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Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients in $k$ (viewed as a trivial module), i.e.,

$$H(G):=\begin{cases}H^*(G,k)&p=2\\H^{ev}(G,k)&p>2\end{cases}$$

For what sorts of groups have we explicitly computed $H(G)$? Or if not an explicit computation, what can we say about $H(G)$ (or at least $|\operatorname{Spec}H(G)|$)? Below is what I'm aware of:

  • If $G$ is elementary abelian of rank $r$, then $H(G)$ is known by using a particularly nice projective resolution of a cyclic group of order $p$ and using the Kunneth formula.
  • Thanks to Quillen, the Krull dimension of $H(G)$ is the $p$-rank of $G$, that is the length of the largest chain of prime ideals in $H(G)$ is equal to the largest rank of an elementary abelian $p$-subgroup of $G$.
  • Jon Carlson has used Magma to compute $H(G)$ for some $2$-groups here.

If you find the question too broad, at the moment I'm particularly interested in finite groups of Lie type, and to be even more specific, $\operatorname{GL}_n(\mathbb{F}_q)$. I recall reading a paper that computed $H(\operatorname{GL}_2(\mathbb{F}_q))$, and I also recall reading another paper showing certain vanishing results of $H(\operatorname{GL}_n(\mathbb{F}_q))$ as one allows $n$ or $q$ to grow independently, but I can't seem to find either of these papers and I don't remember the authors.

What do we know about the commutative algebra $H(\operatorname{GL}_n(\mathbb{F}_q))$? Have any computations been made for certain $n$ and $q$?

Any references are appreciated, even those that address the question for general $G$, and not just $\operatorname{GL}_n(\mathbb{F}_q)$.

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    $\begingroup$ As you recognize, the question is extremely broad. But I did attempt a short survey (with ample references) in Chapter 14 of my 2006 monograph Modular Representations of Finite Groups of Lie Type (LMS Lecture Note Series 326), Cambridge Univ. Press. For example, there are lots of references to Benson and Carlson including the 1983 paper by the latter on cohomology of SL($2,p^n$). Look also at the series by Cline-Parshall-Scott (one with van der Kallen). $\endgroup$ – Jim Humphreys Sep 13 '15 at 19:27
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I'll discuss $H^*(GL_n\mathbb{F}_q; k)$ first, because that is my current area of research.

1) When $\mathbb{F}_q$ and $k$ have different characteristics (although $p$ typically still divides the order of $GL_n\mathbb{F}_q$), the answer is completely computed by Quillen, "On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field." Vaguely speaking, the situation is not too different from the classical computations over $\mathbb{R}$ and $\mathbb{C}$ (though obviously the proof is harder), and the cohomology is detected on a single subgroup "not too far" from the diagonal (a sort of splitting theorem). When $p$ divides $q-1$, it is literally detected on the diagonal.

So from now on I'll assume they have the same characteristic, so $q=p^r$. In this case, things are much more mysterious and complex, and very little is known despite decades of interest in the subject.

2) The computation for $GL_2\mathbb{F}_q$ is in Aguade, "The cohomology of the GL_2 of a finite field." However, be cautious because I seem to recall there is an error in the statements for the case $q=2^r$, $r>1$. The main ideas going in to the proof, though not the final answer, are written up clearly in the paper of Quillen mentioned above. Also, there is an elementary writeup of this computation in the intro chapter of my dissertation.

Vanishing ranges:

3a) There are two sorts of vanishing ranges, one depending on $n$ and the other depending on $q$. The first is due to Friedlander-Parshall, "On the Cohomology of Algebraic and Related Finite Groups," section 7. It says that $H^i(GL_n\mathbb{F}_q;k)$ vanishes for $0<i<r(2p-3)$. (remember we're assuming same characteristic here). This was an improvement on a similar range of Quillens', so he should get some of the credit here as well.

3b) H. Maazen, in his thesis (at Utrecht) proved that $H^i(GL_n\mathbb{F}_q;k)$ vanishes if $0<i<n/2$. (The result is significantly more general.) There is also supposed to be, unfortunately unpublished, an improvement by Quillen in the case $q\neq2$, showing vanishing in the range $0<i<n$.

3c) The papers of Bendel-Nakano-Pillen that someone mentioned above. There are three of them: "On the vanishing ranges for the cohomology of finite groups of Lie type," "On the vanishing ranges for the cohomology of finite groups of Lie type II," "Extensions for finite Chevalley groups III: rational and generic cohomology."

