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Cartier modules on toric varieties


Authors: Jen-Chieh Hsiao, Karl Schwede and Wenliang Zhang
Journal: Trans. Amer. Math. Soc. 366 (2014), 1773-1795
MSC (2010): Primary 14M25, 13A35, 14F18, 14B05
DOI: https://doi.org/10.1090/S0002-9947-2013-05856-4
Published electronically: November 25, 2013
MathSciNet review: 3152712
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Abstract: Assume that $ X$ is an affine toric variety of characteristic $ p > 0$. Let $ \Delta $ be an effective toric $ \mathbb{Q}$-divisor such that $ K_X+\Delta $ is $ \mathbb{Q}$-Cartier with index not divisible by $ p$ and let $ \phi _{\Delta }:F^e_*\mathscr {O}_X\to \mathscr {O}_X$ be the toric map corresponding to $ \Delta $. We identify all ideals $ I$ of $ \mathscr {O}_X$ with $ \phi _{\Delta }(F^e_* I)=I$ combinatorially and also in terms of a log resolution (giving us a version of these ideals which can be defined in characteristic zero). Moreover, given a toric ideal $ \mathfrak{a}$, we identify all ideals $ I$ fixed by the Cartier algebra generated by $ \phi _{\Delta }$ and $ \mathfrak{a}$; this answers a question by Manuel Blickle in the toric setting.


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  • [AE05] Ian M. Aberbach and Florian Enescu, The structure of F-pure rings, Math. Z. 250 (2005), no. 4, 791-806. MR 2180375 (2006m:13009), https://doi.org/10.1007/s00209-005-0776-y
  • [Amb06] Florin Ambro, Basic properties of log canonical centers, Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 39-48. MR 2779466 (2012e:14030), https://doi.org/10.4171/007-1/2
  • [Bli01] Manuel Blickle, The intersection of homology D-module in finite characteristic, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of Michigan. MR 2702619
  • [Bli09] Manuel Blickle, Test ideals via algebras of $ p^{-e}$-linear maps, J. Algebraic Geom. 22 (2013), no. 1, 49-83. MR 2993047
  • [BB09] Manuel Blickle and Gebhard Böckle, Cartier modules: finiteness results, J. Reine Angew. Math. 661 (2011), 85-123. MR 2863904, https://doi.org/10.1515/CRELLE.2011.087
  • [Brø83] Arne Brøndsted, An introduction to convex polytopes, Graduate Texts in Mathematics, vol. 90, Springer-Verlag, New York, 1983. MR 683612 (84d:52009)
  • [Die55] Jean Dieudonné, Lie groups and Lie hyperalgebras over a field of characteristic $ p>0$. II, Amer. J. Math. 77 (1955), 218-244. MR 0067872 (16,789f)
  • [FST11] Osamu Fujino, Karl Schwede, and Shunsuke Takagi, Supplements to non-lc ideal sheaves, Higher dimensional algebraic geometry, RIMS Kôkyûroku Bessatsu, B24, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 1-46. MR 2809647 (2012j:14002)
  • [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)
  • [Gab04] Ofer Gabber, Notes on some $ t$-structures, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 711-734. MR 2099084 (2005m:14025)
  • [HW02] Nobuo Hara and Kei-Ichi Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), no. 2, 363-392. MR 1874118 (2002k:13009), https://doi.org/10.1090/S1056-3911-01-00306-X
  • [HS77] Robin Hartshorne and Robert Speiser, Local cohomological dimension in characteristic $ p$, Ann. of Math. (2) 105 (1977), no. 1, 45-79. MR 0441962 (56 #353)
  • [HS06] Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432 (2008m:13013)
  • [Kaw98] Yujiro Kawamata, Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998), no. 5, 893-899. MR 1646046 (2000d:14020)
  • [KM09] Shrawan Kumar and Vikram B. Mehta, Finiteness of the number of compatibly split subvarieties, Int. Math. Res. Not. IMRN 19 (2009), 3595-3597. MR 2539185 (2010j:13012), https://doi.org/10.1093/imrn/rnp067
  • [Laz04] Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472 (2005k:14001b)
  • [Pay09] Sam Payne, Frobenius splittings of toric varieties, Algebra Number Theory 3 (2009), no. 1, 107-119. MR 2491910 (2010c:14053), https://doi.org/10.2140/ant.2009.3.107
  • [Sch09] Karl Schwede, $ F$-adjunction, Algebra Number Theory 3 (2009), no. 8, 907-950. MR 2587408 (2011b:14006), https://doi.org/10.2140/ant.2009.3.907
  • [Sch10] Karl Schwede, Centers of $ F$-purity, Math. Z. 265 (2010), no. 3, 687-714. MR 2644316 (2011e:13011), https://doi.org/10.1007/s00209-009-0536-5
  • [Sch11a] K. Schwede, A canonical linear system associated to adjoint divisors in characteristic $ p > 0$, arXiv:1107.3833.
  • [Sch11b] Karl Schwede, Test ideals in non- $ \mathbb{Q}$-Gorenstein rings, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5925-5941. MR 2817415 (2012c:13011), https://doi.org/10.1090/S0002-9947-2011-05297-9
  • [ST12] Karl Schwede and Kevin Tucker, A survey of test ideals, Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012, pp. 39-99. MR 2932591

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Additional Information

Jen-Chieh Hsiao
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Address at time of publication: Department of Mathematics, National Cheng Kung University, No. 1 University Road, Tainan 701, Taiwan R.O.C.
Email: jhsiao@math.purdue.edu, jhsiao@mail.ncku.edu.tw

Karl Schwede
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: schwede@math.psu.edu

Wenliang Zhang
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
Email: wzhang15@unl.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05856-4
Received by editor(s): August 18, 2011
Received by editor(s) in revised form: April 10, 2012
Published electronically: November 25, 2013
Additional Notes: The first author was partially supported by the NSF grant DMS #0901123
This research was initiated at the Commutative Algebra MRC held in June 2010. Support for this meeting was provided by the NSF and AMS
The second author was supported by an NSF postdoctoral fellowship and also by NSF grant DMS #1064485
The third author was partially supported by the NSF grant DMS #1068946.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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