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Idea

A bimodule is a module in two compatible ways over two rings.

Definition

Over a ring

With a left action and a right action

Given two rings RR and SS, a RR-SS-bimodule is an abelian group BB with a bilinear left R R -action α R:R×B→B\alpha_R:R \times B \to B and a bilinear right S S -action α S:B×S→B\alpha_S:B \times S \to B such that for all r∈Rr \in R, b∈Bb \in B, and s∈Ss \in S, α R(r,α S(b,s))=α S(α R(r,b),s)\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s).

With a biaction

Equivalently, given two rings RR and SS, a RR-SS-bimodule is an abelian group BB with a trilinear R R - S S -biaction, a function (−)(−)(−):R×B×S→B(-)(-)(-):R \times B \times S \to B such that

  • for all b∈Bb \in B, 1 Rb1 S=b1_R b 1_S = b

  • for all b∈Bb \in B, r 1∈Rr_1 \in R, r 2∈Rr_2 \in R, s 1∈Ss_1 \in S, s 2∈Ss_2 \in S, r 1(r 2bs 1)s 2=(r 1⋅ Rr 2)b(s 1⋅ Ss 2)r_1 (r_2 b s_1) s_2 = (r_1 \cdot_R r_2) b (s_1 \cdot_S s_2)

  • for all r 1∈Rr_1 \in R, r 2∈Rr_2 \in R, b∈Bb \in B, s∈Ss \in S, (r 1+r 2)bs=r 1bs+r 2bs(r_1 + r_2) b s = r_1 b s + r_2 b s

  • for all r∈Rr \in R, b 1∈Bb_1 \in B, b 2∈Bb_2 \in B, s∈Ss \in S, r(b 1+b 2)s=rb 1s+rb 2sr (b_1 + b_2) s = r b_1 s + r b_2 s

  • for all r∈Rr \in R, b∈Bb \in B, s 1∈Ss_1 \in S, s 2∈Ss_2 \in S, rb(s 1+s 2)=rbs 1+rbs 2r b (s_1 + s_2) = r b s_1 + r b s_2

representing simultaneous left multiplication by scalars r∈Rr \in R and right multiplication by scalars s∈Ss \in S.

Over a monoid in a monoidal category

We can define in more generality what is a (A,B)(A,B)-bimodule in a monoidal category (𝒞,⊗,I)(\mathcal{C},\otimes,I) where (A,∇ A,η A)(A,\nabla^{A},\eta^{A}) and (B,∇ B,η B)(B,\nabla^{B},\eta^{B}) are two monoids. It is given by:

  • An object X∈𝒞X \in \mathcal{C}
  • A left-action l:A⊗X→Xl:A \otimes X \rightarrow X
  • A right-action r:X⊗B→Xr:X \otimes B \rightarrow X

such that:

and moreover this diagram commutes:

Properties

Biactions, left actions, and right actions

Let RR and SS be rings, and let BB be a RR-SS-bimodule.

Given a left RR-action α R\alpha_R and a right SS-action α S\alpha_S of a RR-SS-bimodule, the biaction (−)(−)(−):R×B×S→B(-)(-)(-):R \times B \times S \to B is defined as

rbs≔α R(r,α S(b,s))=α S(α R(r,b),s)r b s \coloneqq \alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)

The biaction is trilinear because the left RR-action and right SS-action are bilinear.

On the other hand, given an RR-SS-biaction α\alpha of a RR-SS-bimodule, the left R R -action is defined from the RR-SS-biaction as

α R(r,b)≔rb1 S\alpha_R(r, b) \coloneqq r b 1_S

for all r∈Rr \in R and b∈Bb \in B. It is a left action because

α R(1 R,b)=1 Rb1 S=m\alpha_R(1_R, b) = 1_R b 1_S = m
α R(r 1,α L(r 2,b))=r 1(r 2b1 S)1 S=(r 1⋅ Rr 2)b(1 S⋅ S1 S)=(r 1⋅ Rr 2)b1 S=α R(r 1⋅ Rr 2,b)\alpha_R(r_1, \alpha_L(r_2, b)) = r_1 (r_2 b 1_S) 1_S = (r_1 \cdot_R r_2) b (1_S \cdot_S 1_S) = (r_1 \cdot_R r_2) b 1_S = \alpha_R(r_1 \cdot_R r_2, b)

