Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In the classic paper On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field, Quillen proved (Theorem 6):

$H^i(GL(n,p^d),\mathbb{F}_p)=0$ for $0 < i < d(p-1)$ and all $n$.

On the other hand, the cohomology of a finite group doesn't completely vanish (this is nicely discussed on MO: Non-vanishing of group cohomology in sufficiently high degree). So there is a minimal integer $i_0=i_0(n,d) \ge d(p-1)$ satisfying $$H^{i_0}(GL(n,p^d),\mathbb{F}_p) \neq 0$$ Is Quillen's lower bound $d(p-1)$ sharp ? (I couldn't find any information about sharpness in Quillen's paper). If not, is the precise value of $i_0$ known ?

share|improve this question
add comment

1 Answer 1

up vote 6 down vote accepted

There are some results known in this direction; see 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), 2817-2866). In particular, they show that if $p \geq n+2$, then the first degree in which nonzero cohomology appears is $d(2p-3)$.

Bendel, Nakano, and Pillen also have a sequel paper posted on the arXiv that treats a number of cases for other Lie types that aren't already handled in the reference I gave above.

share|improve this answer
    
Thanks for your answer. I don't have access to the journal, but I there is a paper with same title and result on the arxiv: arxiv.org/pdf/0906.0026.pdf. There the role of $n$ is unclear to me: Isn't $SL_n$ a simple (i.e. finite center) simply connected algebraic group scheme of type $A_{n-1}$ ? Then, according to 6.2 Corollary, $i_0(SL(n,p^d))= d(2p-3)$, provided $p\ge (n-1)+2 = n+1$ and from the Hochschild-Serre spectral sequence, it would follow $i_0(GL(n,p^d))=d(2p-3)$ if $p \ge n+1$ (instead of $p\ge n+2$ in the paper). Am I missing something ? –  Demin Hu Feb 24 '13 at 13:08
    
@Christopher: Getting exact bounds is definitely tricky, as the papers you cite demonstrate (but with a focus on the almost simple groups of Lie type, where there is more unified theory than in the reductive case generally). Their follow-up paper is also posted on arXiv and is to appear in a volume of the AMS Proc. Symp. Pure Math.: front.math.ucdavis.edu/1112.2367 –  Jim Humphreys Feb 24 '13 at 13:18
    
@Demin perhaps this is a typo in their paper. –  Christopher Drupieski Feb 24 '13 at 20:02
1  
@Demin: In Corollary 6.2 (both arXiv and IMRN) they deal with Lie type $A_n$ and the bounds $n \geq 2, p \geq n+2$; this applies to $G=\mathrm{SL}_{n+1}$. It's confusing, but probably consistent throughout. I haven't checked the fine points, but did note that their misprint "indentifies" at the start of Section 6 got changed to "identifies" in IMRN. There may be other minor changes like that. @Christopher: Your use of $n$ is misleading, since the paper uses this for the Lie rank of a simple, simply connected group. –  Jim Humphreys Feb 24 '13 at 20:46
    
Jim, thanks for your help. @Christopher: Maybe it's a typo, but it's peculiar that $p \ge n+2$ is required in 6.15 in the proposition and in the theorem as well. But of course, the assumption $p \ge n+2$ doesn't contradict my guess that $p\ge n+1$ is sufficient. –  Demin Hu Feb 25 '13 at 0:49
show 1 more comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.