They discuss various vanishing ranges for finite groups of Lie type on a case-by-case basis, always with the assumption that $p$ is large compared to the Coxeter number of the group (so, for example, $p=2$ is out...)

Nonvanishing:

4) I have a paper, "Nonvanishing cohomology classes on finite groups of Lie type with Coxeter number at most p" which shows that $GL_n\mathbb{F}_{p^r}$ has nonzero cohomology in degree $r(2p-3)$, provided that $n\leq p$. This was inspired by the results of B-N-P above. The result is actually valid for all (untwisted) finite groups of Lie type (with $n$ replaced by the Coxeter number). But it is also reprinted with some more details in my thesis, so perhaps that is easier to read. The thesis also has more precise information in the case $r=1$. Compare this with item (3a)!

5) Milgram-Priddy, "Invariant theory and $H^*(GL_n\mathbb{F}_p;\mathbb{F}_p)$," give a construction of a family of nonzero, in fact algebraically independent classes, in very high degree. That is, a system of parameters for the cohomology ring. They also ask a question regarding the existence of a particular class in degree $n/2$ ($n$ even), which would be the lowest possible thanks to (3b) --- which I have been trying to answer.


Leaving $GL_n$ now:

There are plenty of small groups and easy families, like finite abelian groups and dihedral groups, etc. whose cohomology you can easily compute. In terms of important families, one very important example where the cohomology is known is the symmetric groups $S_n$, due to Nakaoka.

Another is the "extraspecial 2-groups," that is, central extensions of an elementary abelian $2$-group by the cyclic group of order 2. Quillen, "The mod 2 cohomology rings of extra-special 2-groups and the spinor groups." For extraspecial $p$-groups, I believe the spectrum of cohomology is known. I can't remember whose work that is; if you want I'll try to dig up the reference.

There are plenty more examples, maybe I'll add more later when I have more time.

Also, in terms of qualitative results, you might be interested in the (fairly recent) work of Peter Symonds on the depth, regularity, and local cohomology of the cohomology rings of finite groups. In particular, this has implications for what degrees the generators and relations can live in. It's written up in the book by Totaro, "Group Cohomology and Algebraic Cycles."

In terms of computer computations, beside Jon Carlson's data, there is also David Green, http://users.minet.uni-jena.de/cohomology/ His results are much more extensive, but have a slightly different focus.

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    $\begingroup$ This is a fabulous answer, and just about exactly what I was looking for. I'll certainly be revisiting this to try to unpack everything you've written, along with all of the references. Thank you $\endgroup$ – Jared Sep 14 '15 at 1:35
  • $\begingroup$ You're welcome! By the way, I just posted the thesis to the Arxiv (arxiv.org/abs/1509.03910) to make it less of a pain to track down. $\endgroup$ – David Sprehn Sep 15 '15 at 21:04
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In terms of general results about the spectrum of the cohomology ring, some good places to start would be the papers Support varieties for modules over Chevalley groups and classical Lie algebras by Jon Carlson, Zongzhu Lin, and Daniel Nakano (Trans. Amer. Math. Soc. 360, no. 4 (2008), 1879-1906), and Restrictions to $G(\mathbb{F}_p)$ and $G_{(r)}$ of rational $G$-modules by Eric Friedlander (Compos. Math. 147 (2011), no. 6, 1955-1978), and the references therein.

These papers are concerned with support varieties for finite Chevalley groups. These finite Chevalley groups can be obtained as the groups of $\mathbb{F}_p$-rational points of an ambient connected reductive algebraic group $G$. Most of the results in these papers are directed toward describing support varieties for $G(\mathbb{F}_p)$ in terms of the somewhat better understood support varieties for the scheme-theoretic Frobenius kernels of the ambient algebraic group $G$.

In terms of vanishing results, you may find the reference you are looking for in the introduction to the paper On the vanishing ranges for the cohomology of finite groups of Lie type by Christopher Bendel, Daniel Nakano, and Cornelius Pillen (Int. Math. Res. Not. 2012 (2012), no. 12, 2817-2866).

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There is a comprehensive set of calculations of the homology of classical groups over finite fields in Z. Fiedorowicz and S. Priddy. Homology of classical groups over finite fields and their associated infinite loop spaces. Springer Lecture Notes in Mathematics Volume 674, 1978. I'm a little surprised this has not yet been mentioned, since it seems to give more complete answers than some that are cited above. It is written with a topological slant, following up Quillen's original work that used such homological calculations to determine the K-theory of finite fields, by comparison of Frobenius in algebra with Adams operations in topology.

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