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1. Some Other Type of Grothendieck Groups Let a Be an Abelian Category and S Be a Distinguished Set of Exact Sequences in A

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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 2 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 2 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 2 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 2 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 × : ob A ! Q+ 1=2 by (A) = (dimK A) : Then satisfies: (1) if A is isomorphic to B; (A) = (B); (2) f(A ⊗K B) = f(A) · f(B): Then we obtain a surjective group homomorphism × : K(A; ⊗K ) ! Q+ induced from : Notice that ' : G ! K(A; ⊗K ) defined by p 7! [Mp(K)] defines a section of ; where p is a prime number i.e. ◦ ' = 1 × : Q+ Hence the following exact sequence of abelian groups splits: × 0 ! ker ! K(A; ⊗K ) ! Q+ ! 0: In other words, we have a direct sum decomposition of abelian groups: ∼ × K(A; ⊗K ) = ker ⊕ Q+: Theorem 1.3. The group ker 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 2 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:.
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