Cohomology of finite groups by Alejandro Adem
By Alejandro Adem
Adem A., Milgram R.J. Cohomology of finite teams (Springer, 1994)(ISBN 354057025X)
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Additional info for Cohomology of finite groups
Let a: X -+BA represent a; then, since BA is path connected for n 2: 1, we can suppose that a(xo) = e, the unit of B A. Now, since BA has an abelian multiplication we can extend a to spn(x) by L:'~ Xi f-+ L: a(Xi), so we have factored a through spn(x). o Let H c G be any subgroup. The H-orbit space of Ee is given as pt XH E e , and the projection Ee-+pt XH Ee is a principal H-fibration with contractible total space when H is closed. Consequently pt x H Ee is a model for B H. On the other hand the projection pt x H Ee -+ Be is a fibration with fiber G / H.
This resolution is commonly called the bar construction and is written B(71). It is not hard to extend it to a general resolution of a (left) 7l(G)module M provided M is a free 7l-module. Set B(M) = B(71) 0z M and define the boundary o( 1911 ... 5 Let ¢: M -+ N 80 80 81 82 8i O+--M +--Co +--C 1 +--C 2 +--··· +--Ci +--··· 82 81 82 O+--N +--1)0 +--1)1 +--1)2+-- ... are two resolutions of Mover 7l(G). Then there are 7l(G)-maps ¢i: Ci -+1)i, the dia9ram o ::; i < 00, so o +-- M o N where each Ci is Z( G)-free.
The above is clear except in dimensions less than -1. 7). Note the following simple fact: if F is a free ZG-module, then jj* (G, F) == O. This follows from the fact that F has a trivial projective resolution (itself) and that FG = NF, KerN = IF. As a consequence the same must be true for any projective ZG-module. An additional advantage of Tate cohomology is that it allows us to consider homology and cohomology at the same time. It will enjoy the same general properties as ordinary cohomology with respect to short exact coefficient sequences, restriction and transfer, maps induced by group homomorphisms etc ..