Regularity, singularities and $h$-vector of graded algebras
HTML articles powered by AMS MathViewer
- by Hailong Dao, Linquan Ma and Matteo Varbaro;
- Trans. Amer. Math. Soc. 377 (2024), 2149-2167
- DOI: https://doi.org/10.1090/tran/9089
- Published electronically: January 16, 2024
- HTML | PDF | Request permission
Abstract:
Let $R$ be a standard graded algebra over a field. We investigate how the singularities of $\operatorname {Spec} R$ or $\operatorname {Proj} R$ affect the $h$-vector of $R$, which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if $R$ satisfies Serre’s condition $(S_r)$ and has reasonable singularities (Du Bois on the punctured spectrum or $F$-pure), then $h_0$, …, $h_r\geq 0$. Furthermore the multiplicity of $R$ is at least $h_0+h_1+\dots +h_{r-1}$. We also prove that equality in many cases forces $R$ to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain $\operatorname {Ext}$ modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and $F$-pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others.References
- F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 220–239; English transl., Proc. Steklov Inst. Math. 1(240) (2003), 214–233. MR 1993751
- Luchezar L. Avramov, Ragnar-Olaf Buchweitz, and Judith D. Sally, Laurent coefficients and $\textrm {Ext}$ of finite graded modules, Math. Ann. 307 (1997), no. 3, 401–415. MR 1437046, DOI 10.1007/s002080050041
- Luchezar L. Avramov and Hans-Bjørn Foxby, Grothendieck’s localization problem, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 1–13. MR 1266174, DOI 10.1090/conm/159/01498
- 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
- Aaron Bertram, Lawrence Ein, and Robert Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), no. 3, 587–602. MR 1092845, DOI 10.1090/S0894-0347-1991-1092845-5
- Marc Chardin and Bernd Ulrich, Liaison and Castelnuovo-Mumford regularity, Amer. J. Math. 124 (2002), no. 6, 1103–1124. MR 1939782, DOI 10.1353/ajm.2002.0035
- Hailong Dao, Alessandro De Stefani, and Linquan Ma, Cohomologically full rings, Int. Math. Res. Not. IMRN 17 (2021), 13508–13545. MR 4307694, DOI 10.1093/imrn/rnz203
- Tommaso de Fernex and Roi Docampo, Jacobian discrepancies and rational singularities, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 1, 165–199. MR 3141731, DOI 10.4171/JEMS/430
- Tommaso de Fernex and Lawrence Ein, A vanishing theorem for log canonical pairs, Amer. J. Math. 132 (2010), no. 5, 1205–1221. MR 2732344, DOI 10.1353/ajm.2010.0008
- Lawrence Ein and Shihoko Ishii, Singularities with respect to Mather-Jacobian discrepancies, Commutative algebra and noncommutative algebraic geometry. Vol. II, Math. Sci. Res. Inst. Publ., vol. 68, Cambridge Univ. Press, New York, 2015, pp. 125–168. MR 3496863
- David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
- D. Eisenbud, M. Green, K. Hulek, and S. Popescu, Small schemes and varieties of minimal degree, Amer. J. Math. 128 (2006), no. 6, 1363–1389. MR 2275024, DOI 10.1353/ajm.2006.0043
- David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR 2103875
- A. Goodarzi, M. R. Pournaki, S. A. Seyed Fakhari, and S. Yassemi, On the $h$-vector of a simplicial complex with Serre’s condition, J. Pure Appl. Algebra 216 (2012), no. 1, 91–94. MR 2826421, DOI 10.1016/j.jpaa.2011.05.005
- Robin Hartshorne, Complete intersections and connectedness, Amer. J. Math. 84 (1962), 497–508. MR 142547, DOI 10.2307/2372986
- Melvin Hochster and Joel L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976), no. 2, 117–172. MR 417172, DOI 10.1016/0001-8708(76)90073-6
- Craig Huneke, A remark concerning multiplicities, Proc. Amer. Math. Soc. 85 (1982), no. 3, 331–332. MR 656095, DOI 10.1090/S0002-9939-1982-0656095-2
- Craig Huneke and Karen E. Smith, Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math. 484 (1997), 127–152. MR 1437301
- Shihoko Ishii, Mather discrepancy and the arc spaces, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 1, 89–111 (English, with English and French summaries). MR 3089196, DOI 10.5802/aif.2756
