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Lyubeznik numbers of monomial ideals


Authors: Josep Àlvarez Montaner and Alireza Vahidi
Journal: Trans. Amer. Math. Soc. 366 (2014), 1829-1855
MSC (2010): Primary 13D45, 13N10, 13F55
DOI: https://doi.org/10.1090/S0002-9947-2013-05862-X
Published electronically: November 25, 2013
MathSciNet review: 3152714
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Abstract: Let $ R=k[x_1,...,x_n]$ be the polynomial ring in $ n$ independent variables, where $ k$ is a field. In this work we will study Bass numbers of local cohomology modules $ H^r_I(R)$ supported on a squarefree monomial ideal $ I\subseteq R$. Among them we are mainly interested in Lyubeznik numbers. We build a dictionary between the modules $ H^r_I(R)$ and the minimal free resolution of the Alexander dual ideal $ I^{\vee }$ that allows us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the linear strands of $ I^{\vee }$. The methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support.


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  • [1] Josep Alvarez Montaner, Characteristic cycles of local cohomology modules of monomial ideals, J. Pure Appl. Algebra 150 (2000), no. 1, 1-25. MR 1762917 (2001d:13016), https://doi.org/10.1016/S0022-4049(98)00171-6
  • [2] Josep Àlvarez Montaner, Some numerical invariants of local rings, Proc. Amer. Math. Soc. 132 (2004), no. 4, 981-986 (electronic). MR 2045412 (2005c:13021), https://doi.org/10.1090/S0002-9939-03-07177-6
  • [3] Josep Àlvarez Montaner, Operations with regular holonomic $ \mathcal {D}$-modules with support a normal crossing, J. Symbolic Comput. 40 (2005), no. 2, 999-1012. MR 2167680 (2006e:32013), https://doi.org/10.1016/j.jsc.2005.03.001
  • [4] Josep Àlvarez Montaner, Ricardo García López, and Santiago Zarzuela Armengou, Local cohomology, arrangements of subspaces and monomial ideals, Adv. Math. 174 (2003), no. 1, 35-56. MR 1959890 (2004a:13012), https://doi.org/10.1016/S0001-8708(02)00050-6
  • [5] Josep Àlvarez Montaner and Santiago Zarzuela, Linearization of local cohomology modules, Commutative algebra (Grenoble/Lyon, 2001) Contemp. Math., vol. 331, Amer. Math. Soc., Providence, RI, 2003, pp. 1-11. MR 2011762 (2004j:13022), https://doi.org/10.1090/conm/331/05899
  • [6] F. Barkats,
    Calcul effectif de groupes de cohomologie locale à support dans des idéaux monomiaux,
    Ph.D. Thesis, Univ. Nice-Sophia Antipolis, 1995.
  • [7] J.-E. Björk, Rings of differential operators, North-Holland Mathematical Library, vol. 21, North-Holland Publishing Co., Amsterdam, 1979. MR 549189 (82g:32013)
  • [8] Manuel Blickle, Lyubeznik's invariants for cohomologically isolated singularities, J. Algebra 308 (2007), no. 1, 118-123. MR 2290913 (2008b:13024), https://doi.org/10.1016/j.jalgebra.2006.06.029
  • [9] Manuel Blickle and Raphael Bondu, Local cohomology multiplicities in terms of étale cohomology, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 7, 2239-2256 (English, with English and French summaries). MR 2207383 (2007d:14009)
  • [10] S. C. Coutinho, A primer of algebraic $ D$-modules, London Mathematical Society Student Texts, vol. 33, Cambridge University Press, Cambridge, 1995. MR 1356713 (96j:32011)
  • [11] K. Dalili and M. Kummini,
    Dependence of Betti numbers on characteristic, Comm. Algebra 42 (2014), no. 2, 563-570.
  • [12] John A. Eagon and Victor Reiner, Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), no. 3, 265-275. MR 1633767 (99h:13017), https://doi.org/10.1016/S0022-4049(97)00097-2
  • [13] David Eisenbud, Gunnar Fløystad, and Frank-Olaf Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4397-4426 (electronic). MR 1990756 (2004f:14031), https://doi.org/10.1090/S0002-9947-03-03291-4
  • [14] Hans-Bjørn Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra 15 (1979), no. 2, 149-172. MR 535182 (83c:13008), https://doi.org/10.1016/0022-4049(79)90030-6
  • [15] A. Galligo, M. Granger, and Ph. Maisonobe, $ {\mathcal {D}}$-modules et faisceaux pervers dont le support singulier est un croisement normal, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 1, 1-48 (French). MR 781776 (88b:32027)
  • [16] A. Galligo, M. Granger, and Ph. Maisonobe, $ {\mathcal {D}}$-modules et faisceaux pervers dont le support singulier est un croisement normal. II, Astérisque 130 (1985), 240-259 (French). Differential systems and singularities (Luminy, 1983). MR 804057 (88b:32028)
  • [17] R. García López and C. Sabbah, Topological computation of local cohomology multiplicities, Collect. Math. 49 (1998), no. 2-3, 317-324. Dedicated to the memory of Fernando Serrano. MR 1677136 (2000a:13029)
  • [18] Shiro Goto and Keiichi Watanabe, On graded rings. II. ( $ {\bf Z}^{n}$-graded rings), Tokyo J. Math. 1 (1978), no. 2, 237-261. MR 519194 (81m:13022), https://doi.org/10.3836/tjm/1270216496
  • [19] Hans-Gert Gräbe, The canonical module of a Stanley-Reisner ring, J. Algebra 86 (1984), no. 1, 272-281. MR 727379 (85a:13010), https://doi.org/10.1016/0021-8693(84)90066-8
  • [20] D. Grayson and M. Stillman,
    Macaulay 2, http://www.math.uiuc.edu/Macaulay2.