The right S S -action is defined from the RR-SS-biaction as

α S(b,s)≔1 Rbs\alpha_S(b, s) \coloneqq 1_R b s

for all s∈Ss \in S and b∈Bb \in B. It is a right action because

α S(b,1 S)=1 Rb1 S=m\alpha_S(b, 1_S) = 1_R b 1_S = m
α S(α S(b,s 1),s 2)=1 R(1 R,b,s 1)s 2=(1 R⋅ R1 R)b(s 1⋅ Ss 2)=1 Sb(s 1⋅ Ss 2)=α S(b,s 1⋅ Ss 2)\alpha_S(\alpha_S(b, s_1), s_2) = 1_R (1_R, b, s_1) s_2 = (1_R \cdot_R 1_R) b (s_1 \cdot_S s_2) = 1_S b (s_1 \cdot_S s_2) = \alpha_S(b, s_1 \cdot_S s_2)

The left RR-action and right SS-action satisfy the following identity:

  • for all b∈Bb \in B, r∈Rr \in R and s∈Ss \in S, α R(r,α S(b,s))=α S(α R(r,b),s)\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s).

This is because when expanded out, the identity becomes:

α(r,α(1 R,b,s),1 S)=α(1 R,α(r,b,1 S),s)\alpha(r, \alpha(1_R, b, s), 1_S) = \alpha(1_R, \alpha(r, b, 1_S), s)
(r⋅ R1 R)b(s⋅ S1 S)=(1 R⋅ Rr)b(1 S⋅ Ss)(r \cdot_R 1_R) b (s \cdot_S 1_S) = (1_R \cdot_R r) b (1_S \cdot_S s)
rbs=rbsr b s = r b s

The left RR-action and right SS-action are bilinear because the original biaction is trilinear.

Linear maps

Let RR and SS be rings. A RR-SS-linear map or RR-SS-bimodule homomorphism between two RR-SS-bimodules AA and BB is an abelian group homomorphism f:A→Bf:A \to B such that for all a∈Aa \in A, r∈Rr \in R, and s∈Ss \in S,

f(ras)=rf(a)sf(r a s) = r f(a) s

A RR-SS-linear map f:A→Bf:A \to B is monic or an RR-SS-bimodule monomorphism if for every other RR-SS-bimodule CC and RR-SS-linear maps h:C→Ah:C \to A and k:C→Ak:C \to A, f∘h=f∘kf \circ h = f \circ k implies that h=kh = k.

A sub-RR-SS-bimodule of a RR-SS-bimodule BB is a RR-SS-bimodule AA with a monic linear map i:A↪Bi:A \hookrightarrow B.

A RR-SS-linear map f:A→Bf:A \to B is invertible or an RR-SS-bimodule isomorphism if there exists a RR-SS-linear map g:B→Ag:B \to A such that g∘f=id Ag \circ f = id_A and f∘g=id Bf \circ g = id_B, where id Aid_A and id Bid_B are the identity linear maps on AA and BB respectively.

Tensor product of bimodules

Given rings R,S,TR, S, T and an RR-SS bimodule AA and an SS-TT bimodule BB, the tensor product of AA and BB is formed as a quotient A⊗ NBA \otimes_N B of the tensor product of abelian groups A⊗BA\otimes B. This is a special case of a more general construction:

Given three monoids M,N,PM,N,P in a monoidal category (𝒞,⊗,I)(\mathcal{C},\otimes,I), a MM-NN-bimodules AA and a NN-PP-bimodule BB, we denote the monoid actions as λ A:M⊗A→A\lambda^{A}:M \otimes A \rightarrow A, ρ A:A⊗N→N\rho^{A}:A \otimes N \rightarrow N, λ B:N⊗B→N\lambda^{B}:N \otimes B \rightarrow N and ρ B:B⊗P→P\rho^{B}:B \otimes P \rightarrow P. The tensor product, A⊗ NBA \otimes_{N} B is defined as this coequalizer:

We suppose moreover that this coequalizer is preserved by tensoring on the left by MM and tensoring on the right by PP, meaning that these diagrams are coequalizer diagrams: A⊗BA \otimes B becomes a MM-PP-bimodule with left action defined by the following diagram: and right action defined by the following diagram:

Assuming all requisite (reflective) coequalizers exist, universal property arguments guarantee associativity isomorphisms of type

(A⊗ NB)⊗ PC→A⊗ N(B⊗ PC).(A \otimes_N B) \otimes_P C \to A \otimes_N (B \otimes_P C).

