By K. W. Gruenberg
Xiv + 275 pages, number of casual studies and seminars subject matters contain fastened aspect unfastened motion, cohomology and homology teams, shows and resolutions, unfastened teams, classical extension concept, finite p-groups, cohomological size, extension different types and module idea
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0 G G and take its homology. One checks that this is a possible homological extension in much the same way as in the cohomology case. Ho(G,A) = A G because 0 ~ ~G ~ ~ ~ 0 Note that w h e n tensored w i t h A on the left gives A @ ~ ~ A/Im(A @ ~ ) = A / A ~ . 2 Definition. of A ~ If (Hq(G,); d) is a minlmalLhomologlcal extension AG, we call Hq(G,A) the q-dlmensional (or q-th) homology ~roup of G with coefficients in A. Remark. If one works with a left module A one must use a right resolution.
For if A is given, we embed A in A*: 0 ~ A ~ A* * B ~ O. Then the corresponding exact cohomology sequence 0 ~ AG ~ (A*) G ~ B G ~ Hl(G,A) ~ 0 ~ Hl(G,B) ~ H2(G,A) ~ 0 ~ ... shows that H 1 is unique is unique; (to within an isomorphism); then that H 2 and so on. Existence. Take a projective resolution of ~: "'" ~ P2 ÷ P1 ~ PO ~ L ~ O. Form (HOmzG(Pi,A)) and take the homology of this complex. verify that this functor, call it (Hq(A)), We is a minimal cohomologi- cal extension. (i) HO(A) = H O m G ~ , A ) ~ A G.
This is easily checked to be a G-monomorphism. The finiteness of G is invoked only now to ensure that the mapping is surJectlve. It follows that for a finite group G, we have Hq(G,A) = 0 for all q ~ 0 whenever A is a free ZG-module and therefore also when A is projective. Definition. ), where dq: Hq+I(G,C ) ~ Hq(G,A) whenever 0 ~ A ~ B ~ C ~ 0 is a given exact sequence of G-modules. The resulting "homology sequence" is assumed to be exact and d is to be natural. is induced. 2 23 A natural isomorphism between two such extensions is defined in the expected way: cf.