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Homological properties of pinched Veronese rings

Kyle Maddox, Vaibhav Pandey

Abstract

Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we show that pinched Veronese rings exhibit a variety of F-singularities, including F-regular, F-injective, and F-nilpotent. We also compute upper bounds on the Frobenius test exponents of pinched Veronese rings, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously.

Homological properties of pinched Veronese rings

Abstract

Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we show that pinched Veronese rings exhibit a variety of F-singularities, including F-regular, F-injective, and F-nilpotent. We also compute upper bounds on the Frobenius test exponents of pinched Veronese rings, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously.

Paper Structure

This paper contains 9 sections, 25 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An overview of prime characteristic singularities
  4. Key Combinatorial Results
  5. The Cohen-Macaulay and Gorenstein properties
  6. Pinched Veronese Rings in Prime Characteristic
  7. F-nilpotence of Pinched Veroneses
  8. Frobenius Test Exponents for Pinched Veroneses
  9. Multi-pinched Veronese rings

Key Result

Theorem \oldthetheorem

The pinched Veronese ring $\mathcal{P}_{n,d,\mathbf{m}}$ is Cohen-Macaulay if and only if one of the following three conditions hold. Further, when $\max(\mathbf{m})= d-1$, $\mathcal{P}_{2,d,\mathbf{m}}$ is a Gorenstein ring with $a$-invariant zero, and when $\max(\mathbf{m}) = 1$, $\mathcal{P}_{3,2,\mathbf{m}}$ is a complete intersection ring.

Theorems & Definitions (63)

  • Theorem \oldthetheorem
  • Theorem : Hochster, Hoc
  • Theorem \oldthetheorem
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • ...and 53 more