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Symbolic power containments in singular rings in positive characteristic

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Abstract

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.

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Notes

  1. Here \(I_e(\mathfrak {m})\) is the set of elements \(r \in R\) such that \(\phi (F^e_* r) \in \mathfrak {m}\) for every \(\phi \in {\text {Hom}}_R(F^e_* R, R)\).

  2. In fact, it is well-known that under mild assumptions, the property that \({\text {Hom}}_R(F_*^eR, R)\cong \mathbf {R}{\text {Hom}}_R(F^e_*R, R)\) for all e characterizes Gorenstein rings, see [14].

  3. Note that [15, Theorem 3.4] requires that we enlarge k to k(t), but this is only needed to guarantee that we have an infinite field so that we can pick general elements.

  4. We refer to [26] for the precise definition of asymptotic test ideal. In our context, we are considering the graded family of ideals \(\mathfrak {a}_\bullet {:=}\{\mathfrak {a}_n\}_n\) such that \(\mathfrak {a}_n=I^{(n)}\), and we use \(I^{(\bullet )}\) to abbreviate this notation.

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Acknowledgements

We thank Bernd Ulrich for giving us Example 6.3 and Proposition 6.5. We thank Srikanth Iyengar for very helpful discussions on derived categories and for providing us an alternative approach to Lemma 3.1 in Remark 3.2. We also thank Lucho Avramov, Jack Jeffries, Claudia Miller, and Alexandra Seceleanu for valuable conversations and for giving us several interesting examples. We also thank Javier Carvajal-Rojas, Craig Huneke, Thomas Polstra, and Axel Stäbler for useful conversations and comments. The first author thanks the University of Utah, where she was visiting when part of this project was completed, for their hospitality. Finally, we thank the anonymous referee for their valuable comments, especially Remark 5.3.

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Authors and Affiliations

  1. Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA

    Eloísa Grifo

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Linquan Ma

  3. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Karl Schwede

Authors
  1. Eloísa Grifo
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  2. Linquan Ma
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  3. Karl Schwede
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Correspondence to Eloísa Grifo.

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Grifo was supported in part by NSF Grant DMS #2001445 and #2140355.

Ma was supported in part by NSF Grant DMS #1901672, NSF FRG Grant #1952366, and a fellowship from the Sloan Foundation.

Schwede was supported in part by NSF CAREER Grant DMS #1252860/1501102, NSF Grant #1801849, and a Fellowship from the Simons Foundation.

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Grifo, E., Ma, L. & Schwede, K. Symbolic power containments in singular rings in positive characteristic. manuscripta math. 170, 471–496 (2023). https://doi.org/10.1007/s00229-021-01359-7

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