Group cohomology with compact support - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:49:48Z http://mathoverflow.net/feeds/question/59017 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59017/group-cohomology-with-compact-support Group cohomology with compact support Joël 2011-03-21T00:15:57Z 2011-03-25T08:52:26Z <p>Let $\Gamma$ be a discrete group, $V$ a left $\Gamma$-module. One can define the groups $H^i(\Gamma,V)$ ($i=0,1,2,\dots$) in many ways, and then prove their equivalence: as derived functors of the functor of $\Gamma$-invariants; as the homology of an explicit complex of cochains; or as the usual (Steenrod's) cohomology of the local system $\tilde V$ attached to $V$ on the classifying space $B\Gamma$ of $\Gamma$: $H^i(\Gamma,V) = H^i(B\Gamma,\tilde V)$.</p> <p>Yet I know of only one definition of <em>group cohomology with compact support</em>. One defines $H^i_c(\Gamma,V)$ as $H^i_c(B\Gamma,\tilde V)$. </p> <blockquote> <p>Are there other ways to define group cohomology with compact support, with no reference to the classifying space? is there in particular a definiton with an explicit complex of cochains? </p> </blockquote> <p>Of course, any reference would be welcome.</p> <p>Giving an explicit description in terms of a complex of cochains might be difficult in general, but I would be happy to have one in the following well-known, overstudied example: When $\Gamma$ is a congruence subgroup of $SL_2({\bf Z})$. In this case, one finds in the litterature something close to what I am asking: an explicit description in terms of cochains of the "parabolic cohomology group" $H^1_p(\Gamma,V)$ defined as the image of the natural map $H^1_c(\Gamma,V) \rightarrow H^1(\Gamma,V)$. One shows, under mild assumptions on $\Gamma$, that $H^1_p(\Gamma,V) = Z^1_p(\Gamma,V)/B_1(\Gamma,V)$ where $Z_1(\Gamma,V)$ is the subgroup of the group of cocycles $Z^1(\Gamma,V)={u:\Gamma \rightarrow V,\ u(gg')=u(g)+gu(g')}$ that satisfy $u(p) \in (p-1) V$ for all parabolic elements $p \in \Gamma$. (cf for example Hida, inv. math. 63). Now that's only a description of the $H^1_p$, while the $H^1_c$ is (slightly) bigger. And a similar description of the $H^2_c$ would be handy as well, when computing cup-products. So is it possible to give such a description? Is it done somewhere is the litterature?</p> http://mathoverflow.net/questions/59017/group-cohomology-with-compact-support/59021#59021 Answer by monodromy for Group cohomology with compact support monodromy 2011-03-21T00:45:44Z 2011-03-21T00:45:44Z <p>Please see page 352 (in the Appendix) of Hida's book "Elementary theory of L-functions and Eisenstein series".</p> http://mathoverflow.net/questions/59017/group-cohomology-with-compact-support/59532#59532 Answer by stefan for Group cohomology with compact support stefan 2011-03-25T08:52:26Z 2011-03-25T08:52:26Z <p>I ve got the same problem... nevertheless I found a short description by Kurt Haberland - Perioden von Modulformen in einer Variablen und Gruppenkohomologie 1, section 2.1... if u ve got any problems with german... just tell me... </p>