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.



Potential semi-stability of etale cohomology of etale covers.
Answered to my satisfaction. Well, except for the bit about typesetting $\mathbf{Q}\_p$, that still confuses me.


