Kummer-Artin-Schreier-Witt Theory

1. Problem Statement

The paper addresses the lifting problem for Galois covers of algebraic schemes in positive characteristic $p > 0$. Specifically, it investigates whether a given $G$-cover of a curve in characteristic $p$ can be lifted to characteristic 0.

• Context: While the general lifting problem has negative answers, significant progress has been made for abelian groups. A classical reduction shows that studying abelian covers reduces to studying cyclic isogenies of algebraic groups.
• Specific Goal: The authors focus on lifting the Artin–Schreier–Witt isogeny (which governs cyclic extensions of degree $p^s$ in characteristic $p$) from characteristic $p$ to characteristic 0. This involves constructing a flat group scheme over a mixed-characteristic base ring that interpolates between the Kummer sequence (characteristic 0) and the Artin–Schreier–Witt sequence (characteristic $p$).

2. Methodology

The authors introduce a novel technique that bridges Kummer theory (tamely ramified extensions) and Artin–Schreier–Witt theory (wildly ramified extensions) using Matsuda's Kummer–Artin–Schreier–Witt theory.

• Matsuda's Framework: They utilize the theory developed by Matsuda, which associates a Witt vector $X \in W_s(K)$ with a Kummer class in a mixed-characteristic field $K$ (characteristic 0, residue field characteristic $p$). This association is mediated by a deformation of the Artin–Hasse exponential function, denoted $E_{s,p}$.
• Algebraization: The core strategy involves "algebraizing" the formal power series arising from Matsuda's theory.
  1. They start with a Kummer extension defined by a specific equation involving polynomials $G_i$.
  2. They associate this extension with a Witt vector $X'$ via the morphism $\Psi_s: W_s(R_{int}) \to (R_{int})^\times$, where $R_{int}$ is a Henselian discrete valuation ring.
  3. They construct explicit polynomials $E_i$ and $G_i$ by truncating the infinite power series $E_{s,p}$ and $G_{s,p}$ at a specific valuation threshold ($1/(p-1)$). This ensures the resulting objects are algebraic (polynomials) rather than just formal series.
• Descent and Group Structure:
  • They construct the schemes $W_s$ and $V_s$ as spectra of rings generated by these truncated polynomials.
  • They prove that these schemes carry natural group scheme structures by transporting the Witt vector addition law from the formal completion to the algebraic level using Hopf algebras and descent arguments.
  • They establish that the morphism $\psi_s: W_s \to V_s$ is a homomorphism of group schemes.

3. Key Contributions and Results

A. Explicit Construction of the Lift (Theorem 1.1)

The authors provide an explicit construction of a flat group scheme $W_s$ over the base ring $R = \mathbb{Z}_{(p)}[\zeta_{p^s}]$ that fits into an exact sequence:
$$0 \to \mathbb{Z}/p^s \to W_s \to V_s \to 0$$

• Generic Fiber: Isomorphic to the Kummer exact sequence $0 \to \mu_{p^s} \to \mathbb{G}_m \xrightarrow{(\cdot)^{p^s}} \mathbb{G}_m \to 0$.
• Special Fiber: Isomorphic to the Artin–Schreier–Witt sequence $0 \to \mathbb{Z}/p^s \to W_s \xrightarrow{\wp} W_s \to 0$.
• Explicit Equations: Unlike previous unpublished work by Sekiguchi and Suwa, this paper provides explicit polynomial formulas for the coordinate rings of $W_s$ and $V_s$. These involve polynomials $E_i$ and $G_i$ derived from truncating the deformed Artin–Hasse exponential series.

B. Universality of the Construction (Theorem 1.2)

The paper proves that for any flat local $R$-algebra $B$, any unramified $\mathbb{Z}/p^s$-cover $\text{Spec } C \to \text{Spec } B$ is induced by a Cartesian diagram involving the constructed group schemes $W_s$ and $V_s$. This establishes $W_s$ as a universal object for lifting such covers.

C. Refined Swan Conductor (Proposition 1.4)

The authors apply their framework to the refined Swan conductor, an invariant measuring wild ramification in fields with imperfect residue fields.

• Conjecture: They conjecture that in mixed characteristic, the refined Swan conductor of a cyclic character can be determined by the associated Witt vector $X$, analogous to the equal-characteristic case.
• Result: They prove this for the case $s=1$. Specifically, if a $\mathbb{Z}/p$-extension is associated with a Witt vector $X_0$, the refined Swan conductor is given by:
  $$rsw(F) = -dX_0 \otimes 1$$
  This formula links the differential data of the Kummer extension directly to the Witt vector coordinate, providing a unified view across characteristics.

4. Significance

• Resolution of an Open Problem: The paper provides the first published, explicit construction of the Sekiguchi–Suwa lift, which was previously only known via an unpublished manuscript.
• New Technique: The method of associating Kummer classes with Witt vectors via Matsuda's theory and then algebraizing the result offers a powerful new tool for studying ramification in mixed characteristic.
• Applications:
  • Inseparable Extensions: The technique is suggested to have applications in studying inseparable extensions.
  • Refined Swan Conductors: The work lays the groundwork for a unified theory of refined Swan conductors in mixed characteristic, potentially extending the Grothendieck–Ogg–Shafarevich formula and vanishing cycle formulas to more general settings.
  • Explicit Geometry: By providing explicit polynomial equations, the paper makes these abstract group schemes computable and accessible for further geometric and arithmetic investigations.

In summary, Dang and Nguyen-Dang successfully bridge the gap between Kummer and Artin–Schreier–Witt theories in mixed characteristic, providing explicit algebraic models for lifting cyclic covers and establishing a new formula for the refined Swan conductor in terms of Witt vectors.