# Potential semi-stability of etale cohomology of etale covers.

Let $\mathbf{Q}_p$ denote the field of $p$-adic numbers.

Suppose that $K/\mathbf{Q}_p$ is a finite extension, and let $O_K$ denote the ring of integers of $K$. Suppose that $X$ is proper over $O_K$, with smooth generic fibre. Consider the following three statements:

A. There exists a finite extension $L/K$ such that $X/L$ has a semi-stable model over $O_L$.

B. There exists a finite extension $L/K$ such that, for every proper etale map $Y \rightarrow X$, the etale cohomology $H^i(Y/\overline{K},\mathbf{Q}_p)$ is semi-stable when considered as a representation of $\mathrm{Gal}(\overline{K}/L)$.

C. There exists a finite extension $L/K$ such that $H^i(X/\overline{K},\mathbf{Q}_p)$ is semi-stable when considered as a representation of $\mathrm{Gal}(\overline{K}/L)$.

In light of Tsuji's proof of the semi-stable conjecture of Fontaine and Jannsen, we know that $A \Rightarrow B \Rightarrow C$ (the point being that a semi-stable model for $X$ pulls back to one for $Y$). On the other hand, Tsuji also proved $C$ without proving $A$ by using de Jong's theory of alterations.

My question: Can one also use Tsuji's arguments to prove $B$? One could imagine some argument with alterations being compatible with taking etale covers, but this is not my field so I would rather ask an expert.

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By dJ, get alteration $\phi:X' \rightarrow X$, where $X'_K$ is smooth and there is sst proper $O_L$-model for $X'_L$, with $L/K$ finite. WLOG $L = K$ (via base change).WLOG $X$ and $X'$ conn'd. By properness, smoothness, and conn'dness, there is dense open in $X_K$ over which $\phi_K$ is finite flat of constant degree $d$. Key pt: there is trace relative to $\phi_K$ that is section up to $d$-mult. for $\phi_K$-pullback in cohom. (Need to give ref., as $\phi_K$ can have bad fibers.) Sst inherited by subreps, so wlog $X = X'$. All $Y$ in part B now have sst model (proper + et = finite et). QED –  BCnrd Apr 25 '10 at 22:33
FC, I should have said at the start: the answer to your question (and your idea for how it should go) is a definite "yes". The "wlog X = X'" step in what I said rests on your observation that finite etale cover is compatible with alteration, as well as with property of being sst (in sense defined in dJ's paper, for example, which is sufficient for Tsuji's arguments in that case). –  BCnrd Apr 25 '10 at 22:38
FC, the trace ref: for sep'td f.t. f between qcqs schemes, Exp. XVIII, section 3, SGA4 provides rt adjnt Rf^! to Rf_! (= Rf_ {\ast}) for proper f); for smooth f must be adjnt of Verdier: pullback + Tate twist. Formation local on base, respects composition due to same for Rf_ !. For alterations application with smooth X_K and X'_K of pure dim d (connd!), Q_{\ell}(d)[−d] is "upper−shriek pullback" of Q_{\ell} from Spec(K) so \phi_K^!(Q_{\ell}) = Q_{\ell}, so get trace "section". [If anyone can make LaTeX work using {\rm{R}} and \mathbf{Q} and {\rm{Spec}}, please fix this output!] –  BCnrd Apr 26 '10 at 2:29
FC I think I fixed the typesetting. As far as I can tell the problem is that the automatic preview can't render those, but "one-shot preview" and the actual webpage can handle them fine. BCnrd I couldn't fix yours because one of the many ways that comments don't work as well as answers is that they can't be edited by third parties. –  Noah Snyder Apr 26 '10 at 6:01

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