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Upper Ramification Groups for Arbitrary Valuation Rings

Kazuya Kato, Vaidehee Thatte

Abstract

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring $A$ with field of fractions $K$ and for a finite Galois extension $L$ of $K$, the integral closure $B$ of $A$ in $L$ is a filtered union of subrings of $B$ which are of complete intersection over $A$. By this, we can obtain a ramification theory of Henselian valuation rings as the limit of the ramification theory of Saito. Our theory generalizes the ramification theory of complete discrete valuation rings of Abbes-Saito. We study "defect extensions" which are not treated in these previous works.

Upper Ramification Groups for Arbitrary Valuation Rings

Abstract

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring with field of fractions and for a finite Galois extension of , the integral closure of in is a filtered union of subrings of which are of complete intersection over . By this, we can obtain a ramification theory of Henselian valuation rings as the limit of the ramification theory of Saito. Our theory generalizes the ramification theory of complete discrete valuation rings of Abbes-Saito. We study "defect extensions" which are not treated in these previous works.

Paper Structure

This paper contains 82 sections, 53 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. A brief summary.
  3. Compatibility for DVRs.
  4. Relation with the work Sa.
  5. Relation to the ramification theory in V1V2V3.
  6. Indexing by ideals of $\bar{A}$.
  7. Methodology.
  8. Concerning $G^I_{{\rm nlog}}$.
  9. Concerning $G^I_{\log}$.
  10. Outline.
  11. Notation.
  12. Log smooth extensions of Henselian valuation rings
  13. Preliminaries on commutative monoids.
  14. Valuative monoids.
  15. Associated valuation rings.
  16. ...and 67 more sections

Key Result

Theorem 1.1

Assume that the residue field of $A$ is of characteristic $p>0$. Let $L$ be a cyclic extension of $K$ of degree $p$ and let ${\mathscr{H}}\subset A$ be the associated ideal. Then for a nonzero proper ideal $I$ of $\bar{A}$, the image of $G_{\log}^I$ in $\mathop{\mathrm{Gal}}\nolimits(L/K)$ is $\math

Theorems & Definitions (77)

  • Theorem 1.1
  • Lemma 2.1.2
  • Lemma 2.1.3
  • Proposition 2.1.5
  • Definition 2.2.1
  • Definition 2.2.2
  • Definition 2.2.3
  • Lemma 2.3.1
  • Lemma 2.4.1
  • Lemma 2.4.2
  • ...and 67 more