On the ramification of \'etale cohomology groups

TL;DR

This paper investigates the ramification behavior of étale cohomology groups associated with certain sheaves over schemes with semi-stable reduction, establishing bounds on ramification in terms of the sheaves' properties.

Contribution

It proves that under specific conditions, the ramification bounds of sheaves extend to the étale cohomology groups, linking local ramification to global cohomological properties.

Findings

Ramification of cohomology groups is bounded by that of the sheaves.

The result applies to schemes with semi-stable reduction in positive characteristic.

Provides a precise relation between sheaf ramification and cohomological ramification.

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.