Ramification theory and perfectoid spaces

TL;DR

This paper establishes a correspondence between ramification bounds of finite extensions over complete discrete valuation fields, generalizing Deligne's theorem to imperfect residue fields and connecting with higher field theories.

Contribution

It proves an equivalence of ramification bounds for extensions over different fields and links various ramification theories, extending existing results to imperfect residue fields.

Findings

Ramification bounds are equivalent under certain isomorphisms.

Categories of extensions with bounded ramification are equivalent.

Compatibility of higher fields of norms with ramification theory is demonstrated.

Abstract

Let K and F be complete discrete valuation fields of residue characteristic p>0. Let m be a positive integer no more than their absolute ramification indices. Let s and t be their uniformizers. Let L/K and E/F be finite extensions such that the modulo s^m of the extension O_L/O_K and modulo t^m of O_E/O_F are isomorphic. Let j=<m be a positive rational number. In this paper, we prove that the ramification of L/K is bounded by j if and only if the ramification of E/F is bounded by j. As an application, we prove that the categories of finite separable extensions of K and F 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.