In fact, this tensor product defines composition in a bicategory where objects or 0-cells are monoids in a monoidal category, where 1-cells AA from RR to SS are RR-SS bimodules, and where 2-cells from AA to BB are morphisms of RR-SS bimodules.

This in turn can be seen as a special case of a bicategory of profunctors enriched in a monoidal category with suitably nice cocompleteness properties – see monoidally cocomplete category and Benabou cosmos.

Two-sided ideals of a ring

Every ring RR is a RR-RR-bimodule, with the biaction (−)(−)(−):R×R×R→R(-)(-)(-):R \times R \times R \to R defined by the ternary product abc≔a⋅b⋅ca b c \coloneqq a \cdot b \cdot c for elements a∈Ra \in R, b∈Rb \in R, c∈Rc \in R.

Given a ring RR, a two-sided ideal of RR is a sub-RR-RR-bimodule of RR.

Rings over a ring

Let RR be a ring. An RR-ring SS is a RR-RR-bimodule with a bilinear function (−)⋅(−):S×S→S(-)\cdot(-):S \times S \to S and an element 1∈S1 \in S such that (S,⋅,1)(S, \cdot, 1) forms a monoid.

Categories of bimodules

The 1-category of bimodules and intertwiners

Definition

Write BModBMod for the category whose

  • objects are triples (R,S,B)(R,S,B) where RR and SS are rings and where BB is an RR-SS-bimodule;

  • morphisms are triples (f,g,ϕ)(f,g, \phi) consisting of two ring homomorphisms f:R→R′f \colon R \to R' and g:S→S′g \colon S \to S' and an intertwiner of RR-S′S'-bimodules ϕ:B⋅g→f⋅B′\phi \colon B \cdot g \to f \cdot B'. This we may depict as a

    R →B S f↓ ⇓ ϕ ↓ g R′ →B′ S′. \array{ R &\stackrel{B}{\to}& S \\ {}^{\mathllap{f}}\downarrow &\Downarrow_{\phi}& \downarrow^{\mathrlap{g}} \\ R' &\stackrel{B'}{\to}& S' } \,.
Remark

As this notation suggests, BModBMod is naturally the vertical category of a pseudo double category whose horizontal composition is given by tensor product of bimodules.

The 2-category of rings, bimodules, and intertwiners

Consider bimodules over rings.

Proposition

There is a 2-category whose

The composition of 1-morphisms is given by the tensor product of modules over the middle algebra.

Proposition

There is a 2-functor from the above 2-category of rings and bimodules to Cat which

Proposition

This construction has as its image precisely the colimit-preserving functors between categories of modules.

This is the Eilenberg-Watts theorem.

Remark

In the context of higher category theory/higher algebra one may interpret this as says that the 2-category of those 2-modules over the given ring which are equivalent to a category of modules is that of rings, bimodules and intertwiners. See also at 2-ring.

Remark

The 2-category of rings and bimodules is an archtypical example for a 2-category with proarrow equipment, hence for a pseudo double category with niche-fillers. Or in the language of internal (infinity,1)-category-theory: it naturally induces the structure of a simplicial object in the (2,1)-category CatCat

(⋯→→→X 1→∂ 0→∂ 1X 0)∈Cat Δ op \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0 \right) \in Cat^{\Delta^{op}}

which satisfies the Segal conditions. Here

X 0=Ring X_0 = Ring

is the category of rings and homomorphisms between them, while

X 1=BMod X_1 = BMod

is the category of def. , whose objects are pairs consisting of two rings AA and BB and an AA-BB bimodule NN between them, and whose morphisms are pairs consisting of two ring homomorphisms f:A→A′f \colon A \to A' and g:B→B′g \colon B \to B' and an intertwiner N⋅(g)→(f)⋅N′N \cdot (g) \to (f) \cdot N'.

The (∞,2)(\infty,2)-category of ∞\infty-algebras and ∞\infty-bimodules

The above has a generalization to (infinity,1)-bimodules. See there for more.

References

The 2-category of bimodules in its incarnation as a 2-category with proarrow equipment appears as example 2.3 in

Bimodules in homotopy theory/higher algebra are discussed in section 4.3 of

For more on that see at (∞,1)-bimodule.

Last revised on August 20, 2024 at 13:10:19. See the history of this page for a list of all contributions to it.

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