Timeline for Mod-p cohomology of $GL(n,p^d)$
Current License: CC BY-SA 3.0
8 events
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| Feb 25, 2013 at 11:33 | comment | added | Ralph | @Demin: Concerning your 1st comment: Your spectral sequence argument gives the isomorphism $H^\ast(\text{GL}(n,p^d),\mathbb{F}_p) \cong H^\ast(\text{SL}(n,p^d),\mathbb{F}_p)^{\mathbb{F}^\times_{p^d}}$. But why does $i_0(\text{GL}(n,p^d))=i_0(\text{SL}(n,p^d))$ follow ? But I agree with your observation that it's curious where $p\ge n+2$ in 6.15 results from. | |
| Feb 25, 2013 at 0:50 | vote | accept | Demin Hu | ||
| Feb 25, 2013 at 0:49 | comment | added | Demin Hu | 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. | |
| Feb 24, 2013 at 20:46 | comment | added | Jim Humphreys |
@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.
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| Feb 24, 2013 at 20:02 | comment | added | Christopher Drupieski | @Demin perhaps this is a typo in their paper. | |
| Feb 24, 2013 at 13:18 | comment | added | Jim Humphreys | @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 | |
| Feb 24, 2013 at 13:08 | comment | added | Demin Hu | 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 ? | |
| Feb 23, 2013 at 23:35 | history | answered | Christopher Drupieski | CC BY-SA 3.0 |