Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-07-04T21:51:39.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Lyubeznik numbers

Published online by Cambridge University Press:  11 January 2016

Luis Núñez-Betancourt  and
Emily E. Witt
Luis Núñez-Betancourt
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904, USA, lcn8m@virginia.edu
Emily E. Witt
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA, ewitt@umn.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers but captures finer information. These generalized Lyubeznik numbers are defined in terms of D-modules and are proved well defined using a generalization of the classical version of Kashiwara’s equivalence for smooth varieties; we also give a definition for finitely generated K-algebras. These new invariants are indicators of F-singularities in characteristic p > 0 and have close connections with characteristic cycle multiplicities in characteristic zero. We characterize the generalized Lyubeznik numbers associated to monomial ideals and compute examples of those associated to determinantal ideals.

Keywords

13D4513N1013H99
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 215 , September 2014 , pp. 169 - 201
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[A1] Àlvarez Montaner, J., Characteristic cycles of local cohomology modules of monomial ideals, J. Pure Appl. Algebra 150 (2000), 1–25. MR 1762917. DOI 10.1016/S0022-4049(98)00171–6.Google Scholar
[A2] Àlvarez Montaner, J., Some numerical invariants of local rings, Proc. Amer. Math. Soc. 132 (2004), 981–986. MR 2045412. DOI 10.1090/S0002–9939-03–07177-6.Google Scholar
[AGZ] Àlvarez Montaner, J., García López, R., and Zarzuela Armengou, S., Local coho-mology, arrangements of subspaces and monomial ideals, Adv. Math. 174 (2003), 35–56. MR 1959890. DOI 10.1016/S0001-8708(02)00050–6.Google Scholar
[AV] Àlvarez Montaner, J. and Vahidi, A., Lyubeznik numbers of monomial ideals, Trans. Amer. Math. Soc. 366, no. 4 (2014), 1829–1855. MR 3152714. DOI 10.1090/S0002–9947-2013–05862-X.Google Scholar
[B1] Bj¨ork, J.-E., The global homological dimension of some algebras of differential operators, Invent. Math. 17 (1972), 67–78. MR 0320078.CrossRefGoogle Scholar
[B2] Bj¨ork, J.-E., Rings of Differential Operators, North-Holland Math. Library 21, North-Holland, Amsterdam, 1979. MR 0549189.Google Scholar
[Bl] Blickle, M., The intersection homology D-module in finite characteristic, Math. Ann. 328 (2004), 425–450. MR 2036330. DOI 10.1007/s00208–003-0492-z.Google Scholar
[BlB] Blickle, M. and Bondu, R., Local cohomology multiplicities in terms of étale coho-mology, Ann. Inst. Fourier (Grenoble) 55 (2005), 2239–2256. MR 2207383.Google Scholar
[C] Cohen, I. S., On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106. MR 0016094.Google Scholar
[Co] Coutinho, S. C., A Primer of Algebraic D-Modules, London Math. Soc. Stud. Texts 33, Cambridge University Press, Cambridge, 1995. MR 1356713. DOI 10.1017/CBO9780511623653.Google Scholar
[GS] García Lóopez, R. and Sabbah, C., Topological computation of local cohomology multiplicities, Collect. Math. 49 (1998), 317–324. MR 1677136.Google Scholar
[Gr] Grothendieck, A., Éléments de géométrie algébrique, IV: Etude locale des schémas et des morphismes de schémas, IV, Publ. Math. Inst. Hautes Etudes Sci. 32 (1967). MR 0238860.Google Scholar
[H] Haastert, B., On direct and inverse images of D-modules in prime characteristic, Manuscripta Math. 62 (1988), 341–354. MR 0966631. DOI 10.1007/BF01246838.Google Scholar
