors Ca0 → Ca are affine. Then the map (2.5) is an isomorphism. 2.5. Proof of Proposition 2.4.6. 2.5.1. The assumption that the functors Ca0 → Ca are affine implies that the assignment a 7→ Ca , viewed as a functor A → DGCat canonically factors as A 7→ AssocAlg(Ca0 ) → DGCat, where: –the first arrow is the functor a 7→ FaR0 ,a (1Ca ); –the second arrow is the functor R 7→ R-mod(C0 ),   M 7→ R2 ⊗ M . (R1 → R2 ) R1 In particular, we obtain an A-diagram a 7→ Ra ∈ AssocAlg(Ca0 ). Set R := colim Ra , a where the colimit is taken in AssocAlg(Ca0 ). We have a naturally defined functor colim Ra -mod(Ca0 ) → R-mod(Ca0 ). (2.6) a∈A Lemma 2.5.2. The functor (2.6) is an equivalence. The proof of the lemma is given below. 2.5.3. Thus, returning to the proof of Proposition 2.4.6, we need to show that for M0′ , M0′′ ∈ Ca0 with M0′ compact, the map colim MapsRa -mod(Ca ) (Ra ⊗ M0′ , Ra ⊗ M0′′ ) → MapsR-mod(Ca ) (R ⊗ M0′ , R ⊗ M0′′ ) 0 0 a∈A is an isomorphism. We rewrite the LHS as colim MapsCa (M0′ , oblvAssoc (Ra ) ⊗ M0′′ ), 0 a∈A and the RHS as MapsCa (M0′ , oblvAssoc (R) ⊗ M0′′ ). 0 Since, M0′ was assumed compact, we rewrite the LHS further as     MapsCa M0′ , colim oblvAssoc (Ra ) ⊗ M0′′ ≃ MapsCa M0′ , (colim oblvAssoc (Ra )) ⊗ M0′′ . 0 a∈A 0 a∈A Now, the assumption that A is sifted implies that colim oblvAssoc (Ra ) ≃ oblvAssoc (colim Ra ) = oblvAssoc (R), a∈A a∈A and the assertion follows. [Proposition 2.4.6] A TOY MODEL FOR SHTUKA 33 2.5.4. First proof of Lemma 2.5.2. Recall the equivalence of Lemma 1.4.8. The corresponding category e is the limit C lim Ra -mod(Ca0 ), a∈A where for a1 → a2 , the corresponding functor Ra2 -mod(Ca0 ) → Ra1 -mod(Ca0 ) is given by restriction along Ra1 → Ra2 . In terms of the equivalence e C ≃ C, the functor right adjoint to (2.6), viewed as a functor R-mod(Ca0 ) → lim Ra -mod(Ca0 ), a∈A is given by restrictions along Ra → R. Hence, Lemma 2.5.2 is equivalent to the following statement: Lemma 2.5.5. Let C be a monoidal category, in which the monoidal operation is compatible with colimits in each variable. Then for a sifted colimit diagram of algebras in C, colim Ra ≃ R, a the functor R-mod(C) → lim Ra -mod(C) (2.7) a is an equivalence. Proof of Lemma 2.5.5. Note that functor (2.7) is compatible with the natural forgetful functors of both sides to C. We first show that (2.7) induces an equivalence between the fiber of both categories over a given object of C. For a pair of objects M1 , M2 ∈ C, let Hom(M1 , M2 ) ∈ C be their internal Hom, i.e., MapsC (c, Hom(M1 , M2 )) ≃ Maps(c ⊗ M1 , M2 ) (this object exists by the Brown representability theorem, [Lu1, Prop. 5.5.2.2]). For M1 = M2 = M , the object End(M ) := Hom(M, M ) has a structure of associative algebra. For R′ ∈ AssocAlg(C), the datum of action of R′ on M is equivalent to a map R′ → End(M ) in AssocAlg(C) (see [Lu2, Cor. 4.7.1.40]). Hence, the datum of structure of R-module on M is equivalent to that of a homomorphism R → End(M ), and a compatible system of data of Ra -modules on M is equivalent to that of a point in lim MapsAssocAlg(C) (Ra , End(M )). a We rewrite the latter as   MapsAssocAlg(C) colim Ra , End(M ) , a which is by definition MapsAssocAlg(C) (R, End(M )), implying the assertion. The equivalence of fibers of the two sides of (2.7) over a given M implies that (2.7) induces an equivalence of the underlying groupoids. To show that (2.7) is an equivalence of categories, it remains to show that for M1 , M2 ∈ R-mod, the map (2.8) Hom R-mod(C) (M1 , M2 ) → lim Hom Ra -mod(C) (M1 , M2 ) a∈A 34 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY is an isomorphism. Note that for a given R′ ∈ AssocAlg(C), and M1′ , M2′ ∈ R′ -mod(C), the object Hom R′ -mod(C) (M1′ , M2′ ) is calculated as the limit (a.k.a. totalization) of the co-Bar complex, whose terms are Hom C (oblvAssoc (R′ )⊗n ⊗ M1′ , M2′ ). Note that, since A is sifted, for each n, the map Hom C (oblvAssoc (R)⊗n ⊗ M1′ , M2′ ) → lim Hom C (oblvAssoc (Ra )⊗n ⊗ M1 , M2 ) a∈A is an isomorphism. Interchanging the order of lim and the totalization we obtain that (2.8) is an isomorphism. a∈A  2.5.6. Second proof of Lemma 2.5.2. In fact, Lemma 2.5.2 is a corollary of the following more general assertion (see Corollary 2.5.8 below): Lemma 2.5.7. Let C be a monoidal DG category. Then the functor AssocAlg(C) → (C-modr (DGCat))C/ , R 7→ R-mod(C) is fully faithful and commutes with colimits9. Proof. Same as the proof of Proposition 2.3.11.  Corollary 2.5.8. Let C be a monoidal DG category. Then the functor AssocAlg(C) → DGCat, R 7→ R-mod(C) commutes with colimits over contractible index categories. Proof. Follows from the fact that the forgetful functor C-modr (DGCat) → DGCat commutes with colimits.  2.6. Applying the paradigm. 2.6.1. In order to prove Theorem 2.4.3, it suffices to show that it fits into the paradigm of Proposition 2.4.6. Hence, it suffices to prove the following: Proposition 2.6.2. Let A have an affine diagonal. Then for a finite set I, the functor A = A⊗{∗} ≃ A⊗B(freeE1 (∅)) → A⊗B(freeE1 (I)) is affine. In the process of proving this proposition, we will describe what the category A⊗B(freeE1 (I)) looks like. Remark 2.6.3. Using the argument in the proof of Theorem 1.6.8, one can strengthen the assertion of Proposition 2.6.2 as follows: for a map of spaces Y1 → Y2 which is surjective on π0 , the resulting map AY1 → AY2 is affine. 9In the above formula, the superscript “r” stands for ”right modules”. A TOY MODEL FOR SHTUKA 2.6.4. 35 Since both functors B(−) and freeE1 are left adjoints, we have: B(freeE1 (I)) ≃ ∨ B(freeE1 ({∗})), I where ∨ means push-out with respect to the common base point. Since pushouts in DGCatSymMon are tensor products, we obtain: A⊗B(freeE1 (I)) ≃ A⊗B(freeE1 ({∗})) ⊗ ... ⊗ A⊗B(freeE1 ({∗})) . A A | {z } I-fold product Hence, it suffices to show that A⊗B(freeE1 ({∗})) is affine over A: indeed, for a pair of commutative algebras A1 and A2 in A, we have A1 -mod(A) ⊗ A2 -mod(A) ≃ (A1 ⊗ A2 )-mod(A). A 1 2.6.5. Note that B(freeE1 ({∗})) ≃ S . Indeed, by adjunction (2.9) MapsSpcPtd (B(freeE1 ({∗})), Y ) ≃ MapsE1 (Spc) (freeE1 ({∗}), Ω(Y, y)) ≃ ≃ MapsSpc ({∗}, Ω(Y, y)) ≃ Ω(Y, y) ≃ MapsSpcPtd (S 1 , Y ). 2.6.6. Write S 1 as a push-out {∗} ⊔ {∗}⊔{∗} {∗}. Hence, 1 A⊗B(freeE1 ({∗})) ≃ A⊗S ≃ A ⊗ A. A⊗A 2.6.7. Hence, it remains to show that if A has an affine diagonal, then the functor A→A ⊗ A A⊗A is affine. Note that the above functor is obtained by base change A ⊗ − from the functor A ⊗ A → A. A⊗A Hence, it suffices to prove the following assertion: Lemma 2.6.8. Let A0 → A be an affine functor between symmetric monoidal categories. Then for any diagram of symmetric monoidal categories B′ ← B → A 0 , the resulting functor B′ ⊗ A 0 → B′ ⊗ A B B is also affine. Proof. Follows from the fact that for R ∈ AssocAlg(A0 ), we have B′ ⊗ R-mod(A0 ) ≃ R′ -mod(B′ ⊗ A0 ), B B ′ where R denotes the image of R under A 0 → B′ ⊗ A 0 . B Indeed, the adjunction (id ⊗indR ) : B′ ⊗ A0 ⇄ B′ ⊗ R-mod(A0 ) : (id ⊗oblvR ) B B satisfies the assumption of the Barr-Beck-Lurie theorem (see [Lu2, Theorem 4.7.3.5]) and hence is monadic, with the monad given by R′ ⊗ −.  2.7. Endomorphisms of the unit, term-wise. 36 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 2.7.1. For a finite set I, let I+ denote the pointed finite set I+ = I ⊔ {∗}. Note that we have a canonical identification of pointed spaces B(freeE1 (I)) ≃ {∗} ⊔ {∗} =: Σ(I+ ). I+ Indeed, by adjunction, for a pointed space (Y, y), the following five pieces of data are equivalent: (i) A map of pointed spaces Σ(I+ ) → (Y, y); (ii) A map of pointed spaces (I+ , ∗) → Ω(Y, y); (iii) A map of spaces I → Ω(Y, y); (iv) A map of E1 -objects freeE1 (I) → Ω(Y, y); (v) A map of pointed spaces B(freeE1 (I)) → (Y, y). 2.7.2. In what follows, for γ I ∈ Maps(I, Ω(Y, y)), we let γ I+ denote the corresponding pointed map I+ → Ω(Y, y), i.e., γ I+ = (γ I , γtriv ). Thus, for each γ I we obtain a map Σ(I+ ) → Y and the corresponding map of algebras, to be denoted γ I+ : EndA⊗Σ(I+ ) (1A⊗Σ(I+ ) ) → EndA⊗Y (1A⊗Y ). According to Corollary 2.4.4, the resulting map from the colimit of the terms (2.10) EndA⊗Σ(I+ ) (1A⊗Σ(I+ ) ) to EndA⊗Y (1A⊗Y ), is an isomorphism, provided that A has an affine diagonal. In this subsection we will describe the terms (2.10) more explicitly, assuming that A is rigid (note that according to Remark 2.3.7, a rigid symmetric monoidal category automatically has an affine diagonal). 2.7.3. For a (not necessarily pointed) finite set J, consider commutative diagram mult (2.11) J A⊗J − −−−− →   multJ y A ι A  ιt y s − −−− − → A⊗Σ(J ) , where the two arrows ιs , ιt : A → A⊗Σ(J ) correspond to the two maps {∗} ⇒ Σ(J). Note that since the functor (1.3) preserves colimits, (2.11) is in fact a push-out square in DGCatSymMon . Base change defines a natural transformation (2.12) BCJ : multJ ◦(multJ )R → ιR s ◦ ιt . Lemma 2.7.4. Assume that A is rigid. Then the natural transformation BCJ is an isomorphism. Proof. We will prove the assertion more generally for a push-out diagram of rigid symmetric monoidal categories A − −−−−→ A2     y y A1 − −−−−→ A1 ⊗ A2 . A A TOY MODEL FOR SHTUKA 37 Indeed, we rewrite the latter diagram as (A ⊗ A) ⊗ A − −−−−→ (A ⊗ A2 ) ⊗ A A⊗A A⊗A     y y (A1 ⊗ A) ⊗ A − −−−−→ (A1 ⊗ A2 ) ⊗ A. A⊗A A⊗A Now, the assertion follows using [GR1, Chapter 1, Lemma 9.3.6] from the base change property of the diagram A⊗A − −−−−→ A ⊗ A2     y y A1 ⊗ A − −−−−→ A1 ⊗ A2 . Namely, the assertion of loc.cit. says that the right adjoint of a functor between module categories over a rigid monoidal category, which is a priori only right-lax compatible with the action, is actually strictly compatible. Hence, the resulting (commutative) diagram A⊗A ← −−−− − A ⊗ A2     y y A1 ⊗ A ← −−−− − A1 ⊗ A2 takes place in (A ⊗ A) - mod, and hence stays commutative after applying − ⊗ A. A⊗A  2.7.5. Note that ιs (1A ) ≃ 1A⊗Σ(J) ≃ ιt (1A ). Hence, Hom A (1A , ιR s ◦ ιt (1A )) ≃ End A⊗Σ(J) (1A⊗Σ(J) ). Therefore, as a corollary of Lemma 2.7.4, we obtain: Corollary 2.7.6. For A rigid, we have a canonical isomorphism (in Vect) Hom A (1A , multJ ◦ multR J (1A )) → End A⊗Σ(J) (1A⊗Σ(J) ). 2.7.7. Note that the functor multJ is symmetric monoidal. Hence, the functor multR J has a natural right-lax symmetric monoidal structure. In particular, the object (multJ )R (1A ) has a natural structure of commutative algebra in A⊗J , and hence multJ ◦(multJ )R (1A ) has a natural structure of commutative algebra in A. Hence, MapsA (1A , multJ ◦(multJ )R (1A )) acquires a structure of commutative algebra (in Vect). 2.7.8. For a pair of elements ξ1 , ξ2 ∈ MapsA (1A , multJ ◦ multR J (1A )), let us denote by ξ1 ∗ ξ2 their product in MapsA (1A , multJ ◦ multR J (1A )). Explicitly, ξ1 ∗ ξ2 is given by the composition ξ ⊗ξ 1 2 R 1A ≃ 1A ⊗ 1A −→ multJ ◦ multR J (1A ) ⊗ multJ ◦ multJ (1A ) ≃   R R R ≃ multJ multR J (1A ) ⊗ multJ (1A ) → multJ ◦ multJ (1A ⊗ 1A ) ≃ multJ ◦ multJ (1A ). 38 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 2.7.9. We now claim: Lemma 2.7.10. The isomorphism of Corollary 2.7.6 respects the algebra structures on the two sides. Proof. Since (2.11) is a commutative diagram of symmetric monoidal DG categories, the base change morphism (2.12) respects the right-lax symmetric monoidal structures on the two sides. Hence, the induced morphism R Hom A (1A , multJ ◦ multR J (1A )) → Hom A (1A , ιs ◦ ιr (1A )) ≃ End A⊗Σ(J) (1A⊗Σ(J) ) respects the commutative algebra structures. Finally, the resulting algebra structure on End A⊗Σ(J) (1A⊗Σ(J) ) equals one given by the composition, by the Eckmann-Hilton argument.  2.7.11. In what follows for ξ ∈ MapsA (1A , multJ ◦ multR J (1A )), (2.13) we let Eξ ∈ End A⊗Σ(J) (1A⊗Σ(J) ) (2.14) denote the corresponding element. Let us describe explicitly the product operation on the elements Eξ of (2.14). By Lemma 2.7.10, we have Eξ1 · Eξ2 = Eξ1 ∗ξ2 , where ξ1 ∗ ξ2 is the product of ξ1 and ξ2 in MapsA (1A , multJ ◦ multR J (1A )), which is in turn described explicitly in Sect. 2.7.8. 2.8. Action on a module via excursions. 2.8.1. Let us be given a functor of DG categories SY : A⊗Y → C. In particular, the algebra End A⊗Y (1A⊗Y ) acts on the object SY (1A⊗Y ). Recall (see Sect. 2.7.2) that given a map γ I : I → Ω(Y, y), we have a map γ I+ : EndA⊗Σ(I+ ) (1A⊗Σ(I+ ) ) → EndA⊗Y (1A⊗Y ). Recall that to an element ξ ∈ MapsA (1A , multJ+ ◦(multI+ )R (1A )) we associated an element Eξ ∈ End A⊗Σ(I+ ) (1A⊗Σ(J+ ) ). In this subsection we will give an explicit formula for the action of the element γ I+ (Eξ ) ∈ End A⊗Y (1A⊗Y ) on SY (1A⊗Y ). We will do so in a slightly more general context: instead of the pointed finite set I+ we will consider a non-pointed one. A TOY MODEL FOR SHTUKA 2.8.2. 39 For an object Y ′ ∈ Spc and a point y ′ ∈ Y ′ let evy ′ denote the restriction functor LS(Y ′ ) → Vect corresponding to y′ {∗} ֒→ Y ′ . Given two points y1′ and y2′ and a path γ ′ between them, we have a natural transformation monγ ′ : evy1′ → evy2′ , (2.15) given by “parallel transport” along γ. Here is the formal construction: By definition, a path γ ′ is a point of {∗} × {∗}, where the two maps Y′ {∗} ⇒ Y ′ are given by y1′ and y2′ , respectively. Restriction on LS(−) along the maps in the commutative square −−−−→ {∗} × {∗} − Y′   y y′ {∗}  y ′ y 2 1 − −−− −→ Y ′ , {∗} followed by restriction along γ′ {∗} → {∗} × {∗}, Y′ defines the desired isomorphism of functors (2.15). 2.8.3. Ley y1 and y2 be two points of Y . Let J be a finite set, and let us be given a J-tuple γ J of paths from y1 to y2 . We can consider the points y1J , y2J ∈ Y J and regard γ J as a path from y1J to y2J . Fix a point ξ ∈ MapsA (1A , multJ ◦ multR J (1A )). Recall that according to Proposition 1.7.2, the datum of a functor SY is equivalent to the datum of a collection of functors SI : A⊗I → C ⊗ LS(Y I ), which depends functorially on I ∈ fSet. Define the excursion operator ExcSY (γ J , ξ) to be the following endomorphism of SY (1A⊗Y )

ξ −−−−→ evy1 S{∗} (multJ ◦ multR −−−− → evy1 S{∗} (1A ) − SY (1A⊗Y ) − J (1A )) ∼  ∼ y
evy J SJ (multR J (1A )) 1  mon J γ y
evy J SJ (multR J (1A )) 2  ∼ y

∼ counit −−−− − evy2 S{∗} (1A ) ← SY (1A⊗Y ) ← −−−−− evy2 S{∗} (multJ ◦ multR J (1A )) , where: 40 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY • the first and last isomorphisms are obtained by identifying 1A⊗Y with the image of 1A under A ≃ A{∗} → AY , where {∗} → Y is yi , i = 1, 2; • the third and the fifth isomorphisms are obtained by functoriality with respect to the map J → {∗} in fSet; 2.8.4. In the particular case when C = A⊗Y and SY is the identity functor, we will denote the corresponding ExcSY (γ J , ξ) by Excuniv (γ J , ξ) ∈ End A⊗Y (1A⊗Y ). Explicitly, Excuniv (γ J , ξ) is given by the composition (2.16) monγ J ξ R 1A⊗Y ≃ ιy1 (1A ) → ιy1 ◦ multJ ◦ multR J (1A ) ≃ ιy J ◦ multJ (1A ) −→ 1 counit R → ιy J ◦ multR J (1A ) ≃ ιy2 ◦ multJ ◦ multJ (1A ) −→ ιy2 (1A ) ≃ 1A⊗Y , 2 where yi and yiJ denote the functors A → AY and AJ → AY , corresponding to yJ y i {∗} →i Y and J → Y, respectively. By functoriality, for a general C, the map ExcSY (γ J , ξ), is the image of Excuniv (γ J , ξ) under SY . 2.8.5. Example. Let e be a field of characteristic 0. Take A = Rep(G), and assume that Y has finitely many connected components, so that we have an equivalence Rep(G)⊗Y ≃ QCoh(LocSysG (Y )), see Theorem 1.5.5. Then for (γ J , ξ) as above, we can think of Excuniv (γ J , ξ) as an element of Γ(LocSysG (Y ), OLocSysG (Y ) ). Let us describe this element explicitly for a particular (in fact, a generating family) of choices of ξ. J Let VJ be a representation of GJ . Fix an invariant vector and an invariant covector in ResG G (VJ ), i.e., J J ∗ G v : e → ResG G (VJ ) and v : ResG (VJ ) → e. The datum of v ∗ defines by adjunction a map J R VJ → coIndG G (e) = multJ (e). Let ξv,v∗ denote the composite v∗ J v R e → ResG G (VJ ) = multJ (VJ ) → multJ ◦ multJ (e). Let us describe explicitly the element Excuniv (γ J , ξv,v∗ ) ∈ Γ(LocSysG (Y ), OLocSysG (Y ) ) as a function on LocSysG (Y ). Namely, the value of this function at a point σ of LocSysG (Y ) is the composite v J ∼ v∗ J ∼ −−−− → evy J ((VJ )σ ) e − −−−− → evy1 ((ResG G (VJ ))σ ) − 1  mon J γ y e ← −−−−− evy2 ((ResG −−−− − evy J ((VJ )σ ), G (VJ ))σ ) ← 2 A TOY MODEL FOR SHTUKA 41 ′ ′ where for WJ ′ ∈ Rep(GJ ) we denote by (WJ ′ )σ the corresponding object of LS(Y J ). 2.8.6. We are now ready to state the main result of this subsection: Theorem 2.8.7. For (y1 , y2 , γ J ) as above, consider the corresponding map γ J : Σ(J) → Y. Then the excursion operator ExcSY (γ J , ξ) ∈ End(SY (1A⊗Y )) of Sect. 2.8.3 equals the action of the element of EndA⊗Y (1A⊗Y ) obtained as the image of Eξ (see (2.14)) under the map γJ EndA⊗Σ(J) (1A⊗Σ(J) ) → EndA⊗Y (1A⊗Y ). Remark 2.8.8. We emphasize that the assertion of Theorem 2.8.7 holds for any A: we do not need either A to have an affine diagonal. We only use the existence of the base change map Hom A (1A , multJ ◦ multR J (1A )) → End A⊗Σ(J) (1A⊗Σ(J) ), but we do not need this map to be an isomorphism. 2.9. Proof of Theorem 2.8.7. 2.9.1. First off, since the assertion is functorial in C, it suffices to consider the universal case, namely, C = A⊗Y and SY is the identity functor. Second, since the statement is functorial in Y , we can assume that Y = Σ(J), J of paths and γ J is the tautological J-tuple γtaut ∗s → ∗t , where ∗s , ∗t are the two points of Σ(J). Thus, we need to show that J Excuniv (γtaut , ξ) = Eξ (2.17) as objects in EndA⊗Σ(J) (1A⊗Σ(J) ). 2.9.2. Let q denote the map A⊗J → A⊗Σ(J ) , corresponding to either circuit in (2.11). J The path γcan defines a J-tuple of isomorphisms of functors monjcan : ιs → ιt , j∈J so that the composite monj can q ≃ ιs ◦ multJ −→ ιt ◦ multJ ≃ q is the identity map for all j ∈ J. 2.9.3. Unwinding the definitions, we obtain that the LHS of (2.17) is the endomorphism of 1A⊗Σ(J) given by ∼ ξ ∼ ∼ counit ∼ 1A⊗Σ(J) − −−−− → ιs (1A ) − −−−− → ιs ◦ multJ ◦(multJ )R (1A ) − −−−−→ multJ ◦ι⊗J ◦ (multJ )R (1A ) s N j  yj∈J moncan −−−− − ιt (1A ) ← −−−− − ιt ◦ multJ ◦(multJ )R (1A ) ← −−−−− multJ ◦ι⊗J ◦ (multJ )R (1A ), 1A⊗Σ(J) ← t and the RHS is ξ BC counit J ιs ◦ ιR 1A⊗Σ(J) ≃ ιs (1A ) → ιs ◦ multJ ◦(multJ )R (1A ) −→ s ◦ ιt (1A ) −→ ιt (1A ) ≃ 1A⊗Σ(J) , where BCJ is an is (2.12). 42 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 2.9.4. We obtain that it suffices to show that the following two natural transformations ιs ◦ multJ ◦(multJ )R ⇒ ιt coincide: One is the composite N (2.18) R ιs ◦ multJ ◦(multJ ) ≃ multJ ◦ι⊗J s ◦ (multJ ) R j∈J monjcan multJ ◦ι⊗J ◦ (multJ )R ≃ t ≃ ≃ ιt ◦ multJ ◦(multJ )R → ιt . and the other is the composite BC J ιs ◦ multJ ◦(multJ )R −→ ιs ◦ ιR s ◦ ιt → ιt . (2.19) 2.9.5. However, unwinding the definition of BCJ , we obtain that both (2.18) and (2.19) identify with ιs ◦ multJ ◦(multJ )R ≃ q ◦ (multJ )R ≃ ιt ◦ multJ ◦(multJ )R → ιt . [Theorem 2.8.7] 3. Taking the trace In this section we will approach the central theme of this paper: the operation of taking the trace. The usual trace construction assigns to an endomorphism F of a dualizable object o in a symmetric monoidal category O its trace Tr(F, o), which is an endomorphism of the unit object 1O , see Sect. 3.1.1. However, our primary interest will be the notion of higher trace, when O is actually a symmetric monoidal 2-category. In this case, the trace construction has an additional functoriality, see Sect. 3.2.1. We will apply this formalism in the following two contexts: O = DGCat, in which our traces will be vector spaces, and O = Morita(DGCat), in which our traces will be DG-categories. The main result of this section is Theorem 3.8.5, which describes the interaction between the trace operations at different categorical levels. 3.1. The usual trace. 3.1.1. Let O be a symmetric monoidal category. Given a dualizable object o ∈ O and a point F ∈ EndO (o), we define its trace Tr(F, o) ∈ EndO (1O ) to be the composite unit 1O −→ o ⊗ o∨ 3.1.2. F ⊗ido∨ counit −→ o ⊗ o∨ −→ 1O . The assignment (o, F ) 7→ Tr(F, o) is symmetric monoidal, i.e., we have a canonical isomorphism Tr(F1 ⊗ F2 , o1 ⊗ o2 ) ≃ Tr(F1 , o1 ) · Tr(F2 , o2 ), where · denotes the structure of commutative monoid on End(1O ) induced by the symmetric monoidal structure on O, see [TV, Sect. 2.5], along with higher compatibilities. In particular (3.1) Tr(id1O , 1O ) = id1O = 1EndO (o) . A TOY MODEL FOR SHTUKA 43 3.1.3. For a morphism F : o1 → o2 between dualizable objects let F ∨ denote the dual morphism ∨ o∨ 2 → o1 . Let qF be the point in Maps(1O , o2 ⊗ o∨ 1 ) that represents F , i.e., the composite unit 1O −→ o1 ⊗ o∨ 1 F ⊗ido∨ −→ 1 o2 ⊗ o∨ 1. We have qF = qF ∨ . From here it follows that for every o and F as in Sect. 3.1.1, we have Tr(F, o) = Tr(F ∨ , o∨ ). 3.1.4. Similarly, the trace map has the following cyclicity property: we claim that for morphisms F1,2 : o1 → o2 and F2,1 : o2 → o1 , there is a canonical isomorphism Tr(F1,2 ◦ F2,1 , o2 ) ≃ Tr(F2,1 ◦ F1,2 , o1 ). (3.2) Indeed, let qi,j ∈ Maps(1O , oj ⊗ o∨ i ) be the point that represents Fi,j . Then Fi,j ◦ Fj,i is represented by the map 1O ≃ 1O ⊗ 1O qi,j ⊗qj,i −→ ∨ oj ⊗ o∨ i ⊗ oi ⊗ oj idoj ⊗ counit ⊗ ido∨ j −→ ∨ oj ⊗ 1O ⊗ o∨ j ≃ oj ⊗ oj . Hence, Tr(Fi,j ◦ Fj,i , oj ) is the composite 1O ≃ 1O ⊗ 1O qi,j ⊗qj,i −→ ∨ oj ⊗ o∨ i ⊗ oi ⊗ oj idoj ⊗ counit ⊗ ido∨ −→ j counit ∨ oj ⊗ 1O ⊗ o∨ j ≃ oj ⊗ oj −→ 1O , and the latter expression is manifestly symmetric in i and j. 3.2. Trace in a 2-category. 3.2.1. Let now O be a symmetric monoidal 2-category (we will be assuming the formalism of (∞, 2)categories from [GR1, Chapter 10]). Let o1 and o2 be a pair of dualizable objects, each endowed with an endomorphism Fi . Let t : o1 → o2 be a 1-morphism that admits a right adjoint. This means that there exists a 1-morphism tR : o2 → o1 and 2-morphisms ido1 → tR ◦ t and t ◦ tR → ido2 that satisfy the usual axioms, see [GR1, Chapter 12, Sect. 1]. Let Fi be a 1-endomorphism of oi . In addition, let us be given a 2-morphism α : t ◦ F1 → F2 ◦ t (3.3) F1 / o1 ⑧⑧ ⑧ ⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t t ⑧⑧ α ⑧⑧⑧⑧ ⑧ ⑧ ⑧  { ⑧⑧⑧  / o2 o2 o1 F2 In this case, following [BN1, Definition 2.24] or [KP1, Example 1.2.5], we define the 2-morphism Tr(t, α) : Tr(F1 , o1 ) → Tr(F2 , o2 ) 44 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY to be the composite / o1 ⊗ o∨1 F1 ⊗id/ o1 ⊗ o∨1 counit / 1O ⑧⑧ ⑧⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧⑧ ⑧⑧⑧ ⑧⑧⑧ t⊗top ⑧ t⊗top ⑧ id id ⑧⑧⑧ ⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧  { ⑧⑧⑧  { ⑧⑧⑧   { ⑧⑧⑧ / o2 ⊗ o∨2 / o2 ⊗ o∨2 / 1O , 1O unit 1O F2 ⊗id unit counit where: ∨ R ∨ –top denotes the 1-morphism o∨ 1 → o2 equal to (t ) ; –the 2-morphism in the left square is given by the (t, tR )-adjunction; –the 2-morphism in the middle square is given by α; –the 2-morphism in the right square is given by the (t, tR )-adjunction. Remark 3.2.2. The above construction is equivalent to formula (0.3) given earlier. 3.2.3. The construction of Sect. 3.2.1 is compatible with compositions: For a composition t1,2 t2,3 o1 → o2 → o3 , the 2-morphism Tr(F1 , o1 ) identifies with Tr(t1,3 , α1,3 ), where Tr(t1,2 ,α1,2 ) −→ Tr(F2 , o2 ) Tr(t2,3 ,α2,3 ) −→ Tr(F3 , o3 ) t1,3 = t2,3 ◦ t1,2 , and α1,3 is obtained by composing α1,2 and α2,3 . 