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Linkage and $F$-Regularity of Determinantal Rings

Vaibhav Pandey, Yevgeniya Tarasova

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.

Linkage and $F$-Regularity of Determinantal Rings

Abstract

In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly -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 -rational, then in fact, its generic residual intersections are strongly -regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly -regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong -regularity of determinantal rings defined by maximal minors.
Paper Structure (8 sections, 28 theorems, 60 equations)

This paper contains 8 sections, 28 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linkage and Residual Intersections
  4. Singularities in positive characteristic.
  5. Fedder's Criterion and Linkage
  6. F-Regularity of Determinantal Rings Defined by Maximal Minors
  7. F-Regularity of the Generic Residual Intersections of Graded Complete Intersection Rings
  8. F-Regularity of the Generic Links of Determinantal Rings

Key Result

Theorem 1

(Theorem theoremGenericSFR) Let $X = (x_{i,j})$ be a $t \times n$ matrix of indeterminates for $n\geq t$, $K$ a field, and $R = K[X]$. Let $I_t(X)$ denote the ideal of $R$ generated by the size $t$ minors of $X$. Then,

Theorems & Definitions (56)

  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 46 more