In the classic paper [On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field](http://www.jstor.org/discover/10.2307/1970825?uid=3737864&uid=2129&uid=2134&uid=2&uid=70&uid=4&sid=21101716697401), 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](https://mathoverflow.net/questions/64688/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 ? 



Mod-p cohomology of $GL(n,p^d)$ 
There are some results known in this direction; see the paper [*On the vanishing ranges for the cohomology of finite groups of Lie type*](http://dx.doi.org/10.1093/imrn/rnr130), 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.