3.2.4. The construction of Sect. 3.2.1 has the following cyclicity property: Given a pair of diagrams of 2-morphisms (3.4) F1,2 / o2 ⑧⑧ ⑧ ⑧ ⑧ ⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t1 t2 ⑧ ⑧ ⑧⑧⑧⑧ α ⑧ ⑧ ⑧  { ⑧⑧⑧  ′ / o′2 o1 o1 ′ F1,2 F2,1 / o1 ⑧⑧ ⑧ ⑧ ⑧ ⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t2 t1 ⑧ ⑧ ⑧⑧⑧⑧ β ⑧ ⑧ ⑧   { ⑧⑧⑧ ′ / o′1 , o2 o2 ′ F2,1 we can compose them horizontally in two ways, thus getting diagrams o1 F2,1 ◦F1,2 / o1 ⑧⑧ ⑧ ⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t1 t1 ⑧⑧ ⑧⑧⑧⑧ α◦β ⑧ ⑧ ⑧  { ⑧⑧⑧  ′ / o′1 o1 ′ ′ F2,1 ◦F1,2 o2 F1,2 ◦F2,1 / o2 ⑧⑧ ⑧ ⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t2 t2 ⑧⑧ ⑧⑧⑧⑧ β◦α ⑧ ⑧ ⑧  { ⑧⑧⑧  ′ / o′2 . o2 ′ ′ F1,2 ◦F2,1 Assume now that objects o1 , o2 , o′1 and o′2 are dualizable, while 1-morphisms t1 and t2 admit right adjoints. Then we claim that the trace maps ′ ′ Tr(t1 , α ◦ β) : Tr(F2,1 ◦ F1,2 , o1 ) → Tr(F2,1 ◦ F1,2 , o′1 ) and ′ ′ Tr(t2 , β ◦ α) : Tr(F1,2 ◦ F2,1 , o2 ) → Tr(F1,2 ◦ F2,1 , o′2 ) A TOY MODEL FOR SHTUKA 45 match up under identifications ′ ′ ′ ′ , o′1 ) Tr(F1,2 ◦ F2,1 , o2 ) ≃ Tr(F2,1 ◦ F1,2 , o1 ) and Tr(F1,2 ◦ F2,1 , o′2 ) ≃ Tr(F2,1 ◦ F1,2 of (3.2). More precisely, we claim that there is a canonical isomorphism Tr(t1 , α ◦ β) ≃ Tr(t2 , β ◦ α) (3.5) in ′ ′ ′ ′ Funct(Tr(F2,1 ◦ F1,2 , o1 ), Tr(F2,1 ◦ F1,2 , o′1 )) ≃ Funct(Tr(F1,2 ◦ F2,1 , o2 ), Tr(F1,2 ◦ F2,1 , o′2 )). ′ ′ ′∨ Indeed, let qi,j ∈ Maps(1O , oj ⊗ o∨ i ) and qi,j ∈ Maps(1O , oj ⊗ o i ) be the points that represent Fi,j ′ and Fi,j , respectively. Then arguing as in Sect. 3.1.4 one sees that both maps Tr(t1 , α ◦ β) and Tr(t2 , β ◦ α) are canonically isomorphic to the composite q2,1 ⊗q1,2 / 1O / o1 ⊗ o∨2 ⊗ o2 ⊗ o∨1 counit ♦♦♦ ♦♦ ♦ ♦ ♦ ♦ ♦ ♦♦ ♦♦♦ ♦♦♦ op op ♦♦ ♦ t ⊗t ⊗t ⊗t ♦ id id ♦ 1 2 2 1 ♦ ♦ ♦♦♦ ♦♦♦ ♦ ♦ ♦ ♦  s{ ♦♦   s{ ♦♦♦ / o1 ⊗ o∨2 ⊗ o2 ⊗ o∨1 / 1O , 1O 1O q2,1 ⊗q1,2 counit where: ∨ ′∨ R ∨ –top i denotes the 1-morphism oi → o i equal to (ti ) ; –the 2-morphism in the left square is given by the (ti , tR j )-adjunctions and 2-morphisms α and β; –the 2-morphism in the right square is given by the (ti , tR i )-adjunction. 3.2.5. By functoriality, isomorphisms (3.5) from Sect. 3.2.4 are compatible with vertical compositions: Consider a pair of diagrams of 2-morphisms (3.6) F1,2 / o2 ⑧⑧ ⑧ ⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t1 t2 ⑧⑧ ⑧⑧⑧⑧ α ⑧ ⑧ ⑧ ′  { ⑧⑧⑧ F1,2  ′ / o′2 o1 ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ ⑧⑧ t′2 t′1 ⑧⑧⑧⑧α′ ⑧ ⑧ ⑧ ⑧⑧   { ⑧⑧⑧ ′′ / o′′2 o1 o1 ′′ F1,2 F2,1 / o1 ⑧⑧ ⑧ ⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t2 t1 ⑧⑧ ⑧⑧⑧⑧ β ⑧ ⑧ ⑧ ′  { ⑧⑧⑧ F2,1  ′ / o′1 o2 ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ ⑧⑧ t′1 t′2 ⑧⑧⑧⑧β ′ ⑧ ⑧ ⑧ ⑧⑧   { ⑧⑧⑧ ′′ / o′′1 , o2 o2 ′′ F2,1 in which all objects are dualizable and all vertical 1-morphisms admit right adjoints. Then isomorphisms (3.5), corresponding to the top part, the bottom part and the vertical composition of the diagrams (3.6), respectively, are (3.7) Tr(t1 , α ◦ β) ≃ Tr(t2 , β ◦ α), Tr(t′1 , α′ ◦ β ′ ) ≃ Tr(t′2 , β ′ ◦ α′ ) and (3.8) Tr(t′1 ◦ t1 , (α′ ◦ α) ◦ (β ′ ◦ β)) ≃ Tr(t′2 ◦ t2 , (β ′ ◦ β) ◦ (α′ ◦ α)). 46 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Moreover, by the fuctoriality of (3.5) the following diagram is commutative: Sect. 3.2.3 (3.9) Tr(t′1 ◦ t1 , (α′ ◦ α) ◦ (β ′ ◦ β)) −−−−−−→ Tr(t′1 , α′ ◦ β ′ ) ◦ Tr(t1 , α ◦ β) ∼     (3.7)y∼ (3.8)y∼ Sect. 3.2.3 Tr(t′2 ◦ t2 , (β ′ ◦ β) ◦ (α′ ◦ α)) −−−−−−→ Tr(t′2 , β ′ ◦ α′ ) ◦ Tr(t2 , β ◦ α). ∼ 3.3. Properties of the 2-categorical trace. In this subsection we will explore further functoriality properties of the construction of Sect. 3.2.1. We recommend the reader to skip this subsection and return to it when necessary. 3.3.1. The functoriality mentioned in Sect. 3.2.3 can be promoted to a symmetric monoidal functor between symmetric monoidal (∞, 1)-categories: Consider the category, to be denoted L(O), whose objects are pairs (o, F ), where o ∈ O and F ∈ EndO (o), and whose morphisms are given by diagrams (3.3), see [KP1, Sect. 1.2]10 . The symmetric monoidial structure on O induces one on L(O). Let L(O)rgd ⊂ L(O) be the 1-full subcategory, where we allow as objects those (o, F ) for which o is dualizable as an object of O, and where we restrict 1-morphisms to those pairs (t, α), for which t admits a right adjoint. Then the assignment (o, F ) 7→ Tr(F, o) is a symmetric monoidal functor (3.10) Tr : L(O)rgd → EndO (1O ). The construction of the functor (3.10) can be either performed directly using the definition of ∞categorical symmetric monoidal structures as in [GR1, Chapter 1, Sect. 3.3], or using the device of [HSS, Theorem 1.7]. 3.3.2. Let a ∈ O be an associative/commutative algebra object in O, so that the multiplication map a⊗a→a admits a right adjoint. Assume also that a is dualizable as an object of O. Let Fa be a right-lax monoidal/symmetric monoidal endomorphism of a (see, e.g., Sect. 3.4.4). Then (a, Fa ) acquires a structure of associative/commutative algebra object in L(O)rgd . Hence, we obtain that Tr(Fa , a) acquires a structure of associative/commutative algebra object in EndO (1O ). 3.3.3. Let a be as above, and let m ∈ O be an a-module object. Assume that the action map a⊗m→a admits a right adjoint. Assume also that m is dualizable as an object of O. Let Fm be an endomorphism of m that is right-lax compatible with Fa (see, e.g., Sect. 3.4.4). Then (m, Fm ) acquires a structure of module over (a, Fa ) in L(O)rgd . Hence, we obtain that Tr(Fm , m) ∈ EndO (1O ) acquires a structure of module over Tr(Fa , a). 10To see the full construction of L(O) as an ∞-category see [GR1, Chapter 10, Sect. 4.1]. A TOY MODEL FOR SHTUKA 47 3.3.4. Let a and a′ be a pair of associative/commutative algebra objects in O as in Sect. 3.3.2, each endowed with a right-lax monoidal/symmetric monoidal endomorphism. Let ϕ be a right-lax monoidal/symmetric monoidal map a → a′ , equipped with a 2-morphism in the diagram Fa /a ⑧⑧ ⑧ ⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ϕ ϕ ⑧ ⑧⑧⑧⑧ α ⑧ ⑧ ⑧ ⑧⑧   { ⑧⑧⑧ ′ / a′ , a a Fa′ compatible with the right-lax monoidal/symmetric monoidal structures on the edges. Assume that ϕ admits a right adjoint. We can view the data of (ϕ, α) as a morphism of associative/commutative algebra objects (a, Fa ) → (a′ , Fa′ ) in L(O)rgd . Hence, the above data induces a map Tr(Fa , a) → Tr(Fa′ , a′ ), which is a map of associative/commutative algebras, as follows from the functoriality of the trace construction, see Sect. 3.2.3. 3.3.5. Note that (1O , id) is the unit in L(O)rgd (see Sect. 3.3.1). Since Tr is a symmetric monoidal functor, we can identify Tr(id, 1O ) = 1EndO (1O ) as objects of EndO (1O ). Let now a be as in Sect. 3.3.2. Consider the unit map 1O → a, (3.11) equipped with the 2-morphism (3.12) id / 1O ⑧⑧ ⑧ ⑧ ⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ ⑧ ⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ ⑧   { ⑧⑧⑧ / a, a 1O Fa provided by the right-lax monoidal structure on Fa . Assume that (3.11) admits a right adjoint. It follows from Sect. 3.3.4 that the resulting map 1EndO (1O ) ≃ Tr(id, 1O ) → Tr(Fa , a) is the unit of Tr(Fa , a) as an associative/commutative algebra object in EndO (1O ). 3.4. Trace on DG categories. 3.4.1. The example of primary interest for us is O = DGCat, with its natural symmetric monoidal structure. Note that 1DGCat = Vect, so (3.13) End(1DGCat ) ≃ Vect, as a category, equipped with a symmetric monoidal structure. Hence, for a dualizable DG category C equipped with an endofunctor F , we obtain an object Tr(F, C) ∈ Vect . 48 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Furthermore, if T : C1 → C2 is a morphism in DGCat (i.e., a colimit-preserving e-linear exact functor) between dualizable DG categories that admits a continuous right adjoint, and given a natural transformation α : T ◦ F1 → F2 ◦ T, we obtain a map in Vect Tr(T, α) : Tr(F1 , C1 ) → T r(F2 , C2 ). 3.4.2. Take C = Vect and F = Id. Then by (3.1), we have Tr(Id, Vect) ≃ e, as an object in (3.13). More generally, for V ∈ Vect, the trace of the endofunctor of Vect, given by − ⊗ V , identifies with V as an object in (3.13). 3.4.3. Take C2 = C and C1 = Vect with the functor T corresponding to an object c ∈ C, i.e., the (unique) colimit preserving functor satisfying e 7→ c. (3.14) Note that the condition that T admit a continuous right adjoint is equivalent to the condition that c be compact. Let F2 = F be some endofunctor of C and take F1 = Id. Then the datum of α amounts to a morphism α : c → F (c). The resulting map Tr(T,α) e ≃ Tr(Id, Vect) −→ Tr(F, C) corresponds to a point in Tr(F, C), which we will denote by cl(c, α). 3.4.4. Let R be a monoidal/symmetric monoidal DG category, and M an R-module category. Assume that the functors R ⊗ R → R and R ⊗ M → M, viewed as functors of plain DG categories admit right adjoints (if R and M are compactly generated, the condition of admitting a right adjoint is equivalent to preserving compactness). Let R be endowed with a right-lax monoidal/symmetric monoidal endofunctor FR , i.e., we have a natural transformation FR (a1 ) ⊗ FR (a2 ) → FR (a1 ⊗ a2 ), a1 , a2 ∈ R equipped with higher compatibilities. Assume also that M is dualizable and is endowed with an endofunctor φM that is right-lax compatible with the R-action. I.e., we have a natural transformation FR (a) ⊗ FM (m) → FM (a ⊗ m), a ∈ R, m ∈ M, equipped with higher compatibilities. In this case, applying the construction of Sects. 3.3.2 and 3.3.3, we obtain that Tr(FR , R) acquires a structure of associative/commutative algebra, and Tr(FM , M) acquires a structure of Tr(FR , R)-module. A TOY MODEL FOR SHTUKA 49 3.4.5. Let R and M be as above. Let r ∈ R be a compact object equipped with a map α : r → FR (r). On the one hand, we can consider cl(r, α) ∈ Tr(FR , R). On the other hand, let Hr denote the endofunctor of M given by the action of r (here “H” should be evocative of “Hecke”). The right-lax compatibility of FM with the action defines a natural transformation α Hr ◦ FM → HFR (r) ◦ FM → FM ◦ Hr (3.15) which we denote by αr,M . By Sect. 3.2.1, to the pair (Hr , αr,M ) we can assign the map Tr(Hr , αr,M ) : Tr(FM , M) → Tr(FM , M). We claim: Proposition 3.4.6. The action of cl(r, α) on Tr(FM , M) equals Tr(Hr , αr,M ). Proof. Follows from the functoriality of the trace construction (see Sect. 3.2.3) corresponding to the composition r⊗Id act M ≃ Vect ⊗M −→ R ⊗ M −→ M.  3.5. Examples. The results from this subsection will not be used in the rest of the paper. However, they are meant to provide an intuition for the behavior of the categorical trace construction. 3.5.1. Consider the example C = R-mod, where R ∈ AssocAlg(Vect). We have R1 -mod ⊗ R2 -mod ≃ (R1 ⊗ R2 )-mod, see [Lu2, Theorem 4.8.5.16]. In particular, (R-mod)∨ identifies with Rrev -mod, where Rrev is obtained from R by reversing the multiplication. The unit and counit map are given by the functors Vect → (R ⊗ Rrev )-mod, e 7→ R, and (R ⊗ Rrev )-mod → Vect, Q 7→ R ⊗ R⊗Rrev Q, respectively. Identifying End(R-mod) ≃ R-mod ⊗ (R-mod)∨ ≃ (R ⊗ Rrev )-mod, we obtain that every continuous endofunctor of R-mod is of the form M 7→ FQ (M ) := Q ⊗ M R for Q ∈ (R ⊗ R rev )-mod. The trace of such an endofunctor is given by HH• (R, Q) := R ⊗ R⊗Rrev Q. The identity endofunctor of R-mod corresponds to Q = R. In this case we use a simplified notation HH• (R) := HH• (R, Q). This is the vector space of Hochschild chains on R. 50 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 3.5.2. Let Y be a prestack (see [GR1, Chapter 2, Sect. 1]), such that: –the category QCoh(Y) is dualizable; –the object OY ∈ QCoh(Y) is compact; –the diagonal morphism ∆ : Y → Y × Y is schematic and qsqc (quasi-separated and quasi-compact). For example, these conditions are satisfied for a quasi-compact algebraic stack with affine diagonal, of finite type over a field of characteristic 0 (see [DrGa1, Theorem 1.4.2]). The condition that OY ∈ QCoh(Y) is compact is equivalent to one saying that the functor of global sections Γ(Y, −) : QCoh(Y) → Vect is continuous. The condition that ∆ is schematic and qsqc implies that the direct image functor ∆∗ : QCoh(Y) → QCoh(Y × Y) is continuous and satisfies base change. In this case, the functors e7→O ∆ ∗ QCoh(Y × Y) ≃ QCoh(Y) ⊗ QCoh(Y) Vect → Y QCoh(Y) −→ and ∆∗ QCoh(Y) ⊗ QCoh(Y) ≃ QCoh(Y × Y) −→ QCoh(Y) Γ(Y,−) → Vect define an identification QCoh(Y)∨ ≃ QCoh(Y) (the proof is a diagram chase using base change). 3.5.3. Let φ be an endomorphism of Y and consider the endofunctor of QCoh(Y) given by φ∗ . We claim that Tr(φ∗ , QCoh(Y)) identifies canonically with Γ(Yφ , OYφ ), where Yφ := Y × Y Graphφ ,Y×Y,∆ is the fixed point locus of φ. Indeed, let ι denote the forgetful map Yφ → Y, so that we have ι ◦ φ ≃ ι. We calculate Tr(φ∗ , QCoh(Y)) as the composite QCoh(Y) x  ∗ ∆ ∆ QCoh(Y × Y) x (φ×id)∗  −−−∗−→ QCoh(Y × Y) QCoh(Y) − x  p∗ Y QCoh(pt), where pY : Y → pt is the projection map. (p )∗ − −−Y−−→ QCoh(pt) A TOY MODEL FOR SHTUKA 51 By base change, we rewrite this functor as (p )∗ ι ∗ −−Y−− → QCoh(pt) −−− −→ QCoh(Y) − QCoh(Yφ ) − x  ι∗  QCoh(Y) x  p∗ Y QCoh(pt), which sends e ∈ Vect = QCoh(pt) to Γ(Yφ , OYφ ), as desired. 3.5.4. Let now F ∈ QCoh(Y) be a compact object, equipped with a map α : F → φ∗ (F). Consider the corresponding object cl(F, α) ∈ Tr(φ∗ , QCoh(Y)), see Sect. 3.4.3. We will now describe explicitly the image of cl(F, α) under the identification Tr(φ∗ , QCoh(Y)) ≃ Γ(Yφ , OYφ ). First, we claim: Lemma 3.5.5. Every compact object F ∈ QCoh(Y) is dualizable in the sense of the symmetric monoidal structure on QCoh(Y). Proof. The proof runs parallel to [BFN, Proposition 3.6]: Since the diagonal morphism of Y is schematic and qsqc, any map f : S → Y, where S an affine scheme, is itself schematic and qsqc. In particular, the functor f∗ , right adjoint to f ∗ , is continuous. Hence, the functor f ∗ preserves compactness. This implies that the pullback of every compact object F ∈ QCoh(Y) to every affine scheme S is perfect, and hence dualizable as an object of the symmetric monoidal category QCoh(S). Since QCoh(Y) ≃ lim QCoh(S), S→Y we obtain that F is dualizable in QCoh(Y).  3.5.6. Consider the pullback of α along the map ι : Yφ → Y. Using the fact that ι = φ ◦ ι, we obtain a map α ι∗ (F) → ι∗ ◦ φ∗ (F) = (φ ◦ ι)∗ (F) ≃ ι∗ (F). This is an endomorphism of ι∗ (F), which we denote αφ . We claim: Proposition 3.5.7. The element cl(F, α) ∈ Tr(φ∗ , QCoh(Y)) ≃ Γ(Yφ , OYφ ) identifies with Tr(αφ , ι∗ (F)), where the latter trace is taken in the symmetric monoidal category QCoh(Yφ ), and where we identify EndQCoh(Yφ ) (1QCoh(Yφ ) ) = EndQCoh(Yφ ) (OYφ ) ≃ Γ(Yφ , OYφ ). 52 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY This is proved in [KP1, Prop. 2.2.3]; see Sect. 4.2 below for a proof in a more general context. 3.5.8. Let now Y be a quasi-compact algebraic stack with an affine diagonal over a ground field of characteristic 0. Consider the category D-mod(Y). In this case, the functors ∆dR,∗ e7→ω Vect −→Y D-mod(Y) −→ D-mod(Y × Y) ≃ D-mod(Y) ⊗ D-mod(Y) and ∆! D-mod(Y) ⊗ D-mod(Y) ≃ D-mod(Y × Y) −→ D-mod(Y) ΓdR (Y,−) −→ Vect define an identification D-mod(Y)∨ ≃ D-mod(Y). Let φ be an endomorphism of F. Consider the endofunctor of D-mod(Y), given by φdR,∗ . Then as in Sect. 3.5.3 one shows that there exists a canonical identification Tr(φdR,∗ , D-mod(Y)) ≃ ΓdR (Yφ , ωYφ ), where the latter object is usually called the Borel-Moore homology of Yφ . 3.5.9. Let F be a compact object of D-mod(Y) equipped with a map α : F → φdR,∗ (F). To this object there corresponds an element cl(F, α) ∈ Tr(φdR,∗ , D-mod(Y)) ≃ ΓdR (Yφ , ωYφ ). However, we do not at the moment know how to give an explicit formula for this element, which would be reminiscent of that of Proposition 3.5.7. However, according to [Va], the following particular case is known: Theorem 3.5.10. Assume that Y is a scheme, and let y ∈ Yφ be an isolated fixed point. Assume, moreover, that dφ|Ty (Y ) does not have eigenvalue 1 (i.e., the derived fixed point locus Yφ is smooth at y). Assume that F is holonomic. Then the image of cl(F, α) under the projection on the direct summand corresponding to y ΓdR (Yφ , ωYφ ) → Vect equals Tr(αφ , ιdR,∗ (F)), y y where ιy denotes the embedding pt → Y, and αφ denotes the induced endomorphism of ιdR,∗ (F) equal y to ιdR,∗ (F) ≃ (φ ◦ ιy )dR,∗ (F) = ιdR,∗ ◦ φdR,∗ (F) y y and where α′ : φdR,∗ (F) → F is obtained from α by the (φdR,∗ , φdR,∗ )-adjunction. ιdR,∗ (α′ ) y −→ ιdR,∗ (F), y A TOY MODEL FOR SHTUKA 53 3.5.11. A prototype for the moduli of shtukas. Here we generalize some aspects of the above example and relate it to the construction of the moduli space of shtukas (in the setting of D-modules in characteristic 0). Let Y be a quasi-compact algebraic stack with affine diagonal over a field of characteristic 0. Suppose we have a “Frobenius” endomorphism φ : Y → Y. Moreover, suppose that we have a correspondence p Z /Y q  Y and a sheaf K ∈ D-mod(Z). Let HK denote the functor HK = q∗ (p! (−) ⊗ K) : D-mod(Y) → D-mod(Y). By the same considerations as above, we obtain Tr(HK ◦ φ∗,dR , D-mod(Y)) ≃ ΓdR (Zφ , i! (K)), where Zφ is the pullback Zφ i /Z (p,q)  Y (φ,id)  / Y×Y Analogously, consider Y = BunG (X), where X is an algebraic curve over a finite field Fq , φ is the morphism induced by the Frobenius on X, Z is the Hecke stack (of modifications at some legs xI ∈ X I ) and K is the Satake image of a representation of G∨ . In this case, Zφ is the moduli stack of shtukas and Γ(Zφ , i! (K)) are the corresponding cohomologies of interest on the moduli of shtukas. 3.6. Trace on DG 2-categories. 3.6.1. We will now study a different instance of the abstract formalism of Sect. 3.2.1. Namely, we take O := Morita(DGCat). By definition, objects of this category are parameterized by R ∈ DGCatMon . We will denote the corresponding object of Morita(DGCat) by R - mod. For two objects R1 - mod, R2 - mod ∈ Morita(DGCat), the (∞, 1)-category of 1-morphisms (3.16) MapsMorita(DGCat) (R1 - mod, R2 - mod) is by definition R2 ⊗ Rrev 1 - mod. For an object Q ∈ R2 ⊗ Rrev 1 - mod, we will denote the corresponding object of (3.16) by TQ . The identity 1-morphism is given by TR , where we view R as an R ⊗ Rrev -module category. The composition of two 1-morphisms TQ1,2 TQ2,3 R1 - mod −→ R2 - mod −→ R3 - mod is set to be TQ1,3 , where TQ1,3 := TQ2,3 ⊗ TQ1,2 . R2 54 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 3.6.2. Note that a 1-morphism T : R1 - mod → R2 - mod defines a functor of 2-categories R1 - mod → R2 - mod, (3.17) which, by a slight abuse of notation, we will denote by the same character T. Namely, for T = TQ , the functor (3.17) is given by M 7→ Q ⊗ M. R1 Sometimes, we will refer to this construction as “evaluation of T on M”. Remark 3.6.3. We use Morita(DGCat) as defined above as a substitute for the (∞, 2)-category of “2-DG categories”, hence the title of this subsection. Whatever the latter is, it contains Morita(DGCat) as a full subcategory, which consists of unigenerated 2-DG categories. By way of analogy, Morita(Vect) is a full subcategory of DGCat that consists of DG categories that can be generated by a single compact object. 3.6.4. The symmetric monoidal structure on Morita(DGCat) is given by R1 - mod ⊗ R2 - mod := (R1 ⊗ R2 ) - mod. Note that the unit object of Morita(DGCat) is Vect - mod =: DGCat. 3.6.5. Every object of Morita(DGCat) is dualizable. We have R - mod∨ ≃ Rrev - mod, where the unit and counit map are both given by R ∈ (R ⊗ Rrev ) - mod. Under this identification, for a 1-morphism R1 - mod → R2 - mod corresponding to M ∈ (Rrev 1 ⊗ R2 ) - mod, the dual 1-morphism 3.6.6. Rrev 2 - mod → Rrev 1 - mod is given by the same M. We have EndMorita(DGCat) (1Morita(DGCat) ) ≃ DGCat, as a symmetric monoidal category. Further, MapsMorita(DGCat) (1Morita(DGCat) , R - mod) ≃ R - mod. 3.6.7. Thus, we obtain that for every object C ∈ Morita(DGCat) and its endomorphism F, we can attach Tr(F, C) ∈ DGCat . Furthermore, if T : C1 → C2 is a 1-morphism that admits a right adjoint, and given α : T ◦ F1 → F2 ◦ T, we obtain a functor Tr(T, α) : Tr(F1 , C1 ) → Tr(F2 , C2 ). A TOY MODEL FOR SHTUKA 55 3.6.8. Note that for Q ∈ R2 ⊗ Rrev 1 - mod the condition that FQ admit a right adjoint means that Q is right-dualizable as a bimodule category. By definition, this means that there exists an object QR ∈ R1 ⊗ Rrev 2 - mod equipped the map R1 → QR ⊗ Q R2 in (R1 ⊗ Rrev 1 ) - mod and a map Q ⊗ QR → R2 R1 in (R2 ⊗ Rrev 2 ) - mod that satisfy the usual axioms. 3.6.9. The case of particular interest for us is when the 1-morphism R2 - mod → R1 - mod is given by a monoidal functor Ψ : R1 → R2 , i.e., it is given by the (R1 , R2 )-bimodule QΨ , which is isomorphic to R2 as a DG category, on which R2 acts by right multiplication, and R1 acts by left multiplication via Ψ. Denote this 1-morphism by ResΨ . The corresponding functor ResΨ : R2 - mod → R1 - mod is given by restriction via Ψ. The 1-morphism ResΨ tautologically admits a left adjoint, denoted IndΨ . It is given by the (R2 , R1 )bimodule R2 , on which R2 acts by left multiplication, and R1 acts by right multiplication via Ψ. The unit map for the adjunction is given by R1 → R2 ≃ R2 ⊗ R2 , Ψ R2 and the counit of the adjunction is given by the multiplication map R2 ⊗ R2 → R2 . R1 The corresponding functor IndΨ : R1 - mod → R2 - mod is given by M 7→ R2 ⊗ M. R1 3.6.10. Assume now that R1 and R2 are rigid (see [GR1, Chapter 1, Sect. 9.1]). In this case, the 1-morphism ResΨ admits also a right adjoint, denoted coIndΨ . The corresponding (R2 , R1 )-bimodule is again R2 . The unit map for the adjunction. R2 → R2 ⊗ R2 R1 is the right adjoint to the multiplication functor R2 ⊗ R2 → R2 R1 (the right adjoint is a functor of R2 -bimodule categories due to rigidity). The counit of the adjunction is the functor R2 ⊗ R2 ≃ R2 → R1 , R2 right adjoint to Ψ (again, this right adjoint is a functor of R1 -bimodule categories due to rigidity). Remark 3.6.11. Note that we obtain that in the situation when R1 and R2 are rigid, the left and right adjoints of ResΨ are canonically isomorphic. 56 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 3.7. The 2-categorical trace and (categorical) Hochschild chains. 3.7.1. For R ∈ DGCatMon and Q ∈ (R ⊗ Rrev ) - mod, denote HH• (R, Q) := R ⊗ R⊗Rrev HH• (R) := R Q ∈ DGCat; ⊗ R⊗Rrev R. As in Sect. 3.5.1 we obtain formally that Tr(TQ , R-mod) ≃ HH• (R, Q). (3.18) 3.7.2. Let Q be given by a monoidal endofunctor FR of R, see Sect. 3.6.9, i.e., Q = QFR . In this case, by a slight abuse of notation, we will write HH• (R, FR ) instead of HH• (R, QFR ). 3.7.3. Example. Let R = QCoh(Y) for Y as in Sect. 3.5.2 and with affine diagonal11, and let FR be given by φ∗ for an endomorphism φ of Y. By Corollary 1.6.5, we have: HH• (QCoh(Y), φ∗ ) ≃ QCoh(Yφ ). 3.7.4. Let us consider another example: Let A be a symmetric monoidal category, and for a space Y consider A⊗Y . Let φ be an endomorphism of Y . By functoriality, it induces a symmetric monoidal functor A⊗φ : A⊗Y → A⊗Y . Note that since the monoidal structures involved are symmetric, the category HH• (A⊗Y , A⊗φ ) also acquires a symmetric monoidal structure. Let Y /φ denote the quotient of Y by φ, i.e., Y /φ := Y ⊔ id ⊔ id,Y ⊔Y,φ⊔id Y, where the subscripts indicate the morphisms with respect to which we form the pushout. Note that when φ is an automorphism, Y /φ is isomorphic to the quotient Y /Z, i.e., to the geometric realization of the bar simplicial space (3.19) ...Z × Y ⇒ Y. We claim: Proposition 3.7.5. There exists a canonical equivalence HH• (A⊗Y , A⊗φ ) ≃ A⊗Y /φ . Proof. Follows from the commutation of the functor (1.3) with colimits in Y . 