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

• DOI: 10.48550/arxiv.2307.14958
• Publication Date: 2023-01-01
• Authors: Jiang, Zhan; G, Rebecca R.
• Keywords: 13H10 (Primary) 13D45, 13C14, 13A35 (Secondary); FOS: Mathematics; Mathematics - Commutative Algebra; Commutative Algebra (math.AC)

13 pages, comments welcome, revised to add results on when a cl-rational ring is a normal domain<br/>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.<br/>