On the ramification of \'etale cohomology groups

Abstract

Let $K$ be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let $X$ be a connected, proper scheme over $\mathcal{O}_K$, and let $U$ be the complement in $X$ of a divisor with simple normal crossings. Assume that the pair $(X,U)$ is strictly semi-stable over $\mathcal{O}_K$ of relative dimension one and $K$ is of equal characteristic. We prove that, for any smooth $\ell$-adic sheaf $\mathscr{G}$ on $U$ of rank one, at most tamely ramified on the generic fiber, if the ramification of $\mathscr{G}$ is bounded by $t+$ for the logarithmic upper ramification groups of Abbes-Saito at points of codimension one of $X$, then the ramification of the \'{e}tale cohomology groups with compact support of $\mathscr{G}$ is bounded by $t+$ in the same sense.