Ramification theory and perfectoid spaces

Abstract

Abstract. Let K1 and K2 be complete discrete valuation fields of residue characteristic p > 0. Let πK 1 and πK 2 be their uniformizers. Let L1/K1 and L2/K2 be finite extensions with compatible isomorphisms of rings) for some positive integer m which is no more than the absolute ramification indices of K1 and K2. Let j ≤ m be a positive rational number. In this paper, we prove that the ramification of L1/K1 is bounded by j if and only if the ramification of L2/K2 is bounded by j. As an application, we prove that the categories of finite separable extensions of K1 and K2 whose ramifications are bounded by j are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl's theory of higher fields of norms with the ramification theory of Abbes-Saito, and the integrality of small Artin and Swan conductors of abelian extensions of mixed characteristic.

On differential modules associated to de Rham representations in the imperfect residue field case

“…Note that Shin Hattori ([Hat14]) reproved the above ramification compatibility of Scholl's isomorphism τ by using Peter Scholze's perfectoid spaces ( [Scholze12]), which are a geometric interpretation of Fontaine-Wintenberger theorem. We briefly explain Hattori's proof.…”