By A. I. Kostrikin, I. R. Shafarevich (auth.), A. I. Kostrikin, I. R. Shafarevich (eds.)

This booklet, the 1st printing of which was once released as quantity 38 of the Encyclopaedia of Mathematical Sciences, offers a contemporary method of homological algebra, in response to the systematic use of the terminology and ideas of derived different types and derived functors. The ebook comprises purposes of homological algebra to the idea of sheaves on topological areas, to Hodge idea, and to the idea of modules over earrings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin clarify the entire major principles of the idea of derived different types. either authors are famous researchers and the second one, Manin, is legendary for his paintings in algebraic geometry and mathematical physics. The publication is a wonderful reference for graduate scholars and researchers in arithmetic and likewise for physicists who use tools from algebraic geometry and algebraic topology.

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**Example text**

Theorem. A functor F : C - t 'D is an equivalence of categories if and only if a) F is a fully faithful functor, b) any object Y E Ob'D is isomorphic to an object of the form F(X) for some X E ObC. 14. Examples. A meaningful theorem about an equivalence of categories can often be thought of as the disclosure of two complementary description of a mathematical object. The following small classical theories, each formulated as an equivalence theorem, illustrate this statement. a. Galois theory. Let k be a field, assumed for simplicity to be of characteristic zero.

Some most commonly used derived functors from algebra and topology have standard names. We mention some of these functors. § 4. 1. Functors Ext. Let A be an abelian category. The abelian group E ObA can be interpreted as a functor in three different ways. a. as a functor HomA(M,') : A ~ Ab for a fixed Mj b. as a functor HomA(-,N) : AO ~ Ab for a fixed Nj c. as a functor HomA(-'·) : AO x A ~ Ab. All these functors are left exact. The corresponding right derived functors are denoted Ext~(M,N)j one can show that the groups Ext~(M,N) do not depend on the choice in the interpretation of HomA as a functor.

8. Definition. A category satisfying the axioms AI-A3 is called an additive category. A category satisfying the axioms AI-A4 is called an abelian category. A-D are additive. Moreover, all these categories are abelian. The existence of the canonical decomposition of a morphism cp : 38 Chapter 2. , the isomorphism Im'P <;::! HIKer'P) is guaranteed by the homomorphism theorem for abelian groups; similarly one proves that the category of modules over a fixed ring is abelian. C. is constructed componentwise.