Symbolic power containments in singular rings in positive characteristic

Autor: Eloísa Grifo, Linquan Ma, Karl Schwede
Rok vydání: 2022
Předmět:
Zdroj: manuscripta mathematica. 170:471-496
ISSN: 1432-1785
0025-2611
DOI: 10.1007/s00229-021-01359-7
Popis: The containment problem for symbolic and ordinary powers of ideals asks for what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. 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.
Comment: Final version
Databáze: OpenAIRE
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