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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 $\Gamma$: $H^i(\Gamma,V) = H^i(B\Gamma,\tilde V)$.

Yet I know of only one definition of group cohomology with compact support. One defines $H^i_c(\Gamma,V)$ as $H^i_c(B\Gamma,\tilde V)$.

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?

Of course, any reference would be welcome.

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?

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I think this is in Brown's "Cohomology of groups" but I don't have a copy at hand to check –  Yemon Choi Mar 21 '11 at 3:04
    
(by which I mean: a general definition of group cohomology with compact support. If memory serves correctly, you take coefficients in the integral group ring, with regular left action and trivial right action) –  Yemon Choi Mar 21 '11 at 3:06
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In your main example(s), you seem to have a preferred model for the classifying space. In general, $B\Gamma$ is only well defined up to homotopy, but (unless I'm missing something) your definition is not invariant under $B\Gamma\mapsto B\Gamma\times \mathbb{R}^n$. –  Donu Arapura Mar 25 '11 at 12:14
    
Donu, you're right! The $H^n_c$ of $\mathbb{R}^n$ has dimension $1$. Thanks, you made me realize I don't know even know one definition of $H^i_c(\Gamma,V)$. Yet I have seen people using it in various context without further notice. In each cases I remember of, there was a "natural" $B\Gamma$, but it was not said that $H^i_c$ was defines using this particular $B\Gamma$. Perhaps it was clear for the expert. Well, I have some reading to do: thanks to all for the references. I'll try to edit my question (which right now is meaningless) when I understand. –  Joël Mar 26 '11 at 13:22
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2 Answers 2

Please see page 352 (in the Appendix) of Hida's book "Elementary theory of L-functions and Eisenstein series".

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I have the impression that Joël is looking for a definition without resorting the geometric picture, though I could be wrong. Either way, proposition 2 on page 352 does indicate that $H^1_c=H^1_P$. –  Rob Harron Mar 21 '11 at 4:48
    
Monodromy, I have no access today to a library. Thanks for the reference, I'll look at it when I can (tomorrow I hope). What part of my question does it answer? Rob H., you're right that I'd like some purely group-theoretical description if that exists. But I do not think that $H^1_c = H^1_P$ in the context of a congruence subgroup $\Gamma$ of $\Sl_2$. Modularly speaking, the $H^1_p$ parameterizes two copies of the cuspidal modular forms, while $H^1_c$ also parameterizes Eisenstein series. When $\Gamma$ acts freely, and with trivial coefficients, the dim. of $H^1_c$ (and of $H^1$) is $2g+c$ –  Joël Mar 21 '11 at 12:43
    
while the dimension of $H^1_p$ is $2g$, (where $g$ is the genus of the modular curve $X(\Gamma)$ and $c$ is ne number of cusps, that is the number of points of $X(\Gamma)-Y(\Gamma)$.) Note that we are just talking about computing the cohomology and cohomology with compact support of the modular curve $Y(\Gamma)$, a smooth proj. curve over $\bf C$ minus $c$ points, since it is a classifying space for $\Gamma$. –  Joël Mar 21 '11 at 12:47
    
Yeah, so sorry! That was a typo, the proposition says $H^2_c=H^2_P$! –  Rob Harron Mar 21 '11 at 14:41
    
(and that exclamation mark is just an exclamation mark, not a lower shriek or anything else mathematical...) –  Rob Harron Mar 21 '11 at 14:43
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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...

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