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By A. Parshin,I. Shafarevich

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Example text

We could also define the unbounded derived category D(C) from unbounded complexes Ch(C). Note that there exists a faithful embedding of C in an Abelian category A such that C ⊂ A is closed under extensions and the exact functor C → A reflects exact sequences. So, a complex in Ch(C) is acyclic iff its image in Ch(A) is acyclic. In particular, a morphism in Ch(C) is a quasi-isomorphism iff its image in Ch(A) is a quasi-isomorphism. Hence, the derived category D(C) is the category obtained from Ch(C) formally inverting quasi-isomorphisms.

R EMARKS . (i) If A L V B is an adjoint situation where L is left adjoint to V , then T = LV : B → B is part of a cotriple (T , ε, δ), where ε : LV → idB is the counit of the adjunction. (ii) Given a cotriple T on A, and A ∈ ob A, we have a simplicial object T ∗ A = {T n A} of A with face maps δi = T i ∈ T n−i : T n+1 A → T n A and degeneracy maps si = T i δT n−1 , T n+1 A → T n+2 A. 4. Let Ring be the category of rings and for any ring A, let FA be the free ring on the underlying set of A. Then F is a functor Set → Ring adjoint to the forgetful functor and the adjointness yields a morphism ε : FA → A and a morphism δ : FA → F 2 A such that (F, ε, δ) is a cotriple in Ring.

Let MZ (X) be the Abelian category of OX -modules supported on Z, a an ideal sheaf in OX such that OX /a = OZ . Then every M ∈ MZ (X) has a finite filtration M ⊃ Ma ⊃ Ma 2 ⊃ · · · and so, by devissage, Kn (MZ (X)) ≈ Kn (M(Z)) ≈ Gn (Z). 4. 1. A full subcategory B of an Abelian category A called a Serre subcategory if whenever: 0→M →M →M →0 is an exact sequence in A, then M ∈ B if and only if M , M ∈ B. Given such a B, construct a quotient Abelian category A/B as follows: ob(A/B) = ob A. A/B(M, N ) is defined as follows: If M ⊆ M, N ⊆ N are subobjects such that M/M ∈ ob(B), N ∈ ob(B), then there exists a natural isomorphism B(M, N ) → B(M , N/N ).

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Algebra VII - Combinatorial Group Theory, Applns to Geometry by A. Parshin,I. Shafarevich

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