Test ideals in non-ℚ-Gorenstein rings

Schwede, Karl

doi:10.1090/S0002-9947-2011-05297-9

Test ideals in non-$\mathbb {Q}$-Gorenstein rings
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by Karl Schwede

Trans. Amer. Math. Soc. 363 (2011), 5925-5941

DOI: https://doi.org/10.1090/S0002-9947-2011-05297-9

Published electronically: June 3, 2011

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Abstract:

Suppose that $X = \operatorname {Spec}R$ is an $F$-finite normal variety in characteristic $p > 0$. In this paper we show that the big test ideal $\tau _b(R) = \widetilde {\tau (R)}$ is equal to $\sum _{\Delta } \tau (R; \Delta )$, where the sum is over $\Delta$ such that $K_X + \Delta$ is $\mathbb {Q}$-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 $R$ is not even necessarily normal.

References

Ian M. Aberbach and Brian MacCrimmon, Some results on test elements, Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 3, 541–549. MR 1721770, DOI 10.1017/S0013091500020502
Manuel Blickle, Multiplier ideals and modules on toric varieties, Math. Z. 248 (2004), no. 1, 113–121. MR 2092724, DOI 10.1007/s00209-004-0655-y
Manuel Blickle, Mircea Mustaţǎ, and Karen E. Smith, Discreteness and rationality of $F$-thresholds, Michigan Math. J. 57 (2008), 43–61. Special volume in honor of Melvin Hochster. MR 2492440, DOI 10.1307/mmj/1220879396
Tommaso de Fernex and Christopher D. Hacon, Singularities on normal varieties, Compos. Math. 145 (2009), no. 2, 393–414. MR 2501423, DOI 10.1112/S0010437X09003996
Richard Fedder, $F$-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), no. 2, 461–480. MR 701505, DOI 10.1090/S0002-9947-1983-0701505-0
Nobuo Hara, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1885–1906. MR 1813597, DOI 10.1090/S0002-9947-01-02695-2
Nobuo Hara, A characteristic $p$ analog of multiplier ideals and applications, Comm. Algebra 33 (2005), no. 10, 3375–3388. MR 2175438, DOI 10.1080/AGB-200060022
Nobuo Hara and Shunsuke Takagi, On a generalization of test ideals, Nagoya Math. J. 175 (2004), 59–74. MR 2085311, DOI 10.1017/S0027763000008904
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, DOI 10.1090/S1056-3911-01-00306-X
Nobuo Hara and Ken-Ichi Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3143–3174. MR 1974679, DOI 10.1090/S0002-9947-03-03285-9
M. Hochster, Foundations of tight closure theory, Lecture notes from a course taught on the University of Michigan, Fall 2007.
Melvin Hochster and Craig Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784, DOI 10.1090/S0894-0347-1990-1017784-6
Melvin Hochster and Craig Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62. MR 1273534, DOI 10.1090/S0002-9947-1994-1273534-X
Melvin Hochster and Joel L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115–175. MR 347810, DOI 10.1016/0001-8708(74)90067-X
Ernst Kunz, On Noetherian rings of characteristic $p$, Amer. J. Math. 98 (1976), no. 4, 999–1013. MR 432625, DOI 10.2307/2374038
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, DOI 10.1007/978-3-642-18808-4
Robert Lazarsfeld, Kyungyong Lee, and Karen E. Smith, Syzygies of multiplier ideals on singular varieties, Michigan Math. J. 57 (2008), 511–521. Special volume in honor of Melvin Hochster. MR 2492466, DOI 10.1307/mmj/1220879422
Gennady Lyubeznik and Karen E. Smith, Strong and weak $F$-regularity are equivalent for graded rings, Amer. J. Math. 121 (1999), no. 6, 1279–1290. MR 1719806
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. MR 1828602, DOI 10.1090/S0002-9947-01-02643-5
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, DOI 10.2307/1971368
Karl Schwede and Karen E. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797, DOI 10.1016/j.aim.2009.12.020
Karl Schwede, Centers of $F$-purity, Math. Z. 265 (2010), no. 3, 687–714. MR 2644316, DOI 10.1007/s00209-009-0536-5
Karl Schwede, Generalized test ideals, sharp $F$-purity, and sharp test elements, Math. Res. Lett. 15 (2008), no. 6, 1251–1261. MR 2470398, DOI 10.4310/MRL.2008.v15.n6.a14
K. Schwede, $F$-adjunction, Algebra Number Theory 3 (2009), no. 8, 907–950.
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, DOI 10.1080/00927870008827196
Shunsuke Takagi, F-singularities of pairs and inversion of adjunction of arbitrary codimension, Invent. Math. 157 (2004), no. 1, 123–146. MR 2135186, DOI 10.1007/s00222-003-0350-3
Shunsuke Takagi, An interpretation of multiplier ideals via tight closure, J. Algebraic Geom. 13 (2004), no. 2, 393–415. MR 2047704, DOI 10.1090/S1056-3911-03-00366-7
Shunsuke Takagi, A characteristic $p$ analogue of plt singularities and adjoint ideals, Math. Z. 259 (2008), no. 2, 321–341. MR 2390084, DOI 10.1007/s00209-007-0227-z