Rationality for arbitrary closure operations and the test ideal of full extended plus closure

TL;DR

This paper generalizes the concept of F-rationality to various closure operations, explores conditions linking rationality to parameter ideals, and shows that certain closure types lack big test elements.

Contribution

It introduces a framework for F-rationality across different closure operations and analyzes the properties of full extended plus closure.

Findings

Full extended plus closure has no big test elements.

F-rationality can be characterized by parameter ideals being closed under certain conditions.

Conditions are provided for when cl-rationality aligns with parameter ideals being cl-closed.

Abstract

We extend the notion of F-rationality to other closure operations, inspired by the work of Smith, Epstein and Schwede, and Ma and Schwede, which describe F-rationality in terms of the canonical module and top local cohomology module. We give conditions for a closure operation cl on a Cohen-Macaulay complete local ring under which cl-rationality is equivalent to parameter ideals being cl-closed. We also demonstrate that full extended plus closure as defined by Heitmann and weak full extended plus closure as defined by the first named author have no big test elements.