  • [21] M. Hellus, A note on the injective dimension of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2313-2321. MR 2390497 (2008m:13030), https://doi.org/10.1090/S0002-9939-08-09198-3
  • [22] Jürgen Herzog and Takayuki Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141-153. MR 1684555 (2000i:13019)
  • [23] Jürgen Herzog and Srikanth Iyengar, Koszul modules, J. Pure Appl. Algebra 201 (2005), no. 1-3, 154-188. MR 2158753 (2006d:13013), https://doi.org/10.1016/j.jpaa.2004.12.037
  • [24] Craig L. Huneke and Rodney Y. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765-779. MR 1124167 (93m:13008), https://doi.org/10.2307/2154297
  • [25] S. M. Khoroshkin, $ {\mathcal {D}}$-modules over the arrangements of hyperplanes, Comm. Algebra 23 (1995), no. 9, 3481-3504. MR 1335310 (96h:32050), https://doi.org/10.1080/00927879508825410
  • [26] S. Khoroshkin and A. Varchenko, Quiver $ \mathcal {D}$-modules and homology of local systems over an arrangement of hyperplanes, IMRP Int. Math. Res. Pap. (2006), Art. ID 69590, 116. MR 2282180 (2009d:32030)
  • [27] Gennady Lyubeznik, The minimal non-Cohen-Macaulay monomial ideals, J. Pure Appl. Algebra 51 (1988), no. 3, 261-266. MR 946577 (89h:13031), https://doi.org/10.1016/0022-4049(88)90065-5
  • [28] Gennady Lyubeznik, Finiteness properties of local cohomology modules (an application of $ D$-modules to commutative algebra), Invent. Math. 113 (1993), no. 1, 41-55. MR 1223223 (94e:13032), https://doi.org/10.1007/BF01244301
  • [29] Gennady Lyubeznik, $ F$-modules: applications to local cohomology and $ D$-modules in characteristic $ p>0$, J. Reine Angew. Math. 491 (1997), 65-130. MR 1476089 (99c:13005), https://doi.org/10.1515/crll.1997.491.65
  • [30] Ezra Miller, The Alexander duality functors and local duality with monomial support, J. Algebra 231 (2000), no. 1, 180-234. MR 1779598 (2001k:13028), https://doi.org/10.1006/jabr.2000.8359
  • [31] Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098 (2006d:13001)
  • [32] Mircea Mustaţa, Local cohomology at monomial ideals, J. Symbolic Comput. 29 (2000), no. 4-5, 709-720. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). MR 1769662 (2001i:13020), https://doi.org/10.1006/jsco.1999.0302
  • [33] Irena Peeva and Mauricio Velasco, Frames and degenerations of monomial resolutions, Trans. Amer. Math. Soc. 363 (2011), no. 4, 2029-2046. MR 2746674 (2011k:13021), https://doi.org/10.1090/S0002-9947-2010-04980-3
  • [34] V. Reiner and V. Welker, Linear syzygies of Stanley-Reisner ideals, Math. Scand. 89 (2001), no. 1, 117-132. MR 1856984 (2003a:13016)
  • [35] Victor Reiner, Volkmar Welker, and Kohji Yanagawa, Local cohomology of Stanley-Reisner rings with supports in general monomial ideals, J. Algebra 244 (2001), no. 2, 706-736. MR 1859045 (2002g:13036), https://doi.org/10.1006/jabr.2001.8932
  • [36] T. Römer,
    On minimal graded free resolutions,
    Ph.D. Thesis, Essen (2001).
  • [37] P. Schenzel,
    On the structure of the endomorphism ring of a certain local cohomology module, J. Algebra 344 (2011), 229-245.
  • [38] Anne-Marie Simon, Some homological properties of complete modules, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 231-246. MR 1074711 (91k:13008), https://doi.org/10.1017/S0305004100069103
  • [39] Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston Inc., Boston, MA, 1996. MR 1453579 (98h:05001)
  • [40] N. Terai,
    Local cohomology modules with respect to monomial ideals,
    Preprint 1999.
  • [41] Mauricio Velasco, Minimal free resolutions that are not supported by a CW-complex, J. Algebra 319 (2008), no. 1, 102-114. MR 2378063 (2008j:13028), https://doi.org/10.1016/j.jalgebra.2007.10.011
  • [42] Uli Walther, Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties, J. Pure Appl. Algebra 139 (1999), no. 1-3, 303-321. Effective methods in algebraic geometry (Saint-Malo, 1998). MR 1700548 (2000h:13012), https://doi.org/10.1016/S0022-4049(99)00016-X
  • [43] Kohji Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree $ \mathbf {N}^n$-graded modules, J. Algebra 225 (2000), no. 2, 630-645. MR 1741555 (2000m:13036), https://doi.org/10.1006/jabr.1999.8130
  • [44] Kohji Yanagawa, Bass numbers of local cohomology modules with supports in monomial ideals, Math. Proc. Cambridge Philos. Soc. 131 (2001), no. 1, 45-60. MR 1833073 (2002c:13037), https://doi.org/10.1017/S030500410100514X

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Additional Information

Josep Àlvarez Montaner
Affiliation: Department Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal 647, Barcelona 08028, Spain
Email: Josep.Alvarez@upc.edu

Alireza Vahidi
Affiliation: Department of Mathematics, Payame Noor University, 19395-4697 Tehran, I.R. of Iran
Email: vahidi.ar@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2013-05862-X
Received by editor(s): August 14, 2011
Received by editor(s) in revised form: February 7, 2012, and April 17, 2012
Published electronically: November 25, 2013
Additional Notes: The first author was partially supported by MTM2010-20279-C02-01 and SGR2009-1284
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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