Linkage and $F$-Regularity of Determinantal Rings

TL;DR

This paper demonstrates that generic links of determinantal rings defined by maximal minors are strongly F-regular, providing a new, simpler proof of this property and extending results on residual intersections and F-regularity.

Contribution

It introduces a new, simplified proof that determinantal rings are strongly F-regular and extends the understanding of F-regularity in residual intersections.

Findings

Generic links of determinantal rings are strongly F-regular.

Strengthens results on residual intersections with F-regularity.

Provides a simpler proof of F-regularity for determinantal rings.

Abstract

In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors.