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Patching for étale algebras and the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields

Yidi Wang

TL;DR

This work extends Harbater–Hartmann patching to Hensel semi-global fields and establishes patching for étale algebras and finite-constant-group torsors, using Artin Approximation to descend from semi-global fields. It develops a local-global principle for higher-degree Galois cohomology with respect to patches and applies these tools to derive uniform period-index bounds for $H^i(F,\mu_m^{\otimes i-1})$, including non-prime orders, by relating splitting dimensions to base fields $k$ and $k(x)$. The authors introduce generalized splitting dimensions $\mathrm{sd}$, $\mathrm{ssd}$, and $\mathrm{gssd}$, and prove relative bounds and unramified-splitting results that extend to iterative rank settings $k_r$, yielding explicit iterative bounds. The results provide a systematic framework for understanding splitting fields and indices in higher-degree Galois cohomology over Hensel semi-global fields, with explicit bounds depending on the base residue field and its rational function field. This advances the period-index theory in a broader arithmetic-geometric setting and offers concrete tools for handling non-prime-order coefficient groups through a prime-decomposition approach.

Abstract

In this manuscript, we present a partial generalization of the field patching technique initially proposed by Harbater-Hartmann to Hensel semi-global fields, i.e., function fields of curves over excellent henselian discretely valued fields. More specifically, we show that patching holds for étale algebras over such fields and a suitable set of overfields. Within this new framework, we further establish a local-global principle for higher degree Galois cohomology groups over Hensel semi-global fields. As an application, we extend a recent result regarding a uniform period-index bound for higher degree Galois cohomology classes by Harbater-Hartmann-Krashen to Hensel semi-global fields. Additionally, we prove such a bound for coefficient groups of non-prime orders.

Patching for étale algebras and the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields

TL;DR

This work extends Harbater–Hartmann patching to Hensel semi-global fields and establishes patching for étale algebras and finite-constant-group torsors, using Artin Approximation to descend from semi-global fields. It develops a local-global principle for higher-degree Galois cohomology with respect to patches and applies these tools to derive uniform period-index bounds for , including non-prime orders, by relating splitting dimensions to base fields and . The authors introduce generalized splitting dimensions , , and , and prove relative bounds and unramified-splitting results that extend to iterative rank settings , yielding explicit iterative bounds. The results provide a systematic framework for understanding splitting fields and indices in higher-degree Galois cohomology over Hensel semi-global fields, with explicit bounds depending on the base residue field and its rational function field. This advances the period-index theory in a broader arithmetic-geometric setting and offers concrete tools for handling non-prime-order coefficient groups through a prime-decomposition approach.

Abstract

In this manuscript, we present a partial generalization of the field patching technique initially proposed by Harbater-Hartmann to Hensel semi-global fields, i.e., function fields of curves over excellent henselian discretely valued fields. More specifically, we show that patching holds for étale algebras over such fields and a suitable set of overfields. Within this new framework, we further establish a local-global principle for higher degree Galois cohomology groups over Hensel semi-global fields. As an application, we extend a recent result regarding a uniform period-index bound for higher degree Galois cohomology classes by Harbater-Hartmann-Krashen to Hensel semi-global fields. Additionally, we prove such a bound for coefficient groups of non-prime orders.
Paper Structure (20 sections, 32 theorems, 51 equations)

This paper contains 20 sections, 32 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Henselian rings and the Artin Approximation Theorem
  3. Patching over Hensel Semi-global fields
  4. Patching over semi-global fields
  5. Notation: Hensel semi-global fields and Hensel patches
  6. Patching for torsors under finite constant group
  7. Proof of Theorem \ref{['patching_torsors']}
  8. Some preparatory results from patching
  9. A local-global principle with respect to patches
  10. A comparison between two cohomology sets
  11. A local-global principle with respect to patches
  12. The period-index problem for higher degree Galois cohomology groups
  13. Splitting dimensions and uniform bounds for cohomology classes
  14. Splitting unramified cohomology classes
  15. A relative period-index bound for higher degree Galois cohomology classes
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $F$ be a Hensel semi-global field, $F_1, F_2$ be Hensel patches, and $F_0$ be the Hensel branch field. Let $G$ be a finite group. The map given by base change is an equivalence of categories. In particular, when $G = S_n$, $\beta$ gives an equivalence of categories of étale algebras.

Theorems & Definitions (69)

  • Theorem 1.1: \ref{['patching_torsors']}
  • Theorem 1.2: \ref{['main_theorem']}, \ref{['general_theorem']}
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • ...and 59 more