Ramification theory and perfectoid spaces

Shin Hattori

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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/(πK1 m) ≃ O{script} K2/(πK2 m) and O{script}L1/ (πK1 m) ≃ O{script} L2/(πK2 m) 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
Volume150
Issue number5
DOIs
Publication statusPublished - Jan 1 2014

Fingerprint

Ramification
Integrality
Monodromy
P-adic
Conductor
Imperfect
Valuation
Compatibility
Isomorphism
If and only if
Norm
Ring
Generalise
Integer
Theorem

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Ramification theory and perfectoid spaces. / Hattori, Shin.

In: Compositio Mathematica, Vol. 150, No. 5, 01.01.2014, p. 798-834.

Research output: Contribution to journalArticle

Hattori, Shin. / Ramification theory and perfectoid spaces. In: Compositio Mathematica. 2014 ; Vol. 150, No. 5. pp. 798-834.
@article{6c16f9ffcbfe4b0db970ebb3842dfe83,
title = "Ramification theory and perfectoid spaces",
abstract = "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/(πK1 m) ≃ O{script} K2/(πK2 m) and O{script}L1/ (πK1 m) ≃ O{script} L2/(πK2 m) 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",
author = "Shin Hattori",
year = "2014",
month = "1",
day = "1",
doi = "10.1112/S0010437X1300763X",
language = "English",
volume = "150",
pages = "798--834",
journal = "Compositio Mathematica",
issn = "0010-437X",
publisher = "Cambridge University Press",
number = "5",

}

TY - JOUR

T1 - Ramification theory and perfectoid spaces

AU - Hattori, Shin

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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/(πK1 m) ≃ O{script} K2/(πK2 m) and O{script}L1/ (πK1 m) ≃ O{script} L2/(πK2 m) 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

AB - 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/(πK1 m) ≃ O{script} K2/(πK2 m) and O{script}L1/ (πK1 m) ≃ O{script} L2/(πK2 m) 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

UR - http://www.scopus.com/inward/record.url?scp=84901640098&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901640098&partnerID=8YFLogxK

U2 - 10.1112/S0010437X1300763X

DO - 10.1112/S0010437X1300763X

M3 - Article

AN - SCOPUS:84901640098

VL - 150

SP - 798

EP - 834

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 5

ER -