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On Abelian extensions in mixed characteristic and ramification in codimension one

Daniel Katz, Prashanth Sridhar

Abstract

A theorem of Paul Roberts states that the integral closure of a regular local ring in a generically abelian extension is Cohen-Macaulay, provided the characteristic of the residue field does not divide the order of the Galois group. An example of Koh shows the conclusion is false in the modular case. After a modification to the statement concerning ramification over $p$ in codimension one, we give an extension of Roberts's theorem to the modular case for unramified regular local rings in mixed characteristic when the $p$-torsion of the Galois group is annihilated by $p$.

On Abelian extensions in mixed characteristic and ramification in codimension one

Abstract

A theorem of Paul Roberts states that the integral closure of a regular local ring in a generically abelian extension is Cohen-Macaulay, provided the characteristic of the residue field does not divide the order of the Galois group. An example of Koh shows the conclusion is false in the modular case. After a modification to the statement concerning ramification over in codimension one, we give an extension of Roberts's theorem to the modular case for unramified regular local rings in mixed characteristic when the -torsion of the Galois group is annihilated by .
Paper Structure (6 sections, 19 theorems, 20 equations)

This paper contains 6 sections, 19 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Roberts's Theorem Revisited
  4. Abelian extensions and the tamely $p$-ramified property
  5. Abelian extensions with $p$-torsion annihilated by $p$
  6. Comments on the $p$-ramified case

Key Result

Theorem 1.1

Let $S$ be a regular local ring such that $\mathrm{char}(\mathrm{Frac}(S))=0$. Assume $S$ possesses a primitive $p$-th root of unity for $p\in \mathbb{Z}$ a prime integer. Then the following are equivalent:

Theorems & Definitions (54)

  • Theorem 1.1: \ref{['thm:p_ramification_characterization']}
  • Theorem 1.2: \ref{['thm:roberts_extension']}
  • Definition 2.2: AB
  • Remark 2.3: AB
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • ...and 44 more