Test ideals in non Gorenstein rings
Author:
Karl Schwede
Journal:
Trans. Amer. Math. Soc. 363 (2011), 59255941
MSC (2010):
Primary 13A35, 14F18, 14B05
Published electronically:
June 3, 2011
MathSciNet review:
2817415
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Abstract: Suppose that is an finite normal variety in characteristic . In this paper we show that the big test ideal is equal to , where the sum is over such that is Cartier. This affirmatively answers a question asked by various people, including Blickle, Lazarsfeld, K. Lee and K. Smith. Furthermore, we have a version of this result in the case that is not even necessarily normal.
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Lyubeznik and Karen
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graded rings, Amer. J. Math. 121 (1999), no. 6,
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(2000m:13006)
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Gennady
Lyubeznik and Karen
E. Smith, On the commutation of the test ideal
with localization and completion, Trans. Amer.
Math. Soc. 353 (2001), no. 8, 3149–3180 (electronic). MR 1828602
(2002f:13010), 10.1090/S0002994701026435
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V.
B. Mehta and A.
Ramanathan, Frobenius splitting and cohomology vanishing for
Schubert varieties, Ann. of Math. (2) 122 (1985),
no. 1, 27–40. MR 799251
(86k:14038), 10.2307/1971368
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Karl
Schwede and Karen
E. Smith, Globally 𝐹regular and log Fano varieties,
Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797
(2011e:14076), 10.1016/j.aim.2009.12.020
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Karl
Schwede, Centers of 𝐹purity, Math. Z.
265 (2010), no. 3, 687–714. MR 2644316
(2011e:13011), 10.1007/s0020900905365
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Karl
Schwede, Generalized test ideals, sharp 𝐹purity, and sharp
test elements, Math. Res. Lett. 15 (2008),
no. 6, 1251–1261. MR 2470398
(2010e:13004), 10.4310/MRL.2008.v15.n6.a14
 [Sch09]
K. Schwede, adjunction, Algebra Number Theory 3 (2009), no. 8, 907950.
 [Smi00]
Karen
E. Smith, The multiplier ideal is a universal test ideal,
Comm. Algebra 28 (2000), no. 12, 5915–5929.
Special issue in honor of Robin Hartshorne. MR 1808611
(2002d:13008), 10.1080/00927870008827196
 [Tak04a]
Shunsuke
Takagi, Fsingularities of pairs and inversion of adjunction of
arbitrary codimension, Invent. Math. 157 (2004),
no. 1, 123–146. MR 2135186
(2006g:14028), 10.1007/s0022200303503
 [Tak04b]
Shunsuke
Takagi, An interpretation of multiplier ideals
via tight closure, J. Algebraic Geom.
13 (2004), no. 2,
393–415. MR 2047704
(2005c:13002), 10.1090/S1056391103003667
 [Tak08]
Shunsuke
Takagi, A characteristic 𝑝 analogue of plt singularities
and adjoint ideals, Math. Z. 259 (2008), no. 2,
321–341. MR 2390084
(2009b:13004), 10.1007/s002090070227z
 [AM99]
 I. M. Aberbach and B. MacCrimmon, Some results on test elements, Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 3, 541549. MR 1721770 (2000i:13005)
 [Bli04]
 M. Blickle, Multiplier ideals and modules on toric varieties, Math. Z. 248 (2004), no. 1, 113121. MR 2092724 (2006a:14082)
 [BMS08]
 M. Blickle, M. Mustaţă, and K. Smith, Discreteness and rationality of Fthresholds, Michigan Math. J. 57 (2008), 4361. MR 2492440 (2010c:13003)
 [DH09]
 T. de Fernex and C. Hacon, Singularities on normal varieties, Compos. Math. 145 (2009), no. 2, 393414. MR 2501423 (2010c:14013)
 [Fed83]
 R. Fedder, purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), no. 2, 461480. MR 701505 (84h:13031)
 [Har01]
 N. Hara, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), no. 5, 18851906 (electronic). MR 1813597 (2001m:13009)
 [Har05]
 N. Hara, A characteristic analog of multiplier ideals and applications, Comm. Algebra 33 (2005), no. 10, 33753388. MR 2175438 (2006f:13006)
 [HT04]
 N. Hara and S. Takagi, On a generalization of test ideals, Nagoya Math. J. 175 (2004), 5974. MR 2085311 (2005g:13009)
 [HW02]
 N. Hara and K.I. Watanabe, Fregular and Fpure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), no. 2, 363392. MR 1874118 (2002k:13009)
 [HY03]
 N. Hara and K.I. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), no. 8, 31433174 (electronic). MR 1974679 (2004i:13003)
 [Hoc07]
 M. Hochster, Foundations of tight closure theory, Lecture notes from a course taught on the University of Michigan, Fall 2007.
