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We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

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Expected resurgences and symbolic powers of ideals

Journal of the London Mathematical Society ◽

10.1112/jlms.12324 ◽

2020 ◽

Vol 102 (2) ◽

pp. 453-469

Author(s):

Eloísa Grifo ◽

Craig Huneke ◽

Vivek Mukundan

Keyword(s):

Powers Of Ideals ◽

Symbolic Powers

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Symbolic Powers of Ideals Defining F-pure and Strongly F-regular Rings

International Mathematics Research Notices ◽

10.1093/imrn/rnx213 ◽

2017 ◽

Vol 2019 (10) ◽

pp. 2999-3014 ◽

Cited By ~ 5

Author(s):

Eloísa Grifo ◽

Craig Huneke

Keyword(s):

Regular Ring ◽

Radical Ideal ◽

Containment Problem ◽

Powers Of Ideals ◽

Symbolic Powers ◽

Regular Rings

Abstract Given a radical ideal $I$ in a regular ring $R$, the Containment Problem of symbolic and ordinary powers of $I$ consists of determining when the containment $I^{(a)} \subseteq I^b$ holds. By work of Ein–Lazersfeld–Smith, Hochster–Huneke and Ma–Schwede, there is a uniform answer to this question, but the resulting containments are not necessarily best possible. We show that a conjecture of Harbourne holds when $R/I$ is F-pure, and prove tighter containments in the case when $R/I$ is strongly F-regular.

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