Group cohomology of an abelian group with nontrivial action - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T14:06:00Z http://mathoverflow.net/feeds/question/66473 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66473/group-cohomology-of-an-abelian-group-with-nontrivial-action Group cohomology of an abelian group with nontrivial action Mitja 2011-05-30T17:17:32Z 2011-09-14T10:57:15Z <p>How do I compute the group cohomology $H^2(G,A)$ if G is a finite abelian group acting nontrivially on a finite abelian group A?</p> http://mathoverflow.net/questions/66473/group-cohomology-of-an-abelian-group-with-nontrivial-action/66480#66480 Answer by algori for Group cohomology of an abelian group with nontrivial action algori 2011-05-30T18:39:51Z 2011-05-30T18:39:51Z <p>If $G$ is any group and $A$ is any $G$-module, then $H^2(G,A)$ can be identified with the set of the equivalence classes of extensions $$1\to A\to H\to G\to 1$$</p> <p>such that the action of $G$ on $A$ is the given action. Two extensions $H_1,H_2$ are said to be equivalent if there is an isomorphism $H_1\to H_2$ that makes the extension exact sequences commute. See K. Brown, Group cohomology, chapter 4.</p> http://mathoverflow.net/questions/66473/group-cohomology-of-an-abelian-group-with-nontrivial-action/66481#66481 Answer by norondion for Group cohomology of an abelian group with nontrivial action norondion 2011-05-30T18:48:53Z 2011-05-30T18:48:53Z <p>You can compute it using the Bar resolution, see [Weibel, H-book].</p> http://mathoverflow.net/questions/66473/group-cohomology-of-an-abelian-group-with-nontrivial-action/75386#75386 Answer by Xiao-Gang Wen for Group cohomology of an abelian group with nontrivial action Xiao-Gang Wen 2011-09-14T10:57:15Z 2011-09-14T10:57:15Z <p>One can do the calculation using Kunneth theorem and the cohomology of cyclic group.</p> <p>See eqn J18 and appendix J.6 and J.7 in a physics paper <a href="http://arxiv.org/pdf/1106.4772v2" rel="nofollow">http://arxiv.org/pdf/1106.4772v2</a></p>