11One can show that the extra hypothesis of having an affine diagonal is not necessary here.  A TOY MODEL FOR SHTUKA 3.7.6. 57 Let us specialize further the example considered in Sect. 3.7.4 above. Let e be a field of characteristic 0, and take A = Rep(G). Assume that Y has finitely many connected components; then the same is true for Y /φ. Recall the identifications Rep(G)⊗Y ≃ QCoh(LocSysG (Y )) and Rep(G)⊗Y /φ ≃ QCoh(LocSysG (Y /φ)) of Theorem 1.5.5. With respect to the above identifications, the functor Rep(G)⊗φ corresponds to the functor LocSys(φ)∗ : QCoh(LocSysG (Y )) → QCoh(LocSysG (Y )), where LocSys(φ) : LocSysG (Y ) → LocSysG (Y ) is the map induced by φ. By Sect. 3.7.3, we have HH• (QCoh(LocSysG (Y )), LocSys(φ)∗ ) ≃ QCoh(LocSysG (Y )LocSys(φ) ). Now, LocSysG (Y )LocSys(φ) ≃ LocSysG (Y /φ). To summarize, we obtain HH• (Rep(G)⊗Y , Rep(G)⊗φ ) ≃ HH• (QCoh(LocSysG (Y )), LocSys(φ)∗ ) ≃ ≃ QCoh(LocSysG (Y )LocSys(φ) ) ≃ QCoh(LocSysG (Y /φ)) ≃ Rep(G)⊗Y /φ , which is what Proposition 3.7.5 says in this case. 3.8. The 2-categorical class map. 3.8.1. Let R be a monoidal category, and let M be an R-module. Assume that M is right-dualizable as an R-module (see Sect. 3.6.8). Then the corresponding functor TM : DGCat → R - mod admits a right adjoint. Let Q be an object of (R ⊗ Rrev ) - mod, and let us be given a map α : M→ Q⊗M R in R - mod. Applying the functoriality of 2-categorical trace from 3.6.7 and repeating the construction of Sect. 3.4.3, to the above datum we can assign an object cl(M, α) ∈ Tr(TQ , R-mod) ≃ HH• (R, Q). 58 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 3.8.2. Note that for M ∈ R - mod and Q = QFR from Sect. 3.7.2, the datum of α as in Sect. 3.8.1 amounts to an endofunctor FM , which is compatible with the action of R: FM (a ⊗ m) ≃ FR (a) ⊗ FM (m). We denote this correspondence by FM αFM . We will denote the corresponding object cl(M, αFM ) ∈ HH• (R, FR ) also by Trenh R (FM , M). The reason for the notation Trenh will be explained in Remark 3.8.6 below. R Consider the particular case M = R and FM = FR . We will denote 1HH• (R,FR ) := Trenh R (FR , R) ∈ HH• (R, FR ). Remark 3.8.3. The reason for the notation 1HH• (R,FR ) is the following: Assume for a moment that R is symmetric monoidal. The symmetric monoidal structure on R endows R - mod with a structure of commutative algebra object of Morita(DGCat). Assume that FR is a symmetric monoidal endofunctor of R. Consider the corresponding endomorphism FR of R - mod (see Sect. 3.7.2). This endomorphism is right-lax symmetric monoidal for the above commutative algebra structure on R - mod. Hence, by Sect. 3.3.2, the category HH• (R, FR ) acquires a symmetric monoidal structure. Now, it follows from (3.12) that the object that we have denoted 1HH• (R,FR ) ∈ HH• (R, FR ) is the monoidal unit. 3.8.4. We claim: Theorem 3.8.5. Assume that R is rigid. Then there is a canonical isomorphism of associative algebras (3.20) Tr(FR , R) ≃ End HH• (R,FR ) (1HH• (R,FR ) ), and of modules over these algebras (3.21) Tr(FM , M) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , Trenh R (FM , M)). Remark 3.8.6. The reason for the notation Trenh R (FM , M) is explained by Theorem 3.8.5: this theorem says that the object Tr(FM , M) ∈ Vect upgrades to an object of HH• (R, FR ) (namely, Trenh R (FM , M)), where “upgrades” means that the former is the image of the latter under the functor Hom HH• (R,FR ) (1HH• (R,FR ) , −) : HH• (R, FR ) → Vect, which we can think of as a kind of forgetful functor. Remark 3.8.7. It will follow from the proof that when R is symmetric monoidal, the isomorphism (3.20) that we will construct respects the commutative algebra structure on both sides. A TOY MODEL FOR SHTUKA 59 3.8.8. Example. Let Y be as in Sect. 3.5.2, and take R = QCoh(Y). Note the conditions on Y imply that QCoh(Y) is rigid, see [GR1, Chapter 3, Proposition 3.5.3]. We let FR be given by φ∗ . Let M be an QCoh(Y)-module category, which is dualizable as a plain DG category. Let FM be an endofunctor of M that is compatible with φ∗ , i.e., it makes the diagram QCoh(Y) ⊗ M − −−−−→   φ∗ ⊗FM y M  F y M QCoh(Y) ⊗ M − −−−−→ M commute. Then the construction of Sect. 3.8.1 produces an object φ Trenh QCoh(Y) (FM , M) ∈ QCoh(Y ) and Theorem 3.8.5 says that Γ(Yφ , Trenh QCoh(Y) (FM , M)) ≃ Tr(FM , M). 3.9. A framework for the proof of Theorem 3.8.5. 3.9.1. In order to prove Theorem 3.8.5 we will use the formalism of Sect. 3.3.1, applied to O := Morita(DGCat), EndO (1O ) = DGCat . However, we note that both sides in (3.22) Tr : L(Morita(DGCat))rgd → DGCat are naturally (∞, 2)-categories, and functor (3.22) upgrades to a functor between (∞, 2)-categories. 3.9.2. The structure of (∞, 2)-category on L(Morita(DGCat))rgd comes from the structure of (∞, 3)category on Morita(DGCat). Indeed, note that for a pair of objects Ri - mod ∈ Morita(DGCat), the (∞, 1)-category MapsMorita(DGCat) (R1 - mod, R2 - mod) := (R2 ⊗ Rop 1 ) - mod naturally upgrades to an (∞, 2)-category. Namely, for Q′ , Q′′ ∈ (R2 ⊗ Rop 1 ) - mod, we can consider the (∞, 1)-category Maps(R2 ⊗Rop ) - mod (Q′ , Q′′ ). 1 3.9.3. Let us specify what are the 2-morphisms in L(Morita(DGCat))rgd . By definition, 1-morphisms in L(Morita(DGCat))rgd are diagrams o1 f1 / o1 ⑧⑧ ⑧ ⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t t ⑧⑧⑧⑧ α ⑧ ⑧ ⑧ ⑧⑧   { ⑧⑧⑧ / o2 , o2 f2 where oi ∈ Morita(DGCat), and the edges in the above diagram are 1-morphisms in Morita(DGCat), where we require the 1-morphism t to admit a right adjoint. 60 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Given another 1-morphism represented by f1 / o1 ⑧ ⑧ ⑧ ⑧⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ t′ t′ ⑧⑧⑧⑧ ⑧⑧⑧⑧ α′ ⑧ ⑧ ⑧   { ⑧⑧⑧ / o2 , o2 o1 (3.23) f2 a 2-morphism between them is a diagram ? oEM 1 ⑧⑧✓✓✓ ⑧ id ⑧ ✓ ✓ ⑧ ✓ ⑧⑧ ✓✓✓✓ ⑧ ⑧ ⑧ ✓✓✓✓ t′ o1 ✓✓✓✓✓ ✓✓✓β  ✓✓✓✓✓ ✓✓✓ ⑧o? 2 , t ✓✓ ✓✓ ⑧⑧ ✓✓✓✓✓ ⑧⑧⑧id ✓ ⑧  ✓✓✓✓⑧⑧ o2 ′ where β : t → t is a 2-morphism in Morita(DGCat) that admits a right adjoint 12, and which is equipped with a 3-morphism γ for the cube: / o1 o1 f1 ⑧✓? ✓EM ♥♥♥⑧⑧✓? ✓EM ♥ ⑧ ♥ ⑧ id ⑧⑧ ✓✓✓ ♥♥♥id ⑧⑧⑧ ✓✓✓✓ ⑧ ✓✓✓ ♥♥♥ ♥ ⑧⑧ ✓✓ ⑧ ♥ ✓ ✓ ⑧ α♥♥♥ ⑧⑧ ✓✓✓✓ t′ ♥ ⑧ ⑧⑧ ✓✓✓✓ t′ ♥ ✓✓ /o o1 ♥♥♥ ✓✓✓✓ ✓✓✓✓ ♥♥ 1 f1 ♥♥♥ ♥ ✓ ✓ ♥ ♥ ✓ ✓ β ✓✓ β ✓✓ ♥♥♥ ♥♥♥ ♥♥♥  rz ♥♥♥ ✓✓✓✓✓ ✓✓✓✓ /  ♥ ♥ ✓ ✓ o o2 ′ ♥♥ 2 α ♥♥ ✓✓ ✓✓✓ f2 ⑧? ⑧? ♥ t ✓✓ t ✓✓ ✓✓✓ ⑧⑧⑧ ♥♥♥♥♥ ✓✓✓ ⑧⑧⑧ ✓ ⑧ ✓ ⑧ ♥ ✓✓✓✓✓ ⑧⑧⑧♥id♥♥♥♥ ✓✓✓✓✓ ⑧⑧⑧ id  ✓ ✓⑧⑧  ✓✓✓⑧rz ⑧♥♥♥ / o2✓. o2 f2 I.e., γ is a 3-morphism (3.24) α / f2 ◦ t ⑧⑧ ⑧ ⑧⑧⑧ ⑧ ⑧ ⑧ ⑧⑧⑧ β β ⑧⑧⑧⑧γ ⑧ ⑧ ⑧ ⑧ ⑧  { ⑧⑧⑧  / f2 ◦ t′ . t′ ◦ f1 t ◦ f1 α′ 3.9.4. Let us show how the datum of (β, γ) as above gives rise to a 2-morphism Tr(t, α) → Tr(t′ , α′ ) in EndMorita(DGCat) (DGCat) ≃ DGCat, which is a 3-morphism in DGCat. 12This is a notion that exists in an (∞, 3)-category. A TOY MODEL FOR SHTUKA 61 It is obtained from the diagram cycl / Tr(f2 ◦ t ◦ tR , o2 ) / Tr(f2 , o2 ) / Tr(tR ◦ t ◦ f1 , o1 ) α / Tr(tR ◦ f2 ◦ t, o1 ) q ② ② ♦♦ q ② ② q ②②②②② q ②②② ♦♦♦ q ② ② ♦ ② ② q ② ♦ ② γ qq ②② ②② ♦♦ Id Id ②②②②② ②②②②② qqq ♦♦♦ ② ② q ♦ ② ② q ② ② ♦ ② ② q ♦ ② ② q ② q ♦ ② ② q ② s{ ♦♦  t| qq   x ②②②  x ②②②  ′R ′ ′R ′ ′ ′R / / / Tr(f1 , o2 ) Tr(t ◦ t ◦ f1 , o1 ) Tr(t ◦ f2 ◦ t , o1 ) Tr(f2 ◦ t ◦ t , o2 ) / Tr(f2 , o1 ), Tr(f1 , o1 ) α′ cycl where the middle vertical arrows are induced by the 2-morphisms: • β : t → t′ ; • The 2-morphism tR → t′R , obtained by passing to right adjoints at the level of 1-morphisms from β R : t′ → t, i.e., we consider the partially defined functor “passing to the right adjoint” Maps(o2 , o1 ) → Maps(o1 , o2 ), t 7→ tR and apply it to β R ∈ MapsMaps(o2 ,o1 ) (t′ , t) to obtain an object of MapsMaps(o1 ,o2 ) (tR , t′R ). 3.10. Proof of Theorem 3.8.5: isomorphism of the underlying objects of Vect. 3.10.1. phism As a first step, in the setting of Theorem 3.8.5, we will show that we have a canonical isomor- (3.25) Tr(FM , M) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , Trenh R (FM , M)). as objects of Vect. We will do it in the following general framework. 3.10.2. Let R1 and R2 be a pair of monoidal DG categories, each equipped with a monoidal endofunctor, denoted FR1 and FR2 , respectively. Let Ψ : R1 → R2 be a monoidal functor, equipped with an isomorphism (3.26) FR2 ◦ Ψ ≃ Ψ ◦ FR1 . Restriction and induction along Ψ define an adjoint pair of 1-morphisms in Morita(DGCat) (3.27) IndΨ : R1 - mod ⇄ R2 - mod : ResΨ , see Sect. 3.6.9. The isomorphism (3.26) give rise to a commutative square (3.28) ResF R ResF R 2 R2 - mod −−−−−→ R2 - mod   Res  ResΨ y y Ψ 1 R1 - mod −−−−−→ R1 - mod. 62 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY By passing to left adjoints along the vertical arrows in (3.28), we obtain a lax-commutative square: (i.e., a square that commutes up to a 2-morphism) (3.29) ResF R1 / R1 - mod ttt t tttt t t t tttt IndΨ IndΨ ttttt t t t t t t  v~ tttt  / R2 - mod. R2 - mod R1 - mod ResF R2 3.10.3. Assume now that R1 and R2 are rigid. In this case, by Sect. 3.6.10, the 1-morphism ResΨ admits also a right adjoint. Furthermore, in this case the unit and the counit of the adjunction (IndΨ , ResΨ ), which are 2morphisms Id → ResΨ ◦ IndΨ and IndΨ ◦ ResΨ → Id in Morita(DGCat), admit right adjoints. Indeed, the above 2-morphisms are given by the functors Ψ mult R1 → R2 and R2 ⊗ R2 → R2 R1 (as (R1 , R1 )- and (R2 , R2 )-bimodule categories-bimodule categories, respectively) and the rigidity assumption implies that these functors admit right adjoints that respect the bimodule structures. From here it follows that the adjunction (3.27) and the diagrams (3.28) and (3.29) give rise to an adjunction (3.30) IndΨ : (R1 - mod, ResFR1 ) ⇄ (R2 - mod, ResFR2 ) : ResΨ in the (∞, 2)-category L(Morita(DGCat))rgd . In the unit and counit 2-morphisms that encode the adjunction (3.30), the corresponding 3-morphisms (denoted γ in Sect. 3.9.3, see (3.24)) are actually isomorphisms. 3.10.4. Hence, by Sect. 3.9, from the adjunction (3.30), by applying the functor Tr of (3.22), we obtain an adjunction (3.31) Tr(IndΨ ) : HH• (R1 , FR1 ) ⇄ HH• (R2 , FR2 ) : Tr(ResΨ ). 3.10.5. Let M be an object of R2 - mod; assume that it is dualizable as a DG category. Since R2 was assumed rigid, we obtain that M is dualizable as an object of R2 - mod. Let FM be its endofunctor as in Sect. 3.8.2. Then to it there corresponds an object Trenh R2 (FM , M) ∈ HH• (R2 , FR2 ). Consider now the object ResΨ (M) ∈ R1 - mod. Since R1 is rigid and M is dualizable as a DG category, we obtain that ResΨ (M) is dualizable as an object of R1 - mod. The datum of FM defines the corresponding datum for ResΨ (M) ∈ R1 - mod. We will deduce Theorem 3.8.5 (at the level of the underlying objects of Vect) from the following more general relative statement: Theorem 3.10.6. In the situation above, there is a canonical isomorphism enh Tr(IndΨ )R (Trenh R2 (FM , M)) ≃ TrR1 (FM , ResΨ (M)). A TOY MODEL FOR SHTUKA 63 3.10.7. Proof of Theorem 3.10.6. We can view the pair (M, FM ) as a 1-morphism (DGCat, Id) → (R2 - mod, ResFR2 ) (3.32) in L(Morita(DGCat))rgd . By construction, Trenh R2 (FM , M) ∈ Tr(ResFR2 , R2 - mod) := HH• (R2 , FR2 ) is obtained by applying the functor Tr of (3.22) to (3.32). The pair (ResΨ (M), FM ) is obtained from (3.32) as the composition (M,F ) Res (DGCat, Id) −→M (R2 - mod, ResFR2 ) −→Ψ (R1 - mod, ResFR1 ). (3.33) Hence, Trenh R1 (FM , ResΨ (M)) ∈ Tr(ResFR1 , R1 - mod) := HH• (R1 , FR1 ) is obtained by applying the functor Tr of (3.22) to (3.33). 3.10.8. From here we obtain a canonical identification enh Tr(ResΨ )(Trenh R2 (FM , M)) ≃ TrR1 (FM , ResΨ (M)). (3.34) Note that by that by (3.31), we have: Tr(ResΨ ) ≃ Tr(IndΨ )R . Hence, we can rewrite (3.34) as: enh Tr(IndΨ )R (Trenh R2 (FM , M)) ≃ TrR1 (FM , ResΨ (M)). (3.35) [Theorem 3.10.6] Remark 3.10.9. Note that for the construction of the equivalence (3.35) we use less than the full force of the assumption that both R1 and R2 be rigid. What we actually use is that the symmetric monoidal functor Ψ : R1 → R2 is rigid (we leave it to the reader to work out what this means). For example, when R1 is symmetric monoidal and Ψ makes R2 into a R1 -algebra object in DGCat, the assumption we need is that R2 be rigid over R1 . For R1 = Vect and R2 = R, this just means that R is rigid. 3.10.10. Let us now deduce the isomorphism (3.25). Take (R2 , FR2 ) = (R, FR ) and (R1 , FR1 ) = (Vect, Id) with Ψ being the unit functor Vect → R. Note that by construction, the functor Tr(IndΨ ) : Vect ≃ Tr(Id, DGCat) → Tr(ResFR , R - mod) := HH• (R, FR ) sends e 7→ 1HH• (R,FR ) , see Remark 3.8.3. Hence, Tr(IndΨ )R is given by Hom HH• (R,FR ) (1HH• (R,FR ) , −). Finally, apply (3.35). 3.11. Proof of Theorem 3.8.5: algebra and module structure. 64 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 3.11.1. The functor Tr of (3.22) induces a functor Tr : EndL(Morita(DGCat))rgd (DGCat, Id) → EndDGCat (Vect) ≃ Vect . (3.36) We note that we have a canonical identification of symmetric monoidal categories L(DGCat)rgd ≃ EndL(Morita(DGCat))rgd (DGCat, Id), (3.37) and the resulting functor L(DGCat)rgd → Vect is the functor (3.10) for O = DGCat. 3.11.2. Let Ψ be the unit functor DGCat → R. The adjunction IndΨ : (DGCat, Id) ⇆ (R - mod, ResFR ) : ResΨ (3.38) defines a monad on (DGCat, Id) ∈ L(Morita(DGCat))rgd , i.e., an associative algebra in the (symmetric) monoidal category EndL(Morita(DGCat))rgd (DGCat, Id). Under the identification (3.37) this algebra is given by (R, FR ) ∈ L(DGCat)rgd . Applying the functor Tr of (3.36) (which, by the above, is the same as the functor Tr of (3.10) for O = DGCat) we recover Tr(FR , R) ∈ Vect with its associative algebra structure. 3.11.3. Now, by the functoriality of (3.22), this associative algebra, regarded as a monad on Vect, identifies with the monad corresponding to the adjunction Tr(IndΨ ) : Tr((DGCat, Id)) ⇆ Tr((R - mod, ResFR )) : Tr(ResΨ ), (3.39) i.e., the adjunction obtained from (3.38) by applying the functor Tr. We identify Tr((DGCat, Id)) ≃ Vect and Tr((R - mod, ResFR )) ≃ HH• (R, FR ), where the 1-morphism Tr(IndΨ ) identifies with e 7→ 1HH• (R,FR ) . Hence, we obtain that the associative algebra Tr(FR , R) ∈ Vect corresponds to the adjunction Vect ⇆ HH• (R, FR ), e 7→ 1HH• (R,FR ) . This establishes the isomorphism (3.20) as associative algebras. 3.11.4. Similarly, a pair (M, FM ) can be viewed as a 1-morphism (DGCat, Id) → (R - mod, ResFR ). Composing with ResΨ we obtain an object in EndL(Morita(DGCat))rgd (DGCat, Id), which is a module over the monad ResΨ ◦ IndΨ , and which under the identification (3.37) corresponds to (M, FM ) ∈ (R, FR )-mod(L(DGCat)rgd ). Applying the functor Tr of (3.10) for O = DGCat, we recover Tr(FM , M) as a module over Tr(FR , R). Now, by the functoriality of (3.22), the pair Tr(FR , R), Tr(FM , M) ∈ Tr(FR , R)-mod is the same as one obtained from (R, FR ), (M, FM ) ∈ (R, FR )-mod(L(DGCat)rgd ) by applying the functor Tr of (3.22). A TOY MODEL FOR SHTUKA 65 This establishes the isomorphism (3.21) as modules over the two sides of (3.20). [Theorem 3.8.5] 3.12. A more elementary proof of Theorem 3.8.5. For the convenience of the reader and future reference, in this subsection we will outline a more elementary proof (of a particular case) of Theorem 3.8.5, which does not use the machinery of (∞, 3)-categories. We first establish the stated isomorphism for the underlying objects of Vect. 3.12.1. Consider the following composition of 1-morphisms in Morita(DGCat) oblv M DGCat −→ R - mod −→ DGCat, where the second arrow is the forgetful map, i.e., given by TR . The composite is the map DGCat → DGCat corresponding to oblv(M) ∈ DGCat, i.e., M, viewed as a plain DG category. We have the following diagram of 2-morphisms Id / DGCat tt t t t ttt t t t tttt M M ttttαtFM t t t ttt t t t  v~ ttt  / R - mod R - mod FR ttt ttttt t t t ttttt oblv oblv tttttaut t t tttt t t t  v~ ttt  Id / DGCat. DGCat DGCat The composite 2-morphism identifies with Id / DGCat ⑧⑧ ⑧⑧⑧⑧ ⑧ ⑧ ⑧⑧ ⑧⑧⑧⑧ M M ⑧ ⑧ FM ⑧ ⑧⑧⑧ ⑧ ⑧  { ⑧⑧  / DGCat. DGCat DGCat Id By unwinding the definitions, it is easy to see that the resulting map Tr(Id, DGCat) → Tr(Id, DGCat), viewed as a functor DGCat → DGCat, is given by Tr(FM , M) ∈ DGCat. 3.12.2. Hence, to prove the isomorphism (3.21) as vector spaces, it suffices to show that the map Tr(FR , R - mod) → Tr(Id, DGCat), 66 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY corresponding to the diagram / R - mod t tttt ttttt t t t tttt oblv oblv ttttttaut t t t t t t  v~ tttt  Id / DGCat, DGCat R - mod FR viewed as a functor HH• (R, FR ) → Vect, (3.40) is given by Hom HH• (R,FR ) (1HH• (R,FR ) , −). (3.41) 3.12.3. Consider the corresponding diagram FR ⊗Id counit / R⊗2 - mod / R⊗2 - mod / DGCat ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦♦ ♦♦ ♦♦ ♦♦♦ ♦♦♦ ♦♦♦ taut ♦ ♦ ♦ ♦ ♦ ♦ Id Id ♦♦ ♦♦ ♦♦ ♦♦♦ ♦♦♦ ♦♦♦ ♦ ♦ ♦ ♦ ♦ ♦  s{ ♦♦  s{ ♦♦  s{ ♦♦  / DGCat / DGCat. / DGCat DGCat (3.42) DGCat unit unit counit Id We need to calculate the 2-morphism from the clockwise circuit, which corresponds to HH• (R, FR ) ∈ DGCat ≃ End(DGCat), to the counterclockwise circuit, which corresponds to Vect ∈ DGCat ≃ End(DGCat). 3.12.4. We now recall that for a rigid symmetric monoidal category R′ , the right adjoint to (3.43) R′ - mod −→ DGCat oblv identifies with (3.44) R′ DGCat −→ R′ - mod, see Sect. 3.6.10. Thus, we obtain that the two middle vertical arrows in diagram (3.42) are given by the forgetful map (R ⊗ R) - mod oblv ⊗ oblv −→ DGCat. Furthermore, we obtain that the 2-morphism in the left square, viewed as a functor R → Vect, is given by Hom R (1R , −). The 2-morphism in the right square, evaluated on Q ∈ (R ⊗ R) - mod (see Sect. 3.6.2 for what we mean by “evaluate”), is the map HH• (R, Q) = R ⊗ Q R⊗R multR ⊗ IdQ −→ (R ⊗ R) ⊗ Q ≃ Q, R⊗R A TOY MODEL FOR SHTUKA 67 which is the right adjoint to the map Q ≃ (R ⊗ R) ⊗ Q mult ⊗ IdQ −→ 3.12.5. R ⊗ Q = HH• (R, Q). R⊗R R⊗R We obtain that the functor in (3.40) equals the composite HH• (R, FR ) = R ⊗ R (mult)R ⊗Id −→ mult,R⊗R,mult ◦(FR ⊗Id) (R ⊗ R) ⊗ R⊗R,mult ◦(FR ⊗Id) R≃R Hom R (1R ,−) −→ Vect . By adjunction, this composite is the same as HH• (R, FR ) Hom HH• (R,F ) ((mult ⊗ Id)(1R ),−) R −→ Vect, i.e., (3.41). Remark 3.12.6. Let us contrast the computation of the map (3.40) with the computation of the map Tr(Id, QCoh(Y)) → e (3.45) corresponding to the functor Γ(Y, −) : QCoh(Y) → Vect, where Y is a smooth proper scheme. We identity Tr(Id, QCoh(Y)) ≃ ⊕ Γ(Y, Ωi (Y))[i], i see Sect. 4.3.3. The computation performed in [KP2] amounts to saying that the resulting map ⊕ Γ(Y, Ωi (Y))[i] → e i is the projection ⊕ Γ(Y, Ωi (Y))[i] → Γ(Y, Ωtop (Y))[top] i Serre duality −→ e, precomposed with the operation of multiplication by the Todd class. So, it is highly non-trivial. By contrast, in the setting of Sect. 3.12.2, the map (3.40) is something very simple, namely, the map (3.41). This my be viewed as an incarnation of the fact that for a rigid symmetric monoidal category, the 2-category R - mod is 0-Calabi-Yau, in the sense that the left and right adjoints to the functor DGCat → R - mod, C 7→ R ⊗ C are canonically isomorphic, see Remark 3.6.11. 3.12.7. Thus, we have established the isomorphism between the two sides of (3.21) as objects of Vect. In particular, we obtain an isomorphism between the two sides of (3.20), also as objects of Vect. We will now assume that R is symmetric monoidal, and upgrade these isomorphisms to isomorphisms of algebras (resp., modules over them). This will be achieved by an Eckmann-Hilton argument. Let Q denote the category, whose objects are quadruples (R, M, FR , FM ), where R is a rigid symmetric monoidal DG category, and M is an R-module, dualizable as a plain DG category. For a pair of objects (R, M, FR , FM ) and (R′ , M′ , FR′ , FM′ ), the space of morphisms between them consists of a symmetric monoidal functor ϕR : R → R′ , intertwining FR with FR′ , and a functor of R-module categories ϕM : M → M′ , intertwining FM with FM′ , such that the induced functor R′ ⊗ M → M′ R is an equivalence. 68 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY The assignments (R, M, FR , FM ) 7→ Tr(FM , M) (3.46) and (R, M, FR , FM ) 7→ Hom HC• (R,FR ) (1HC• (R,FR ) , Trenh R (FM , M)) (3.47) are both functors Q → Vect. Moreover, the category Q carries a naturally defined symmetric monoidal structure: (R1 , M1 , FR1 , FM1 ) ⊗ (R2 , M2 , FR2 , FM2 ) := (R1 ⊗ R2 , M1 ⊗ M2 , FR1 ⊗ FR2 , FM1 ⊗ FM2 ), and functors (3.46) and (3.47) are symmetric monoidal. Furthermore, by the construction of the isomorphism of (3.21), it upgrades to an isomorphism of symmetric monoidal functors (3.46) and (3.47), as symmetric monoidal functors. Note now that (R, R, FR , FR ) is naturally a commutative algebra in Q. Hence, both sides of (3.20) have a structure of associative algebras, and (3.20) respects these structures. By construction, the above commutative algebra structure on Tr(FR , R) is the same one as given by the construction of Sect. 3.4.4. Furthermore, by the Eckmann-Hilton argument, the above commutative algebra structure on End HH• (R,FR ) (1HH• (R,FR ) ) goes over under the forgetful functor ComAlg(Vect) → AssocAlg(Vect), to the structure of associative algebra on End . This implies the assertion that (3.20) is an algebra isomorphism. 3.12.8. Finally, for (M, FM ) as in Sect. 3.8.2, the object (R, M, FR , FM ) ∈ Q is a module over the algebra object (R, R, FR , FR ). The construction in Sect. 3.12.7 gives each side of (3.21) a structure of module over the corresponding side of (3.20), and (3.21) respects these structures. The resulting action of Tr(FR , R) on Tr(FM , M) is the same one as given by the construction of Sect. 3.4.4. Again, by the Eckmann-Hilton argument, the resulting action of End HH• (R,FR ) (1HH• (R,FR ) ) on Hom HH• (R,FR ) (1HH• (R,FR ) , Trenh R (FM , M)) coincides with one coming from the action of End on Hom. This implies that (3.21) is an isomorphism of modules, as desired. 4. A few mind-twisters In this section we will study some particular cases and generalizations of Theorem 3.8.5. We recommend the reader to skip this section on the first pass, because the assertions contained therein may appear abstract and un-motivated, and return to it when necessary. That said, the results discussed in this section will all acquire a transparent meaning in the context of shtukas, which will be introduced in Sect. 5. 4.1. The class of a class. A TOY MODEL FOR SHTUKA 69 4.1.1. Let R be a rigid symmetric monoidal category, and let FR : R → R be a symmetric monoidal endofunctor. Consider the category (4.1) HH• (R, FR ) := R ⊗ R. mult,R⊗R,mult ◦(FR ⊗Id) Since the tensor product in the right-hand side of (4.1) involves symmetric monoidal categories and functors, we obtain that HH• (R, FR ) acquires a symmetric monoidal structure. On the other hand, by (3.18), we have HH• (R, FR ) ≃ Tr(ResFR , R - mod), and hence it acquires a symmetric monoidal structure by Sect. 3.3.2. However, it is easy to see that these two ways of defining a symmetric monoidal structure on HH• (R, FR ) are equivalent. 