[Ha] Hartshorne, R., Local Cohomology, Lecture Notes in Math. 41, Springer, Berlin, 1967. MR 0224620.Google Scholar
[HH] Hochster, M. and Huneke, C., Tight closure of parameter ideals and splitting in module-finite extensions. J. Algebraic Geom. 3 (1994), 599–670. MR 1297848.Google Scholar
[K1] Kawasaki, K., On the Lyubeznik number of local cohomology modules, Bull. Nara Univ. Ed. Natur. Sci. 49 (2000), 5–7. MR 1814657.Google Scholar
[K2] Kawasaki, K., On the highest Lyubeznik number, Math. Proc. Cambridge Philos. Soc. 132 (2002), 409–417. MR 1891679. DOI 10.1017/S0305004101005722.Google Scholar
[L1] Lyubeznik, G., Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. Math. 113 (1993), 41–55. MR 1223223. DOI 10.1007/BF01244301.CrossRefGoogle Scholar
[L2] Lyubeznik, G., Injective dimension of D-modules: A characteristic-free approach, J. Pure Appl. Algebra 149 (2000), 205–212. MR 1757731. DOI 10.1016/S0022-4049(98)00175–3.Google Scholar
[L3] Lyubeznik, G., Finiteness properties of local cohomology modules: A characteristic-free approach, J. Pure Appl. Algebra 151 (2000), 43–50. MR 1770642. DOI 10.1016/S0022-4049(99)00080–8.Google Scholar
[L4] Lyubeznik, G., “A partial survey of local cohomology” in Local Cohomology and Its Appli cations (Guanajuato, 1999), Lect. Notes Pure Appl. Math. 226, Dekker, New York, 2002, 121–154. MR 1888197.Google Scholar
[MN] Mebkhout, Z. and Narváez-Macarro, L., La théorie du polynôme de Bernstein-Sato pour les algebres de Tate et de Dwork-Monsky-Washnitzer, Ann. Sci. Ec. Norm. Super. (4) 24 (1991), 227–256. MR 1097693.Google Scholar
[MS] Miller, E. and Sturmfels, B., Combinatorial Commutative Algebra, Grad. Texts in Math. 227, Springer, New York, 2005. MR 2110098.Google Scholar
[Mu] Mustaă, M., Local cohomology at monomial ideals, symbolic computation in algebra, analysis, and geometry (Berkeley, 1998), J. Symbolic Comput. 29 (2000), 709–720. MR 1769662. DOI 10.1006/jsco.1999.0302.Google Scholar
[NP] Núñez-Betancourt, L. and Pérez, F., F-jumping and F-Jacobian ideals for hypersur-faces, preprint, arXiv:1302.3327v2 [math.AG]. DOI 10.1016/j.jpaa.2012.02.010.Google Scholar
[PS] Peskine, C. and Szpiro, L., Dimension projective finie et cohomologie locale: Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Publ. Math. Inst. Hautes Etudes Sci. (1973), 47–119. MR 0374130.Google Scholar
[V] Vassilev, J. C., Test ideals in quotients of F-finite regular local rings, Trans. Amer. Math. Soc. 350, no. 10 (1998), 4041–4051. MR 1458336. DOI 10.1090/S0002–9947-98–02128-X.Google Scholar
[W1] Walther, U., On the Lyubeznik numbers of a local ring, Proc. Amer. Math. Soc. 129 (2001), 1631–1634. MR 1814090. DOI 10.1090/S0002–9939-00–05755-5.Google Scholar
[W2] Walther, U., Bernstein-Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements, Compos. Math. 141 (2005), 121–145. MR 2099772. DOI 10.1112/S0010437X04001149.Google Scholar
[Wi] Witt, E. E., Local cohomology with support in ideals of maximal minors, Adv. Math. 231 (2012), 1998–2012. MR 2964631. DOI 10.1016/j.aim.2012.07.001.Google Scholar
[Y1] Yanagawa, K., Alexander duality for Stanley–Reisner rings and square-free Nn-graded modules, J. Algebra 225 (2000), 630–645. MR 1741555. DOI 10.1006/jabr.1999.8130.Google Scholar
[Y2] Yanagawa, K., Bass numbers of local cohomology modules with supports in monomial ideals, Math. Proc. Cambridge Philos. Soc. 131 (2001), 45–60. MR 1833073. DOI 10.1017/S030500410100514X.CrossRefGoogle Scholar
[Z] Zhang, W., On the highest Lyubeznik number of a local ring, Compos. Math. 143 (2007), 82–88. MR 2295196. DOI 10.1112/S0010437X06002387.Google Scholar