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Kummer-Artin-Schreier-Witt Theory

Huy Dang, Khai-Hoan Nguyen-Dang

Abstract

We study the problem of lifting the Artin--Schreier--Witt isogeny from characteristic $p>0$ to characteristic $0$, which is central to the lifting problem for Galois covers of algebraic schemes in positive characteristic. We introduce a new technique that associates a Kummer class, representing a tamely ramified cyclic extension, to a Witt vector via Matsuda's Kummer--Artin--Schreier--Witt theory. This viewpoint leads to an explicit construction of a lift of the isogeny over a concrete base ring. Our results lay the groundwork for further applications, including the study of inseparable extensions and Kato's refined Swan conductor.

Kummer-Artin-Schreier-Witt Theory

Abstract

We study the problem of lifting the Artin--Schreier--Witt isogeny from characteristic to characteristic , which is central to the lifting problem for Galois covers of algebraic schemes in positive characteristic. We introduce a new technique that associates a Kummer class, representing a tamely ramified cyclic extension, to a Witt vector via Matsuda's Kummer--Artin--Schreier--Witt theory. This viewpoint leads to an explicit construction of a lift of the isogeny over a concrete base ring. Our results lay the groundwork for further applications, including the study of inseparable extensions and Kato's refined Swan conductor.

Paper Structure

This paper contains 26 sections, 51 theorems, 306 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Notations
  4. Background
  5. Artin--Schreier--Witt theory
  6. Kummer Theory
  7. Kummer--Artin--Schreier--Witt theory
  8. Deformed Artin--Hasse exponential function
  9. Matsuda's Kummer--Artin--Schreier--Witt theory
  10. Constructing the Kummer extension
  11. The Witt Vector Associated to a Kummer Extension
  12. Constructing the Kummer Extension
  13. Defining the Étale Cover
  14. Computing the Ws
  15. The case s=1
  16. ...and 11 more sections

Key Result

Theorem 1.1

For each positive integer $s$, there exists a flat group scheme $\mathcal{W}_s$ over $R := \mathbb{Z}_{(p)}[\zeta_{p^s}]$ that fits into an exact sequence of group schemes over $\overline{R}$. The generic fiber of eqnKASW is isomorphic to the Kummer exact sequence over $\overline{K}$, where $K = \mathop{\mathrm{Frac}}\nolimits(R)$. Its special fiber is isomorphic to the Artin--Schreier--Witt seq

Figures (2)

  • Figure 1: Construction of $\mathcal{W}_1$
  • Figure 2: Construction of $\mathcal{W}_s$

Theorems & Definitions (104)

  • Theorem 1.1: cf. SS:kasw2
  • Theorem 1.2: cf. SS:kasw2
  • Remark 1.3
  • Proposition 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 94 more