- Mitra Koley and Matteo Varbaro, Gröbner deformations and $F$-singularities, Math. Nachr. 296 (2023), no. 7, 2903–2917. MR 4626865, DOI 10.1002/mana.202100459
- János Kollár and Sándor J. Kovács, Deformations of log canonical and $F$-pure singularities, Algebr. Geom. 7 (2020), no. 6, 758–780. MR 4156425, DOI 10.14231/ag-2020-027
- Sándor J. Kovács and Karl E. Schwede, Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities, Topology of stratified spaces, Math. Sci. Res. Inst. Publ., vol. 58, Cambridge Univ. Press, Cambridge, 2011, pp. 51–94. MR 2796408
- Sándor J. Kovács and Karl Schwede, Du Bois singularities deform, Minimal models and extremal rays (Kyoto, 2011) Adv. Stud. Pure Math., vol. 70, Math. Soc. Japan, [Tokyo], 2016, pp. 49–65. MR 3617778, DOI 10.2969/aspm/07010049
- Sándor J. Kovács, Karl Schwede, and Karen E. Smith, The canonical sheaf of Du Bois singularities, Adv. Math. 224 (2010), no. 4, 1618–1640. MR 2646306, DOI 10.1016/j.aim.2010.01.020
- Manoj Kummini and Satoshi Murai, Regularity of canonical and deficiency modules for monomial ideals, Pacific J. Math. 249 (2011), no. 2, 377–383. MR 2782675, DOI 10.2140/pjm.2011.249.377
- Gennady Lyubeznik, Étale cohomological dimension and the topology of algebraic varieties, Ann. of Math. (2) 137 (1993), no. 1, 71–128. MR 1200077, DOI 10.2307/2946619
- Linquan Ma, $F$-injectivity and Buchsbaum singularities, Math. Ann. 362 (2015), no. 1-2, 25–42. MR 3343868, DOI 10.1007/s00208-014-1098-3
- Linquan Ma, Karl Schwede, and Kazuma Shimomoto, Local cohomology of Du Bois singularities and applications to families, Compos. Math. 153 (2017), no. 10, 2147–2170. MR 3705286, DOI 10.1112/S0010437X17007321
- Linquan Ma and Wenliang Zhang, Eulerian graded $\scr D$-modules, Math. Res. Lett. 21 (2014), no. 1, 149–167. MR 3247047, DOI 10.4310/MRL.2014.v21.n1.a13
- Satoshi Murai and Naoki Terai, $h$-vectors of simplicial complexes with Serre’s conditions, Math. Res. Lett. 16 (2009), no. 6, 1015–1028. MR 2576690, DOI 10.4310/MRL.2009.v16.n6.a10
- Wenbo Niu, Mather-Jacobian singularities under generic linkage, Trans. Amer. Math. Soc. 370 (2018), no. 6, 4015–4028. MR 3811518, DOI 10.1090/tran/7065
- Peter Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe, Lecture Notes in Mathematics, vol. 907, Springer-Verlag, Berlin-New York, 1982 (German). With an English summary. MR 654151, DOI 10.1007/BFb0094123
- Karl Schwede, A simple characterization of Du Bois singularities, Compos. Math. 143 (2007), no. 4, 813–828. MR 2339829, DOI 10.1112/S0010437X07003004
- Wolmer Vasconcelos, Integral closure, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. Rees algebras, multiplicities, algorithms. MR 2153889
- 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, DOI 10.1006/jabr.1999.8130
Bibliographic Information
- Hailong Dao
- Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-7523
- MR Author ID: 828268
- ORCID: 0000-0001-8109-9724
- Email: hdao@ku.edu
- Linquan Ma
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906
- MR Author ID: 1050700
- ORCID: 0000-0002-7452-8639
- Email: ma326@purdue.edu
- Matteo Varbaro
- Affiliation: Dipartimento di Matematica, Università di Genova Via Dodecaneso, 35 16146 Genova, Italy
- MR Author ID: 873871
- Email: varbaro@dima.unige.it
- Received by editor(s): February 9, 2023
- Received by editor(s) in revised form: July 5, 2023, and September 30, 2023
- Published electronically: January 16, 2024
- Additional Notes: The first author was partially supported by Simons Foundation Collaboration Grant 527316. The second author was partially supported by NSF Grant DMS #2302430, NSF FRG Grant #1952366, and a fellowship from the Sloan Foundation, and was partially supported by NSF Grant DMS #1836867/1600198 and #1901672 when working on this article. The third author was supported by MIUR Grant PRIN #2020355B8Y
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 2149-2167
- MSC (2020): Primary 13A02, 13A35, 14B05, 14B15, 13D45
- DOI: https://doi.org/10.1090/tran/9089
- MathSciNet review: 4744753