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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 degreeNon-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 ?

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 ?

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 ?

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Demin Hu
Demin Hu
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Mod-p cohomology of $GL(n,p^d)$

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 ?