1. Grothendieck Group of Abelian categories

Roughly speaking, an abelian category is an additive category such that finite direct sum exists, the kernel and the cokernel of a morphism exist, and the coimage of a morphism is isomorphic to its image (first isomorphism theorem holds). For example, if A is a noetherian ring, the category of finitely generated (left) A-modules is an abelian category. A morphism f : A → B in an abelian category A is a monomorphism if ker f = 0 and a epimorphism if coker f = 0. We say that the a sequence

A −−−−→ B^{f} −−−−→ C^{g}
is exact at B if ker g = Im f. A sequence of morphisms

· · · → A_{i−1}→ A_{i} → A_{i+1}→ · · ·
is said to be exact if it is exact at all A_{i}.

The Grothendieck group K(A) of an abelian category A is an abelian group generated by the set {[A]} of symbols [A] of objects of A subject to the relations

[A] = [A^{0}] + [A^{00}]
whenever 0 → A^{0} → A → A^{00} → 0 is exact.

Lemma 1.1. Let A be an abelian category and K(A) be its Grothendieck group. Then (1) [0] = 0, and

(2) if A ∼= B, [A] = [B], and (3) [A ⊕ B] = [A] + [B].

Proof. To prove (1), we consider the exact sequence 0 → 0 → 0 → 0 → 0. To prove (2), we consider the exact sequence 0 → 0 → A → B → 0. To prove (3), we consider

0 → A → A ⊕ B → B → 0.

The Grothendick group K(A) of A can be constructed as follows. Let F (A) be the
free abelian group generated by the isomorphism classes of objects of A and R(A) be the
subgroup generated by [A] − [A^{0}] − [A^{00}] whenever 0 → A^{0} → A → A^{00} → 0 is exact. The
quotient group F (A)/R(A) is the Grothendick group K(A).

Theorem 1.1. Let A and B be an abelian category A. Then [A] = [B] in K(A) if and only
if there exist short exact sequence 0 → C^{0} → C → C^{00} → 0 and 0 → D^{0} → D → D^{00} → 0
such that A ⊕ C ⊕ C^{00}⊕ D is isomorphic to B ⊕ C ⊕ D ⊕ D^{00}.

An exact functor F : A → B on abelian categories is an additive functor such that for any
exact sequence 0 → A^{0} → A → A^{00} → 0, the sequence 0 → F (A^{0}) → F (A) → F (A^{00}) → 0 is
also exact.

Lemma 1.2. Let F : A → B be an exact functor of abelian categories. Then F induces a group homomorphism F∗: K(A) → K(B).

Proof. Let us define a map F : F (A) → F (B) by F ([A]) = [F (A)] and extend it additively to all elements of F (A). Since F is additive,

F ([A] − [A^{0}] − [A^{00}]) = F ([A]) − F ([A^{0}]) − F ([A^{00}]) = [F (A)] − [F (A^{0})] − [F (A^{00})].

Since F is exact, the above identity implies that F (R(A)) ⊂ R(B). This shows that the map F∗ : K(A) → K(B) sending [A] + R(A) → [F (A)] + R(B) is well-defined. This map

F∗ can be extend to a group homomorphism.

1

2

Let us abuse the use of the notation [A] : the image of [A] in K(A) will also be denoted by [A].

Lemma 1.3. Let F : A → B be an exact equivalence of categories. Then F∗ : K(A) → K(B) is an isomorphism of abelian groups.

Proof. Let G : B → A be the inverse of F. Then G ◦ F = 1_{A} and F ◦ G = 1_{B}. One can
check that G∗ : K(B) → K(A) is the inverse of F∗ : K(A) → K(B).
Definition 1.1. Let A be a noetherian scheme and Coh(X) be the category of coherent
sheaves on X. The category Coh(X) is an abelian category. We define the K^{0}-group of the
scheme X to be the Grothendieck group of Coh(X) :

K^{0}(X) = K(Coh(X)).

Let A be a commutative noetherian ring and X = Spec A be its corresponding noetherian affine scheme. We know that the global section functor

Γ : Coh(X) → Mod(A)

sending F → Γ(F ) gives an exact equivalence of abelian categories whose inverse is given by M 7→ fM . Lemma 1.3 implies that:

Corollary 1.1. Let A be a noetherian ring and X = Spec A be its corresponding affine acheme. Then we have a group isomorphism:

K^{0}(X) ∼= K(Mod(A)).

Now, let us define the Grothendieck group of a triangulated category.

Let C be a triangulated category. The Grothendieck group K(C) of C is the abelian group
generated by isomorphism classes of objects of C subject to the relation [A] = [A^{0}] + [A^{00}]
whenever we have a distinguish triangle A^{0}→ A → A^{00}→ A^{0}[1].

Theorem 1.2. Let A be an abelian category and D^{b}(A) be its bounded derived category.

Then the natural map

K(A) → K(D^{b}(A))
is an isomorphism of abelian groups.

Proof. This will be discussed later.

Let X be a noetherian scheme and D^{b}(X) be the bounded derived category of coherent
sheaves on X. Theorem 1.1 implies:

Corollary 1.2. We have a natural isomorphism if abelian groups: K^{0}(X) = K(D^{b}(X)).

Notice that furthermore, if X is regular, K^{0}(X) ∼= K^{0}(X). In this case, we obtain
K^{0}(X) ∼= K(D^{b}(X)).