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

This publication, the 1st printing of which used to be released as quantity 38 of the Encyclopaedia of Mathematical Sciences, offers a contemporary method of homological algebra, in accordance with the systematic use of the terminology and concepts of derived different types and derived functors. The e-book comprises purposes of homological algebra to the idea of sheaves on topological areas, to Hodge thought, and to the idea of modules over earrings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin clarify all of the major rules 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 e-book is a superb reference for graduate scholars and researchers in arithmetic and likewise for physicists who use equipment from algebraic geometry and algebraic topology.

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**Example text**

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. Denote by G the Galois group of the algebraic closure k/ k with the Krull topology. A major part of the classical Galois theory can be formulated as follows: the category (k-Alg)O dual to the category of commutative finite-dimensional semisimple k-algebras is equivalent to the category G-Set of finite topological G-sets.

This object is defined together with a morphism k : K -+ X such that cp 0 k = O. The diagram K ~ X ~ Y satisfies the following universality condition: for any morphism k' : K' -+ X such that cp 0 k' = 0 there exists a unique morphism h : K' -+ K such that k' = k 0 h. We call the morphism k or the pair (K, k) the kernel of cp; sometimes by the kernel we mean also the object K. One can easily verify that if the kernel (K, k) of a morphism cp exists it is defined uniquely. A-D we have the set-theoretical construction of the kernel: cp-I(O) for groups and for modules; the family cp;I(O) for coefficient systems; the family CPi'/(O) for a morphism of sheaves cp : F -+ 9 represented by the family CPu : F(U) -+ g(U) for all open sets U.

We repeat some of these definitions. a. A (cochain) complex in an additive category C is a sequence of objects and morphisms • dn - 1 n dn X: ... -+X-+X n+l dn +1 -+ ... with the property a,n 0 dn - 1 = 0 for all n. h. Let the category C be abelian. ) c. A complex X· in an abelian category C is said to be acyclic at xn if Hn(x·) = o. 41 § 2. Additive and Abelian Categories d. A complex X' in an abelian category C is said to be exact (or an exact sequence) if it is acyclic at all terms. In a similar way one can generalize other notions from § 1 of Chap.