4.1.2. Denote by ι the functor R→R ⊗ mult,R⊗R,mult ◦(FR ⊗Id) R ≃ HH• (R, FR ), corresponding to the left copy of R. By (4.1), the functor ι is symmetric monoidal. Note that by construction ι ◦ FR ≃ ι. (4.2) 4.1.3. Let r ∈ R be a compact object equipped with a map αr : r → FR (r). On the one hand, to the pair (r, αr ) we attach its class cl(r, αr ) ∈ Tr(FR , R). 4.1.4. The data of αr gives rise to a map α (4.2) r ι ◦ FR (r) ≃ ι(r); ι(r) −→ F denote this map by ar R . Since r ∈ R is compact and R is rigid, we obtain that r is dualizable as an object of R as a monoidal category. Since ι is symmetric monoidal, we obtain that ι(r) is dualizable as an object of HH• (R, FR ). So, on the other hand, we can consider the element R Tr(aF r , ι(r)) ∈ End HH• (R,FR ) (1HH• (R,FR ) ). 4.1.5. We claim: R Proposition 4.1.6. The elements cl(r, αr ) and Tr(aF r , ι(r)) coincide under the identification (4.3) Tr(FR , R) ≃ End HH• (R,FR ) (1HH• (R,FR ) ) of Theorem 3.8.5. Remark 4.1.7. Note that in the particular case of R = QCoh(Y) for a prestack Y as in Sect. 3.5.2, and FR given by φ∗ for an endomorphism φ of Y, the assertion of Proposition 4.1.6. coincides with that of Proposition 3.5.7. 4.2. Proof of Proposition 4.1.6. The proof is a word-for-word repetition of the proof of [KP1, Proposition 2.2.3]. We include it for the sake of completeness. 70 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 4.2.1. First, let us make the isomorphism (4.3) explicit (this will imitate the manipulation in Sect. 3.5.3). Recall that R is self-dual as a DG category, with the duality datum given by multR 1 R R −→ R ⊗ R Vect → and mult R ⊗ R −→ R Hom R (1R ,−) −→ Vect . Hence, on the one hand, Tr(FR , R) is the composition Hom (1 ,−) R R −−−−− −−R−−→ Vect x  mult (4.4) R x  1R  multR R⊗R x F ⊗Id  R − −−−− → R⊗R Vect . Consider the commutative diagram ι ⊗ x  ι R mult,R⊗R,mult ◦(FR ⊗Id) (4.5) R ← −−−−− mult R x mult ◦(F ⊗Id)  R ← −−−−− R ⊗ R. R By rigidity, the diagram obtained by passing to right adjoints along the horizontal arrows is also commutative: R (4.6) ιR ⊗ R − −−−− → mult,R⊗R,mult ◦(FR ⊗Id) x  ι multR R x mult ◦(F ⊗Id)  R − −−−− → R ⊗ R. R Hence, the composite in (4.4) identifies with R ⊗ x  ι mult,R⊗R,mult ◦(FR ⊗Id) (4.7) ιR Hom (1 ,−) R R − −−−−→ R −−−−− −−R−−→ Vect R x  1R  Vect . In the latter diagram, the composite horizontal arrow is the right adjoint of the composite vertical arrow, and the latter is e 7→ 1HH• (R,FR ) ∈ HH• (R, FR ) ≃ R ⊗ R. mult,R⊗R,mult ◦(FR ⊗Id) Hence, the resulting functor Vect → Vect is given by End HH• (R,FR ) (1HH• (R,FR ) ). A TOY MODEL FOR SHTUKA 71 By unwinding the constructions (see Sect. 3.12), one shows that the identification Tr(FR , R) ≃ End HH• (R,FR ) (1HH• (R,FR ) ) (4.8) just constructed is equivalent to one in Theorem 3.8.5. 4.2.2. For a compact object r ∈ R, let r ∨ ∈ R be its monoidal dual; this is also its formal dual with respect to the identification R∨ ≃ R. The class cl(r, αr ) ∈ Tr(FR , R) corresponds to the 2-morphism from the clockwise circuit to the counter-clockwise circuit in the following diagram Id Id Id / Vect / / Vect t t ttVect t t t t t t t t t t t tttt tttt tttt ttttt ttttt ttttt taut t t t ∨ ∨ t t t t t t Id Id r⊗r r⊗r tttt tttt tttt ttttt ttttt ttttt t t t t t t t t  v~ ttt  v~ ttt  v~ tttt  / R⊗R / Vect . / R⊗R Vect Vect FR ⊗Id unit counit In this diagram, the 2-morphism in the left square is the map r ⊠ r ∨ → unit(e) = multR (1R ) equal to r ⊠ r ∨ → multR ◦ mult(r ⊠ r ∨ ) = multR (r ⊗ r ∨ ) → multR (1R ). The 2-morphism in the middle square is obtained from the map αr : r → FR (r). The 2-morphism in the right square is map e → counit(r ⊠ r ∨ ) = Hom R (1R , mult(r ⊠ r ∨ )) = Hom R (1R , r ⊗ r ∨ ), corresponding to the canonical map 1R → r ⊗ r ∨ . 4.2.3. (4.9) Hence, cl(r, αr ) ∈ Tr(FR , R) is the composite unit α ⊗id r e → Hom R (1R , r ⊗ r ∨ ) −→ Hom R (1R , FR (r) ⊗ r ∨ ) ≃ ≃ Hom R (1R , mult ◦(FR ⊗ Id)(r ⊠ r ∨ )) → Hom R (1R , mult ◦(FR ⊗ Id) ◦ multR (1R )). Using the commutative diagram (4.6), we identify mult ◦(FR ⊗ Id) ◦ multR ≃ ιR ◦ ι. Hence, we can rewrite the composition in (4.9) as (4.10) unit α ⊗id r e → Hom R (1R , r ⊗ r ∨ ) −→ Hom R (1R , FR (r) ⊗ r ∨ ) ≃ ≃ Hom R (1R , mult ◦(FR ⊗ Id)(r ⊠ r ∨ )) → Hom R (1R , mult ◦(FR ⊗ Id) ◦ multR (r ⊗ r ∨ )) ≃ ≃ Hom R (1R , ιR ◦ ι(r ⊗ r ∨ )) → Hom R (1R , ιR ◦ ι(1R )). 72 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 4.2.4. We have a commutative diagram ∼ Hom R (1R , FR (r) ⊗ r ∨ )   y Hom R (1R , mult ◦(FR ⊗ Id)(r ⊠ r ∨ ))   y − −−−− → Hom HH• (R,FR ) (1HH• (R,FR ) , ι(FR (r) ⊗ r ∨ ))   (4.2)y∼ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r ⊗ r ∨ )) Hom R (1R , ιR ◦ ι(r ⊗ r ∨ ))  ∼ y ∼ − −−−− → Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r ⊗ r ∨ )). Hence, the composition in (4.10) can be rewritten as α ⊗id unit r e → Hom R (1R , r ⊗ r ∨ ) −→ Hom R (1R , FR (r) ⊗ r ∨ ) → (4.2) counit → Hom HH• (R,FR ) (1HH• (R,FR ) , ι(FR (r) ⊗ r ∨ )) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r ⊗ r ∨ )) → → Hom HH• (R,FR ) (1HH• (R,FR ) , ι(1R )) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , 1HH• (R,FR ) ), and further as (4.2) α ⊗id unit r e → Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r)⊗ι(r ∨ )) −→ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(FR (r))⊗ι(r ∨ )) ≃ counit ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r) ⊗ ι(r ∨ )) → Hom HH• (R,FR ) (1HH• (R,FR ) , 1HH• (R,FR ) ), F while the latter is the right-hand side is by definition Tr(ar R , ι(r)). [Proposition 4.1.6] 4.3. The “trivial” case and excursions. We will now specialize further to the case when FR is the identity map. 4.3.1. Let us take α to be the identity endomorphism of r. Consider the corresponding endomorphism (4.11) R ∈ End HH• (R) (ι(r)), idId r see Sect. 4.1.4. Denote R , ι(r)) ∈ End HH• (R) (1HH• (R) ). ch(r) := Tr(idId r Note that according to Proposition 4.1.6, we have (4.12) cl(r, idr ) = ch(r) under the identification Tr(IdR , R) ≃ End HH• (R) (1HH• (R) ) of Theorem 3.8.5. Remark 4.3.2. We emphasize that despite the fact that we plugged in the identity map everywhere, Id the endomorphism idr R of ι(r) is not the identity map (for one thing, if it were the identity, formula (4.12) would fail). IdR See also the last line in the proof of Proposition 4.3.5 for another interpretation of the element idr . A TOY MODEL FOR SHTUKA 73 4.3.3. An example. Let R = QCoh(Y), where Y is as in Sect. 3.5.2. In [KP2, Sect. 1.2] it is explained Id that for r = F ∈ QCoh(Y), the map idr R can be interpreted as the action of L(Y) := Y × Y, Y×Y thought of as the group object over Y, on F. Furthermore, ch(r) ∈ Γ(L(Y), OL(Y) ) can be identified with the character of this action. Let now Y be a smooth scheme over a field of characteristic zero. Consider the derivative of the above action of L(Y) on F, which is an action of the Lie algebra Lie(L(Y )) ≃ TY [−1] on F. It is shown in [KP2, Sect. 1.3] that the resulting map TY [−1] ⊗ F → F is the Atiyah class of F. Further, the Hochschild-Kostant-Rosenberg theorem identifies Γ(L(Y), OL(Y) ) ≃ ⊕ Γ(Y, Ωi (Y))[i]. i Under this identification, the element ch(r) corresponds to the classical Chern character ch(F) ∈ ⊕ Γ(Y, Ωi (Y))[i]. i 4.3.4. We will now give one more interpretation of the above element ch(r) ∈ End HH• (R) (1HH• (R) ), in terms of excursion operators. Take Y = S 1 = {∗} ⊔ {∗}⊔{∗} {∗} so that 1 R⊗S ≃ HH• (R). (4.13) + denote the pair (γtaut , γtriv ). Consider the tautological point γtaut ∈ Ω(S 1 , ∗). Let γtaut Let ξr be the map 1R → mult ◦ multR (1R ) equal to the composition (4.14) 1R → r ⊗ r ∨ ≃ mult(r ⊠ r ∨ ) → mult ◦ multR (1R ), where: –the map 1R → r ⊗ r ∨ is the unit of the duality; –r ⊠ r ∨ denotes the corresponding object of R ⊗ R; –the map r ⊠ r ∨ → multR (1R ) is obtained by adjunction from the map counit mult(r ⊠ r ∨ ) =: r ⊗ r ∨ −→ 1R . We claim: Proposition 4.3.5. The element ch(r) ∈ End HH• (R) (1HH• (R) ) corresponds to the universal excursion element (see Sect. 2.8.4) + Excuniv (γtaut , ξr ) ∈ End R⊗S 1 (1R⊗S 1 ) under the identification (4.13). 74 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Proof. The proof is essentially an application of the definitions: + By formula (2.16), the element Excuniv (γtaut , ξr ) is the composition unit 1HH• (R) −→ ι(r) ⊗ ι(r ∨ ) monγtaut ⊗ id −→ counit ι(r) ⊗ ι(r ∨ ) −→ 1HH• (R) , where monγtaut denotes the automorphism of the functor ι, i.e., 1 R ≃ R⊗{∗} → R⊗S , corresponding to the loop γtaut . However, by definition, the above automorphism monγtaut of ι, when evaluated on r ∈ R, identifies with idrIdR .  Combining Propositions 4.1.6 and 4.3.5, we obtain: Corollary 4.3.6. Under the identification Tr(Id, R) ≃ End HH• (R) (1HH• (R) ) ≃ End R⊗S 1 (1R⊗S 1 ) of Theorem 3.8.5, the element cl(r, id) ∈ Tr(Id, R) goes over to + Excuniv (γtaut , ξr ) ∈ End R⊗S 1 (1R⊗S 1 ). Remark 4.3.7. Note that on the one hand, Tr(IdR , R) is explicitly given by the composition multR 1 mult R R −→ R ⊗ R −→ R Vect → Hom R (1R ,−) → Vect, i.e., the resulting vector space is Hom R (1R , mult ◦ multR (1R )). On the other hand, by Lemma 2.7.4, we have End HH• (R) (1HH• (R) ) ≃ End R⊗S 1 (1R⊗S 1 ) = Hom R⊗S 1 (ι(1R ), ι(1R )) ≃ ≃ Hom R (1R , ιR ◦ ι(1R )) ≃ Hom R (1R , mult ◦ multR (1R )). Thus, we obtain an identification Tr(IdR , R) ≃ Hom R (1R , mult ◦ multR (1R )) ≃ End HH• (R) (1HH• (R) ). By unwinding the definitions, one can show that this is the same identification as one given by Theorem 3.8.5. Assuming this, one can obtain the assertion of Corollary 4.3.6 by combining (4.12) and Theorem 2.8.7. Indeed, this follows from the fact that the element cl(r, idr ), thought of as an element in Tr(IdR , R) ≃ Hom R (1R , mult ◦ multR (1R )), equals ξr . 4.4. Introducing observables. In this subsection we will study a certain generalization of Theorem 3.8.5, where we modify both sides by inserting an object r ∈ R. 4.4.1. Let R be a rigid symmetric monoidal category. Let M be a dualizable R-module category. On the one hand, we consider the endofunctor of M, given by Hr ◦ FM , where Hr (m) := r ⊗ m denotes the action of the object r on M as an R-module category. A TOY MODEL FOR SHTUKA 4.4.2. 75 On the other hand, consider ι(r) ∈ HH• (R, FR ), where we recall that ι denotes the functor R→R ⊗ mult,R⊗R,mult ◦(FR ⊗Id) R =: HH• (R, FR ), corresponding to the left copy of R in the tensor product. Recall also that HH• (R, FR ) is itself a symmetric monoidal category, so for any F ∈ HH• (R, FR ) it makes sense to consider ι(r) ⊗ F ∈ HH• (R, FR ). 4.4.3. We claim: Theorem 4.4.4. There exists a canonical isomorphism (4.15) Tr(Hr ◦ FM , M) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r) ⊗ Trenh R (FM , M)), functorial in r ∈ R. This isomorphism is compatible with the actions of the two sides of (3.20). The rest of this subsection is devoted to the proof of this theorem. 4.4.5. For M as in the theorem, denote by FM,r the composite Hr ◦FM . Note that since R is symmetric monoidal, FM,r is also compatible with the action of FR on R. By Theorem 3.8.5, we have Tr(Hr ◦ FM , M) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , Trenh R (FM,r , M)). Hence, in order to prove Theorem 4.4.4, it suffices to establish the following: Theorem 4.4.6. There exists a canonical isomorphism (4.16) enh ι(r) ⊗ Trenh R (FM , M) ≃ TrR (FM,r , M) as objects of HH• (R, FR ). [Theorem 4.4.4] 4.4.7. Proof of Theorem 4.4.6. Consider the diagram (4.17) FR / R - mod ttt ttttt t t t tttt Id Id ttttαtr t t t ttt t t t  v~ ttt  / R - mod, R - mod R - mod FR where αr is the 2-morphism, which, when evaluated on M′ ∈ R - mod (see Sect. 3.6.2 for what evaluation means), acts as Hr : M′ → M′ , viewed as a map of R-module categories. Concatenating with the diagram Id / DGCat tt t t tttt t t t tttt M M ttttαtFM t t t ttt t t t  v~ ttt  / R - mod, R - mod DGCat FR 76 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY which produces Trenh R (FM , M), we obtain the diagram Id / DGCat tt t t tttt t t t tttt M M ttttαtFM,r t t tttt t t  v~ tttt  / R - mod, R - mod DGCat FR which produces Trenh R (FM,r , M). Since the formation of trace is compatible with compositions, it suffices to prove the following: Proposition 4.4.8. The map HH• (R, FR ) → HH• (R, FR ), (4.18) induced by (4.17), is given by ι(r) ⊗ −. [Theorem 4.4.6] Proof of Proposition 4.4.8. Consider the corresponding diagram / R⊗2 - mod FR ⊗Id / R⊗2 - mod counit / DGCat tt tt tt t t t t t t t ttt tttt tttt t t t t t t t t t αr t idtt id t tt ttt ttt Id Id Id Id ttttt ttttt ttttt t t t t t t t t t t t t t t t t t tt  v~ tttt FR ⊗Id  v~ ttt counit  v~ tttt unit  ⊗2 ⊗2 / / DGCat, / DGCat R - mod R - mod unit DGCat which gives rise to (4.18). The 2-morphism in the inner square, when evaluated on Q ∈ (R ⊗ R) - mod, acts as Hr ⊗ Id . This makes the assertion concerning (4.18) manifest.  Remark 4.4.9. The proof of Theorem 4.4.6 can be reformulated as the combination of the following two assertions. Consider the R-module which is R itself, equipped with the endofunctor FR,r := Hr ◦ FR . The first assertion is that there is a canonical isomorphism in HH• (R, FR ) Trenh R (FR,r , R) ≃ ι(r). This is a particular case of Proposition 4.4.8. For the second assertion, consider (R - mod, ResFR ) as a commutative algebra object in the symmetric monoidal category L(Morita(DGCat))rgd , and the monoid (4.19) MapsL(Morita(DGCat))rgd ((DGCat, Id), (R - mod, ResFR )), where we recall that (DGCat, Id) is the unit in L(Morita(DGCat))rgd . The second assertion is that Trenh is a symmetric monoidal functor R (4.20) MapsL(Morita(DGCat))rgd ((DGCat, Id), (R - mod, ResFR )) → HH• (R, FR ). This follows from the fact that the functor Tr of (3.10) is symmetric monoidal. To deduce Theorem 4.4.6, we note that for any (M, FM ) we have (M, FM,r ) ≃ (R, FR,r ) ⊗ (M, FM ) as objects in (4.19). A TOY MODEL FOR SHTUKA 77 Note also that for this manipulation, we regarded L(Morita(DGCat))rgd as an (∞, 1)-category (i.e., we did not need to consider non-invertible 3-morphisms, as in Sect. 3.9). More generally, Trenh can be R considered as a functor of symmetric monoidal categories (4.21) MapsL(Morita(DGCat))rgd ((DGCat, Id), (R - mod, ResFR )) → HH• (R, FR ), whose source is the symmetric monoidal category (4.22) MapsL(Morita(DGCat))rgd ((DGCat, Id), (R - mod, ResFR )), where Maps(−, −) denotes the (∞, 1)-category of maps between objects in a given (∞, 2)-category. 4.5. Cyclicity and observables. We return to the setting of Theorem 4.4.4. Let us now fix two objects r1 , r2 ∈ R. 4.5.1. On the one hand, consider the following two modules over Tr(FR , R): Tr(Hr1 ◦ Hr2 ◦ FM , M) and Tr(HFR (r1 ) ◦ Hr2 ◦ FM , M). We claim that there exists a canonical isomorphism Tr(Hr1 ◦ Hr2 ◦ FM , M) ≃ Tr(HFR (r1 ) ◦ Hr2 ◦ FM , M) (4.23) Indeed, it is obtained as the composition cyclicity of trace Tr(Hr1 ◦ Hr2 ◦ FM , M) −−−−−−−−−−→ Tr(HFR (r1 ) ◦ Hr2 ◦ FM , M) ← −−−−− ∼ (4.24) ∼ Tr(Hr2 ◦ FM ◦ Hr1 , M)  ∼ y Tr(Hr2 ◦ HFR (r1 ) ◦ FM , M), where the second arrow uses the isomorphism FM ◦ Hr1 ≃ HFR (r1 ) ◦ FM , (4.25) and the last arrow uses the fact that the product on R is commutative. 4.5.2. On the other hand, consider the following two modules over End HH• (R,FR ) (1HH• (R,FR ) ): Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r1 ⊗ r2 ) ⊗ Trenh R (FM , M)) and Hom HH• (R,FR ) (1HH• (R,FR ) , ι(FR (r1 ) ⊗ r2 ) ⊗ Trenh R (FM , M)). We claim that there is a canonical isomorphism Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r1 ⊗ r2 ) ⊗ Trenh R (FM , M))   ∼y (4.26) Hom HH• (R,FR ) (1HH• (R,FR ) , ι(FR (r1 ) ⊗ r2 ) ⊗ Trenh R (FM , M)). Indeed, this follows from the fact that ι(r) ≃ ι(FR (r)). (4.27) 4.5.3. We now claim: Lemma 4.5.4. The isomorphisms (4.23) and (4.26) match up under the isomorphisms Tr(Hr1 ◦ Hr2 ◦ FM , M) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r1 ⊗ r2 ) ⊗ Trenh R (FM , M)) and Tr(HFR (r1 ) ◦ Hr2 ◦ FM , M) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(FR (r1 ) ⊗ r2 ) ⊗ Trenh R (FM , M)) of Theorem 4.4.4. 78 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Proof. Note first that it suffices to show the assertion in the particular case r2 = 1R and r1 = r. (Namely, the general case would follow from this particular case by replacing FM by FM,r2 ). Thus, we have to show that under the isomorphisms Tr(Hr ◦ FM , M) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(r) ⊗ Trenh R (FM , M)) and Tr(HFR (r) ◦ FM , M) ≃ Hom HH• (R,FR ) (1HH• (R,FR ) , ι(FR (r)) ⊗ Trenh R (FM , M)) of Theorem 4.4.4, the composition (4.28) cyclicity of trace (4.25) ∼ ∼ Tr(Hr ◦ FM , M) −−−−−−−−−−→ Tr(FM ◦ Hr , M) − −−−− → Tr(HFR (r) ◦ FM , M), of the left hand sides corresponds to the isomorphism induced by (4.27) of the right hand side. Note that isomorphism (3.5), corresponding to the pair of diagrams (4.29) Id / DGCat tttt t t t tttt ttttt t M M t t tttttαFM t t t t  v~ tttt  / R - mod R - mod DGCat Id / DGCat tttt t t t tttt ttttt t M M t t tttttαidM ,r t t t t  v~ tttt  / R - mod R - mod DGCat FR Id induces an isomorphism enh Trenh R (Hr ◦ FM , M) ≃ TrR (FM ◦ Hr , M) (4.30) of objects of HH• (R, FR ). To finish the proof, it suffices to show that diagrams (3.21) (4.31) Tr(Hr ◦ FM , M) − −−−− → Hom HH• (R,FR ) (1HH• (R,FR ) , Trenh R (Hr ◦ FM , M)) ∼     ∼ycyclicity of trace ∼y(4.30) (3.21) Tr(FM ◦ Hr , M) − −−−− → Hom HH• (R,FR ) (1HH• (R,FR ) , Trenh R (FM ◦ Hr , M)) ∼ and Trenh R (Hr ◦ FM , M)   (4.25)◦(4.30)y∼ (4.32) (4.16) − −−−− → ∼ ∼ ι(r) ⊗ Trenh R (FM , M))   ∼y(4.27) Trenh −−−− → ι(FR (r)) ⊗ Trenh R (HFR (r) ◦ FM , M) − R (FM , M)) (4.16) are commutative. We will deduce both assertions from the commutativity of the diagram (3.9). To show the commutativity of (4.31), consider the following pair of diagrams: (4.33) FR / R - mod ttt t tttt t t t tt tttt oblv oblv ttttttaut t t t t t t  v~ tttt  / DGCat DGCat R - mod Id Id / R - mod ttt t tttt t t t tt tttt oblv oblv ttttttaut t t t t t t  v~ tttt  / DGCat. DGCat R - mod Id Arguing as in Sect. 3.10.10 or Sect. 3.12.2, the isomorphism (3.5) corresponding to the diagrams in (4.33) can be identified with the identity endomorphism on Hom HH• (R,FR ) (1HH• (R,FR ) , −). A TOY MODEL FOR SHTUKA 79 Since the vertical compositions of the diagrams appearing in (4.29) and (4.33) are isomorphic, respectively, to (4.34) Id / DGCat tt t t t ttt t t t tt tttt M M tttttαFM t t t t t t  v~ tttt  / DGCat DGCat DGCat Id Id / DGCat tt t t t ttt t t t tt tttt M M tttttαidM ,r t t t t t t  v~ tttt  / DGCat, DGCat DGCat Id the commutative diagram (3.9), corresponding to (4.29) and (4.33), naturally identifies with (4.31). To show the commutativity of (4.32), note that the diagrams in (4.29) can be written as vertical compositions of the diagrams (4.35) Id / DGCat tt t t t ttttt t t t tttt M M tttttαFM t t t t t t  v~ tttt  / R - mod R - mod FR ttt ttttt t t t ttttt Id Id ttttId t t t t t t t t   v~ tttt / R - mod R - mod DGCat FR Id / DGCat tt t t t ttttt t t t tttt M M tttttId t t t t t t  v~ tttt  / R - mod R - mod Id ttt ttttt t t t tttt Id Id ttttαtr t t t t t t t t   v~ tttt / R - mod. R - mod DGCat Id Arguing as in the proof of Proposition 4.4.8, the isomorphism (3.5) corresponding to the bottom part of (4.35) can be identified with the isomorphism of functors ι(r) ⊗ − ≃ ι(FR (r)) ⊗ −, induced by isomorphism (4.27). Since the isomorphism (3.5) corresponding to the top part of (4.35) can be identified with the identity endomorphism of Trenh R (FM , M), the commutative diagram (3.9), corresponding to the diagrams in (4.35), naturally identifies with (4.32).  5. The “shtuka” construction In this section we combine all the ingredients developed in the previous sections to obtain our the toy model for the shtuka construction. 5.1. The universal shtuka. 5.1.1. Let A be a rigid symmetric monoidal category and Y an object of Spc. Consider the symmetric monoidal category A⊗Y . Let φ be an endomorphism of Y , and let A⊗φ be the induced (symmetric monoidal) endofunctor of A . ⊗Y Let M be a dualizable DG category, equipped with an action of A⊗Y . Let FM be an endofunctor of M compatible with the action of A⊗φ on A⊗Y . 80 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 5.1.2. According to Sect. 3.8.2, to this data we can attach an object ⊗Y Trenh , A⊗φ ). A⊗Y (FM , M) ∈ HH• (A We identify HH• (A⊗Y , A⊗φ ) ≃ A⊗Y /φ by Proposition 3.7.5. We will use yet another notation for Trenh A⊗Y (FM , M), namely ShtM,FM ,univ ∈ A⊗Y /φ , and call it “the universal shtuka”. 5.1.3. According to Theorem 3.8.5, we have an identification End A⊗Y /φ (1A⊗Y /φ ) ≃ Tr(A⊗φ , A⊗Y ) (5.1) and Hom A⊗Y /φ (1A⊗Y /φ , ShtM,FM ,univ ) ≃ Tr(FM , M). (5.2) The isomorphisms (5.1) and (5.2) are compatible with the action of End A⊗Y /φ (1A⊗Y /φ ) on Hom A⊗Y /φ (1A⊗Y /φ , ShtM,FM ,univ ) and the action of Tr(A⊗φ , A⊗Y ) on Tr(FM , M). In particular, Tr(FM , M) carries an action of End A⊗Y /φ (1A⊗Y /φ ). 5.1.4. Example. Let e be a field of characteristic zero. Let A = Rep(G) and assume that Y has finitely many connected components. Then according to Theorem 1.5.5, A⊗Y /φ ≃ QCoh(LocSysG (Y /φ)), and hence End A⊗Y /φ (1A⊗Y /φ ) ≃ Γ(LocSysG (Y /φ), OLocSysG (Y /φ) ). We obtain that in this case ShtM,FM ,univ ∈ QCoh(LocSysG (Y /φ)), and the vector space Tr(FM , M) ≃ Γ(LocSysG (Y /φ), ShtM,FM ,univ ) carries an action of the algebra Γ(LocSysG (Y /φ), OLocSysG (Y /φ) ). 5.1.5. According to Sect. 1.8.5, we can think of an object of A⊗Y /φ as a compatible family of functors A⊗I → LS((Y /φ)I ), I ∈ fSet . Applying this to the object ShtM,FM ,univ ∈ A⊗Y /φ we obtain a family of functors denoted ShtM,FM ,Y /φ,I : A⊗I → LS((Y /φ)I ). We will denote by ShtM,FM ,Y,I the composite of ShtM,FM ,Y /φ,I with the pullback functor LS((Y /φ)I ) → LS(Y I ), corresponding to the projection Y → Y /φ. By Sect. 1.8.6, the object ShtM,FM ,Y /φ,∅ ∈ Vect can be canonically identified with Hom A⊗Y /φ (1A⊗Y /φ , ShtM,FM ,univ ) ≃ Tr(FM , M). A TOY MODEL FOR SHTUKA 5.1.6. 81 The goals of the present section are the following: –Describe the functors ShtM,FM ,Y,I explicitly via the usual (i.e., 1-categorical) trace construction; –Describe the descent of ShtM,FM ,Y,I to ShtM,FM ,Y /φ,I via the action of “partial Frobeniuses”; –Describe the action of End A⊗Y /φ (1A⊗Y /φ ) on Tr(FM , M) in terms of the functors ShtM,FM ,Y /φ,I via the “excursion operators”; –Prove the “S=T” identity (see Sect. 5.5 for what this means). All of the above will amount to an application of the constructions of the previous subsections. 5.1.7. Before we proceed further, let us note the following functoriality property of the shtuka construction in Y : Let us be given another space Y ′ , equipped with an endomorphism φ′ , and a map of spaces ψ : ′ Y ′ → Y that intertwines φ′ and φ. Let Resψ (M) denote the A⊗Y -module category, obtained from M by restricting along ψ. Consider the resulting objects ShtM,FM ,univ ∈ A⊗Y /φ and ShtResψ (M),FM ,univ ∈ A⊗Y ′ /φ′ . From (3.35) and we obtain: Corollary 5.1.8. The object ShtResψ (M),FM ,univ is obtained from ShtM,FM ,univ by applying the right adjoint of the functor ′ ′ A⊗ψ : A⊗Y /φ → A⊗Y /φ . Remark 5.1.9. Note that the equivalences A⊗Y /φ ≃ (A⊗Y /φ )∨ and A⊗Y from Sect. 1.8.5 identify the right adjoint (A Lemma 9.2.6]). ⊗ψ R ) of A ′ /φ′ ⊗ψ ≃ (A⊗Y ′ /φ′ ∨ ) with the dual (A⊗ψ )∨ (see [GR1, Section 1, Therefore it follows from Corollary 5.1.8 that the functor SY ′ /φ′ : A⊗Y corresponding to ShtResψ (M),FM ,univ ∈ A ⊗Y ′ /φ′ ′ /φ′ → Vect, , is obtained from the functor SY /φ : A⊗Y /φ → Vect, corresponding to ShtM,FM ,univ ∈ A⊗Y /φ , by precomposition with A⊗ψ . Remark 5.1.10. Note that in the example of Sect. 5.1.4, the map ψ induces a map LocSysG (ψ) : LocSysG (Y /φ) → LocSysG (Y ′ /φ′ ), and the functors A⊗ψ and (A⊗ψ )R identify with LocSysG (ψ)∗ and LocSysG (ψ)∗ , respectively. 