Symbolic power containments in singular rings in positive characteristic

Authors

Eloísa Grifo, University of Nebraska-LincolnFollow
Linquan Ma, Purdue UniversityFollow
Karl Schwede, University of UtahFollow

Document Type

Article

Date of this Version

8-2022

Citation

arXiv:1911.06307v3 [math.AC] 14 Aug 2022

Comments

Copyright © 2022 ELO´ISA GRIFO, LINQUAN MA, AND KARL SCHWEDE

Abstract

The containment problem for symbolic and ordinary powers of ideals asks for what values of a and b we have I(a)⊆Ib. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for F-purity (respectively, strong F-regularity) for ideals of finite projective dimension over F-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi