On delta invariants and indices of ideals

TL;DR

This paper explores delta invariants and indices of ideals in Cohen-Macaulay rings, providing new criteria for parameter ideals, bounds for Ulrich ideals, and extending index definitions to ideals with related regularity results.

Contribution

It introduces conditions characterizing parameter ideals via delta invariants and generalizes the relationship between index and regularity to ideal cases.

Findings

Conditions for an ideal to be a parameter ideal using delta invariants

Upper bounds for orders of Ulrich ideals in Gorenstein rings

Extension of index definitions and their relation to regularity

Abstract

Let R be a Cohen-Macaulay local ring with a canonical module. We consider Auslander's (higher) delta invariants of powers of certain ideals of R. Firstly, we shall provide some conditions for an ideal to be a parameter ideal in terms of delta invarints. As an application of this result, we give upper bounds for orders of Ulrich ideals of R when R has Gorenstein punctured spectrum. Secondly, we extend the definition of indices to the ideal case, and generalize the result of Avramov-Buchweitz-Iyengar-Miller on the relationship between the index and regularity.