5.2. Explicit description of I-legged shtukas. We retain the setting of Sect. 5.1.1. 5.2.1. Fix a finite set I. The evaluation map YI ×I →Y defines a map from Y I to the space of symmetric monoidal functors A⊗I → A⊗Y , and hence to the space of actions of A⊗I on M. Define the functor Sht′M,FM ,Y,I : Y I × A⊗I → Vect as follows: It sends y ∈ Y I , r ∈ A⊗I 7→ Tr(Hry ◦ FM , M), 82 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY where ry denotes the image of r along the functor y A⊗I −→ A⊗Y . See Sect. 3.5.11 for an explanation of how this relates to the usual notion of shtukas. Remark 5.2.2. We can tautologically rewrite the functor Y I × A⊗I → End(M), (5.3) which appears in the definition of Sht′M,FM ,Y,I , (y, r) 7→ Hry , as follows: Recall that according to Sect. 1.7.3, a datum of action of A⊗Y on M gives rise to a map A⊗I → End(M) ⊗ LS(Y I ), or, equivalently by Proposition 1.4.10, to a map Y I → Maps(A⊗I , End(M)). (5.4) The map (5.3) is the one corresponding to (5.4). 5.2.3. We claim: Proposition 5.2.4. The functor A⊗I → LS(Y I ), corresponding to Sht′M,FM ,Y,I , identifies canonically with ShtM,FM ,Y,I . Proof. Taking into account Sect. 1.8.5, we have to establish the isomorphism Hom A⊗Y /φ (1A⊗Y /φ , rȳ ⊗ ShtM,FM ,univ ) ≃ Tr(Hry ◦ FM , M), functorial in y ∈ Y I , where ȳ is the image of y under the projection Y → Y /φ. Now the assertion follows from Theorem 4.4.4 using the fact that rȳ ≃ ι(ry ).  5.3. Partial Frobeniuses. 5.3.1. Since the object is the image of the object under the pullback functor   ShtM,FM ,Y,I ∈ Funct A⊗I , LS(Y I )   ShtM,FM ,Y /φ,I ∈ Funct A⊗I , LS((Y /φ)I ) LS((Y /φ)I ) → LS(Y I ), the former should carry the structure of equivariance with respect to the endomorphisms that act as φ along each of the Y factors in Y I . Fix an element i1 ∈ I and write I = {i1 } ⊔ I ′ . We will now write down explicitly the structure of equivariance corresponding to this endomorphism of Y I , to be denoted φi1 . Using the identification of Proposition 5.2.4, this amounts to describing the isomorphism (5.5) Sht′M,FM ,Y,I ◦(φi1 × IdA⊗I ) ≃ Sht′M,FM ,Y,I as functors Y I × A⊗I → Vect . A TOY MODEL FOR SHTUKA 5.3.2. 83 By definition, the datum of (5.5) amounts to a system of isomorphisms Tr(Hrφ 1 (y) ◦ FM , M) ≃ Tr(Hry ◦ FM , M), (5.6) i r ∈ A⊗I . Write y as y = y1 ⊔ y′, ′ y 1 ∈ Y, y ′ ∈ Y I . It is enough to establish (5.6) for r of the form r1 ⊗ r′ , ′ r 1 ∈ A, r ′ ∈ A⊗I . Proposition 5.3.3. The map (5.6) is given by the composition ∼ (5.7) −−−−− Tr(Hr1 ◦ Hr′ ′ ◦ FM , M) ← y y1   ∼ycyclicity of trace Tr(Hry ◦ FM , M) ∼ Tr(Hr′ ′ ◦ FM ◦ Hr1 , M) − −−−−→ Tr(Hr′ ′ ◦ Hr1 ◦ FM , M) y y y1 φ(y1 )  ∼ y Tr(Hrφ 1 (y) ◦ FM , M) i ∼ ← −−−−− Tr(Hr1 φ(y1 ) ◦ Hr′ ′ ◦ FM , M). y The proof follows from Lemma 4.5.4. 5.4. Description of the action via excursions. 5.4.1. Choose a pair of points ȳ1 , ȳ2 in Y /φ. Fix finite set J and a J-tuple γ J of paths γ i from ȳ1 to ȳ2 . Choose an element ξ ∈ Hom(1A , multJ ◦ multR J (1A )). Let Excuniv (γ J , ξ) denote the resulting endomorphism of 1A⊗Y /φ , see Sect. 2.8.4. 5.4.2. The next assertion follows from the definition of the excursion operators in Sect. 2.8.3: Proposition 5.4.3. The action of Excuniv (γ J , ξ) on Tr(FM , M) ≃ Hom A⊗Y /φ (1A⊗Y /φ , ShtM,FM ,univ ) ≃ ShtM,FM ,Y /φ,∅ is given by the excursion ShtM,FM ,Y /φ,∅   ∼y (5.8)

ξ −−−− → evȳ1 ShtM,FM ,Y /φ,{∗} (multJ ◦ multR evȳ1 ShtM,FM ,Y /φ,{∗} (1A ) − J (1A ))  ∼ y
evȳ J ShtM,FM ,Y /φ,J (multR J (1A )) 1  mon J γ y
evȳ J ShtM,FM ,Y /φ,J (multR J (1A )) 2  ∼ y

counit −−−− − evȳ2 ShtM,FM ,Y /φ,{∗} (multJ ◦ multR evȳ2 ShtM,FM ,Y /φ,{∗} (1A ) ← J (1A ))   ∼y ShtM,FM ,Y /φ,∅ . 84 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 5.5. The “S=T” identity, vacuum case. We are now coming to what is perhaps in the most interesting part in the entire story. 5.5.1. Let y0 ∈ Y be a φ-fixed point. Fix a compact object a ∈ A. Note that the object ay0 ∈ A⊗Y is equipped with a natural isomorphism ∼ αtaut : ay0 → φ∗ (ay0 ). Consider the corresponding natural transformation (in fact, an isomorphism) ∼ αay0 ,M,FM ,taut : Hay0 ◦ FM → FM ◦ Hay0 , (5.9) see (3.15). Hence we can consider the endomorphism Tr(Hay0 , αay0 ,M,FM ,taut ) : Tr(FM , M) → Tr(FM , M). Remark 5.5.2. We should think of Tr(Hay0 , αay0 ,M,FM ,taut ) as an analogue of the Hecke operator acting on the space of automorphic functions, corresponding to a finite-dimensional representation of Ǧ (thought of as a ∈ Rep(Ǧ)) applied at a rational point of the curve (thought of as y0 ). So, this is V. Lafforgue’s “T” operator. 5.5.3. point Let ȳ0 be the projection of y0 to Y /φ. The fact that y0 was φ-invariant defines a tautological γtaut ∈ Ω(Y /φ, ȳ0 ). Let ξa be the map 1A → mult ◦ multR (1A ) defined by a, see (4.14). 5.5.4. We claim: Theorem 5.5.5. The endomorphism Tr(Hay0 , αay0 ,M,FM ,taut ) of Tr(FM , M) ≃ ShtM,FM ,Y /φ,∅ + equals the operator (5.8) for (ȳ0 , J = {∗} ⊔ {∗}, γtaut , ξa ), where + γtaut = (γtaut , γtriv ). + Remark 5.5.6. We observe that by Proposition 5.3.3, the operator (5.8) for (ȳ0 , J = {∗}⊔{∗}, γtaut , ξa ), appearing in Theorem 5.5.5, is given explicitly by ξa Tr(FM , M) − −−−− → Tr(H(mult ◦ multR (1A ))y0 ◦ FM , M)  ∼ y Tr(H(multR (1A ))y0 ,y0 ◦ FM , M)  partial Frobenius, i.e., isomorphism (5.7) y (5.10) counit Tr(H(multR (1A ))y0 ,y0 ◦ FM , M)  ∼ y Tr(FM , M) ← −−−−− Tr(H(mult ◦ multR (1A ))y0 ◦ FM , M). Remark 5.5.7. Note that the operator (5.10) is an analog of V. Lafforgue’s “S” operator. For this reason, we view Theorem 5.5.5 as the toy model for V. Lafforgue’s “S=T” statement, i.e., [Laf, Proposition 6.2]. A TOY MODEL FOR SHTUKA 85 Proof of Theorem 5.5.5. First, we note that when considering the endomorphism Tr(Hay0 , αay0 ,M,FM ,taut ) of Tr(FM , M), we can replace the original Y by {∗} (with the necessarily trivial endomorphism φ) via y 0 {∗} −→ Y. Next, we claim that when considering the operator (5.8) for + (ȳ0 , J = {∗} ⊔ {∗}, γtaut , ξa ), we can also replace Y by {∗}. Indeed, this immediately follows either from Remark 5.1.9 or from Remark 5.5.6. Hence, in proving the theorem, we can assume that Y = {∗}. Note that {∗}/ id ≃ S 1 . On the one hand, by Proposition 3.4.6, the endomorphism Tr(Hay0 , αay0 ,M,FM ,taut ) of Tr(FM , M) equals the action of the element cl(ay0 , αa,taut ) ∈ Tr(A⊗φ , A⊗Y ), which in the case Y = {∗} amounts to the element cl(a, id) ∈ Tr(Id, A). + On the other hand, by Proposition 5.4.3, the operator (5.8) for (ȳ0 , J = {∗} ⊔ {∗}, γtaut , ξa ) is given by the action of + , ξa ) ∈ End A⊗Y /φ (1A⊗Y /φ ), Excuniv (γtaut which in our case is the element + Excuniv (γtaut , ξa ) ∈ End A⊗S 1 (1A⊗S 1 ) in the notations of Proposition 4.3.5. Hence, we need to show to show that the above two elements match under the isomorphism (5.11) Tr(A⊗φ , A⊗Y ) ≃ End A⊗Y /φ (1A⊗Y /φ ) of Theorem 3.8.5, where we identify A⊗Y /φ ≃ HH• (A⊗Y , A⊗φ ). Thus, in our case, the identification (5.11) amounts to Tr(Id, A) ≃ End A⊗S 1 (1A⊗S 1 ). Now, the desired identity of elements follows from Corollary 4.3.6.  5.6. The “S=T” identity, general case. As was explained in Remark 5.5.7, the assertion of Theorem 5.5.5 is an analog of V. Lafforgue’s S = T identity as operators acting on the space of automorphic functions, i.e., empty-legged shtukas. We will now discuss its generalization, which extends the S = T identity as operators on I-legged shtukas. 86 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY 5.6.1. Let y0 and a be as in Sect. 5.5.1 above. Fix a finite set J, a point y ∈ Y J and an object r ∈ A⊗J . Denote J+ := {∗} ⊔ J, y + := y0 ⊔ y, J++ := {∗} ⊔ {∗} ⊔ J, y ++ := y0 ⊔ y0 ⊔ y. The natural transformation (5.9) induces a natural transformation Hay0 ◦ Hry ◦ FM ≃ Hry ◦ Hay0 ◦ FM → Hry ◦ FM ◦ Hay0 , to be denoted αa,M,FM ,taut,ry . Then we can consider the resulting endomorphism (5.12) Tr(Hay0 , αa,M,FM ,taut,ry ) : Tr(Hry ◦ FM , M) → Tr(Hry ◦ FM , M). Remark 5.6.2. The map (5.12) is an analog of V. Lafforgue’s “T” operator acting on the cohomology of shtukas. 5.6.3. Let ȳ be the projection of y to (Y /φ)J . By Proposition 5.2.4, we have an identification

Tr(Hry ◦ FM , M) ≃ evy Sht′M,FM ,Y,J (r) ≃ evȳ ShtM,FM ,Y /φ,J (r) . 5.6.4. Set ȳ + := ȳ0 ⊔ ȳ and ȳ++ := ȳ0 ⊔ ȳ0 ⊔ ȳ. Then we can consider the endomorphism of evȳ (ShtM,FM ,Y,J (r)) equal to the composite

∼ −−−−→ evȳ+ ShtM,FM ,Y /φ,J+ (r ⊗ 1A ) evȳ ShtM,FM ,Y /φ,J (r) −   ξa y
evȳ+ ShtM,FM ,Y /φ,J+ (r ⊗ mult ◦ multR (1A ))  ∼ y
evȳ ++ ShtM,FM ,Y /φ,J++ (r ⊗ multR (1A ))  monγ ,γ ,γ J (5.13) y taut triv triv
evȳ ++ ShtM,FM ,Y /φ J++ (r ⊗ multR (1A ))  ∼ y
evȳ+ ShtM,FM ,Y /φ,J+ (r ⊗ mult ◦ multR (1A ))   counity

∼ −−−− − evȳ+ ShtM,FM ,Y /φ,J+ (r ⊗ 1A ) evȳ ShtM,FM ,Y /φ,J (r) ← Remark 5.6.5. Note that by Propositions 5.2.4 and 5.3.3, the operator (5.13) identifies with ξ a Tr(Hry ◦ FM , M) − −−− − → Tr(H(mult ◦ multR (1A ))y0 ◦ Hry ◦ FM , M)  ∼ y Tr(H(multR (1A ))y0 ,y0 ◦ Hry ◦ FM , M)  partial Frobenius, i.e., isomorphism (5.7) y (5.14) counit Tr(H(multR (1A ))y0 ,y0 ◦ Hry ◦ FM , M)  ∼ y −−−−− Tr(H(mult ◦ multR (1A ))y0 ◦ Hry ◦ FM , M). Tr(Hry ◦ FM , M) ← So the operator (5.13) is indeed an analog of V. Lafforgue’s “S” operator in the presence of other Hecke functors. A TOY MODEL FOR SHTUKA 5.6.6. 87 We claim: Theorem 5.6.7. The endomorphism Tr(Hay0 , αa,M,FM ,taut,ry ) of Tr(Hry ◦ FM , M) ≃ evȳ ShtM,FM ,Y /φ,J (r) equals the operator (5.13).
Proof. We claim that Theorem 5.6.7 is in fact a formal corollary of Theorem 5.5.5. Namely, for our given M set FM,ry := Hry ◦ FM . We claim that the identity stated in Theorem 5.6.7 specializes to the identity of Theorem 5.5.5 for the endomorphism FM,ry . Indeed, we have an isomorphism Tr(Hry ◦ FM , M) ≃ Tr(FM,ry , M), and under this identification, the endomorphism Tr(Hay0 , αa,M,FM ,taut,ry ) of the LHS corresponds to the endomorphism Tr(Hay0 , αa,M,FM,ry ,taut ) of the RHS (see Sect. 5.5.1 for the notation). It therefore suffices to show that there exists an isomorphism
(5.15) evȳ ShtM,FM ,Y /φ,J (r) ≃ ShtM,FM,ry ,Y /φ,∅ such that the endomorphism (5.13) of the LHS corresponds to the endomorphism (5.8) with + (ȳ0 , J = {∗} ⊔ {∗}, γtaut , ξa ) of the RHS. It follows from Theorem 4.4.6 that there exists a canonical isomorphism ι(ry ) ⊗ ShtM,FM ,univ ≃ ShtM,FM,ry ,univ of objects of A⊗Y /φ . Therefore using explicit formulas of Sect. 1.8.5, for every finite set I, a point ȳ ′ ∈ Y I and an object r ′ ∈ A⊗I , we have an isomorphism  
evȳ⊔ȳ′ ShtM,FM ,Y /φ,J ⊔I (r ⊗ r ′ ) ≃ evȳ ′ ShtM,FM,ry ,Y /φ,I (r ′ ) , functorial in ȳ′ and r ′ . Applying this isomorphism in the particular case I = ∅ and r ′ = 1Vect , we get the isomorphism (5.15) we need.  Appendix A. Sheaves and singular support This appendix is included for the sake of completeness. We will review the notion of singular support in different sheaf-theoretic contexts. The main result of this section is Theorem A.3.9, which says the following: In the product situation Y × X, where Y is an algebraic stack and X a proper scheme, the category of sheaves on Y × X whose singular support is of the form N′ := N × {zero section} ⊂ T ∗ (Y) × T ∗ (X) = T ∗ (Y × X), is equivalent to the tensor product category ShvN (Y) ⊗ Shvlisse (X). A.1. Sheaf-theoretic contexts. 88 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY A.1.1. In this section and the next, we will take Shv(−) to be any of the following sheaf-theoretic contexts: (a) Shvcl (S), the category of all sheaves in the classical topology with coefficients in e, for S an affine scheme over C; (a’) Shv(S) = Ind(Shvcl,constr (S)), where Shvcl,constr (S) ⊂ Shvcl (S) is the (small) subcategory consisting of constructible sheaves; (b) Shv(S) = D-mod(S), for S an affine scheme over a ground field k of characteristic 0; (b’) Shv(S) = Ind(D-modhol (S)), where D-modhol (S) ⊂ D-mod(S) is the (small) subcategory consisting of holonomic D-modules; (b”) Shv(S) = Ind(D-modhol,RS (S)), where D-modhol,RS (S) ⊂ D-mod(S) is the (small) subcategory consisting of holonomic D-modules with regular singularities; (c) Shv(S) := ShvZ/ℓn ,et (S), the category of all étale sheaves on S with coefficients in Z/ℓn Z, for S an affine scheme over any ground field of characteristic prime to ℓ. Note that we have ShvZ/ℓn ,et (S) ≃ Ind(ShvZ/ℓn ,et,constr (S)), where ShvZ/ℓn ,et,constr (S) ⊂ ShvZ/ℓn ,et (S) is the full subcategory of constructible sheaves, see [GaLu, Proposition 2.2.6.2]; (d) Shv(S) := Ind(ShvZℓ ,et,constr (S)), where ShvZℓ ,et,constr (S) is the category of constructible ℓ-adic sheaves on S (see [GaLu, Defn. 2.3.2.1]), which is equivalent to lim ShvZ/ℓn ,et,constr (S); n (d’) Shv(S) := Ind(ShvQℓ ,et,constr (S)), where ShvQℓ ,et,constr (S) is obtained from ShvZℓ ,et,constr (S) by inverting ℓ. Note that Ind(ShvQℓ ,et,constr (S)) ≃ Ind(ShvZℓ ,et,constr (S)) ⊗ Qℓ . Zℓ A.1.2. In what follows, we will refer to the cases (a’) and (b’)-(d’) as “ind-constructible”. Note that, by definition, in these cases, the category Shv(S) is compactly generated. In particular, it is dualizable. A.1.3. Let Y be a prestack. In all cases apart from (a), we define Shv(Y) as (A.1) lim S∈(Schaff )op /Y Shv(S), f where for S1 → S2 , the corresponding functor Shv(S2 ) → Shv(S1 ) is f ! . In the above formula, Schaff /Y is the category of affine schemes over Y. Note that in all cases apart from (b), the functor f ! admits a left adjoint, namely f! . Hence, from Lemma 1.4.8(b), we obtain that in all of these cases, we can write Shv(Y) also as (A.2) colim Shv(S), S∈Schaff /Y f where for S1 → S2 , the corresponding functor Shv(S1 ) → Shv(S2 ) is f! . Note also that in the ind-constructible contexts (i.e., cases (a’) and (b’)-(d’)), we obtain from (A.2) that the category Shv(Y) is compactly generated (since each Shv(S) is). In particular, Shv(Y) is dualizable. A TOY MODEL FOR SHTUKA 89 A.1.4. Let us now consider case (a). (The slight glitch here is that in this case the functors f ! are no longer continuous.) We define Shv(Y) by formula (A.2), where the colimit is taken DGCat. Let DGCatdiscont denote the category whose objects are not necessarily cocomplete DG categories, and we allow all (exact) functors between such categories. Specifically, DGCatdiscont is the category of stable idempotent-complete ∞-categories tensored over Vectf.d. and exact Vectf.d. -linear functors. We have a tautological functor DGCat → DGCatdiscont (A.3) We can also form the limit (A.1), taking place in DGCatdiscont . Now, according to [GR1, Corollary 5.3.4(b)] (which is a generalization of Lemma 1.4.8 to the case when the right adjoints are not necessarily continuous), the image of Shv(Y), which is defined by formula (A.2), under the forgetful functor (A.3) identifies with the limit (A.1). In other words, we have a canonical isomorphism Shvcl (Y) ≃ lim S∈Schaff /Y Shvcl (S) as objects of DGCatdiscont . A.1.5. Let now Y be an algebraic stack. In this case, we can consider the category Schaff /Y,sm , consisting of affine schemes equipped with a smooth map to Y, and whose morphisms are smooth maps between affine schemes over Y. A smooth descent argument shows that the restriction functor lim S∈Schaff /Y Shv(S) → lim S∈Schaff /Y,sm Shv(S) is an equivalence. Hence, for an algebraic stack Y, we have (A.4) Shv(Y) ≃ lim S∈Schaff /Y,sm Shv(S) where this isomorphism takes place in DGCatdiscont in context (a) and in DGCat in other contexts. Remark A.1.6. The presentation of Shv(Y) given by (A.4) as an object of DGCat is valid also in case (a): indeed, since smooth pullbacks f ! are continuous also in case (a), the assertion follows from (A.4) and the fact that the tautological functor DGCat → DGCatdiscont preserves limits, and for C1 , C2 ∈ DGCat, the map induced by forgetful functor MapsDGCat (C1 , C2 ) → MapsDGCatdiscont (C1 , C2 ) induces an isomorphism on the connected components corresponding to equivalences. A.1.7. Let Corr(PreStk)ind-sch,all be the category of correspondences as in [GR2, Chapter 3, Sect. 5.4], whose objects are prestacks Y, and where morphisms from Y1 to Y2 are diagrams g (A.5) −−−− → Y1 Y1,2 −   fy Y2 , where g any map, and f is required to be ind-schematic. The composition of (A.5) and Y2,3 − −−−− → Y2   y Y3 90 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY is given by Y2,3 × Y1,2 − −−−−→ Y1 Y2   y Y3 . Let us first exclude the context (a). Then the construction of loc.cit. applies, and we can extend Shv(−) to a functor ShvCorr : Corr(PreStk)ind-sch,all → DGCat . At the level of objects this functor sends Y 7→ Shv(Y). At the level of morphisms, this functor sends a morphism (A.5) to the functor f∗ ◦ g ! : Shv(Y1 ) → Shv(Y2 ). Compatibility with compositions is insured by base change. Furthermore, the functor ShvCorr possesses a natural right-lax symmetric monoidal structure, see [GR2, Chapter 3, Sect. 6.1], where Corr(PreStk)ind-sch,all is a symmetric monoidal category with respect to the level-wise product. A.1.8. Let us now consider the context (a). In this case we will consider ShvCorr as taking values in DGCatdiscont . We will regard ShvCorr as equipped with the right-lax symmetric monoidal structure, with respect to the following symmetric monoidal structure on DGCatdiscont : For C ∈ DGCatdiscont , we let FunctDGCatdiscont (C1 discont ⊗ C2 , C) consist of all bi-exact bi-e-linear (but not necessarily bi-continuous) functors C1 × C2 → C. More formally, we apply the construction of [GR1, Chapter 1, Sect. 6.1.1], but dropping the continuity condition. Remark A.1.9. Note that given an object C ∈ DGCat, the space of structures of associative (resp., commutative) algebras on it within DGCat embeds fully faithfully into the space of such structures within DGCatdiscont . The same applies to actions of a given monoidal DG category on another DG category. A.2. Sheaves on a product. A.2.1. Note that for a pair of affine schemes, we have a naturally defined functor, given by external tensor product (A.6) Shv(S1 ) ⊗ Shv(S2 ) → Shv(S1 × S2 ), F1 , F2 7→ F1 ⊠ F2 . This functor is an equivalence in case (b): for a pair of associative algebras A1 and A2 , the functor A1 -mod ⊗ A2 -mod → (A1 ⊗ A2 )-mod, is an equivalence (see [Lu2, Theorem 4.8.5.16]). M1 , M2 7→ M1 ⊗ M2 A TOY MODEL FOR SHTUKA 91 A.2.2. The following is known (the assertion is valid for any pair of locally compact Hausdorff topological spaces, see [Lu1, Theorem 7.3.3.9, Prop. 7.3.1.11] and [Lu2, Prop. 4.8.1.17]): Theorem A.2.3. The functor (A.6) is an equivalence in case (a). Remark A.2.4. It follows from Theorem A.2.3 that for a locally compact topological space M , the functors e ∆ M Vect → Shv(M ) →! Shv(M × M ) ≃ Shv(M ) ⊗ Shv(M ) and ∆∗ Shv(M ) ⊗ Shv(M ) → Shv(M × M ) → Shv(M ) C• c (M,−) −→ Vect define an identification Shv(M )∨ ≃ Shv(M ). Note, however, that as was shown by A. Neeman (see [Ne1]), for a topological manifold M , the category Shv(M ) is not compactly generated, unless M is discrete. So, Shv(M ) is an example of a dualizable but not compactly generated category. A.2.5. have: In the ind-constructible contexts, the functor (A.6) fails to be an equivalence. However, we Lemma A.2.6. In the constructible contexts the functor (A.6) is fully faithful. Proof. For a pair of DG categories C1 , C2 and c′i ∈ Cci , c′′i ∈ Ci , i = 1, 2, the map Hom C1 (c′1 , c′′1 ) ⊗ Hom C2 (c′2 , c′′2 ) → Hom C1 ⊗C2 (c′1 ⊗ c′2 , c′′1 ⊗ c′′2 ) is an isomorphism (see [GR1, Chapter 1, Proposition 7.4.2]). Since the tensor product of compactly generated categories is compactly generated by tensor products of compact objects (see [GR1, Chapter 1, Proposition 7.4.2]), in order to prove the lemma, it suffices to show that for Fi′ , Fi′′ ∈ Shv(Si )c , i = 1, 2 the map Hom Shv(S1 ) (F1′ , F1′′ ) ⊗ Hom Shv(S2 ) (F2′ , F2′′ ) → Hom Shv(S1 ×S2 ) (F1′ ⊠ F2 , F1′′ ⊠ F2′′ ) is an isomorphism. However, this follows from Kunneth’s formula. A.2.7.  We now consider the case of prestacks. Again, external tensor product gives rise to a functor (A.7) Shv(Y1 ) ⊗ Shv(Y2 ) → Shv(Y1 × Y2 ), F1 , F2 7→ F1 ⊠ F2 . The argument in [GR1, Chapter 3, Theorem 3.1.7] shows: Lemma A.2.8. In case (b), if one of the categories Shv(Yi ) is dualizable, then the functor (A.7) is an equivalence. In addition, from Theorem A.2.3 and [GR1, Chapter 3, Equation (3.4)], we obtain: Corollary A.2.9. The functor (A.7) is an equivalence in case (a). Finally, we claim: Proposition A.2.10. The functor (A.7) is fully faithful in the ind-constructible contexts. 92 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Proof. Since Shv(Yi ) are dualizable (see Sect. A.1.3), the argument in [GR1, Chapter 3, Theorem 3.1.7] shows that we can write Shv(Y1 ) ⊗ Shv(Y2 ) as lim S1 →Y1 ,S2 →Y2 Shv(S1 ) ⊗ Shv(S2 ), and by [GR1, Chapter 3, Equation (3.4)], we have Shv(Y1 × Y2 ) ≃ lim S1 →Y1 ,S2 →Y2 Shv(S1 × S2 ). Hence, the assertion follows from Lemma A.2.6.  