Faculty Publications, Department of MathematicsCopyright (c) 2023 University of Nebraska - Lincoln All rights reserved.
https://digitalcommons.unl.edu/mathfacpub
Recent documents in Faculty Publications, Department of Mathematicsen-usThu, 13 Jul 2023 02:30:11 PDT3600Symbolic power containments in singular rings in positive characteristic
https://digitalcommons.unl.edu/mathfacpub/287
https://digitalcommons.unl.edu/mathfacpub/287Tue, 11 Jul 2023 20:48:53 PDT
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
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Eloísa Grifo et al.Expected resurgences and symbolic powers of ideals
https://digitalcommons.unl.edu/mathfacpub/286
https://digitalcommons.unl.edu/mathfacpub/286Tue, 11 Jul 2023 20:48:48 PDT
We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves
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Eloísa Grifo et al.A stable version of Harbourne's Conjecture and the containment problem for space monomial curves
https://digitalcommons.unl.edu/mathfacpub/285
https://digitalcommons.unl.edu/mathfacpub/285Tue, 11 Jul 2023 20:48:43 PDT
The symbolic powers I(^{n}^{)} of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers In, but it is natural to compare the two notions. The containment problem consists of determining the values of n and m for which I(^{n}^{)}⊆Im holds. When I is an ideal of height 2 in a regular ring, I(^{3}^{)}⊆I2 may fail, but we show that this containment does hold for the defining ideal of the space monomial curve (t^{a},t^{b},t^{c}). More generally, given a radical ideal I of big height h, while the containment I(^{hn}^{−}^{h}^{+}^{1}^{)}⊆In conjectured by Harbourne does not necessarily hold for all n, we give sufficient conditions to guarantee such containments for n≫0.
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Eloísa GrifoDemailly's Conjecture and the Containment Problem
https://digitalcommons.unl.edu/mathfacpub/284
https://digitalcommons.unl.edu/mathfacpub/284Tue, 11 Jul 2023 10:51:36 PDT
We investigate Demailly’s Conjecture for a general set of sufficiently many points. Demailly’s Conjecture generalizes Chudnovsky’s Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly’s bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.
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Sankhaneel Bisui et al.Expected resurgence of ideals defining Gorenstein rings
https://digitalcommons.unl.edu/mathfacpub/283
https://digitalcommons.unl.edu/mathfacpub/283Tue, 11 Jul 2023 10:51:31 PDT
Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is noetherian.
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Eloísa Grifo et al.Chudnovsky's Conjecture and the stable Harbourne-Huneke containment
https://digitalcommons.unl.edu/mathfacpub/282
https://digitalcommons.unl.edu/mathfacpub/282Tue, 11 Jul 2023 10:51:27 PDT
We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points.
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Sankhaneel Bisui et al.Symbolic Rees algebras
https://digitalcommons.unl.edu/mathfacpub/281
https://digitalcommons.unl.edu/mathfacpub/281Tue, 11 Jul 2023 09:22:32 PDT
We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.
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Eloísa Grifo et al.Constructing non-proxy small test modules for the complete intersection property
https://digitalcommons.unl.edu/mathfacpub/280
https://digitalcommons.unl.edu/mathfacpub/280Tue, 11 Jul 2023 09:22:28 PDT
A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category Df(R), which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in Df(R) is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley-Reisner rings.
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Benjamin Briggs et al.Bounds on cohomological support varieties
https://digitalcommons.unl.edu/mathfacpub/279
https://digitalcommons.unl.edu/mathfacpub/279Tue, 11 Jul 2023 09:12:27 PDT
Over a local ring R, the theory of cohomological support varieties attaches to any bounded complex M of finitely generated R-modules an algebraic variety VR(M) that encodes homological properties of M. We give lower bounds for the dimension of VR(M) in terms of classical invariants of R. In particular, when R is Cohen-Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When M has finite projective dimension, we also give an upper bound for dimVR(M) in terms of the dimension of the radical of the homotopy Lie algebra of R. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of R. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.
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Benjamin Briggs et al.Lower bounds on Betti numbers
https://digitalcommons.unl.edu/mathfacpub/278
https://digitalcommons.unl.edu/mathfacpub/278Tue, 11 Jul 2023 09:12:22 PDT
We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.
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Adam Boocher et al.A uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic
https://digitalcommons.unl.edu/mathfacpub/277
https://digitalcommons.unl.edu/mathfacpub/277Tue, 11 Jul 2023 09:12:17 PDT
We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers that does not require the existence of a p-derivation on the direct summand.
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Alessandro De Stefani et al.