Global parameter test ideals

TL;DR

This paper establishes the existence of global parameter test ideals in various classes of rings, providing constructive methods and algorithms for their computation, and applies these to analyze HSL numbers of local cohomology modules.

Contribution

It introduces constructive techniques to prove the existence of global parameter test ideals and develops algorithms for their computation in different ring settings.

Findings

Existence of ideals whose localizations and completions are parameter test ideals.

Algorithms for computing global parameter test ideals.

Analysis of HSL numbers of local cohomology modules.

Abstract

This paper shows the existence of ideals whose localizations and completions at prime ideals are parameter test ideals of the localized and completed rings. We do this for Cohen-Macaulay localizations (resp., completions) of non-local rings, for generalized Cohen-Macaulay rings, and for non-local rings with isolated non Cohen-Macaulay points, each being an isolated non $F$-rational point. The tools used to prove this results are constructive in nature and as a consequence our results yield algorithms for the computation of these global parameter test ideals. Finally, we illustrate the power of our methods by analyzing the HSL numbers of local cohomology modules with support at any prime ideal.