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Purity and quasi-split torsors over Prüfer bases

Ning Guo, Fei Liu

Abstract

We establish an analogue of the Zariski--Nagata purity theorem for finite étale covers on smooth schemes over Prüfer rings by demonstrating Auslander's flatness criterion in this non-Noetherian context. We derive an Auslander--Buchsbaum formula for general local rings, which provides a useful tool for studying the algebraic structures involved in our work. Through analysis of reflexive sheaves, we prove various purity theorems for torsors under certain group algebraic spaces, such as the reductive ones. Specifically, using results from EGAIV4 on parafactoriality on smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type groups at this level of generality. Subsequently, we leverage the aforementioned purity results to resolve the Grothendieck--Serre conjecture for torsors under a quasi-split reductive group scheme over schemes smooth over Prüfer rings. Along the way, we also prove a version of the Nisnevich purity conjecture for quasi-split reductive group schemes in our Prüferian context, inspired by the recent work of Cesnavicius.

Purity and quasi-split torsors over Prüfer bases

Abstract

We establish an analogue of the Zariski--Nagata purity theorem for finite étale covers on smooth schemes over Prüfer rings by demonstrating Auslander's flatness criterion in this non-Noetherian context. We derive an Auslander--Buchsbaum formula for general local rings, which provides a useful tool for studying the algebraic structures involved in our work. Through analysis of reflexive sheaves, we prove various purity theorems for torsors under certain group algebraic spaces, such as the reductive ones. Specifically, using results from EGAIV4 on parafactoriality on smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type groups at this level of generality. Subsequently, we leverage the aforementioned purity results to resolve the Grothendieck--Serre conjecture for torsors under a quasi-split reductive group scheme over schemes smooth over Prüfer rings. Along the way, we also prove a version of the Nisnevich purity conjecture for quasi-split reductive group schemes in our Prüferian context, inspired by the recent work of Cesnavicius.
Paper Structure (16 sections, 62 theorems, 160 equations)

This paper contains 16 sections, 62 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Reflexive sheaves and depths
  3. Geometry of schemes over Prüfer bases
  6. Auslander's flatness criterion
  7. Generalities on torsors over algebraic spaces
  8. Purity for torsors and finite étale covers
  11. Cohomology of groups of multiplicative type
  15. Generically trivial torsors under a quasi-split group
  16. ...and 1 more sections

Key Result

Lemma 1

Let $X$ be a scheme and let $\mathscr{F}$ and $\mathscr{G}$ be coherent $\mathscr{O}_X$-modules.

Theorems & Definitions (127)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Lemma 4
  • proof
  • Lemma 7
  • proof
  • ...and 117 more