Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians

Li, Jiamin; Perlman, Michael

doi:10.1090/proc/16645

Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians
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by Jiamin Li and Michael Perlman;

Proc. Amer. Math. Soc. 152 (2024), 599-615

DOI: https://doi.org/10.1090/proc/16645

Published electronically: December 7, 2023

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Abstract:

Let $S$ be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let $I$ be the determinantal ideal of maximal minors or $Pf$ the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the $S$-module structures of $Ext^j_S(S/I^t, S)$ and $Ext^j_S(S/Pf^t, S)$, from which we get the degrees of generators of these $Ext$ modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form $H^j_\mathfrak {m}(S/I^t)$.

References

Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207–278. MR 658729, DOI 10.1016/0001-8708(82)90039-1
S. Abeasis and A. Del Fra, Young diagrams and ideals of Pfaffians, Adv. in Math. 35 (1980), no. 2, 158–178. MR 560133, DOI 10.1016/0001-8708(80)90046-8
Bhargav Bhatt, Manuel Blickle, Gennady Lyubeznik, Anurag K. Singh, and Wenliang Zhang, Stabilization of the cohomology of thickenings, Amer. J. Math. 141 (2019), no. 2, 531–561. MR 3928045, DOI 10.1353/ajm.2019.0013
Bhargav Bhatt, Manuel Blickle, Gennady Lyubeznik, Anurag K. Singh, and Wenliang Zhang, An asymptotic vanishing theorem for the cohomology of thickenings, Math. Ann. 380 (2021), no. 1-2, 161–173. MR 4263681, DOI 10.1007/s00208-020-02140-z
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
C. de Concini, David Eisenbud, and C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), no. 2, 129–165. MR 558865, DOI 10.1007/BF01392548
Hailong Dao and Jonathan Montaño, On asymptotic vanishing behavior of local cohomology, Math. Z. 295 (2020), no. 1-2, 73–86. MR 4100017, DOI 10.1007/s00209-019-02353-2
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 463157, DOI 10.1007/978-1-4757-3849-0
Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361, DOI 10.1007/978-0-8176-4523-6
Craig Huneke, Lectures on local cohomology, Interactions between homotopy theory and algebra, Contemp. Math., vol. 436, Amer. Math. Soc., Providence, RI, 2007, pp. 51–99. Appendix 1 by Amelia Taylor. MR 2355770, DOI 10.1090/conm/436/08404
Jennifer Kenkel, Lengths of Local Cohomology of Thickenings, arXiv:1912.02917 (2019).
Jiamin Li, On the generalized multiplicities of maximal minors and sub-maximal Pfaffians, arXiv:2205.09657 (2022).
Michael Perlman, Lyubeznik numbers for Pfaffian rings, J. Pure Appl. Algebra 224 (2020), no. 5, 106247, 24. MR 4046247, DOI 10.1016/j.jpaa.2019.106247
Michael Perlman, Regularity and cohomology of Pfaffian thickenings, J. Commut. Algebra 13 (2021), no. 4, 523–548. MR 4366836, DOI 10.1216/jca.2021.13.523
Claudiu Raicu, Characters of equivariant $\mathcal {D}$-modules on spaces of matrices, Compos. Math. 152 (2016), no. 9, 1935–1965. MR 3568944, DOI 10.1112/S0010437X16007521
Claudiu Raicu, Regularity and cohomology of determinantal thickenings, Proc. Lond. Math. Soc. (3) 116 (2018), no. 2, 248–280. MR 3764061, DOI 10.1112/plms.12071
Claudiu Raicu and Jerzy Weyman, Local cohomology with support in generic determinantal ideals, Algebra Number Theory 8 (2014), no. 5, 1231–1257. MR 3263142, DOI 10.2140/ant.2014.8.1231
Claudiu Raicu and Jerzy Weyman, Local cohomology with support in ideals of symmetric minors and Pfaffians, J. Lond. Math. Soc. (2) 94 (2016), no. 3, 709–725. MR 3614925, DOI 10.1112/jlms/jdw056
Claudiu Raicu, Jerzy Weyman, and Emily E. Witt, Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians, Adv. Math. 250 (2014), 596–610. MR 3122178, DOI 10.1016/j.aim.2013.10.005
Hunter Simper, Local cohomology of certain determinantal thickenings, arXiv:2205.06738 (2022).
Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR 1988690, DOI 10.1017/CBO9780511546556
Wenliang Zhang, On asymptotic socle degrees of local cohomology modules, J. Pure Appl. Algebra 225 (2021), no. 12, Paper No. 106789, 12. MR 4260035, DOI 10.1016/j.jpaa.2021.106789