1. Some Other Type of Grothendieck Groups Let a Be an Abelian Category and S Be a Distinguished Set of 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 generated 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 0 0 [L] of invertible sheaves L on X with the multiplication defined by [L][L ] = [L ⊗OX L ].

It follows from the deﬁnition that Pic(X) = K(inv, ⊗OX ), where inv is the category of invertible sheaves on X. Let Φ : A → invX be the functor M 7→ M,f Then Φ is an exact equivalence and

Φ(M ⊗A M) = Φ(M) ⊗OX Φ(N) ∼ for any M,N ∈ ob A. This implies that K(A, ⊗A) = K(inv, ⊗O ). We conclude that ∼ X 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 ⊗K Mn(K) = B ⊗K Mm(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 ⊗K B]. 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+ 1/2 by ψ(A) = (dimK A) . Then ψ satisfies: (1) if A is isomorphic to B, ψ(A) = ψ(B), (2) f(A ⊗K B) = 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.