Large modules with non-trivial cohomology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:24:52Z http://mathoverflow.net/feeds/question/78122 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78122/large-modules-with-non-trivial-cohomology Large modules with non-trivial cohomology unknown (yahoo) 2011-10-14T11:50:19Z 2011-10-15T21:22:12Z <p>Let $p$ be a prime and $F$ algebraic closer of $F_p$. I want to know if it is possible to construct family of groups ${G_i}_{i=1}^{\infty}$ and a family of simple modules $V_i$ over $F[G_i]$ of large dimension i.e., $dim V_i \geq |G_i|^{0.01}$ such that $H^1(G_i,V_i)$ is non trivial? </p> http://mathoverflow.net/questions/78122/large-modules-with-non-trivial-cohomology/78158#78158 Answer by Chris Gerig for Large modules with non-trivial cohomology Chris Gerig 2011-10-14T18:26:05Z 2011-10-15T21:22:12Z <p>This isn't a complete answer, but I believe it is possible by the <em>Kan-Thurston Theorem</em>, which says that every path connected space has the (co)homology of a $K(G,1)$. You can build a space with prescribed nontrivial cohomology (with $V_i$-coefficients), and then that must be isomorphic to the cohomology of some $BG_i$-space (hence group $G_i)$.</p> <p>The reason I am not going to say this is a complete answer, is because due to the construction of the theorem, there might be a small problem getting the desired coefficients. The paper is entitled <em>Every Connected Space has the Homology of a $K(\pi,1)$</em>, by Kan and Thurston.</p>