### Abstract

Let K_{1} and K_{2} be complete discrete valuation fields of residue characteristic p > 0. Let π_{K1} and π_{K2} be their uniformizers. Let L_{1}/K_{1} and L_{2}/K _{2} be finite extensions with compatible isomorphisms of rings O{script}_{K1}/(π_{K1} ^{m}) ≃ O{script} _{K2}/(π_{K2} ^{m}) and O{script}L_{1}/ (π_{K1} ^{m}) ≃ O{script} L_{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. copyright

Original language | English |
---|---|

Pages (from-to) | 798-834 |

Number of pages | 37 |

Journal | Compositio Mathematica |

Volume | 150 |

Issue number | 5 |

DOIs | |

Publication status | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

*150*(5), 798-834. https://doi.org/10.1112/S0010437X1300763X

**Ramification theory and perfectoid spaces.** / Hattori, Shin.

Research output: Contribution to journal › Article

*Compositio Mathematica*, vol. 150, no. 5, pp. 798-834. https://doi.org/10.1112/S0010437X1300763X

}

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 -