# Étale cohomology of linear groups

This is in a sense a follow up question to the answer here Analytic tools in algebraic geometry

Let $k$ be an algebraically closed field of positive characteristic and let $R$ be the result of applying the Witt vector procedure to $k$ (see e.g. http://eom.springer.de/W/w098100.htm) This is a complete local ring with residue field $k$ and fraction field of characteristic 0. Let $L$ be an algebraic closure of the fraction field.

Using the procedure explained in SGA 4 1/2, pp. 54-56 one can construct, given a smooth scheme over $R$, a comparison morphism $H^*(X_1,A)\to H^*(X_0,A)$ where $X_0$ and $X_1$ are the fibers of $X$ over $Spec(k)$, respectively, $Spec(L)$, and $A$ is a finite abelian group of order prime to $char(k)$. The construction is as follows: we have the maps $X_1\to X\gets X_0$; due to the smoothness of $X$, the map $H^{\ast}(X,A)\to H^{\ast}(X_1,A)$ is an isomorphism, which we invert and then take the restriction to $X_0$.

These comparison maps are in general not isomorphisms, but they are if we take $X=GL_n(R)$ or $SL_n(R)$. I think I know how to prove this, but I also think this should be standard material. So I'd like to ask if somebody knows a reference for that.

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Dear algori: I presume in the final paragraph you really meant to refer to the $R$-schemes ${\rm{GL}}_n$ and ${\rm{SL}}_n$ (which are quite different beasts from the groups ${\rm{GL}}_n(R)$ and ${\rm{SL}}_n(R)$). Anyway, to deal with a general (necessarily split) reductive group scheme (with connected fibers) over a strictly henselian dvr, probably a good place to look is the paper "A topological property of quasi-reductive group schemes" by Fakhruddin and Srinivas in volume 2, issue 2, of Algebra and Number Theory Journal (2008). Just cross out "quasi-" everywhere. :) – BCnrd Aug 8 '10 at 15:27
Dear BCnrd -- thanks. The paper you mention doesn't seem to be on the arxiv, so I guess I'll have to check it in the library tomorrow. Re $GL$ and $SL$ -- yes, I did mean schemes over $R$, but what is the standard notation for them? – algori Aug 8 '10 at 16:12
There is a proof by Deligne in SGA 4 1/2 passing through G/B which is smooth and proper and then G is an iterated $G_a$- and $G_m$-torsor over G/B. – Torsten Ekedahl Aug 8 '10 at 16:57
Thanks, Torsten! Did you mean p. 230, end of Applications to trigonometric sums? (Actually I was thinking of another Leray spectral sequence, namely that of the map that takes a matrix to its last column; this makes sense for GL and SL and at least some other reductive groups as well.) – algori Aug 8 '10 at 18:41
@algori: The standard notation is either $(GL_n)_R$ or $GL_{n,R}$. – Kevin Ventullo Aug 8 '10 at 19:03