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Relative cohomology of groups. (English) Zbl 0544.20047

For a group G and a G-module A denote by \(C^*(G,A)\) the cochain complex given by the bar resolution. Let N be a normal subgroup of G, \(Q=G/N\), and let B be a Q-submodule of \(A^ N\). Then the natural projection \(G\to Q\) induces an injection \(C^*(Q,B)\to C^*(G,A).\) Let \(C^*(Q,G;B,A)\) denote the quotient complex. Its homology \(H^*(Q,G;B,A)\) deserves to be called the relative cohomology since there is a long exact sequence \(...\to H^*(Q,B)\to H^*(G,A)\to H^*(Q,G;B,A)\to....\) This paper is devoted to the study of this relative cohomology. First a spectral sequence is obtained which converges to the relative cohomology and which is close to the Lyndon-Hochschild-Serre spectral sequence in ordinary cohomology. Secondly a complex \(L^*(G,N;A,B)\) is described, which has been introduced by Ortiz in his thesis. The authors show that its homology is \(H^*(Q,G;B,A)\) thus proving a conjecture of Ortiz. The complex \(L^*(G,N;A,B)\) is somewhat easier to handle than \(C^*(Q,G;B,A)\). This fact allows the authors to give some applications concerning low dimensions.
Reviewer: U.Stammbach

MSC:

20J05 Homological methods in group theory
18G35 Chain complexes (category-theoretic aspects), dg categories
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
18G40 Spectral sequences, hypercohomology