Cohomology of Groups
نویسنده
چکیده
Notes on Kenneth Brown’s book Cohomology of Groups. 1. Some Homological Algebra 1.1. Review of Chain Complexes. Let R be a ring, and let (C, d) and (C ′, d′) be two chain complexes of left R-modules. Define a complex of abelian groups HR(C,C ′) as follows. Let HR(C,C )n = ∏ q∈Z HomR(Cq, C ′ q+n) and define the boundary map Dn by Dn(f) = d ′f − (−1)fd. 1.5. The Standard Resolution. For any group G, we can always form the following free resolution of Z over ZG. Let Fn be the free ZG module with basis given by the (n+1)-tuples of elements of G whose first component is 1: (1, g1, g2, . . . , gn). The G-action on Fn is defined on basis elements component-wise. We introduce the shorthand bar notation: [g1|g2| . . . |gn] = (1, g1, g1g2, . . . , g1g2 . . . gn) If n = 0, there is only one basis element which we denote by [ ]. Define the boundary morphisms by
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