Potential semi-stability of etale cohomology of etale covers. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:28:49Z http://mathoverflow.net/feeds/question/22533 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22533/potential-semi-stability-of-etale-cohomology-of-etale-covers Potential semi-stability of etale cohomology of etale covers. Lavender Honey 2010-04-25T20:25:40Z 2010-05-02T22:19:23Z <p>Let $\mathbf{Q}_p$ denote the field of $p$-adic numbers.</p> <p>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:</p> <p>A. There exists a finite extension $L/K$ such that $X/L$ has a semi-stable model over $O_L$.</p> <p>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 $\mathrm{Gal}(\overline{K}/L)$.</p> <p>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 $\mathrm{Gal}(\overline{K}/L)$.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/22533/potential-semi-stability-of-etale-cohomology-of-etale-covers/23289#23289 Answer by Lavender Honey for Potential semi-stability of etale cohomology of etale covers. Lavender Honey 2010-05-02T22:19:03Z 2010-05-02T22:19:03Z <p>Answered to my satisfaction. Well, except for the bit about typesetting $\mathbf{Q}_p$, that still confuses me.</p>