Let A be an abelian category and S be the set of split exact sequences in A

COMPUTATION OF SOME EXAMPLES OF GROTHENDIECK GROUPS

1. Some Other type of Grothendieck Groups

Let A be an abelian category and S be a distinguished set of exact sequences in A. We define K(A, S) to be the abelian group generated by [A] with A ∈ ob A with the relation [A] = [A⁰] + [A⁰⁰] whenever 0 → A⁰ → A → A⁰⁰→ 0 belongs to S.

Example 1.1. Let A be an abelian category and S be the set of split exact sequences in A.

Denote K(A, S) by K(A, +) in this case. Then K(A, +) is characterized by the following properties.

(1) [A] = [A⁰] + [A⁰⁰] whenever A is isomorphic to A⁰⊕ A⁰⁰, (2) A isomorphic to C implies [A] = [C].

Theorem 1.1. Let A, B be objects in an abelian category A. If [A] = [B] in K(A, ⊕) then there exists an object C in A so that A ⊕ C ∼= B ⊕ C.

This construction can be generalized as follows. Let F : A × A → A be a bifunctor.

The Grothendieck group K(A, F ) is the abelian group generated by the symbols [A] with A ∈ ob A subject to the relation:

(1) A isomorphic to B implies [A] = [B], (2) [F (A, B)] = [A] + [B].

The Grothendieck group K(A, F ) can be constructed as follows. Let F be the free abelian group generated by isomorphism classes of objects of A and R be the subgroup of F gen- erated by [F (A, B)] − [A] − [B] with A, B ∈ ob A. Then the Grothendieck group K(A, F ) is the quotient group F /R.

Theorem 1.2. Let A be a commutative ring with identity and A be the category of finitely generated projective A-modules of rank one. Consider the bifunctor ⊗A : A × A → A defined by the tensor product of A-modules. Then K(A, ⊗_A) ∼= Pic(X), where X = Spec A and Pic(X) is the Picard group of X.

Proof. The Picard group of the affine scheme X = Spec A is the group of isomorphism classes [L] of invertible sheaves L on X with the multiplication defined by [L][L⁰] = [L ⊗OX L⁰].

It follows from the definition that Pic(X) = K(inv, ⊗O_X), where inv is the category of invertible sheaves on X.

Let Φ : A → inv_X be the functor M 7→ fM , Then Φ is an exact equivalence and Φ(M ⊗_AM ) = Φ(M ) ⊗O_X Φ(N )

for any M, N ∈ ob A. This implies that K(A, ⊗_A) ∼= K(inv, ⊗O_X). We conclude that K(A, ⊗A) ∼= Pic(X).

 Let K be a field. Assume that A, B are finite dimensional central simple algebras over K. We say that A ∼ B if there exist n, m ≥ 1 so that

A ⊗KMn(K) ∼= B ⊗KMm(K),

1

2 COMPUTATION OF SOME EXAMPLES OF GROTHENDIECK GROUPS

where Mp(K) is the algebra of p × p matrices over K. The Brauer group Br(K) is the set of all equivalent classes with multiplication defined by

[A] · [B] = [A ⊗KB].

The class [K] is the unit element and [A^◦] is the inverse of [A]. Here A^◦ is the opposite ring of A.

Let A be the category of finite dimensional central simple algebras over K. Define a map ψ : ob A → Q^×+

by ψ(A) = (dim_KA)^1/2. Then ψ satisfies:

(1) if A is isomorphic to B, ψ(A) = ψ(B), (2) f (A ⊗_KB) = f (A) · f (B).

Then we obtain a surjective group homomorphism ψ : K(A, ⊗_K) → Q^×₊ induced from ψ. Notice that

ϕ : G → K(A, ⊗_K)

defined by p 7→ [Mp(K)] defines a section of ψ, where p is a prime number i.e. ψ ◦ ϕ = 1

Q^×+. Hence the following exact sequence of abelian groups splits:

0 → ker ψ → K(A, ⊗_K) → Q^×₊→ 0.

In other words, we have a direct sum decomposition of abelian groups:

K(A, ⊗K) ∼= ker ψ ⊕ Q^×₊.

Theorem 1.3. The group ker ψ is isomorphic to the Brauer group Br(K).

Let A be an abelian category and B be a full subcategory of A. We define K(B) to be the abelian group generated by [B] with B ∈ ob B subject to the relation [B] = [B⁰] + [B⁰⁰] whenever 0 → B⁰→ B → B⁰⁰→ 0 is an exact sequence in A. Notice that the Grothendieck group K(B) depends on the category A.

Example 1.2. Let A be an abelian category of P be the full subcategory of A consisting of projective objects of A. Then K(P) = K(P, ⊕). This follows from the fact that every short exact sequences in P splits.

When A is the category of finitely generated modules over a noetherian ring A, P is the category of finitely generated projective modules over A. Then K(P) is exactly the Grothendieck group defined in the beginning of the lecture.

Let D be an additive category (not necessarily abelian). Suppose A, B, C, D are objects of D. We say that (A, B) ∼ (C, D) if there exist objects E, F of D such that A ⊕ E ∼= C ⊕ F and B ⊕ E ∼= D ⊕ F. Denote G the set of equivalent classes [A, B] defined by this relation.

We define

[A, B] + [C, D] = [A ⊕ C, B ⊕ D].

Then (G, +) forms an abelian group denoted by K(D, ⊕), Definition 1.1. We call K(D, ⊕) the Grothendieck group of D.