Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite torsors over strongly $F$-regular singularities

Javier Carvajal-Rojas

TL;DR

The paper addresses finite torsors over big opens of spectra of strongly $F$-regular germs, proving existence of a finite local cover $R o R^{igstar}$ that preserves strong $F$-regularity and ensures extension of all solvable-component torsors to the whole spectrum. A generalized transformation rule for the $F$-signature under finite local extensions drives a bound on divisor class torsion, namely $ ext{Cl}(R)$ is torsion-free when $s(R)>1/2$ and Picard groups of globally $F$-regular varieties are torsion-free via cone arguments. It systematically treats unipotent and linearly reductive cases, establishing maximal covers through chains of quasitorsors and explaining their relation to the local Nori fundamental group-scheme, with consequences for canonical and index-$1$ cyclic covers. The results illuminate how $F$-signature governs torsor extension, linking singularity theory in positive characteristic to divisor-class geometry and offering a framework for further exploration beyond solvable groups.

Abstract

We investigate finite torsors over big opens of spectra of strongly $F$-regular germs that do not extend to torsors over the whole spectrum. Let $(R,\mathfrak{m},k)$ be a strongly $F$-regular $k$-germ where $k$ is an algebraically closed field of characteristic $p>0$. We prove the existence of a finite local cover $R \subset R^{\star}$ so that $R^{\star}$ is a strongly $F$-regular $k$-germ and: for all finite algebraic groups $G/k$ with solvable neutral component, every $G$-torsor over a big open of $\mathrm{Spec} R^{\star}$ extends to a $G$-torsor everywhere. To achieve this, we obtain a generalized transformation rule for the $F$-signature under finite local extensions. Such formula is used to show that that the torsion of $\mathrm{Cl} R$ is bounded by $1/s(R)$. By taking cones, we conclude that the Picard group of globally $F$-regular varieties is torsion-free. Likewise, it shows that canonical covers of $\mathbb{Q}$-Gorenstein strongly $F$-regular singularities are strongly $F$-regular.

Finite torsors over strongly $F$-regular singularities

TL;DR

The paper addresses finite torsors over big opens of spectra of strongly -regular germs, proving existence of a finite local cover that preserves strong -regularity and ensures extension of all solvable-component torsors to the whole spectrum. A generalized transformation rule for the -signature under finite local extensions drives a bound on divisor class torsion, namely is torsion-free when and Picard groups of globally -regular varieties are torsion-free via cone arguments. It systematically treats unipotent and linearly reductive cases, establishing maximal covers through chains of quasitorsors and explaining their relation to the local Nori fundamental group-scheme, with consequences for canonical and index- cyclic covers. The results illuminate how -signature governs torsor extension, linking singularity theory in positive characteristic to divisor-class geometry and offering a framework for further exploration beyond solvable groups.

Abstract

We investigate finite torsors over big opens of spectra of strongly -regular germs that do not extend to torsors over the whole spectrum. Let be a strongly -regular -germ where is an algebraically closed field of characteristic . We prove the existence of a finite local cover so that is a strongly -regular -germ and: for all finite algebraic groups with solvable neutral component, every -torsor over a big open of extends to a -torsor everywhere. To achieve this, we obtain a generalized transformation rule for the -signature under finite local extensions. Such formula is used to show that that the torsion of is bounded by . By taking cones, we conclude that the Picard group of globally -regular varieties is torsion-free. Likewise, it shows that canonical covers of -Gorenstein strongly -regular singularities are strongly -regular.

Paper Structure

This paper contains 23 sections, 22 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Notations, conventions, and preliminaries
  3. $F$-signatures and strong $F$-regularity
  4. Linear and bilinear forms
  5. Group-schemes and their actions on schemes
  6. Examples of affine algebraic groups
  7. Trigonalizable groups
  8. Actions, quotients, and torsors
  9. Traces of quotients by finite group-schemes
  10. Construction of the trace
  11. Initial properties of the trace
  12. Cohomological tameness and total integrals
  13. On the existence of a maximal cover
  14. Local finite torsors
  15. The generalized transformation rule for the $F$-signature
  16. ...and 8 more sections

Key Result

Theorem 3.1

Let $H$ be a Hopf algebra. An element $t \in H$ is a left integral if $ht=e(h)t$ for all $h \in H$. Left integrals form a $\mathcal{k}$-submodule of $H$ denoted by $\int_H$. If $H/\mathcal{k}$ is finite, $\dim_{\mathcal{k}} \int_H =1$.

Theorems & Definitions (78)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 3.1: MontgomeryHopfAlgebras
  • Remark 4: Geometric description of integrals
  • Remark 5
  • Remark 6
  • Example 1
  • proof
  • Remark 7
  • ...and 68 more