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On homotopy classes of cochain maps. (English) Zbl 0583.18005

The author proves the following theorem. Let \(A\) and \(B\) be cochain complexes of modules over a commutative ring R. Let \(A_{\#}\) be the complex with \(A^ n_{\#}=A^{n+1}\) and \(\delta_{\#}=-\delta\). Let \([A,B]\) denote the abelian group of homotopy classes of cochain maps from \(A\) to \(B\) and \(E_ R(A_{\#},B)\) the abelian group of equivalence classes of weakly splitting extensions of \(B\) by \(A_{\#}\). Then there is an isomorphism \(\Phi\) : [A,B]\(\cong E_ R(A_{\#},B)\) given by \(\Phi ([f])=\{C(f)\}\), where \(C(f)\) is the mapping cone of \(f\).
Reviewer: T.W.Hungerford

MSC:

18G35 Chain complexes (category-theoretic aspects), dg categories
13D25 Complexes (MSC2000)
55U15 Chain complexes in algebraic topology
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References:

[1] K. S. Brown: Cohomology of Groups. Springer-Verlag, New York (1982). · Zbl 0584.20036
[2] H. Cartan and S. Eilenberg: Homological Algebra. Princeton, N. J. (1956). · Zbl 0075.24305
[3] A. Hattori: Some arithmetical applications of groups HQ(R, G). Tohoku Math. J., 33, 35-63 (1981). · Zbl 0476.12009
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