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Acta Mathematica Academiae Seientiarum Hungarieae Tomus 26 (1--2), (1975), 1--2. CALCULATING SOME RELATIVE HOMOLOGY OVER GROUP RINGS By J. F. CARLSON (Athens) All definitions and preliminaries can be found in [1]. Let R be a commutative ring with unit. Suppose H is a normal subgroup of a group G such that G/H is cyclic of order n < ~. Let x be an element of G such that x" c H but x k ~ H for 1 <-_ k< n. Let M and N be left RG-modules and let L be a right RG-module. Then the rela- tive homology groups Tor(~Ra'RB)(L,M) and the relative cohomology groups Ext"(Ra, Rm(M, N) can be computed using the following THEOREM. Let M be a left RG-module. Let Xi=RG| for i=0, 1 ..... Define to: Xo~m by to(X~| x~m (r=0,1,...,n-l,m~M). For all r=0 .... , n-l, rnCM let tk: Xk~Xk_ 1 be defined by [ xr@m--xr-1| if k is odd, t~ (x ~ | m) = I "-1 xi| if k is even. t i~O Tken the complex .... 2(2 t~-L X~ t~_ Xo t_o M-~ 0 is an (RG, RH)-projeetive resolution of M. PROOF. The proof that h_l.h=0 is straightforward. The RH-homotopy
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: May 21, 2016
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