Ramification theory and perfectoid spaces

摘要

Let K_1 and K_2 be complete discrete valuation fields of residue characteristic p > 0. Let πK_1 and πK_2 be their uniformizers. Let L_1/K_1 and L_2/K_2 be finite extensions with compatible isomorphisms of rings OK_1/(π_K_1~m) = OK_2 /(π_K_2~m) and O_L_1/(π_K_1~m) = OL_2/(π_K2~m) for some positive integer m which is no more than the absolute ramification indices of K_1 and K_2. Let. j ≤ m be a positive rational number. In this paper, we prove that the ramification of L_1/K_1 is bounded by j if and only if the ramification of L_2/K_2 is bounded by j. As an application, we prove that the categories of finite separable extensions of K_1 and K_2 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 p-adic representations with finite local monodromy.