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.