Grothendieck Groups: Ariyan Javan Peykar

Grothendieck groups
Ariyan Javan Peykar
Summary
This talk intends to introduce one aspect of the Grothendieck-Riemann-Roch theorem: K-theory.
We define the Grothendieck group K0 (X) associated to a projective variety X. We shall study
some of its properties, such as its ringstructure, and give some elementary examples. There is
another Grothendieck group which is easier to construct, namely K 0 (X). The main goal will
consist of establishing a natural group isomorphism between the groups K0 (X) and K 0 (X) when
X is nonsingular, quasi-projective and irreducible. We will finish with a generalization of the
Riemann-Roch theorem for nonsingular curves.
Varieties will always be quasi-projective over an algebraically closed field.
K-theory
Let C be an additive category embedded in an abelian category A. Let Ob(C) denote the class of
objects and let Ob(C)/
= be the set of isomorphism classes1 . Let F (C) be the free abelian group
on Ob(C)/
=, i.e. an element T F (C) is a finite formal sum
X
nX [X],
where [X] denotes the isomorphism class of X Ob(C) and nX is an integer which is almost
always zero.
Definition 1.1. To any sequence
(E)
/A
/ A0
/ A00
/0
in C, which is exact in A, we associate the element Q(E) = [A] [A0 ] [A00 ] in F (C). Let H(C) be
the subgroup generated by the elements Q(E) where E runs through all short exact sequences.
Definition 1.2. We define the Grothendieck group, denoted by K(C), as the quotient group
K(C) = F (C)/H(C).
Remark 1.3. Note that C has finite direct sums. The fact that the sequence
0
/A
/ AB
/B
/0
is exact (in A) shows that the addition is given by [A B] = [A] + [B].

Examples 1.4. Let R be a commutative ring.
1. Let C be the category of R-modules. (This is not a small category. To avoid this the reader
may consider only countably generated R-modules.) Let
L us show that
L K(C) = (0). To this
extent, L
let M be an L
R-module and note that M nN M
=
nN M . We see that
[M ] + [ nN M ] = [ nN M ], which shows that [M ] = 0 in K(C).
1 The
reader should actually only consider small categories.
2. Let R be a principal ideal domain and C be the category of finitely generated R-modules.
By the structure theorem of R-modules, any R-module is isomorphic to the direct sum
of a free module and a torsion part which is the direct sum of cyclic modules. The rank
of an R-module is defined as the rank of its free part. That gives us a surjective map
rk : Ob(C)/
= Z which induces a surjective homomorphism from F (C) to Z. Since the
e from K(C) to Z. Note
rank is trivial on the elements Q(E), it induces a group morphism rk
that for any nonzero ideal I = (x), we have a short exact sequence
0
/R
/R
/ R/I
/0,
e is trivial. Furthermore,
which shows that [R/I] = 0 in K(C). Therefore the kernel of rk
since any short exact sequence of free R-modules is split and rk([R]) = 1, the rank induces
an isomorphism from K(C) to Z.
3. The above example actually shows that if R is a ring and C is the category of finitely
=
/Z.
generated free R-modules, we have an isomorphism rk : K(C)
4. When C is the category of finitely generated projective R-modules, the reader may look at
Chapter II of Weibels book on K-theory.
OX -modules
The reader is referred to Chapter II, paragraph 5 of [HAG] for the theory of coherent sheafs on
affine and projective varieties.
Let X be a quasi-projective variety and let OX be its structure sheaf. Recall that the affine open
subsets of X form a basis for the topology on X and that OX is determined by the rule
OX (U ) = (U, OX ) = (U, OU ) = k[T1 , . . . , Tn ]/I,
if U X is isomorphic to the affine variety determined by the prime ideal I k[T1 , . . . , Tn ].
Definition 2.1. A coherent sheaf is a sheaf of abelian groups F on X endowed with a multiplication OX F F such that the following properties hold.
OX -module structure: For each open U X, the abelian group of sections F(U ) becomes a
module over OX (U ).
Quasi-coherence: For every open affine subsets U V X, F(U ) = F(V ) OX (V ) OX (U ).
Coherence: For each open affine U X the module F(U ) is finitely generated over OX (U ).
Let Coh(X) be the category of coherent sheaves on X. (A morphism of coherent sheaves is a
morphism of sheaves which respects the module structure.)
Remark 2.2. The category Coh(X) is abelian.
Definition 2.3. A vector bundle (of rank r) is a coherent sheaf F where every point x X has
an affine neighborhood U X such that F(U ) is a free OX (U )-module of rank r. A line bundle
is a vector bundle of rank 1. Let Vect(X) be the category of vector bundles on X.
Remark 2.4. The category Vect(X) is additive and embedded in Coh(X). It is not an abelian
category in general. The following theorem states why this is the case.
Theorem 2.5. For X = Spec(A) and A noetherian, Vect(X) is equivalent to the category of projective finitely generated A-modules and Coh(X) is equivalent to the category of finitely generated
A-modules.
ii
K-theory of a variety
Let X be a quasi-projective variety.

