Welcome to MathOverflow

Question

6

1

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?

flag	asked Mar 21 2011 at 0:15 Joël 10.4k●1●20●56

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 2011 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 2011 at 3:06

1

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 2011 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 2011 at 13:22

stefan · Answer 1 · 2011-03-25 08:52:26Z UTC

1

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...

answered Mar 25 2011 at 8:52

stefan
11●1

monodromy · Answer 2 · 2011-03-21 00:45:44Z UTC

1

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

answered Mar 21 2011 at 0:45

monodromy
538●5●9

	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 2011 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 2011 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 2011 at 12:47
	Yeah, so sorry! That was a typo, the proposition says $H^2_c=H^2_P$! – Rob Harron Mar 21 2011 at 14:41
	(and that exclamation mark is just an exclamation mark, not a lower shriek or anything else mathematical...) – Rob Harron Mar 21 2011 at 14:43

Group cohomology with compact support

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.