Finite torsors over strongly $F$-regular singularities

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.

30 pages. Final version accepted to EPIGA

References in corpus (2)

• Fundamental groups of $F$-regular singularities via $F$-signature
• The number of torsion divisors in a strongly $F$-regular ring is bounded by the reciprocal of $F$-signature

Cited by in corpus (10)

• An analog of adjoint ideals and PLT singularities in mixed characteristic
• The number of torsion divisors in a strongly $F$-regular ring is bounded by the reciprocal of $F$-signature
• The weak Beauville--Bogomolov decomposition in characteristic $p\ge 0$
• On the behavior of $F$-signatures, splitting primes, and test modules under finite covers
• Tame fundamental groups of pure pairs and Abhyankar's lemma
• $F$-purity deforms in $\mathbb{Q}$-Gorenstein rings
• Inversion of adjunction for $F$-signature
• Torsors over the Rational Double Points in Characteristic $\mathbf{p}$
• Compatible ideals in Gorenstein rings
• On the local étale fundamental group of KLT threefold singularities