A.3. Singular support. A.3.1. Let S be an affine scheme. First, we assume that S is smooth. Let N ⊂ T ∗ (S) be a conical Zariski-closed subset. In each of our sheaf-theoretic contexts we can single out a full subcategory ShvN (S) ⊂ Shv(S), consisting of objects with singular support contained in N: –In case (a), we require that each cohomology sheaf belongs to Shvcl,N (S), where the latter is defined in [KS, Sect. 8]; –In case (a’), we inherit the definition from case (a) for Shvcl,constr (S), and then ind-extend (alternatively, transfer the definition via the Riemann-Hilbert from case (b”)). –In case (b), we take the ind-completion of the category D-modN (S)f.g ⊂ D-mod(S)f.g , obtained by requiring that each cohomology belong to D-modN (S)f.g.,♥ , where the latter is the standard D-module notion; –In case (b’) (resp., (b”)), we inherit the definition for D-modhol (S) (resp., D-modhol,RS (S)), then ind-extend. –In cases (c), we give the definition for ShvZ/ℓn ,et,constr (S) following [Be], then ind-extend. –In case (d) we will say that an object of ShvZℓ ,et,constr (S) has singular support in N if its projection mod ℓ does (alternatively, we apply the definition of [Be] directly to ShvZℓ ,et,constr (S)), then ind-extend; –In case (d’) we apply the definition of [Be] to ShvQℓ ,et,constr (S)), then ind-extend. Remark A.3.2. Note that in all cases apart from (a), there exists another (a priori, larger) full subcategory of Shv(S), to be denoted Shv∧ N (S), that one could call “sheaves with singular support in N”. Namely, an object belongs to Shv∧ N (S) if its cohomology sheaves (with respect to the perverse t-structure) belong to ShvN (S). One can show that Shv∧ N (S) identifies with the left completion of ShvN (S) with respect to its tstructure. What is not clear, however, is whether the category Shv∧ N (S) is compactly generated. ∧ Note that the embedding ShvN (S) ֒→ Shv∧ N (S) is not always an equivalence (equivalently, ShvN (S) is not always generated by objects that are compact in Shv(S)). For example, this occurs in the example of S = P1 and N being the zero-section. A TOY MODEL FOR SHTUKA 93 A.3.3. We will now show how to extend the above definition to the case when S is not necessarily smooth. Let F be a coherent sheaf on S (in cohomological degree 0). Represent F as coker(E1 → E0 ), (A.8) where E1 and E0 are locally free. Consider the total spaces of Ei as group-schemes over S, Tot(Ei ) := SpecS (SymOS (E∨ i )). (A.9) Consider the algebraic stack Tot(F) := Tot(E0 )/ Tot(E1 ). The object Tot(F), viewed as an algebraic stack, depends on the presentation (A.8). But it is well-defined in the localization of the category of algebraic stacks, where we invert morphisms that are smooth, surjective and whose fibers are of the form pt /H where H is a vector group. We have a well-defined notion of a Zariski-closed subset of Tot(F). For a choice of a presentation (A.8), they bijectively correspond to Zariski-closed subsets of Tot(E0 ) that are invariant under the action of Tot(E1 ). This notion does not depend on the presentation (A.8). A.3.4. Taking F := Ω1 (S), we thus obtain an object T ∗ (S) := Tot(Ω1 (S)) in the above localization of the category of algebraic stacks. For a closed embedding f : S ֒→ S ′ with S ′ smooth, the codifferential map S × T ∗ (S ′ ) → T ∗ (S) (A.10) S′ ∗ ∗ ′ realizes T (S) as a quotient of S × T (S ) by a vector group. S′ A.3.5. Thus, we have a well-defined notion of (conical) Zariski-closed subset of T ∗ (S). We emphasize that although T ∗ (S) as an algebraic stack depends on some choices, the set of its Zariski-closed subsets does not. Note also that the cotangent fibers, i.e., the sets Ts∗ (S) for s ∈ S(k), underlying the fibers of T ∗ (S), are the classical cotangent spaces, and as such do not depend on any choices. For N ⊂ T ∗ (S) let N′ ⊂ S × T ∗ (S ′ ) ⊂ T ∗ (S ′ ) S′ be the preimage of N under the map (A.10). A subset N as above is completely determined by its fibers Ns := N ∩ Ts∗ (S) ⊂ Ts∗ (S), s ∈ S(k). We say that an object F ∈ Shv(S) has singular support in N if f∗ (F) has singular support in N′ . It is not difficult to verify that this definition does not depend in the choice of the embedding f : S → S ′ . We shall say that N ⊂ T ∗ (S) is half-dimensional if N′ ⊂ T ∗ (S ′ ) is such for some/any smooth S ′ . A.3.6. Let Y be an algebraic stack. For a coherent sheaf F on Y we can talk about Zariski-closed subsets of Tot(F). Namely, they correspond bijectively to compatible families of Zariski-closed subsets aff of Tot(F|S ) for S ∈ Schaff /Y (or, equivalently, S ∈ Sch/Y,sm ). Taking F := Ω1 (Y), we thus obtain a well-defined notion of (conical) Zariski-closed subset of T ∗ (Y) (note that we are not even trying to define T ∗ (Y) itself; that said, as in the case of schemes, the cotangent fibers Ty∗ (Y), y ∈ Y(k), are the classical cotangent spaces, and thus are well-defined). Thus, we can talk about Zariski-closed subsets N ⊂ T ∗ (Y). As in the case of schemes, such N is completely determined by the subsets Ny := N ∩ Ty∗ (Y) ⊂ Ty∗ (Y), y ∈ Y(k). To a conical Zariski-closed subset N ⊂ T ∗ (Y) we associate a full subcategory ShvN (Y) ⊂ Shv(Y). 94 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Namely, an object belongs to ShvN (Y) if for any smooth map S → Y (for S an affine scheme), its pullback to S belongs to ShvNS (S) ⊂ Shv(S), where NS is the image of N × S under the co-differential Y T ∗ (Y) × S → T ∗ (S). (A.11) Y Here we are using the presentation of Shv(Y) as in Sect. A.1.5 (see also Remark A.1.6 in case (a)). In what follows we shall say that N ⊂ T ∗ (Y) is half-dimensional if its image is such under (A.11) for some/any smooth cover S → Y. A.3.7. Let Y be a smooth. In this case we define a full subcategory Shvlisse (Y) ⊂ Shv(Y). We first give the definition for affine schemes; for stacks, lisse would mean that the pullback to affine schemes under smooth maps is lisse. For an affine scheme, we set: –In case (a), lisse means that each cohomology sheaf is locally constant; –In case (a’), lisse means a colimit of constructible locally constant objects; –In cases (b), (b’) and (b”), lisse means a colimit of O-coherent objects; –In case (c), lisse means a colimit of constructible locally constant objects; –In case (d), lisse means a colimit of objects that are constructible and locally constant (i.e., ones whose reduction mod ℓ is constructible and locally constant); –In case (d’), lisse means a colimit of objects that are constructible and locally constant. We note that in all of the above case, being lisse is equivalent to belonging to ShvN (Y), where N is the zero-section. A.3.8. We are going to prove: Theorem A.3.9. Let N ⊂ T ∗ (Y) be half-dimensional. Let X be a smooth scheme, assumed proper in all cases apart from (a), (a’) and (b”). Set N′ := N × {zero-section} ⊂ T ∗ (Y × X). Then the functor ShvN (Y) ⊗ Shvlisse (X) → ShvN′ (Y × X) is an equivalence. Remark A.3.10. Note that if our sheaf-theoretic context is (b), the assumption that N ⊂ T ∗ (Y) is half-dimensional implies that objects from D-modN (Y) are automatically holonomic, i.e., this puts us into context (b’). A.4. Proof of Theorem A.3.9 in case (a). A TOY MODEL FOR SHTUKA A.4.1. 95 The initial observation is the following: Let Z 7→ Z top , Sch → Top denote the functor that associates to a scheme over C the topological space underlying the corresponding analytic space. Let Y → Ysing , Top → Spc denote the functor of singular chains. We will denote the composite functor Sch → Spc by Z 7→ Z top,sing . Taking fibers at the points of Z top defines a functor Z top,sing × Shvlisse (Z) → Vect, i.e., a functor r : Shvlisse (Z) → LS(Z top,sing ). (A.12) We have: Lemma A.4.2. The functor (A.12) is an equivalence. Proof. Both categories are equipped with t-structures, in which they are left and right complete, and the functor r is t-exact. Hence, it is enough to show that the functor (A.12) is fully faithful on the bounded subcategories and essentially surjective on the hearts. The former is the expression of the fact that sheaf cohomology of a complex with locally constant cohomology sheaves can be computed via singular cochains. The latter follows from the fact that both abelian categories in question identify with modules over the fundamental groupoid of Z top .  A.4.3. We proceed with the proof of Theorem A.3.9. Since Shvlisse (X) is dualizable (e.g., by Lemma A.4.2 and Proposition 1.4.5(b)), by the argument of [GR1, Chapter 3, Theorem 3.1.7], we have: ShvN (Y) ⊗ Shvlisse (X) ≃ lim S∈(Schaff )op /Y,sm ShvNS (S) ⊗ Shvlisse (X). Similarly, a smooth descent argument shows that the functor ShvN′ (Y × X) → lim S∈(Schaff )op /Y,sm ShvNS′ (S × X) is an equivalence. Hence, the assertion of the theorem reduces to the case when Y is an (affine) scheme. 96 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY A.4.4. Note that if C1 is dualizable, and C′2 ֒→ C2 is a fully faithful embedding, then C1 ⊗ C′2 → C1 ⊗ C2 is also fully faithful (indeed, interpret C1 ⊗ − as Functcont (C∨ 1 , −)). Hence, the functors ShvN (Y) ⊗ Shvlisse (X) → Shv(Y) ⊗ Shvlisse (X) → Shv(Y) ⊗ Shv(X) are both fully faithful. Combined with Theorem A.2.3, we obtain that the functor ShvN (Y) ⊗ Shvlisse (X) → ShvN′ (Y × X) (A.13) is also fully faithful. Hence, to prove Theorem A.3.9, it remains to prove that (A.13) is essentially surjective. A.4.5. From this point, the proof is essentially borrowed from [NY, Page 20]. According to [KS, Corollary 8.3.22], we can choose a µ-stratification of Y = ∪ Yα , such that N is α contained in the union of the conormals to the strata. Consider the corresponding stratification Yα × X of Y × X. This is still a µ-stratification. Let Shvstr (Y) ⊂ Shv(Y) and Shvstr′ (Y × X) ⊂ Shv(Y × X) denote the full subcategories consisting of objects locally constant along the strata. By [KS, Proposition 8.4.1], we have ShvN (Y) ⊂ Shvstr (Y) and ShvN′ (Y × X) ⊂ Shvstr (Y × X). A.4.6. First, we claim that the functor (A.14) Shvstr (Y) ⊗ Shvlisse (X) → Shvstr′ (Y × X) is an equivalence. By Sect. A.4.4, this functor is fully faithful. To prove that it is essentially surjective, by a Cousin argument, it reduces to the assertion that for each α, the functor Shvlisse (Yα ) ⊗ Shvlisse (X) → Shvlisse (Yα × X) is an equivalence. However, this follows from Lemma A.4.2, and the fact that for any two Y1 , Y2 ∈ Spc, the functor (A.15) LS(Y1 ) ⊗ LS(Y2 ) → LS(Y1 × Y2 ) is an equivalence (say, by Proposition 1.4.10). A.4.7. Thus, it remains to show that if F ∈ Shvstr (Y) ⊗ Shvlisse (X) is an object whose image along (A.14) belongs to ShvN′ (Y × X), then F itself belongs to ShvN (Y) ⊗ Shvlisse (X). With no restriction of generality, we can assume that X is connected, and let us choose a base point x ∈ X. Interpreting Shvlisse (X) via Lemma A.4.2, we obtain that if C′ ֒→ C is a fully faithful map in DGCat, then C′ ⊗ Shvlisse (X) → C ⊗ Shvlisse (X) is also fully faithful, and an object cX ∈ C ⊗ Shvlisse (X) belongs to C′ ⊗ Shvlisse (X) if and only if its essential image under the evaluation functor Id ⊗ ev C ⊗ Shvlisse (X) −→ x C belongs to C′ . A TOY MODEL FOR SHTUKA 97 Consider the commutative diagram ShvN (Y) ⊗ Shvlisse (X) − −−−−→ Shvstr (Y) ⊗ Shvlisse (X) − −−−−→ Shvstr′ (Y × X)       Id ⊗ evx y y yId ⊗ evx ShvN (Y) − −−−−→ Shvstr (Y) Id − −−−−→ Shvstr (Y), where the right vertical arrow is given by *-restriction along {x} ֒→ X. We obtain that it suffices to show that the functor −−−−− ShvN′ (Y × X) Shvstr′ (Y × X) ←   y Shvstr (Y) takes values in ShvN (Y). This is a standard fact, but let us prove it for completeness. A.4.8. Let Dx be an open disc around x. We have a commutative diagram Shvstr′ (Y × X) ← −−−− − ShvN′ (Y × X)     y y −−−− − ShvN′ (Y × Dx ), Shvstr′ (Y × Dx ) ← so it suffices to show that the functor Shvstr′ (Y × Dx ) ← −−−−− ShvN′ (Y × Dx )   y Shvstr (Y) takes values in ShvN (Y). However, by (A.14), the functor Shvstr′ (Y × Dx ) → Shvstr (Y) is an equivalence, whose inverse is given by pullback. In particular, this inverse functor preserves singular support and hence defines an equivalence ShvN (Y) → ShvN′ (Y × Dx ), as desired. A.5. Proof of Theorem A.3.9 in the ind-constructible contexts. A.5.1. The proof of Theorem A.3.9 in case (a’) follows verbatim the argument in case (a). Case (b”) follows from case (a’) by Lefschetz principle and Riemann-Hilbert. We will now prove Theorem A.3.9 in cases (a’) and (b’)-(d’) assuming that X is proper. With no restriction of generality, we can assume that X is connected. A.5.2. As in Sect. A.4.3, we reduce the assertion to the case when Y is an affine scheme. By Proposition A.2.10 (e.g., using the argument in Sect. A.4.4), we know that the functor in question is fully faithful. Hence, it remains to show that it is essentially surjective. Let F be an object in ShvN′ (Y × X). Since our functor preserves colimits, we can assume that F is constructible. We will argue by Noetherian induction on Y, so we will assume that the support of F is dominant over Y (otherwise, replace Y by the closure of the image of the support of F). 98 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY A.5.3. We claim that in the constructible case, F belongs to ShvN′ (Y × X) if and only if this holds for all of its perverse cohomology sheaves. Indeed: –In case (a’) this follows from the fact that SingSupp is measured by the functor of vanishing cycles, which is t-exact for the perverse t-structure. –In cases (b’), (b”) this follows from the definition. –In cases (c), (d) and (d’), this follows from the definition of SingSupp in [Be] and the corresponding fact for the ULA property, see [Ga6]. For the same reason, for a short exact sequence of perverse sheaves 0 → F1 → F → F2 → 0, we have SingSupp(F) = SingSupp(F1 ) ∪ SingSupp(F2 ). A.5.4. Hence, we can assume that F is of the form ′ j!∗ (FU ), for a smooth locally closed j′ U ֒→ Y × X and FU ∈ Shvlisse (U) is perverse and lisse. With no restriction of generality we can assume that U is connected. By the assumption in Sect. A.5.2, our U is dominant over Y. We claim that we can assume that ◦ ◦ j U is of the form Y × X, where Y ֒→ Y is an open subset. This reduction will be carried out in Sects. A.5.5-A.5.6. A.5.5. By the transitivity property of j!∗ , we can replace U by any of its non-empty open subsets. Let Y1 ⊂ Y be a non-empty smooth open subset contained in the image of U. Let U1 be the preimage of Y1 in U. We claim that U1 is dense in Y1 × X. Indeed, let U1 denote the closure of U1 in Y1 × X. If U1 were not dense in Y1 × X, the object F|Y1 ×X would be the direct image under the closed embedding U1 ֒→ Y1 × X, and hence SingSupp(F) would contain the conormals to U1 at each of its generic points. However, since U1 → Y1 is surjective, these conormals are not contained in T ∗ (Y1 ) × {zero-section}, contradicting the assumption on SingSupp(F). A.5.6. ◦ Let U be an open subset ◦ U1 ⊆ U ⊆ Y1 × X, ◦ maximal with respect to the property that the restriction to it of F|Y1 ×X is lisse. We claim that U2 ◦ is all of Y1 × X. Once we prove this, we will be able to take Y := Y1 , and thus achieve the reduction claimed in Sect. A.5.4. To prove the desired form of U2 we argue as follows. By purity, ◦ D′ := Y1 × X − U ′ is a divisor. We want to show that each irreducible component Dα of D′ is the preimage of a divisor ◦ in Y1 . Since U → Y1 is surjective, this would imply that D′ is empty. A TOY MODEL FOR SHTUKA 99 Since F|Y1 ×X is ramified around D′ , its singular support is not contained in the zero-section of ′ ′ T (Y1 × X) near the generic point of each irreducible component Dα of D′ . Hence, for every such Dα , ′ there exists an irreducible component of Nα ⊂ SingSupp(F) such that the projection ∗ ′ Nα → T ∗ (Y1 × X) → Y1 × X ′ maps to Dα with positive-dimensional fibers. We now use the assumption that SingSupp(F) ⊂ N × {zero-section}. We obtain that for each α there exists an irreducible component Nα of N such that ′ Nα ⊂ Nα × X. Let Dα be the (closure of the) image of Nα along the map Nα ֒→ T ∗ (Y1 ) → Y1 . We obtain a commutative diagram ′ Nα − −−−− → Nα × X     y y ′ Dα − −−−− → Dα × X. In particular, we have an inclusion ′ Dα ⊆ Dα × X. (A.16) Thus, it suffices to show that Dα is a divisor. Suppose not. Then Dα would be all of Y1 . Since N was assumed half-dimensional, we would obtain ′ that Nα is the zero-section. However, this would imply that Nα is contained in the zero-section of ′ ′ are positive-dimensional. → Dα Y1 × X. However, this contradicts the fact that the fibers of Nα A.5.7. ◦ We will now explicitly exhibit F := F| ◦ Y×X as lying in the essential image of the functor ◦ ◦ Shvlisse (Y) ⊗ Shvlisse (X) → Shvlisse (Y × X). (A.17) ◦ ◦ Pick a point y ∈ Y and let FX be the !-restriction of F to y × X ⊂ Y × X. Let πY and πX denote ◦ ◦ ◦ the projections from Y × X to Y and X; let π Y and π X denote their respective restrictions to Y × X. Consider the object ◦ ◦ ! ◦ ◦ ◦ F Y := (π Y )∗ (F ⊗ π!X (D(FX ))) ∈ Shv(Y). ◦ ◦ Since X is proper, FY is also lisse. The object FY is acted on the left by the associative algebra ∨ End (FX ) ≃ End (FX )rev . By adjunction, we have a map ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ π!Y (FY ) ⊗ π !X (FX ) ≃ FY ⊠ FX ≃ π∗Y (F Y ) ⊗ π ∗X (FX ) → F. Moreover, this map factors via a map (A.18) ◦ FY ⊠ End(FX ) ◦ ◦ FX := π!Y (FY ) ⊗ End(FX ) ◦ ◦ π !X (FX ) → F. We claim that (A.18) is an isomorphism. Indeed, the !-fiber the map (A.18) over y identifies with ! C• (X, FX ⊗ D(FX )) ⊗ End(FX ) FX ≃ End (FX ) ⊗ End(FX ) FX → FX , and hence is an isomorphism (in the above formula C• (X, −) denotes the functor of global sections, i.e., sheaf cohomology at the cochain level). 100 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY ◦ Since both sides in (A.18) are lisse sheaves and Y × X is connected, we obtain that (A.18) is an isomorphism. A.5.8. Set ! ! FY := (πY )∗ (F ⊗ πX (D(FX ))) ∈ Shv(Y). Since singular support is preserved under direct images along proper maps, we have: FY ∈ ShvN (Y). Consider the object F ′ := FY ⊠ End(FX ) FX ∈ ShvN′ (Y × X) := πY∗ (FY ) ⊗ End(FX ) ∗ πX (FX ) ∈ ShvN′ (Y × X). By construction, it belongs to the essential image of the functor ShvN (Y) ⊗ Shvlisse (X) → ShvN′ (Y × X). Moreover, by adjunction, we have a map F ′ → F, ◦ which becomes an isomorphism when restricted to Y×X. Passing to the fiber of this map we accomplish the induction step (in our Noetherian induction on Y). Appendix B. Spectral action in the context of Geometric Langlands (after [NY]) In this section we will reprove a result from [NY] that says that the subcategory of Shv(BunG ) consisting of objects whose singular support belongs to the nilpotent cone carries a canonical action of top,sing Rep(Ǧ)⊗X , where X top,sing is the object of Spc corresponding to X. The proof we present will apply to any sheaf-theoretic context (see Theorem B.4.2). B.1. The players. B.1.1. For the duration of this section we let X be a smooth projective curve over k. Let G be a reductive group (over k). Let BunG denote the moduli stack of G-bundles on X. B.1.2. Recall that T ∗ (BunG ) is the moduli space of pairs (P, ξ), where P is a G-bundle on X, and ξ is an element of Γ(X, g∗P ⊗ ωX ) where g∗P is the vector bundle on X associated to P and the co-adjoint representation g∗ of G. Let Nilp ⊂ T ∗ (BunG ) be the nilpotent cone, defined to be the locus of pairs (P, ξ), where ξ is nilpotent, i.e., maps to zero under the Chevalley map g∗ → g∗ // Ad(G) ≃ t∗ //W. B.1.3. Let Shv(−) be any of the sheaf-theoretic contexts from Sect. A.1.1 in which the ring e of coefficients is a field of characteristic 013. Our interest in this section is the category Shv(BunG ) and its full subcategory ShvNilp (BunG ) ⊂ Shv(BunG ). Let Ǧ be the Langlands dual of G, thought of as an algebraic group over e. We let Rep(Ǧ) denote the symmetric monoidal category of representations of Ǧ. 13The results of this section can be applied to any sheaf theoretic from Sect. A.1.1, but one will need to modify the category Rep(Ǧ) appropriately. A TOY MODEL FOR SHTUKA 101 B.2. The Hecke action. In this subsection we will discuss the general formalism of Hecke action. Convention: for the duration of this subsection, when working in the sheaf-theoretic context (a), when we write DGCat we actually mean DGCatdiscont , and when we write ⊗, we actually mean B.2.1. discont ⊗ . Let A be an index category. Let a 7→ Ma and a 7→ Ca be functors MA : A → DGCat and CA : A → DGCatMon , respectively. Then we can talk about an action of CA on MA . Indeed, we can view CA as an associative algebra object in the category Funct(A, DGCat), equipped with the level-wise (symmetric) monoidal structure. Let Act(CA , MA ) denote the space of such actions. In other words, Act(CA , MA ) = MapsE1 (Funct(A,DGCat)) (CA , End(MA )). Suppose now that we are given yet another functor C′A : A → DGCatMon , a 7→ C′a , equipped with a natural transformation C′A → CA . Given an action of C′A on MA we can talk about an extension of this action to an action of CA . The space of such extensions is by definition Act(CA , MA ) × Act(C′A ,MA ) {∗}, where {∗} → Act(C′A , MA ) is the initial action. B.2.2. We take A := fSet. Take C′A to be the functor I 7→ Shv(X I ), (B.1) ! where Shv(X I ) is viewed as a (symmetric) monoidal category with respect to the ⊗ operation. Take CA to be the functor I 7→ Rep(Ǧ)⊗I ⊗ Shv(X I ). (B.2) Take MA to be the functor I 7→ Shv(BunG ×X I ), (B.3) equipped with a natural action of C′A (see Sect. B.2.5). We will prove the following: Proposition-Construction B.2.3. There exists a canonical extension of the action of (B.1) on (B.3) to an action of (B.2). Remark B.2.4. It would follow from the construction, that in the sheaf-theoretic context (a), the restriction of the action in Proposition B.2.3 along Rep(Ǧ)⊗I ⊗ Shvlisse (X I ) ֒→ Rep(Ǧ)⊗I ⊗ Shv(X I ) is given by continuous functors, see Remark A.1.9. The rest of this subsection is devoted to the proof of Proposition B.2.3. It will be carried out using a certain formalism explained below. 102 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY B.2.5. Let PreStk / Sch be the category of pairs (Y, Z), where Z is a scheme, Y is prestack over Z. Take A = PreStkop / Schop . Take MA to be the functor (Y, Z) 7→ Shv(Y). (B.4) Take C′A to be the functor (Y, Z) 7→ Shv(Z). (B.5) Then we have a natural action of B.2.6. C′A on MA , given by !-pullback along Y → Z. Let Grpds / PreStk / Sch be the category of triples (H, Y, Z), where (Y, Z) are as above, and where H → Y × Y is a groupoid acting on Y over Z, such that the Z projections H ⇒ Y are ind-schematic. Note that Shv(H) is a monoidal category with respect to a convolution, defined to be the indextension of the ∗-convolution from compact objects. Take A = Grpdsop / PreStkop / Schop . We take MA to be the composition of the forgetful functor Grpds / PreStk / Sch → PreStk / Sch, (B.6) (H, Y, Z) 7→ (Y, Z) and the functor (B.4) We take C′A to be the composition of the forgetful functor (B.6) with (B.5). We take CA to be the functor that sends (H, Y, Z) 7→ Shv(H). Note that we have a canonical action of CA on MA . We also have a canonical natural transformation C′A → CA , given, level-wise, by the unit section of H. The induced action of C′A on MA is one from Sect. B.2.5. Remark B.2.7. More formally, the above constructions should be spelled out as follows: we have the functors (H, Y, Z) 7→ Y, (H, Y, Z) 7→ Z, (H, Y, Z) 7→ H that map Grpds / PreStk / Sch to Corr(PreStk)ind-sch,all , ComAlg(Corr(PreStk)ind-sch,all ), AssocAlg(Corr(PreStk)ind-sch,all ), respectively, and (MA , C′A , CA ) are obtained by composing these functors with the functor ShvCorr of Sect. A.1.7. B.2.8. (B.7) We have a canonically defined functor fSet → Grpds / PreStk / Sch, I 7→ HeckeI / BunG ×X I /X I , where HeckeI is the I-legged Hecke stack. (Note that HeckeI and BunG are ordinary prestacks, so the construction of the functor (B.7) takes place in (2, 1)-categories, i.e., involves finitely many pieces of data.) Hence, in order to perform the construction in Proposition B.2.3, it suffices to construct a map (in the category Funct(fSet, DGCatMon )) from (B.2) to (B.8) extending the map from (B.1). I 7→ Shv(HeckeI ), A TOY MODEL FOR SHTUKA B.2.9. 103 Note, however, that in the context of Sect. B.2.6, the natural transformation (H, Y, Z) 7→ (Shv(Z) → Shv(H)) as functors Grpds / PreStk / Sch → DGCatMon , extends naturally to a natural transformation (H, Y, Z) 7→ (Shv(H) ⊗ Shv(Z) → Shv(H)), expressing the fact that Shv(Z) maps to the center of Shv(H). Hence, it suffices to construct a map from the functor I 7→ Rep(Ǧ)⊗I . (B.9) to (B.8). B.2.10. The required natural transformation Rep(Ǧ)⊗I → Shv(HeckeI ), (B.10) I ∈ fSet is given by what is known as the naive geometric Satake functor. For completeness, we will recall its construction. B.3. Digression: naive geometric Satake. B.3.1. Recall that S-points of the Hecke stack HeckeI are quadruples (xI , P′ , P′′ , α), where –xI = {xi , i ∈ I} is an I-tuple of S-points of X; –P′ and P′′ are G-bundles on S × X; –α is an identification between P′ and P′′ on S × X − ∪ Graphxi . i B.3.2. We introduce the local Hecke stack Heckeloc as follows. Its S-points are quadruples I (xI , P′ , P′′ , α), where: –xI is an I-tuple of S-points of X; –P′ and P′′ are G-bundles on DxI –the parameterized formal disc around xI (i.e., the completion of the graph of S × X along ∪ Graphxi ); i ◦ –α is an identification between P′ and P′′ on DxI –the parameterized formal punctured disc around xI (see [Ga4, Sect. 6.4.3]). B.3.3. Convolution defines on Shv(Heckeloc I ) a structure of monoidal category, and the assignment (B.11) I 7→ Shv(Heckeloc I ) is a functor fSet → DGCatMon . Restriction along DxI → S × X defines a map rI : HeckeI → Heckeloc I . The functors r!I give rise to a natural transformation from (B.11) to (B.8). Thus, it is sufficient to construct a natural transformation (B.12) Rep(Ǧ)⊗I → Shv(Heckeloc I ), I ∈ fSet . 