 [HH90]
 M. Hochster and C. Huneke, Tight closure, invariant theory, and the BriançonSkoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31116. MR 1017784 (91g:13010)
 [HH94]
 M. Hochster and C. Huneke, regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 162. MR 1273534 (95d:13007)
 [HR74]
 M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are CohenMacaulay, Advances in Math. 13 (1974), 115175. MR 0347810 (50:311)
 [Kun76]
 E. Kunz, On Noetherian rings of characteristic , Amer. J. Math. 98 (1976), no. 4, 9991013. MR 0432625 (55:5612)
 [Laz04]
 R. 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, SpringerVerlag, Berlin, 2004, Positivity for vector bundles, and multiplier ideals. MR 2095472 (2005k:14001b)
 [LLS08]
 R. Lazarsfeld, K. Lee, and K. E. Smith, Syzygies of multiplier ideals on singular varieties, Michigan Math. J. 57 (2008), 511521, Special volume in honor of Melvin Hochster. MR 2492466
 [LS99]
 G. Lyubeznik and K. E. Smith, Strong and weak regularity are equivalent for graded rings, Amer. J. Math. 121 (1999), no. 6, 12791290. MR 1719806 (2000m:13006)
 [LS01]
 G. Lyubeznik and K. E. Smith, On the commutation of the test ideal with localization and completion, Trans. Amer. Math. Soc. 353 (2001), no. 8, 31493180 (electronic). MR 1828602 (2002f:13010)
 [MR85]
 V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 2740. MR 799251 (86k:14038)
 [SS09]
 K. Schwede and K. Smith, Globally regular and log Fano varieties, Advances in Mathematics 224 (2010), no. 3, 863894. MR 2628797
 [Sch08a]
 K. Schwede, Centers of purity, Mathematische Zeitschrift 265 (2010), no. 3, 687714. MR 2644316
 [Sch08b]
 K. Schwede, Generalized test ideals, sharp purity, and sharp test elements, Math. Res. Lett. 15 (2008), no. 6, 12511261. MR 2470398
 [Sch09]
 K. Schwede, adjunction, Algebra Number Theory 3 (2009), no. 8, 907950.
 [Smi00]
 K. E. Smith, The multiplier ideal is a universal test ideal, Comm. Algebra 28 (2000), no. 12, 59155929, Special issue in honor of Robin Hartshorne. MR 1808611 (2002d:13008)
 [Tak04a]
 S. Takagi, Fsingularities of pairs and inversion of adjunction of arbitrary codimension, Invent. Math. 157 (2004), no. 1, 123146. MR 2135186
 [Tak04b]
 S. Takagi, An interpretation of multiplier ideals via tight closure, J. Algebraic Geom. 13 (2004), no. 2, 393415. MR 2047704 (2005c:13002)
 [Tak08]
 S. Takagi, A characteristic analogue of plt singularities and adjoint ideals, Math. Z. 259 (2008), no. 2, 321341. MR 2390084 (2009b:13004)
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Additional Information
Karl Schwede
Affiliation:
Department of Mathematics, The Pennsylvania State University, 318C McAllister Building, University Park, Pennsylvania 16802
Email:
schwede@math.psu.edu
DOI:
http://dx.doi.org/10.1090/S000299472011052979
Keywords:
Tight closure,
test ideal,
$\mathbb{Q}$Gorenstein,
log $\mathbb{Q}$Gorenstein,
multiplier ideal,
$F$singularities
Received by editor(s):
June 24, 2009
Received by editor(s) in revised form:
November 30, 2009
Published electronically:
June 3, 2011
Additional Notes:
The author was partially supported by a National Science Foundation postdoctoral fellowship and by RTG grant number 0502170.
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
