On the ramification of étale cohomology groups

Abstract

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 𝒪 K \mathcal{O}_{K} , and let U be the complement in X of a divisor with simple normal crossings. Assume that the pair ( X , U ) (X,U) is strictly semi-stable over 𝒪 K \mathcal{O}_{K} of relative dimension one and K is of equal characteristic. We prove that, for any smooth ℓ \ell -adic sheaf 𝒢 \mathcal{G} on U of rank one, at most tamely ramified on the generic fiber, if the ramification of 𝒢 \mathcal{G} is bounded by t + t+ for the logarithmic upper ramification groups of Abbes–Saito at points of codimension one of X, then the ramification of the étale cohomology groups with compact support of 𝒢 \mathcal{G} is bounded by t + t+ in the same sense.