104 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Remark B.3.4. For the reader unwilling to consider the category Shv(Heckeloc I ) “as-is” (for reasons that the prestack Heckeloc is not locally of finite type), it can be equivalently defined as I Shv(GrG,I )L + (G)I , where –GrG,I is the I-legged version of the affine Grassmannian; –L+ (G)I is the group-scheme over X I of arcs into G. B.3.5. Each of the categories Rep(Ǧ)⊗I is endowed with a t-structure, for which the monoidal structure is t-exact. For a map I1 → I2 in fSet, the corresponding functor Rep(Ǧ)⊗I1 → Rep(Ǧ)⊗I2 is t-exact. Each of the categories Shv(Heckeloc I ) is also endowed with a t-structure: this is the perverse tstructure shifted so that the dualizaing sheaf on the unit section X I → Heckeloc I lies in the heart. loc ♥ The monoidal operation on Shv(Heckeloc I ) is right t-exact, making (Shv(HeckeI )) into a monoidal abelian category. For a map I1 → I2 in fSet, the corresponding functor loc Shv(Heckeloc I1 ) → Shv(HeckeI2 ), which is given by !-pullback, is right t-exact. B.3.6. The starting point for the construction of the natural transformation (B.12) is a natural transformation (B.13) ♥ (Rep(Ǧ)⊗I )♥ ≃ Rep(ǦI )♥ → (Shv(Heckeloc I )) , as functors fSet → AbCatMon . The natural transformation (B.13) is given by the geometric Satake functor of [MV]. Remark B.3.7. Although [MV] was stated in the context of (a’), it applies in any (ind)-constructible context. The construction in cases (a) and (b) follows from that in cases (a’) and (b’), respectively. B.3.8. Let us now explain how to use (B.13) to obtain the desired natural transformation (B.10). The first key property of (B.13) is that for each I ∈ fSet, the corresponding functor is t-exact. Hence, (B.13) gives rise to a system of t-exact functors (B.14) (Rep(Ǧ)⊗I )b ≃ D((Rep(Ǧ)⊗I )♥ )b → Shv(Heckeloc I ), equipped with a right-lax monoidal structure (here D(−)b stands for the bounded derived category of a given abelian category, equipped with its natural DG structure, see [Lu2, Sect. 1.3.3]; we are using its universal property, which is a variant of [Lu2, Theorem 1.3.3.2] for the bounded derived category). The second key property of (B.13) is that the right-lax monoidal structure on (B.14) is strict. Hence, precomposing with (Rep(Ǧ)⊗I )c ֒→ (Rep(Ǧ)⊗I )b and ind-extending, we obtain the desired natural transformation (B.12). B.4. Hecke action on the subcategory with nilpotent singular support. A TOY MODEL FOR SHTUKA B.4.1. 105 Let Nilp ⊂ T ∗ (BunG ) be the nilpotent cone. Consider the full subcategory ShvNilp (BunG ) ⊂ Shv(BunG ). Our goal is to prove the following: Theorem-Construction B.4.2. There exists a natural transformation between the following two functors fSet → DGCatMon : from the functor I 7→ Rep(Ǧ)⊗I to the functor I 7→ End(ShvNilp (BunG )) ⊗ Shvlisse (X I ). B.4.3. Before we prove Theorem B.4.2, let us explain how it recovers the result of [NY]. Let us specialize to the sheaf-theoretic context (a) from Sect. A.1.1. Recall (see Lemma A.4.2) that in this case, we have a functorial identification Shvlisse (X I ) ≃ LS((X top,sing )I ). Hence, combining with Sect. 1.7.3, we obtain: Corollary B.4.4. We have a canonically defined action of Rep(Ǧ)⊗X top,sing on ShvNilp (BunG ). Finally, combining with Theorem 1.5.5, we obtain: Corollary B.4.5. There is a canonically defined action of the (symmetric) monoidal category QCoh(LocSysǦ (X top,sing )) on ShvNilp (BunG ). B.4.6. Note that by Lemma A.4.2, the stack LocSysǦ (X top,sing ) that appears in Corollary B.4.5 identifies canonically with the Betti version of the stack of Ǧ-local systems on X. B.5. Proof of Theorem B.4.2. B.5.1. Consider the full subcategory ShvNilp′ (BunG ×X I ) ⊂ Shv(BunG ×X I ), where Nilp′ := Nilp ×{zero-section} ⊂ T ∗ (BunG ×X I ). The crucial ingredient for the proof is the following geometric assertion: Theorem B.5.2 ([NY]). For every I ∈ fSet, the action of Rep(Ǧ)⊗I on Shv(BunG ×X I ) preserves the full subcategory ShvNilp′ (BunG ×X I ) ⊂ Shv(BunG ×X I ). We will prove Theorem B.5.2 in Sect. B.6 below. Let us show how Theorem B.5.2 leads to the construction of the natural transformation in Theorem B.4.2. B.5.3. Let us return to the situation of Sect. B.2.1. Let us be given an action of CA on MA , and let M′A : A → DGCat, a 7→ M′a be another functor. Let M′A be equipped with a natural transformation to MA , such that for every a ∈ A, the corresponding functor M′a → Ma is fully faithful. Assume being given an action of CA on MA so that for every a ∈ A, the action of Ca on Ma preserves the subcategory M′a . In this case we obtain an action of CA on M′A . 106 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY B.5.4. Let us take A = fSet, and CA be the functor I 7→ Rep(Ǧ)⊗I ⊗ Shvlisse (X I ). (B.15) We take MA to be the functor (B.3). We let CA act on MA by precomposing the action of Proposition B.2.3 with the natural transformation Rep(Ǧ)⊗I ⊗ Shvlisse (X I ) → Rep(Ǧ)⊗I ⊗ Shv(X I ), I ∈ fSet . We take M′A to be the family of full subcategories ShvNilp′ (BunG ×X I ) ⊂ Shv(BunG ×X I ), I ∈ fSet . Thus, by Sect. B.5.3 and Theorem B.5.2 we obtain an action of (B.15) on I 7→ ShvNilp′ (BunG ×X I ). (B.16) B.5.5. Let us be again in the situation of Sect. B.2.1, and let us be given an action of CA on MA . Assume now that CA arises as a tensor product C1A ⊗ C2A , Assume also that MA arises as as M1 ⊗ C2A , for some fixed DG category M1 . Consider the action of C2A on M given by right multiplication along the second factor. Then the datum of extension of this action of C2A to an action of CA is equivalent to that of a natural transformation C1A → End(M1 ) ⊗ (C2A )rev of functors A → DGCatMon . B.5.6. We apply this to C1A being the functor (B.9) and C2A being the functor I 7→ Shvlisse (X I ). Take M1 = ShvNilp (BunG ). Now, by Theorem A.3.9, the natural transformation ShvNilp (BunG ) ⊗ Shvlisse (X I ) → ShvNilp′ (BunG ×X I ) is an isomorphism. Hence, by Sect. B.5.5, the action of (B.15) on (B.16) gives rise to a sought-for natural transformation Rep(Ǧ)⊗I → End(ShvNilp (BunG )) ⊗ Shvlisse (X I ), I ∈ fSet . [Theorem B.4.2] B.6. Preservation of nilpotence of singular support. The goal of this subsection is to prove Theorem B.5.2. The proof is essentially a paraphrase of [NY]. B.6.1. have (B.17) By Theorem A.3.9, we have to show that for F ∈ ShvNilp (BunG ) and any V ∈ Rep(Ǧ)⊗I , we H(V, F) ∈ ShvNilp′ (BunG ×X I ) ⊂ Shv(BunG ×X I ), where H(−, −) denotes the Hecke action Rep(Ǧ)⊗I ⊗ Shv(BunG ) → Shv(BunG ×X I ). By the associativity of the Hecke action, we can assume that I = {∗}. A TOY MODEL FOR SHTUKA 107 B.6.2. Let HeckeX denote the 1-legged global Hecke stack. Let Heckeloc X denote the 1-legged local Hecke stack, defined in Sect. B.3.2. Let r denote the restriction map HeckeX → Heckeloc X . We have the diagram X x π  Heckeloc X x r  (B.18) ← h → h −−−−→ BunG . −−−−− HeckeX − BunG ← Denote also s := π ◦ r : HeckeX → X. With these notations, we have: → ← H(V, F) = ( h × s)∗ ◦ ( h × r)! (F ⊠ SV ), where SV ∈ Shv(Heckeloc X ) corresponds to V ∈ Rep(Ǧ) by geometric Satake. B.6.3. Note that the map ← h × s : HeckeX → BunG × Heckeloc X is (pro)-smooth, while the map → h × s : HeckeX → BunG ×X is (ind)-proper. For a given point (x, P′ , P′′ , α) ∈ HeckeX consider the corresponding diagram of cotangent spaces Tx∗ (X)  (dπ)∗ y (B.19) ← (d h )∗ TP∗′ (BunG ) − −−−−→ ∗ (Heckeloc T(x,P ′ ,P′′ ,α)| X ) D x (dr)∗ y ∗ T(x,P ′ ,P′′ ,α) (HeckeX ) → (d h )∗ ← −−−−− TP∗′′ (BunG ). Hence, in order to prove (B.17), it suffices to show the following. Consider a quadruple of elements ∗ ∗ (Heckeloc ξ ′ ∈ TP∗′ (BunG ), ξ ′′ ∈ TP∗′′ (BunG ), ξH ∈ T(x,P ′ ,P′′ ,α)| X ), ξX ∈ Tx (X), Dx so that • ξ ′ is nilpotent; • ξH ∈ SingSupp(SV ); ← → • (d h )∗ (ξ ′ ) + (dr)∗ (ξH ) = (d h )∗ (ξ ′′ ) + (ds)∗ (ξX ). We need to prove that in this case: • ξ ′′ is also nilpotent; • ξX = 0. 108 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Indeed, once we show this, (B.17) would follow by combining the following two assertions: –For a smooth map f : Y1 → Y2 and F ∈ Shv(Y2 ), the subset SingSupp(f ! (F)) ⊂ T ∗ (Y1 ) equals the image of SingSupp(F) × Y1 ⊂ T ∗ (Y2 ) × Y1 Y2 Y2 along the codifferential map T ∗ (Y2 ) × Y1 → T ∗ (Y1 ). (B.20) Y2 –For a proper map f : Y1 → Y2 and F ∈ Shv(Y1 ), the subset SingSupp(f∗ (F)) ⊂ T ∗ (Y2 ) is contained in the image along the projection T ∗ (Y2 ) × Y1 → T ∗ (Y2 ) Y2 of the preimage of SingSupp(F) along (B.20). loc B.6.4. Let Heckex (resp., Heckeloc x ) be the fiber of HeckeX (resp., HeckeX ) over x. Along with the diagrams (B.18) and (B.19) consider the diagrams ← h Heckeloc x x r x → h x BunG ← −−− −− Heckex − −−−x− → BunG and ← (d h x )∗ TP∗′ (BunG ) −−−−−→ ∗ (Heckeloc T(P ′ ,P′′ ,α)| x ) Dx  (dr )∗ y x ∗ T(P ′ ,P′′ ,α) (Heckex ) → (d h x )∗ ←−−−−− TP∗′′ (BunG ). Let ξHx denote the image of ξH under the restriction map ∗ loc ∗ loc T(x,P ′ ,P′′ ,α) (HeckeX ) → T(P′ ,P′′ ,α) (Heckex ). We have a commutative diagram (drx )∗ ∗ −−−− → (Heckeloc T(P ′ ,P′′ ,α)| x ) − Dx x   (dr)∗ ∗ T(P ′ ,P′′ ,α) (Heckex ) x   ∗ ∗ −−−− → T(x,P (Heckeloc T(x,P ′ ,P′′ ,α) (HeckeX ). ′ ,P′′ ,α)| X ) − D x ′ ′′ Hence, the assumption on (ξ , ξ , ξH ) implies ← B.6.5. → (d h x )∗ (ξ ′ ) + (drx )∗ (ξHx ) = (d h x )∗ (ξ ′′ ). (B.21) We identify TP∗′ (BunG ) := Γ(X, g∗P′ ⊗ ωX ) and TP∗′′ (BunG ) := Γ(X, g∗P′′ ⊗ ωX ). ∗ (Heckex ) with the dual of R1 Γ(X, K) where K is the complex Further, we identify T(P ′ ,P′′ ,α)| Dx   α (B.22) K := Cone gP′ ⊕ gP′′ → j∗ (j ∗ (gP′ ) ≃ j ∗ (gP′′ )) , where j denotes the embedding X − x ֒→ X. ◦ 1 ∗ (Heckeloc Finally, we identify T(P ′ ,P′′ ,α)| x ) with the dual of R Γ(Dx , K) where Dx (resp., Dx ) is the Dx formal (resp., formal punctured disc) around x (see Sect. B.3.2). A TOY MODEL FOR SHTUKA 109 Since Dx is affine, the above R1 Γ(Dx , K) identifies with Γ(Dx , H 1 (K)), which is the same as Γ(X, H 1 (K)), since H 1 (K) is set-theoretically supported at x ∈ X. ∗ (Heckeloc From here we obtain that T(P ′ ,P′′ ,α)| x ) identifies with the set of pairs Dx
′ ∗ ′′ (B.23) ξloc ∈ Γ(Dx , gP′ ⊗ ωX ), ξloc ∈ Γ(Dx , g∗P′′ ⊗ ωX ) , such that ′ ′′ α(ξloc ) = ξloc ◦ as elements of Γ(Dx , g∗P′′ ⊗ ωX ). B.6.6. Equation (B.21) translates into α(ξ ′ | ◦ ) = ξ ′′ | ◦ . Dx Dx In particular, we obtain: ξ ′ is nilpotent ⇒ ξ ′ | ◦ Dx is nilpotent ⇒ ξ ′′ | ◦ Dx is nilpotent ⇒ ξ ′′ is nilpotent. Thus, it remains to show that ξX = 0. B.6.7. Note that the short exact sequences (B.24) ∗ ∗ 0 → Tx∗ (X) → T(x,P ′ ,P′′ ,α) (HeckeX ) → T(P′ ,P′′ ,α) (Hecke) → 0 (B.25) loc ∗ ∗ (Heckeloc ′ ,P′′ ,α)| 0 → Tx∗ (X) → T(x,P X ) → T(P′ ,P′′ ,α)|Dx (Heckex ) → 0 Dx admit a canonical splittings (compatible with the map between these short exact sequences induced by r): These splittings are a consequence of the fact that the prestacks HeckeX and Heckeloc X each carries a canonical structure of crystal along X. Indeed, for a test-scheme S and two points x1 , x2 : S ⇒ X such that that x1 |Sred = x2 |Sred , (B.26) the data of lifting of these points to points of HeckeX (resp., Heckeloc X ) coincide. This is because (B.26) implies ◦ ◦ S × X − Graphx1 = S × X − Graphx2 and Dx1 = Dx2 , Dx1 = Dx2 . B.6.8. Consider the resulting maps (B.27) ∗ ∗ T(x,P ′ ,P′′ ,α) (HeckeX ) → Tx (X) and (B.28) ∗ ∗ (Heckeloc T(x,P ′ ,P′′ ,α)| X ) → Tx (X). Dx It follows from the definition of the above crystal structure that the composition of (B.27) with the maps ← (d h )∗ → (d h )∗ ∗ ∗ TP∗′ (BunG ) −→ T(x,P ′ ,P′′ ,α) (HeckeX ) ←− TP′′ (BunG ) vanishes. ∗ (Heckeloc Therefore, to prove that ξX = 0, it suffices to show the following. Let ξH ∈ T(x,P ′ ,P′′ ,α)| X ) Dx be a vector that belongs to SingSupp(SV ). Suppose that ∗ (Heckeloc ξHx ∈ T(P ′ ,P′′ ,α)| x ) Dx ′ ′′ is nilpotent, i.e., the corresponding pair (ξloc , ξloc ) in (B.23) consists of nilpotent elements. 110 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Then we need to prove that the image of ξH along (B.28) is zero. B.6.9. To prove the latter assertion, we can assume that X = A1 and x = 0. In this case, we identify loc 1 Heckeloc A1 ≃ Hecke0 ×A . (B.29) With respect to (B.29), the object SV is the pullback along the projection loc Heckeloc A1 → Hecke0 . In addition, from (B.29), we obtain a different splitting of (B.25). The discrepancy between the two different splittings of (B.25) is a map ∗ loc ∗ 1 T(P ′ ,P′′ ,α) (Hecke0 ) → T0 (A ) ≃ k. (B.30) It suffices to show that this map vanishes on nilpotent elements (we refer to the description of ∗ loc T(P ′ ,P′′ ,α) (Heckex ) given by (B.23); when we say “nilpotent”, we mean that the corresponding el′ ′′ ements ξloc and ξloc are nilpotent). B.6.10. Let us calculate the map (B.30). Let L(G)X be the version of the loop group spread over X, and let L(G)x be its fiber at x ∈ X. The group ind-scheme L(G)X also has a crystal structure along X, which gives rise to a splitting of the short exact sequence 0 → Tg (L(G)x ) → Tg (L(G)X ) → Tx (X) → 0, (B.31) g ∈ L(G)x . 1 For X = A , we have an identification L(G)A1 ≃ L(G)0 × A1 , and hence a different splitting of the short exact sequence (B.31). The discrepancy between these two splittings is a map k ≃ T0 (A1 ) → Tg (L(G)0 ). (B.32) Let u denote the coordinate on A1 . We will identify the tangent space Tg (L(G)0 ) with g((u)) by means of the left translation by g. Under this identification the map (B.32), thought of as an element of g((u)) is dg , du i.e., the (left) logarithmic derivative of g with respect to the coordinate u. g −1 · (B.33) B.6.11. Let (P′ , P′′ , α) be a point of Heckeloc 0 ≃ L(G)0 \L(G)/L(G)0 , represented by an element g ∈ L(G)0 . By the Cartan decomposition, we can assume that g = uλ for a dominant coweight λ. Note that the corresponding element (B.33) equals λ · u−1 ⊂ t((u)) ⊂ g((u)). We need to show that for a nilpotent element ∗ loc ∗ ∗ ∗ ξ ∈ T(x,P ′ ,P′′ ,α) (Hecke0 ) ≃ g [[u]]du ∩ Aduλ (g [[u]]du) ⊂ g ((u))du, the residue pairing g((u)) × g∗ ((u))du → k against λ · u−1 evaluates to 0. Set ξ0 := ξ mod u ∈ g∗ du ≃ g∗ . This is a nilpotent element of g∗ . We need to show that its pairing with λ ∈ t ⊂ g gives zero. A TOY MODEL FOR SHTUKA B.6.12. 111 Let − g = n+ λ ⊕ mλ ⊕ nλ be the triangular decomposition of g corresponding to λ, i.e., mλ is the centralizer of λ and n+ λ (resp., n− ) corresponds to positive (resp., negative) eigenvalues of λ for the adjoint action. λ Consider the corresponding decomposition ξ0 = ξ0+ + ξ00 + ξ0− of ξ0 . Note, however, that the condition that ξ ∈ Aduλ (g∗ [[u]]du) implies that ξ0− is zero. Knowing that, the fact that ξ0 is nilpotent implies that ξ00 is a nilpotent element of mλ . We have hλ, ξ0 i = hλ, ξ0+ i + hλ, ξ00 i = hλ, ξ00 i. However, the latter is zero, being the pairing of a central element (in the reductive Lie algebra mλ ) with a nilpotent one.  Appendix C. Integrated actions in the context of D-modules In Proposition 1.7.2 we described what it takes to have an action of A⊗Y on a DG category M for Y ∈ Spc. In this section we will use the RHS of Proposition 1.7.2 to give a definition of integrated action in the context of D-modules. We will then study how our definition plays out in the context of geometric Langlands. C.1. Definition of integrated action. C.1.1. Let us place ourselves in the sheaf-theoretic context (b) from Sect. A.1.1. In this case our ring of coefficients e is the same as the ground field k. Thus, we will work with k-linear DG categories. C.1.2. Let X be a scheme over k (in the applications to geometric Langlands, X will be a smooth projective curve). For a symmetric monoidal DG category A and a DG category M, an action of A⊗X on M is by definition a natural transformation between the following two functors fSet → DGCatmon : From the functor (C.1) I 7→ A⊗I , to the functor (C.2) I 7→ End(M) ⊗ D-mod(X I ). The totality of categories equipped with an action of A⊗X forms a 2-category, which we will denote A⊗X - mod. Remark C.1.3. It is easy to see that the datum of action of A⊗X on M is equivalent to that of action on M of the (non-unital) symmetric monoidal category Fact(A)Ran(X) , defined as in [Ga3, Sect. 2.5]. C.1.4. Example. Let X be a curve and let G be a reductive group. Take A = Rep(Ǧ) and M = D-mod(BunG ). Then Proposition B.2.3 says that we have an action of Rep(Ǧ)⊗X on D-mod(BunG ). 112 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY C.1.5. Let us be given an action of A⊗X on M. For a finite set I and r ∈ A⊗I , let SI (r) denote the resulting functor M → M ⊗ D-mod(X I ). Lemma C.1.6. Let r be dualizable. Then the functor SI (r) admits a continuous right adjoint. Proof. The adjoint in question is explicitly given by M ⊗ D-mod(X I ) SI (r ∨ )⊗IdD-mod(X I ) −→ M ⊗ D-mod(X I ) ⊗ D-mod(X I ) IdM ⊗∆! −→ → M ⊗ D-mod(X I ) IdM ⊗ΓdR (X I ,−) −→ M.  C.1.7. The ULA property. Let us assume that A is compactly generated and rigid, and that M is compactly generated. For a pair of compact objects r ∈ A⊗I and m ∈ M consider again the object SI (r)(m) ∈ M ⊗ D-mod(X I ) Note that by Lemma C.1.6, this object is automatically compact. We claim: Proposition C.1.8. The object SI (r)(m) is ULA over X I . We refer the reader to Sect. D, where the notion of ULA is reviewed. Proof. Note that the dual of M ⊗ D-mod(X I ) as a D-mod(X I )-module category identifies with M∨ ⊗ D-mod(X I ). Let m∨ ∈ M∨ be the abstract Verdier dual of m ∈ M. Let r ∨ be the monoidal dual of r. Then the object SI (r ∨ )(m∨ ) ∈ M∨ ⊗ D-mod(X I ) satisfies the requirements from Sect. D.1.4.  C.2. Lisse actions. C.2.1. (C.3) Note that for a smooth scheme Y , the functor D-modlisse (Y ) → D-mod(Y ) is a fully faithful embedding that admits a continuous right adjoint. Hence, for another DG category M, the functor M ⊗ D-modlisse (Y ) → M ⊗ D-mod(Y ) is also fully faithful and admits a continuous right adjoint given by tensoring the right adjoint to (C.3) with IdM . Further, if M → M′ is conservative, an object of M ⊗ D-mod(Y ) belongs to M ⊗ D-modlisse (Y ) if and only if its image in M′ ⊗ D-mod(Y ) belongs to M′ ⊗ D-modlisse (Y ). A TOY MODEL FOR SHTUKA C.2.2. 113 For a pair of objects r ∈ A⊗I and m ∈ M consider the object SI (r)(m) ∈ M ⊗ D-mod(X I ). (C.4) We shall say that the action of A⊗X on M is lisse if for every r and m as above, the object (C.4) belongs to the full subcategory. M ⊗ D-modlisse (X I ) ⊂ M ⊗ D-mod(X I ). Note also that by associativity, it is enough to check this condition for I = {∗}. Let A⊗X - modlisse ⊂ A⊗X - mod denote the corresponding full subcategory. C.2.3. Example. The statement of Theorem B.5.2 is equivalent to the assertion that the action of Rep(Ǧ)⊗X on D-modNilp (BunG ) is lisse. C.2.4. Given an action of A⊗X on M, let Mlisse ⊂ M be the full subcategory consisting of objects m, for which (C.4) with I = {∗} belongs to the subcategory M ⊗ D-modlisse (X) for all r. Proposition C.2.5. For an object m ∈ Mlisse , the object (C.4) with I = {∗} belongs to Mlisse ⊗ D-modlisse (X) ⊂ M ⊗ D-modlisse (X). Proof. By the last remark in Sect. C.2.1, it suffices to show that the functors S{∗} (r) send Mlisse to Mlisse ⊗ D-mod(X). For any DG category C, consider the full subcategory (M ⊗ C)lisse ⊂ M ⊗ C. I.e., we apply the definition of (−)lisse to M ⊗ C on which A⊗X acts via the first factor. Explicitly, this subcategory consists of objects m′ for which (S{∗} (r ′ ) ⊗ IdC )(m′ ) ∈ M ⊗ D-modlisse (X) ⊗ C ⊂ M ⊗ D-mod(X) ⊗ C for all r ′ ∈ A. The functor Mlisse ⊗ C → M ⊗ C factors via a functor Mlisse ⊗ C → (M ⊗ C)lisse . (C.5) We claim that if C is dualizable, then (C.5) is an equivalence. Indeed, this follows by interpreting M ⊗ C as Functcont (C∨ , M). Hence, applying this to C = D-mod(X), it suffices to show that for any r and r ′ , we have (S{∗} (r ′ ) ⊗ IdC ) ◦ S{∗} (r)(m) ∈ M ⊗ D-modlisse (X) ⊗ D-mod(X). We have (S{∗} (r ′ ) ⊗ IdC ) ◦ S{∗} (r)(m) = S{∗}⊔{∗}(r ′ ⊗ r)(m) ≃ σ(S{∗}⊔{∗}(r ⊗ r ′ )(m)), where σ denotes the transposition acting on X × X. Hence, it suffices to show that S{∗}⊔{∗} (r ⊗ r ′ )(m) ⊂ M ⊗ D-mod(X) ⊗ D-modlisse (X). However, S{∗}⊔{∗}(r ⊗ r ′ )(m) ≃ (S{∗} (r) ⊗ IdC ) ◦ S{∗} (r ′ )(m), 114 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY and the assertion follows from the fact that S{∗} (r ′ )(m) ⊂ M ⊗ D-modlisse (X), by the assumption on m.  Corollary C.2.6. The subcategory Mlisse carries an action of A⊗X , and this action is lisse. C.2.7. Thus, Theorem B.5.2 says that we have an inclusion: D-modNilp (BunG ) ⊂ (D-mod(BunG ))lisse . (C.6) We propose the following: Conjecture C.2.8. The inclusion (C.6) is an equality. C.2.9. Example. Let us call an object F ∈ D-mod(BunG ) a loose Hecke eigensheaf, if for every V ∈ Rep(Ǧ) we have an isomorphism H(V, F) ≃ F ′ ⊗ EV , where EV is some object of D-modlisse (X) and F ′ ∈ D-mod(BunG ). Clearly, a loose Hecke eigensheaf belongs to (D-mod(BunG ))lisse . Hence, Conjecture C.2.8 contains as a special case the following conjecture, first proposed by G. Laumon (for actual Hecke eigensheaves, rather than loose ones): Conjecture C.2.10. A loose Hecke eigensheaf has a nilpotent singular support. C.3. A spectral characterization of lisse actions. In this subsection we specialize to the case when A = Rep(G) for an algebraic group G, and X is a smooth and proper curve. C.3.1. Let LocSysG (X) be the stack of de Rham G-local systems on X, defined as in [AG, Sect. 10.1]. It is easy to see that one can write LocSysG (X) as a quotient of a quasi-compact (derived) scheme by an action of the algebraic group. From this it follows that QCoh(LocSysG (X)) is compactly generated and rigid. In particular, the object OLocSysG (X) ∈ QCoh(LocSysG (X)) is compact. C.3.2. Recall (see [Ga1, Sect. 4.3]) that there exists a canonically defined symmetric monoidal functor Fact(Rep(G))Ran(X) → QCoh(LocSysG (X)), (C.7) which admits a continuous and fully faithful right adjoint. The functor (C.7) defines a fully faithful embedding QCoh(LocSysG (X)) - mod → A⊗X - mod. (C.8) C.3.3. Let M be a compactly generated category equipped with an action of QCoh(LocSysG (X)). To a compact object m ∈ M one can attach its set-theoretic support supp(m) ⊂ LocSysG (X). This is the smallest among closed subsets Y ⊂ LocSysG (X) such that the image of m under the functor M ≃ QCoh(LocSysG (X)) is zero. ⊗ QCoh(LocSysG (X)) M → QCoh(LocSysG (X) − Y ) ⊗ QCoh(LocSysG (X)) M A TOY MODEL FOR SHTUKA C.3.4. 