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ISSN 1088-6850(e) ISSN 0002-9947(p)

-module structure of local cohomology modules of toric algebras

Author: Jen-Chieh Hsiao
Journal: Trans. Amer. Math. Soc. 364 (2012), 2461-2478
MSC (2010): Primary 13D45, 13N10, 14M25
Posted: January 13, 2012
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Abstract: Let be a toric algebra over a field $\mathbb{K}$ of characteristic 0 and let be a monomial ideal of . We show that the local cohomology modules are of finite length over the ring of differential operators $D(S;\mathbb{K})$ , generalizing the classical case of a polynomial algebra . As an application, we compute the characteristic cycles of some local cohomology modules.

References

[ÁM00] 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), http://dx.doi.org/10.1016/S0022-4049(98)00171-6
[ÁM04] Josep Àlvarez Montaner, Characteristic cycles of local cohomology modules of monomial ideals. II, J. Pure Appl. Algebra 192 (2004), no. 1-3, 1–20. MR 2067186, http://dx.doi.org/10.1016/j.jpaa.2004.01.003
[ÁMBL05] Josep Alvarez-Montaner, Manuel Blickle, and Gennady Lyubeznik, Generators of 𝐷-modules in positive characteristic, Math. Res. Lett. 12 (2005), no. 4, 459–473. MR 2155224 (2006m:13024)
[Bjö79] J.-E. Björk, Rings of differential operators, North-Holland Mathematical Library, vol. 21, North-Holland Publishing Co., Amsterdam, 1979. MR 549189 (82g:32013)
[Bøg95] Rikard Bøgvad, Some results on 𝒟-modules on Borel varieties in characteristic 𝓅>0, J. Algebra 173 (1995), no. 3, 638–667. MR 1327873 (97a:14015), http://dx.doi.org/10.1006/jabr.1995.1107
[Bøg02] Rikard Bögvad, An analogue of holonomic 𝒟-modules on smooth varieties in positive characteristics, Homology Homotopy Appl. 4 (2002), no. 2, 83–116. The Roos Festschrift volume, 1. MR 1918185 (2003h:14030)
[BS98] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cambridge, 1998. MR 1613627 (99h:13020)
[BW99] Yuri Berest and George Wilson, Classification of rings of differential operators on affine curves, Internat. Math. Res. Notices 2 (1999), 105–109. MR 1670188 (2000f:14025), http://dx.doi.org/10.1155/S1073792899000057
[Ful93] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037 (94g:14028)
[Gin86] V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), no. 2, 327–402. MR 833194 (87j:32030), http://dx.doi.org/10.1007/BF01388811
[Har70] Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145–164. MR 0257096 (41 #1750)
[HM03] David Helm and Ezra Miller, Bass numbers of semigroup-graded local cohomology, Pacific J. Math. 209 (2003), no. 1, 41–66. MR 1973933 (2004c:13028), http://dx.doi.org/10.2140/pjm.2003.209.41
[HM05] David Helm and Ezra Miller, Algorithms for graded injective resolutions and local cohomology over semigroup rings, J. Symbolic Comput. 39 (2005), no. 3-4, 373–395. MR 2168288 (2007d:13025), http://dx.doi.org/10.1016/j.jsc.2004.11.009
[HS93] 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), http://dx.doi.org/10.1090/S0002-9947-1993-1124167-6
[ILL+07] Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, RI, 2007. MR 2355715 (2009a:13025)
[Ish88] Masa-Nori Ishida, The local cohomology groups of an affine semigroup ring, Algebraic geometry and commutative algebra, vol. I, Kinokuniya, Tokyo, 1988, pp. 141-153.
[Jon94] A. G. Jones, Rings of differential operators on toric varieties, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 143–160. MR 1258039 (95d:16030), http://dx.doi.org/10.1017/S0013091500018770
[Lyu93] 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), http://dx.doi.org/10.1007/BF01244301
[Lyu97] 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), http://dx.doi.org/10.1515/crll.1997.491.65
[Lyu00] Gennady Lyubeznik, Finiteness properties of local cohomology modules: a characteristic-free approach, J. Pure Appl. Algebra 151 (2000), no. 1, 43–50. MR 1770642 (2001g:13038), http://dx.doi.org/10.1016/S0022-4049(99)00080-8
[MM06] Laura Felicia Matusevich and Ezra Miller, Combinatorics of rank jumps in simplicial hypergeometric systems, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1375–1381 (electronic). MR 2199183 (2006j:33016), http://dx.doi.org/10.1090/S0002-9939-05-08245-6
[MS05] Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098 (2006d:13001)
[Mus87] Ian M. Musson, Rings of differential operators on invariant rings of tori, Trans. Amer. Math. Soc. 303 (1987), no. 2, 805–827. MR 902799 (88m:32019), http://dx.doi.org/10.1090/S0002-9947-1987-0902799-2
[Mus94] Ian M. Musson, Differential operators on toric varieties, J. Pure Appl. Algebra 95 (1994), no. 3, 303–315. MR 1295963 (95i:16026), http://dx.doi.org/10.1016/0022-4049(94)90064-7
[Sai10] Mutsumi Saito, Critical modules of the ring of differential operators of an affine semigroup algebra, Comm. Algebra 38 (2010), no. 2, 618–631. MR 2598902 (2011c:13054), http://dx.doi.org/10.1080/00927870902828603
[SS88] S. P. Smith and J. T. Stafford, Differential operators on an affine curve, Proc. London Math. Soc. (3) 56 (1988), no. 2, 229–259. MR 922654 (89d:14039), http://dx.doi.org/10.1112/plms/s3-56.2.229
[SS90] Uwe Schäfer and Peter Schenzel, Dualizing complexes of affine semigroup rings, Trans. Amer. Math. Soc. 322 (1990), no. 2, 561–582. MR 1076179 (92a:13012), http://dx.doi.org/10.1090/S0002-9947-1990-1076179-6
[SS04] Anurag K. Singh and Irena Swanson, Associated primes of local cohomology modules and of Frobenius powers, Int. Math. Res. Not. 33 (2004), 1703–1733. MR 2058025 (2005d:13030), http://dx.doi.org/10.1155/S1073792804133424
[ST01] Mutsumi Saito and William N. Traves, Differential algebras on semigroup algebras, engineering (South Hadley, MA, 2000) Contemp. Math., vol. 286, Amer. Math. Soc., Providence, RI, 2001, pp. 207–226. MR 1874281 (2003d:16034)
[ST04] Mutsumi Saito and William N. Traves, Finite generation of rings of differential operators of semigroup algebras, J. Algebra 278 (2004), no. 1, 76–103. MR 2068067 (2005e:16041), http://dx.doi.org/10.1016/j.jalgebra.2004.01.009
[ST09] Mutsumi Saito and Ken Takahashi, Noetherian properties of rings of differential operators of affine semigroup algebras, Osaka J. Math. 46 (2009), no. 2, 529–556. MR 2549600 (2011c:16065)
[TT08] Shunsuke Takagi and Ryo Takahashi, 𝐷-modules over rings with finite 𝐹-representation type, Math. Res. Lett. 15 (2008), no. 3, 563–581. MR 2407232 (2009e:13003)

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