Ramification theory and perfectoid spaces

Shin Hattori

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1 Citation (Scopus)


Let K1 and K2 be complete discrete valuation fields of residue characteristic p > 0. Let πK1 and πK2 be their uniformizers. Let L1/K1 and L2/K 2 be finite extensions with compatible isomorphisms of rings O{script}K1/(πK1m) ≃ O{script} K2/(πK2m) and O{script}L1/ (πK1m) ≃ O{script} L2/(πK2m) for some positive integer m which is no more than the absolute ramification indices of K1 and K 2. 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 p-adic representations with finite local monodromy. copyright

Original languageEnglish
Pages (from-to)798-834
Number of pages37
JournalCompositio Mathematica
Issue number5
Publication statusPublished - May 2014

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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