Definition 3.1. We define the Grothendieck group of vector bundles on X, denoted by K 0 (X),
as
K 0 (X) = K(Vect(X)).
Proposition 3.2. The tensor product (over OX ) defines a commutative ringstructure on F (Vect(X)).
Proof. The tensor product of vector bundles is a vector bundle. The tensor product is associative
and commutative as follows from its universal property and OX is clearly the identity element.
The tensor product is also distributive with respect to the direct sum.
Proposition 3.3. The tensor product defines a commutative ring structure on K 0 (X).
Proof. We need to show that the subgroup H(Vect(X)) is an ideal of F (Vect(X)). But this follows
from the fact that any vector bundle is flat.
Definition 3.4. The Grothendieck group of coherent sheaves, denoted by K0 (X), is defined as
K0 (X) = K(Coh(X)).
Remark 3.5. The embedding Vect(X) Coh(X) of categories induces a natural homomorphism K 0 (X) K0 (X).
Theorem 3.6. If X is nonsingular, quasi-projective and irreducible, the canonical homomorphism
K 0 (X) K0 (X) is an isomorphism of groups.
Proof. From standard considerations on projective varieties it follows that any coherent sheaf F
is the quotient of some vector bundle: for n 0, the twisted sheaf F(n) is generated by its
global sections. Since X is quasi-compact, we may cover X with a finite number of open affine
subsets Ui (i = 1, . . . , d). On each Ui , F(n)(Ui ) is generated by a finite number of global sections
and therefore there exist a finite number of global sections s1 , . . . , sr F(n)(X) which generate
r
F(n). Since the tensor
F(n) on every open Ui . Therefore there is a surjective morphism OX
r
product is right exact, tensoring this with OX (n) gives a surjective morphism OX
(n) F.
Replacing a quasi-projective variety by its closure in some projective and extending our sheaf to
this closure shows that any coherent sheaf on X is the quotient of a vector bundle. This allows
one to always construct a (not necessarily finite) resolution of vector bundles for a coherent sheaf.
Let n = dim(X). Since X is nonsingular projective, any coherent sheaf F has a finite resolution
of vector bundles E0 , . . . , En . That is, we have a complex
/0
/ En
/ ...
/ E0
/0
/ En
/ ...
/ E0
/F
/0
...
such that the augmented complex
0
Pdim X
is exact. This means that we can define an inverse to the above map by [F] 7 i=1 (1)i [Ei ].
One can show that this map is well-defined, i.e. independent of the chosen resolution and that it
an additive map. (See Lemma 11 and Lemma 12 in [BorSer].)
Let us illustrate the importance of nonsingularity.
iii
Example 3.7. Let k be a field, A = k[x] and I = (x) A. (Picture) The A-module k = A/I has
a finite resolution of free A-modules
/A
/A
/k
/0.
Here f : s 7 sx. We see that the (Krull) dimension of A is equal to the length of this (minimal)
resolution.
Example 3.8. Let k be a field, A = k[x, y] and I = (x, y) A. (Picture) The ring A is regular
and the A-module k = A/I has a finite resolution of free A-modules
/A
/ A2
/k
/A
/0.
Here g : s 7 (sy, sx) and f : (s, t) 7 sx + ty. Again the dimension of A is equal to the length
of this (minimal) resolution.
Example 3.9. Let k be a field, A = k[x, y]/(xy) and I = (x, y) A. (Picture) Note that A is
nonregular. Consider the infinite resolution of free A-modules for k = A/I given by
...
/ A2
/ A2
/ A2
/ A2
/ A2
/A
/k
/0.
Here
f : (s, t) 7 sx + ty, g : (s, t) 7 (sy, tx) and h : (s, t) 7 (sx, ty).
It is easy to see that
T oriA (k, k) =
k
k2
if i = 0
.
if i > 0
To this extent it suffices to note that after tensoring the above resolution with k A all the maps
are zero. This shows that there can not be a finite (projective) resolution of A-modules for k.
(Since then the T oriA (k, ) functors would be identically zero for i 0.) Now, in contrast with
the above examples, we see that the dimension of A is strictly smaller than the length of any flat
resolution, i.e., the flat dimension of k is bigger than the dimension of A. The reader may took
a look at Chapter 4 of Weibels Homological Algebra for a readable account on dimension theory
for rings.
Corollary 3.10. Using the isomorphism in Theorem 3.6 we get a ringstructure on K0 (X). We can
Pdim X
show that the product of two coherent sheaves F and G equals F G = i=0 (1)i T oriOX (F, G)
in K0 (X).
The next talk will be on a generalization of the Riemann-Roch theorem for non-singular curves.
iv
References
[BorSer] A. Borel, J.P. Serre Le theor`eme de Riemann-Roch Bull. Soc. math. France, 86, 1985, p.97136.
[Har] R. Hartshorne Algebraic geometry Springer Science 2006.
[FAC] J.P. Serre Faisceaux algebriques coherents The Annals of Mathematics, 2nd Ser., Vol. 61, No.
2. (Mar., 1955), pp. 197-278.

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Grothendieck groups

Ariyan Javan Peykar

is exact (in A) shows that the addition is given by [A B] = [A] + [B].

reader should actually only consider small categories.

Let X be a quasi-projective variety.

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