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Homological Algebra by S.I. Gelfand, Yu.I. Manin, S.I. Gelfand, Yu.I. Manin, A.I.

By S.I. Gelfand, Yu.I. Manin, S.I. Gelfand, Yu.I. Manin, A.I. Kostrikin, I.R. Shafarevich

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A. For every pair n,n, where n is a natural number, n is a group (abelian if n > 2) there casts a CW-complex K(n,n) such that H n ( X , n ) = [X,K(n,n)] as functors on the homotopy category of CWcomplexes. b. The object K(n,n) of this category can be uniquely characterized by the following properties: n,(K(n,n)) = 0 for i # n, nn(K(n,n)) = n. 3. Torsors. If G is a non-commutative group one can define a set H1(X, G), Properties of this set constitute the most simple and useful part of the still fragmentary noncommutative cohomology theory.

14. About Proofs in Abelian Categories. The main thesis in working with abelian categories is that if a statement involving a finite number of objects and a fmite number of morphisms is true in the category of modules over a ring, the it remains true in an arbitrary abelian category. The thesis is based on the following embedding theorem. 1. Theorem. Let A be an abelian categorg whose objects form a set. Then there exists a ring R and an exact functor F : A + R-mod, which is an embedding on objects and an isomorphism on Hom's.

Torsors. If G is a non-commutative group one can define a set H1(X, G), Properties of this set constitute the most simple and useful part of the still fragmentary noncommutative cohomology theory. a. Let G be a topological group, X a topological space. A principal Gfibration, or simply G-torsor on X consists of the following data: a continuous map n : P + X; a continuous action G x P -+ P, (g,p) -+ gp, such that n(p) = n(gp) and locally n is isomorphic to a projection G x U -+ U with the fiberwise action g(h,p) = (gh,p).

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