115 Let P be a parabolic in G with Levi quotient M, and consider the diagram p q P P LocSysM (X). LocSysP (X) −→ LocSysG (X) ←− Let σ be an irreducible M-local system on X. Note that we have a closed embedding iσ : pt / Aut(σ) ֒→ LocSysM (X). In the above formula, Aut(σ) is the group of automorphisms of σ. It contains the Z(M) (the center of M), and by the irreducibility assumption, Aut(σ)/M is finite. Denote LocSysP,σ (X) := LocSysP (X) × pt / Aut(σ). LocSysM (X) Let pP,σ denote the projection LocSysP,σ (X) → LocSysG (X). Let YP,σ ⊂ LocSysG (X) be a closed subset equal to the image of pP,σ . Proposition C.3.5. Let m ∈ M be a compact object such that supp(m) is contained in some YP,σ . Then m ∈ Mlisse . Proof. It is enough to prove the proposition for M := QCoh(LocSysG (X))YP,σ , where the latter is the full subcategory of QCoh(LocSysG (X)) consisting of objects with set-theoretic support on YP,σ . Note that the functor p∗P,σ : QCoh(LocSysG (X))YP,σ → QCoh(LocSysP,σ (X)) is conservative. Hence, by the last remark in Sect. C.2.1, we can further replace the category QCoh(LocSysG (X))P,Yσ by QCoh(LocSysP,σ (X)). The action of Rep(G)⊗X on QCoh(LocSysP,σ (X)) factors through the restriction Rep(G)⊗X → Rep(P)⊗X . Hence, it is enough to show that the action of Rep(P)⊗X on QCoh(LocSysP,σ (X)) is lisse. Next, we note that the essential image of the restriction functor Rep(M) → Rep(P) generates Rep(P). Hence, it is enough to show that the action of Rep(M)⊗X on QCoh(LocSysP,σ (X)) is lisse. Next, we note that the map qP,σ : LocSysP,σ (X) → pt / Aut(σ) has the property that its base change with respect to pt → pt / Aut(σ) yields an algebraic stack of the form derived affine scheme/An for some n. Hence, the direct image functor QCoh(LocSysP,σ (X)) → QCoh(pt / Aut(σ)) is conservative. Hence, it is enough to show that the action of Rep(M)⊗X on QCoh(pt / Aut(σ)) is lisse. Finally, applying the (conservative) pullback functor along pt → pt / Aut(σ), we obtain that it is enough to show that the action of Rep(M)⊗X on Vect, corresponding to the M-local system σ, is lisse. 116 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY However, the latter is evident: for V ∈ Rep(M)c and k ∈ Vect, the corresponding object S{∗} (V )(k) ∈ Vect ⊗ D-mod(X) ≃ D-mod(X) is just Vσ , the twist of V by σ.  lisse C.3.6. The next conjecture proposes to describe the subcategory M support: ⊂ M in terms of set-theoretic Conjecture C.3.7. A compact object m ∈ M belongs to Mlisse if and only if it can be written as a colimit of objects mα , for each of which supp(mα ) is contained in a finite union of closed subsets of the form YP,σ . Appendix D. The notion of universal local acyclicity (ULA) In this section we review the notion of universal local acyclicity, for a general sheaf-theoretic context with the exclusion of (a). In the particular case of D-modules we will express it via the forgetful functor to quasi-coherent sheaves. D.1. The abstract ULA property. D.1.1. In this subsection C will be an abstract symmetric monoidal DG category. The example to ! keep in mind is C = D-mod(Y ) for a scheme Y with respect to ⊗, so C is very far from being rigid. D.1.2. Let M be a C-module category, which is dualizable as a C-module. This means that there exists another C-module category M∨,C and functors unitC M C −→ M∨,C ⊗ M and M ⊗ M∨,C C counitC M −→ C C that satisfy the usual duality axioms. Note in this case we have a natural identification M∨,C ⊗ N ≃ FunctC - mod (M, N), (D.1) N ∈ C - mod. C D.1.3. Example. Take M = C ⊗ M0 where M0 is a plain DG category. If M0 is dualizable as a DG category, then M is dualizable as a C-module and M∨,C ≃ C ⊗ M∨ 0. D.1.4. Let m ∈ M be an object. We shall say that m is ULA over C if there exists an object m∨,C ∈ M∨,C equipped with a functorial identification MapsM (c ⊗ m, m′ ) ≃ MapsC (c, counitCM (m′ ⊗ m∨,C )), C c ∈ C, m′ ∈ M. The datum of such an identification is equivalent to that of a pair of maps µ ǫ 1C → counitCM (m ⊗ m∨,C ) and m∨,A ⊗ m → unitCM (1C ) C C such that the composite (D.2) µ ǫ m → counitCM (m ⊗ m∨,C ) ⊗ m ≃ (counitCM ⊗ Id)(m ⊗ m∨,C ⊗ m) → C C C C C → (counitCM ⊗ Id)(m ⊗ unitCM (1C )) ≃ m C is the identity map, and similarly for m ∨,C C . Note that if an object m ∈ M is ULA, then so is the corresponding m∨,C ∈ M∨,C with (m∨,C )∨,C ≃ m. A TOY MODEL FOR SHTUKA 117 D.1.5. In what follows we will assume that C and M are compactly generated (as DG categories). We will also assume that the unit object 1C is compact. (But we do not assume that the tensor product operation on C or the action of C on M preserve compactness.) Note that for a pair of objects m, m′ ∈ M , we have a well-defined Hom(m, m′ ) ∈ C, satisfying MapsC (c, Hom(m, m′ )) := MapsM (c ⊗ m, m′ ). For a fixed m, the functor m′ 7→ Hom(m, m′ ) preserves colimits if and only if c ⊗ m are compact for all compact c ∈ C; in particular in this case m itself is compact. For c ∈ C, we have a tautological map c ⊗ Hom(m, m′ ) → Hom(m, c ⊗ m′ ). (D.3) We have: Lemma D.1.6. An object m ∈ M is ULA if and only if the functor m′ 7→ Hom(m, m′ ), M→C preserves colimits and (D.3) is an isomorphism for all c ∈ C. Proof. If m is ULA we have Hom(m, m′ ) ≃ counitCM (m′ ⊗ m∨,C ). C The other direction follows from (D.1) for N = C.  Corollary D.1.7. If C is rigid, any compact m ∈ M is ULA. Proof. It is enough to check that the objects c ⊗ m are compact and the maps (D.3) are isomorphisms for c that are dualizable in C. In the rigid case this is automatic.  D.2. Adding self-duality. Let C be as above (a compactly generated symmetric monoidal DG category with a compact unit). D.2.1. eC , so that the pairing Assume that C contains a compact object, to be denoted 1 eC , c1 ⊗ c2 ) C ⊗ C 7→ Vect, c1 , c2 7→ Hom C (1 defines a self-duality (D.4) C∨ ≃ C. Let DC denote the corresponding contravariant self-equivalence on Cc , i.e., eC , c1 ⊗ c2 ), c1 ∈ Cc . Hom C (DC (c1 ), c2 )) ≃ Hom C (1 By construction eC ≃ DC (1C ). 1 D.2.2. Example. Let C = D-mod(Y ), where Y is a scheme of finite type over a ground field k of ! characteristic 0. We regard C as a symmetric monoidal category via the ⊗ tensor product. eC to be the “constant sheaf” kY ∈ D-mod(Y ), i.e., the Verdier dual of the Then we can take 1 dualizing sheaf ωY . The functor DC is the Verdier duality self-equivalence of D-mod(Y )c , see Sect. D.3.1. D.2.3. Let M be a C-module category. Note that for a pair of objects m, m′ ∈ M and c ∈ Cc we have a canonically defined map eC ⊗ m, DC (c) ⊗ m′ ), (D.5) Hom(c ⊗ m, m′ ) → Hom M (DC (c) ⊗ c ⊗ m, DC (c) ⊗ m′ ) → Hom M (1 where the last arrow comes from the canonical map eC → DC (c) ⊗ c. 1 118 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY D.2.4. Assume that M is compactly generated as a DG category. Let DM denote the canonical contravariant equivalence Mc → (M∨ )c . Let h−, −i denote the tautological pairing M × M∨ → Vect . Consider M∨ as a C-module category, where the action of c ∈ C on M∨ is the functor dual to that of the action of c on M. Define a functor (D.6) h−, −iC : M × M∨ → C, Hom C (c, hm, m′ iC ) := hDC (c) ⊗ m, m′ i ≃ hm, DC (c) ⊗ m′ i, m ∈ M, m′ ∈ M∨ . By construction, (D.6) factors via a functor M ⊗ M∨ → C. (D.7) C D.2.5. Assume that (D.7) is the counit of a duality, so we find ourselves in the context of Sect. D.1. In particular, we can identify M∨,C ≃ M∨ . D.2.6. We claim: Proposition D.2.7. For m ∈ Mc the following conditions are equivalent: (i) The object m is ULA over C; (ii) The object m is ULA, with and µ being the canonical map eC ⊗ m) m∨,C = DM (1 eC ⊗ m)iC . 1C → hm, DM (1 eC ⊗ m is compact, and the map (D.5) is an isomorphism for every c ∈ Cc and m′ ∈ M. (iii) The object 1 Proof. The implication (ii) ⇒ (i) is tautological. To prove (iii) ⇒ (ii) we note that if (D.5) is an eC ⊗ m) satisfies the requirements of m∨,C . It remains prove (i) ⇒ isomorphism, then the object DM (1 (iii). Let m be ULA over C, and let m∨,C be the corresponding object of M∨,C , which we now identify with M∨ . We have: Hom M (c ⊗ m, m′ ) ≃ MapsC (c, hm′ , m∨,C iC ) ≃ hDC (c) ⊗ m′ , m∨,C i, eC , we obtain In particular, taking c = 1 eC ⊗ m, m′ ) ≃ hm′ , m∨,C i, Hom M (1 c ∈ Cc , m′ ∈ M. m′ ∈ M. Hence, we obtain that m∨,C is compact and eC ⊗ m is compact. in particular 1 DM (m∨,C ) ≃ e 1C ⊗ m; With respect to the last identification, the map (D.5) becomes the map Hom M (c ⊗ m, m′ ) ≃ MapsC (c, hm′ , m∨,C iC ) ≃ hDC (c) ⊗ m′ , m∨,C i ≃ and hence is an isomorphism. eC ⊗ m, DC (c) ⊗ m′ ), ≃ Hom M (DM (m∨,C ), DC (c) ⊗ m′ ) ≃ Hom M (1  A TOY MODEL FOR SHTUKA 119 D.2.8. Example. Let M be of the form C ⊗ M0 for a compactly generated DG category M0 . Then the pairing (D.7) does indeed define the counit of a duality. The composite identification ∨ ∨,C C∨ ⊗ M∨ ≃ C ⊗ M∨ 0 ≃ (C ⊗ M0 ) ≃ (C ⊗ M0 ) 0 is given by the self-duality (D.4). D.3. The ULA condition in the geometric situation. In this subsection we will take C to be D-mod(Y ), where Y is a scheme of finite type. Let f : Z → Y be a scheme over Y ; take M := D-mod(Z), which is acted on by D-mod(Y ) via ! FY , FZ 7→ f ! (FY ) ⊗ FZ . We will bring the concept of ULA developed above in contact with the more familiar geometric geometric notion. D.3.1. First, we note that D-mod(Y ) fits into the paradigm of Sect. D.2.1 with DD-mod(Y ) := DVerdier . Y eD-mod(Y ) is eY , the “constant sheaf”, i.e., The corresponding object 1 Hom(eY , F) = ΓdR (Y, F). Similarly, we use Verdier duality on Z to identify D-mod(Z)∨ with D-mod(Z). In this case, the pairing (D.6) identifies with ! hF1 , F2 iD-mod(Y ) = f∗ (F1 ⊗ F2 ). D.3.2. We claim that the resulting pairing (D.7) provides the counit of the adjunction. Indeed, the unit map D-mod(Y ) → D-mod(Z) ⊗ D-mod(Y ) D-mod(Z) is determined by the condition that the corresponding object D-mod(Y ) unitD-mod(Z) (1D-mod(Y ) ) ∈ D-mod(Z) ⊗ D-mod(Z) ≃ D-mod(Z × Z) Y D-mod(Y ) equals (∆Z/Y )∗ (ωZ ), where ∆Z/Y : Z → Z × Z Y is the relative diagonal map. D.3.3. Thus, from Proposition D.2.7 we obtain: Corollary D.3.4. For a compact object F ∈ D-mod(Z) the following conditions are equivalent: (i) There exists an object F ∨ ∈ D-mod(Z)c equipped with maps ! µ : ωY → f∗ (F ⊗ F ∨ ) and ǫ : F ∨ ⊠ F → (∆Z/Y )∗ (ωZ ) Y (here − ⊠ − stands for the !-pullback of − ⊠ − along Z × Z → Z × Z) such that the composite Y (D.8) Y ! µ ! ! ǫ F ≃ f ! (ωY ) ⊗ F → f ! (f∗ (F ⊗ F ∨ )) ⊗ F ≃ (p2 )∗ ◦ (∆Z/Y × idZ )! (F ⊠ F ∨ ⊠ F) → Y Y Y → (p2 )∗ ◦ (∆Z/Y × idZ )! (F ⊠ (∆Z/Y )∗ (ωZ )) ≃ F Y Y (here p2 denotes the second projection Z × Z → Z) is the identity map, and similarly for F ∨ . Y 120 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY ! ! (ii) Same as (i), but for F ∨ = DVerdier (f ! (eY ) ⊗ F) (in particular, f ! (eY ) ⊗ F is compact), with µ being Z the map ! ! ωY → f∗ (F ⊗ DVerdier (f ! (eY ) ⊗ F)), Z corresponding to the identity element under     ! ! ! ! ! Verdier ! Verdier ! Hom D-mod(Y ) ωY , f∗ (F ⊗ DZ (f (eY ) ⊗ F)) ≃ (f (eY ) ⊗ F)) ≃ ΓdR Y, eY ⊗ f∗ (F ⊗ DZ   ! ! ! ! ! projection formula (f ! (eY ) ⊗ F) ≃ Hom D-mod(Z) (f ! (eY ) ⊗F, f ! (eY ) ⊗F)). ≃ ΓdR Z, f ! (eY ) ⊗ F ⊗ DVerdier Z (iii) For any FY ∈ D-mod(Y )c and F ′ ∈ D-mod(Z), the map (D.9) ! Hom D-mod(Z) (f ! (FY ) ⊗ F, F ′ ) → ! ! ! (FY )) ⊗ F ′ ) → (FY )) ⊗ f ! (FY ) ⊗ F, f ! (DVerdier → Hom D-mod(Z) (f ! (DVerdier Y Y ! ! (FY )) ⊗ F ′ ) → Hom D-mod(Z) (f ! (eY ) ⊗ F, f ! (DVerdier Y is an isomorphism. D.3.5. We will call objects of D-mod(Z) satisfying the equivalent conditions of Corollary D.3.4 ULA over Y (or ULA with respect to f ). Remark D.3.6. Note that condition (iii) in Corollary D.3.4 can be rephrased as follows: For FY ∈ D-mod(Y ), the object ∗ ! f ∗ (FY ) ⊗ (f ! (eY ) ⊗ F) ∈ D-mod(Z) is defined and the canonical map ∗ ! ! f ∗ (FY ) ⊗ (f ! (eY ) ⊗ F) → f ! (FY ) ⊗ F is an isomorphism. D.3.7. When we view this definition from the point of view of condition (i), we obtain the following assertion that reads that “local acyclicity implies universal local acyclicity”: Corollary D.3.8. Let g : Y ′ → Y be a morphism of schemes, and let set Z ′ := Y ′ × Z. If F is ULA Y over Y , then its !-pullback to Z ′ is ULA over Y ′ . In the context of ℓ-adic sheaves, an assertion parallel to Corollary D.3.8 had been proved by O. Gabber, see [LZ, Corollary 6.6]. D.4. An aside: the notion of ULA in other sheaf-theoretic contexts. We will now place ourselves in a general sheaf-theoretic context of Sect. A.1.1, excluding (a) (the exclusion is because we want our categories to be compactly generated). D.4.1. Let f : Z → Y be a map of schemes. We consider Shv(Z) as a module category over Shv(Y ) via the operation ! f ! (−) ⊗ −. However, the notion of ULA developed from Sect. D.1 is not directly applicable in this context, since Shv(Z) is not in general dualizable as a Shv(Y )-module category. However, we claim that the following analog of Corollary D.3.4 holds: A TOY MODEL FOR SHTUKA 121 Proposition D.4.2. For a compact object F ∈ Shv(Z) the following conditions are equivalent: (i) There exists an object F ∨ ∈ Shv(Z)c equipped with maps ! µ : ωY → f∗ (F ⊗ F ∨ ) and ǫ : F ∨ ⊠ F → (∆Z/Y )∗ (ωZ ) Y such that the composite (D.10) ! ! µ ! ǫ F ≃ f ! (ωY ) ⊗ F → f ! (f∗ (F ⊗ F ∨ )) ⊗ F ≃ (p2 )∗ ◦ (∆Z/Y × idZ )! (F ⊠ F ∨ ⊠ F) → Y Y Y → (p2 )∗ ◦ (∆Z/Y × idZ )! (F ⊠ (∆Z/Y )∗ (ωZ )) ≃ F Y Y ∨ is the identity map, and similarly for F . ! (ii) Same as (i), but for F ∨ = DVerdier (f ! (eY ) ⊗ F) with µ being the canonical map Z ! ! ωY → f∗ (F ⊗ DVerdier (f ! (eY ) ⊗ F)). Z (iii) For any base change gZ e − Z −−− − →   fey (D.11) Z  f y gY −−− − → Y, Ye − ! e := gZ e ′ ∈ Shv(Z), e the map and F (F), for any FYe ∈ Shv(Ye )c and F (D.12) ! ! ! e F e ′ ) → Hom e fe! (DVerdier e′) e! e ) ⊗ F, e! e ) ⊗ F, Hom Shv(Z) (FYe )) ⊗ F e (f (FY e (f (eY e Shv(Z) Y is an isomorphism. Proof. Clearly, (ii) implies (i). Let us show that (iii) implies (ii). First, note that the RHS in (D.9) a priori identifies with   ! ! ! e ′ ⊗ DVerdier e e F e , fe∗ (F ( f (e ) ⊗ F)) . (D.13) Hom e e e Shv(Y ) Y Y Z Take Ye := Z; so that the square (D.11) becomes p 2 Z ×Z − −−− −→ Y   p1 y Z Z  f y f − −−−−→ Y. ! e ′ := (∆Z/Y )∗ (ωZ ) and and F e := DVerdier Take F (f ! (eY ) ⊗ F). The expression in (D.13) identifies with Z Y   ! ! (D.14) Hom Shv(Z) DVerdier (f ! (eY ) ⊗ F), DVerdier (f ! (eY ) ⊗ F)) . Z Z The required map ǫ is obtained via (D.9) from the identity element in (D.14). Finally, let us assume (i) and deduce (iii). If we have a datum as in (i) for f : Z → Y , its !-pullback e → Ye . So we can assume that g = id. defines a similar data for fe : Z 122 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY The datum of µ defines a map (D.15) ! ! ! ! Hom Shv(Z) (f ! (FY ) ⊗ F, F ′ ) → Hom Shv(Z) (f ! (FY ) ⊗ F ⊗ F ∨ , F ′ ⊗ F ∨ ) →   ! ! ! → Hom Shv(Y ) f∗ (f ! (FY ) ⊗ F ⊗ F ∨ ), f∗ (F ′ ⊗ F ∨ ) ≃   ! ! ! ! µ ≃ Hom Shv(Y ) FY ⊗ f∗ (F ⊗ F ∨ ), f∗ (F ′ ⊗ F ∨ ) → Hom Shv(Y ) (FY , f∗ (F ′ ⊗ F ∨ )). The datum of ǫ defines an inverse map ! ! Hom Shv(Y ) (FY , f∗ (F ′ ⊗ F ∨ )) → Hom Shv(Z) (f ! (FY ), f ! (f∗ (F ′ ⊗ F ∨ ))) ≃     ! ! ! ! ′ ! ∨ ! ! ′ ! ∨ ≃ Hom Shv(Z) f (FY ), p2∗ (p1 (F ⊗ F )) → Hom Shv(Z) f (FY ) ⊗ F, p2∗ (p1 (F ⊗ F )) ⊗ F ≃   ! ! ! ≃ Hom Shv(Z) f ! (FY ) ⊗ F, p2∗ (p!1 (F ′ ⊗ F ∨ ) ⊗ p!2 (F)) ≃   ! ! ! ǫ ≃ Hom Shv(Z) f ! (FY ) ⊗ F, p2∗ (p!1 (F ∨ ) ⊗ p!2 (F)) ⊗ p!1 (F ′ )) →   ! ! ! ! ! ′ → Hom Shv(Z) f (FY ) ⊗ F, p2∗ ((∆Z/Y )∗ (ωZ ) ⊗ p1 (F )) ≃ Hom Shv(Z) (f ! (FY ) ⊗ F, F ′ ). In particular, taking FY = eY , we obtain an identification between ! ! ! (F), F ′ ) ≃ ΓdR (Z, F ′ ⊗ F ∨ ) ≃ ΓdR (Y, f∗ (F ′ ⊗ F ∨ )) ≃ Hom Shv(Y ) (eY , f∗ (F ′ ⊗ F ∨ )) Hom Shv(Z) (DVerdier Z and ! Hom Shv(Z) (f ! (eY ) ⊗ F, F ′ ), functorial in F ′ . Hence, ! (f ! (eY ) ⊗ F). F ∨ ≃ DVerdier Z Under this identification, the map (D.12) goes over to the composition of (D.15) and ! ! (FY )) ⊗ F ′ ), (F ∨ ), f ! (DVerdier Hom Shv(Y ) (FY , f∗ (F ′ ⊗ F ∨ )) ≃ Hom Shv(Z) (DVerdier Y Z hence it is an isomorphism.  D.4.3. We will call objects of D-mod(Z) satisfying the equivalent conditions of Proposition D.4.2 ULA over Y (or ULA with respect to f ). Remark D.4.4. Let us compare the notion of ULA in the three contexts (b), (b’) and (b”). Note that we have fully faithful embeddings of the corresponding categories (b”) → (b’) → (b). We note that if F is an object of a smaller category, then, if we view the from the point of view of condition (ii) of Proposition D.3.4, it is ULA in the smaller category if and only if it is such from the point of view of a bigger category. Note that this property is non-obvious from the point of view of condition (iii) of Proposition D.3.4, since for the larger category we are testing the isomorphism on a larger collection of objects (one that e ′ ). are denoted F Remark D.4.5. Note that the property of being ULA is by design stable under base change: this is obvious from the point of view of each of the conditions (i), (ii), (iii) in Proposition D.3.4. That said, the proof of Proposition D.3.4 shows that it is enough to require that the map (D.12) be an isomorphism for (Ye , gY ) being the pair (Z, f ). A TOY MODEL FOR SHTUKA 123 D.5. The ULA property for D-modules. In this subsection we will come back to the ULA property in the context of D-modules. We will be working over a ground field k of characteristic 0. Let Y be a smooth scheme of finite type over k. Consider the symmetric monoidal category C := D-mod(Y ). In this subsection we will give a hands-on criterion for objects (in some class of D-mod(Y )-module categories) to be ULA. D.5.1. We will regard the category D-mod(Y ) as QCoh(YdR ), where YdR is the de Rham prestack of Y , i.e., Maps(S, YdR ) = Maps(Sred , Y ). A key observation observation is that YdR is 1-affine (see [Ga5, Definition 1.3.7] for what this means, and [Ga5, Theorem 2.6.3] for the statement of the result). In other words, a datum of a QCoh(YdR )-module category M is equivalent to a datum of a sheaf of categories over YdR , i.e., an assignment y (S → YdR ) S ∈ Schaff , MS,y ∈ QCoh(S) - mod, f (S ′ → S) MS ′ ,y◦f ≃ QCoh(S ′ ) ⊗ QCoh(S) MS,y , equipped with a data of homotopy-compatibility for compositions. Explicitly, each MS,y is recovered as MS,y ≃ QCoh(S) ⊗ M. QCoh(YdR ) Vice versa, M can be identified with MS,y , lim y S →YdR equipped with an action of lim QCoh(S) = QCoh(YdR ). y S →YdR We will denote by MY the value of this sheaf categories on Y , where we regard Y as equipped with the canonical projection to YdR . Explicitly, MY ≃ QCoh(Y ) ⊗ M. QCoh(YdR ) D.5.2. Examples. If M = M0 ⊗ D-mod(Y ), then MY = M0 ⊗ QCoh(Y ). Let now M = QCoh(ZdR ). Then MY = QCoh(ZdR × Y ). YdR If Z → Y is a smooth map, one can identify QCoh(ZdR × Y ) with the derived category of (quasiYdR coherent sheaves of) modules over the ring DZ/Y of vertical differential operators; this is a subring of DZ , generated by functions and vertical vector fields, i.e., vector fields along the fibers of the map Z →Y. D.5.3. The adjoint pair ind : QCoh(Y ) ⇄ QCoh(YdR ) : oblv induces an adjoint pair (D.16) ind : MY ⇄ M : oblv. The functor oblv is conservative (indeed, since Y is smooth, any map S → YdR can be lifted to Y ). In what follows we will assume that MY is compactly generated as a DG category. The conservativity of oblv implies that in this case M is also compactly generated. 124 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY D.5.4. Since MY is compactly generated, and hence dualizable, all the categories MS,y are dualizable (as DG categories, or equivalently, as QCoh(S)-module categories). The assignment (S, y) 7→ M∨ S,y is also a sheaf of categories over YdR . Set M∨,YdR := lim M∨ S,y . y S →YdR The 1-affineness of YdR implies that M∨,YdR is the dual of M as a QCoh(YdR )-module category. Denote by h−, −iYdR the resulting pairing M × M∨,YdR → QCoh(YdR ). Define the pairing h−, −i : M × M∨,YdR → Vect by h−, −i := Γdr (Y, −) ◦ h−, −iYdR , where Γdr (Y, −) is the functor of de Rham cohomology QCoh(YdR ) ≃ D-mod(Y ) → Vect . Lemma D.5.5. The functor h−, −i defines an equivalence M∨,YdR ≃ M∨ . Proof. The adjunction (∆Y )dR,∗ : D-mod(Y ) ⇄ D-mod(Y ) ⊗ D-mod(Y ) : ∆!Y (here we identify D-mod(Y ) ⊗ D-mod(Y ) ≃ D-mod(Y × Y )) defines an adjunction M∨,YdR ⊗ D-mod(Y ) M ≃ (M∨,YdR ⊗ M) ⊗ D-mod(Y )⊗D-mod(Y ) ⇄ (M∨,YdR ⊗ M) D-mod(Y ) ⇄ (D-mod(Y ) ⊗ D-mod(Y )) ≃ M∨,YdR ⊗ M ⊗ D-mod(Y )⊗D-mod(Y ) We define the unit object unitM ∈ M∨,YdR ⊗ M to be the image of the unit unitYMdR ∈ M∨,YdR under the above left adjoint M∨,YdR ⊗ D-mod(Y ) ⊗ M D-mod(Y ) M → M∨,YdR ⊗ M. The fact that unitM and h−, −i satisfy the duality axioms follows by diagram chase.  D.5.6. Let eQCoh(Y ) ∈ QCoh(YdR ) 1 dR be the object corresponding to kY ∈ D-mod(Y ). Note that it satisfies the assumption of Sect. D.2.1, and the corresponding self-duality of D-mod(Y ) is DVerdier . Y It follows from Lemma D.5.5 that any M as above satisfies the assumption of Sect. D.2.5. D.5.7. We now claim: Theorem D.5.8. For m ∈ Mc the following conditions are equivalent: (i) The object m is ULA over D-mod(Y ); (ii) For every F ∈ D-mod(Y )c , the object F ⊗ m ∈ M is compact; (iii) The object oblv(m) ∈ MY is compact. The rest of this subsection is devoted to the proof of this theorem. A TOY MODEL FOR SHTUKA 125 D.5.9. The implication (i) ⇒ (ii) follows from Sect. D.1.5 and Lemma D.1.6. To prove (iii) ⇒ (ii), it suffices to show that for F0 ∈ QCoh(Y )c , we have ind(F0 ) ⊗ m ∈ Mc . However, ind(F0 ) ⊗ m ≃ ind(F0 ⊗ oblv(m)), and the assertion follows from the fact that the functor ind preserves compactness. Proof of (iii) ⇒ (i). This is in fact a general assertion. Let φ : C → C0 be a symmetric monoidal functor that admits a C-linear left adjoint. Let M be a dualizable C-module category and set M0 := C0 ⊗ M. C Consider the corresponding functor φM := (φ ⊗ IdM ) : M → M0 ; L it admits a left adjoint, given by (φ ⊗ IdM ). Consider the inner Hom object Hom(m, m′ ) ∈ C. Note that we have φ(Hom(m, m′ )) ≃ Hom0 (φM (m), φM (m′ )), (D.17) where Hom0 (−, −) denotes inner Hom in C0 . Assume now that φ is conservative. It follows from (D.17) and Lemma D.1.6 that if φM (m) ∈ M0 is ULA over C0 , then m is ULA over over C. We apply this to φ being oblv : D-mod(Y ) → QCoh(Y ). Assumption (iii) says that oblvY (m) is compact. It is then automatically ULA over QCoh(Y ) by Corollary D.1.7.  The implication (ii) ⇒ (iii) follows by ind ◦ oblv(m) ≃ DY ⊗m from the following more general assertion: Proposition D.5.10. Let m ∈ MY be such that ind(m) ∈ M is compact. Then m is compact. Proof. We need to show that the functor m′ 7→ Hom MY (m, m′ ) preserves colimits. We will do so by expressing it via the functor m′ 7→ Hom M (ind(m), ind(m′ )) ≃ Hom MY (m, oblv ◦ ind(m′ )), while the latter preserves colimits by assumption. With no restriction of generality, we can assume that Y is affine. Consider the formal completion of Y in Y × Y p p 1 2 Y ← Y∧ → Y. Let MY ∧ be the value of M over Y ∧ . Set M′Y ∧ := IndCoh(Y ∧ ) ⊗ QCoh(Y ∧ ) The maps p1 , p2 define the functors p∗i : MY → MY ∧ and (pIndCoh )∗ : M′Y ∧ → MY . i MY ∧ 126 D. GAITSGORY, D. KAZHDAN, N. ROZENBLYUM AND Y. VARSHAVSKY Note that the functor identifies with where we regard DY m′ 7→ oblv ◦ ind(m′ ) m′ 7→ (pIndCoh )∗ (DY ⊗p∗2 (m′ )), 1 ∧ as an object of IndCoh(Y ). From here it follows oblv ◦ ind(m′ ) carries an action of the algebra OY ×Y of functions on Y × Y (via its action on DY ), and we have a functorial identification m′ ≃ Hom OY ×Y (OY , oblv ◦ ind(m′ )). From here, we obtain that Hom MY (m, oblv ◦ ind(m′ )) carries an action of OY ×Y and
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D.G.14: Department of Mathematics, Harvard University; D.K.: Einstein School of Mathematics, Hebrew University of Jerusalem; N.R.: Department of Mathematics, The University of Chicago; Y.V.: Einstein School of Mathematics, Hebrew University of Jerusalem. 14 Corresponding author. Full address: Department of Mathematics, Harvard University, 1 Oxford str, Cambridge MA 02138, USA