Tight closure of products and F-rational singularities
Alessandro De Stefani, Ilya Smirnov
TL;DR
The paper investigates how tight closure can characterize F-rational singularities in positive characteristic by focusing on products of parameter ideals, mirroring Lipman–Teissier style results for rational surface singularities. The authors prove that for an excellent $F$-injective normal domain $R$ that is F-rational on the punctured spectrum, $R$ is F-rational if and only if $( rakq_1 rakq_2)^*= rakq_1^* rakq_2^*$ for all parameter ideals $ rakq_1 eq rakq_2$ with $ rakq_1 eseq rakq_2$, with a broader statement using Frobenius powers. They further show that the stronger condition $(IJ)^*=I^*J^*$ for all ideals $I,J$ implies weak F-regularity, illustrating limits of a universal product-tight-closure criterion. The results illuminate the interplay between tight closure, parameter ideals, and rational-like singularities, while also highlighting natural open questions, such as whether squared products universally behave under tight closure in dimension two. These findings contribute a sharp, ideal-theoretic lens on F-rationality in positive characteristic.
Abstract
We prove a characterization of F-rationality in terms of tight closure of products of parameter ideals. Our results are inspired by the theory of complete ideals for surfaces and, in particular, the fundamental results of Lipman-Teissier and Cutkosky characterizing rational surface singularities in terms of products of complete ideals, but are valid also in higher dimensions.