Authors: Javier Carvajal-Rojas

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.

Comment: 30 pages. Final version accepted to EPIGA

• 13A35 - Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
• 13A50 - Actions of groups on commutative rings; invariant theory
• 14F35 - Homotopy theory and fundamental groups in algebraic geometry
• 14G17 - Positive characteristic ground fields in algebraic geometry
• 14L15 - Group schemes
• 16T05 - Hopf algebras and their applications