Homological properties of pinched Veronese rings

TL;DR

This paper investigates the homological and F-singularity properties of pinched Veronese rings, providing new proofs, corrections, and bounds related to their Cohen-Macaulayness, Gorenstein, and complete intersection conditions.

Contribution

It offers a new proof of Cohen-Macaulayness classification, corrects previous omissions, and computes bounds on Frobenius test exponents for pinched Veronese rings.

Findings

Re-proved Cohen-Macaulayness classification using semigroup methods.

Corrected classification by identifying a small overlooked class.

Computed upper bounds on Frobenius test exponents.

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.