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Creators/Authors contains: "Quinlan-Gallego, Eamon"

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  1. ; ; ; ; (, Journal of Commutative Algebra)
    Log-canonical and F-pure thresholds of pairs in equal characteristic admit an analog in the recent theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this paper we study plus-pure thresholds for singularities of the form pa+xb∈ℤp[[x]], showing that in a number of cases this plus-pure threshold agrees with the F-pure threshold of the singularity ta+xb∈𝔽p[[t,x]]. We also discuss a few other sporadic examples. 
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    Free, publicly-accessible full text available October 7, 2027
  2. ; ; ; ; (, Journal of Commutative Algebra)
    Free, publicly-accessible full text available June 9, 2027
  3. ; (, Épijournal de Géométrie Algébrique)
    For fixed prime integer $p > 0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $$\mathbb{Z} / p^m$$-coefficients, compatible with existing theory in the case $m = 1$. We show that the ``roots" of such polynomials are rational and we show that the negative roots agree with those of the mod-$$p$$ reduction. We give examples to show that, surprisingly, roots may be positive in this context. Moreover, our construction allows us to define a notion of ``strength" for roots by measuring $$p$$-torsion, and we show that ``strong" roots give rise to roots in characteristic zero through mod-$$p$$ reduction. Comments welcome. v3: final version. v2: fixed typos, small changes in notation, and additional example following suggestions from the referee 
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  4. ; ; (, Revista Matemática Iberoamericana)
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  5. (, International Mathematics Research Notices)
    Abstract We develop a formalism of unit $$F$$-modules in the style of Lyubeznik and Emerton-Kisin for rings that have finite $$F$$-representation type after localization and completion at every prime ideal. As applications, we show that if $$R$$ is such a ring then the iterated local cohomology modules $$H^{n_{1}}_{I_{1}} \circ \cdots \circ H^{n_{s}}_{I_{s}}(R)$$ have finitely many associated primes, and that all local cohomology modules $$H^{n}_{I}(R / gR)$$ have closed support when $$g$$ is a nonzerodivisor on $$R$$. 
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  6. ; ; (, Transactions of the American Mathematical Society)
    The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case. 
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  7. (, Journal of Algebra)
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  8. (, Transactions of the American Mathematical Society)
    Mustaţă defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the F F -jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to F F -jumping numbers. 
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