Cohomology of cyclic groups of prime square order

Abstract

Let G be a cyclic group of order p 2 , p a prime, and let U be its unique proper subgroup.If A is any G-module, then the four cohomology groups H%G, A) HKG, A) H\U, A) H\U, A) determine all the cohomology groups of A with respect to G and to U.This article determines what values this ordered set of four groups takes on as A runs through all finitely generated G-modules.Reduction* Let G be any finite group.A finitely generated Gmodule M is quotient of a finitely generated G-free module L. The kernel K is Z-ΐΐee, and since the cohomology of L is zero with respect to all subgroups of G, K is a dimension shift of M. The standard dimension shifting module P = ZG/(S a ) is Z-free, so K(g)P is a Z-ϊree G-module having the same cohomology as M with respect to all subgroups of G.PROPOSITION !•If G is any finite p-group and M any ^-free Gmodule, the cohomology of M is that of R 0 M where R is the ring of p-adic integers.

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF