Centers of \(F\)-purity. (English) Zbl 1213.13014

Let \(R\) be a Noetherian excellent ring of positive characteristic \(p\), where \(p\) is prime. The paper under review introduces, in positive characteristic, the notion of a center of \(F\)-purity for a pair \((X=\text{Spec}(R), \Delta)\) that is analogous to the zero characteristic notion of a center of log canonicity for a pair \((X, \Delta)\), where \(X\) is a normal affine scheme and \(\Delta\) an effective \(\mathbb{Q}\)-divisor on it. In fact, the author develops the theory for the context of triples \((R, \Delta, \mathfrak{a}_{\bullet})\), where \(R\) is a ring as above (normal when \(\Delta \neq 0\)), \(\Delta\) an effective \(\mathbb{R}\)-divisor and \(\mathfrak{a}_{\bullet}\) a graded system of ideals that satisfy some natural conditions with regard to \(R\). The author defines the concept of sharply \(F\)-pure and strong \(F\)-regular triples, which extends his earlier work [Math. Res. Lett. 15, No. 5-6, 1251–1261 (2008; Zbl 1185.13010)], the notion of big test ideal for triples, and proves basic results on them.
The core of the paper consists of the theory of ideals of \(R\) that are \((\Delta, \mathfrak{a}_{\bullet}, F)\)-compatible. This notion is related to that of \(F\)-compatible ideals of V. B. Mehta and A. Ramanathan [Ann. Math. (2) 122(1), 27–40 (1985; Zbl 0601.14043)] and leads naturally to the concept of centers of (sharp) \(F\)-purity which are the characteristic zero analog of centers of log canonicity. Important results connected to the work of G. Lyubeznik and K. E. Smith [Trans. Am. Math. Soc. 353, No. 8, 3149–3180 (2001; Zbl 0977.13002) (electronic)] on one hand and to work of I. M. Aberbach and F. Enescu [Math. Z. 250, No. 4, 791–806 (2005; Zbl 1102.13001)] on the other hand are developed. Also, finiteness results on uniformly \(F\)-compatible ideals and centers of \(F\)-purity inspired by work of F. Enescu and M. Hochster [Algebra Number Theory 2, No. 7, 721–754 (2008; Zbl 1190.13003)], and R. Y. Sharp [Trans. Am. Math. Soc. 359, No. 9, 4237–4258 (2007; Zbl 1130.13002) (electronic)] are obtained. Relations to big test ideals, multiplier ideals, and local geometric properties of the centers of \(F\)-purity are pursued in the final two sections of the paper.


13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14B05 Singularities in algebraic geometry
Full Text: DOI arXiv


[1] Aberbach I.M., Enescu F.: The structure of F-pure rings. Math. Z. 250(4), 791–806 (2005) (MR2180375) · Zbl 1102.13001
[2] Ambro, F.: The locus of log canonical singularities. available at arXiv:math.AG/9806067 (1998, preprint)
[3] Blickle M., Mustaţă M., Smith K.: Discreteness and rationality of F-thresholds, Michigan Math. J. 57, 43–61 (2008) (Dedicated to Mel Hochster on the occasion of his 65th birthday) · Zbl 1177.13013
[4] Brion, M., Kumar, S.: Frobenius splitting methods in geometry and representation theory. In: Progress in Mathematics, vol. 231. Birkhäuser Boston Inc., Boston (2005) [MR2107324 (2005k:14104)] · Zbl 1072.14066
[5] De Fernex, T., Hacon, C.: Singularities on normal varieties. Compositio Math. (2008, to appear) arXiv:0805.1767 · Zbl 1179.14003
[6] Enescu F., Hochster M.: The Frobenius structure of local cohomology. Algebra Number Theory 2(7), 721–754 (2008) (MR2460693) · Zbl 1190.13003
[7] Fedder R.: F-purity and rational singularity. Trans. Am. Math. Soc. 278(2), 461–480 (1983) [MR701505 (84h:13031)] · Zbl 0519.13017
[8] Greco S., Traverso C.: On seminormal schemes. Compos. Math. 40(3), 325–365 (1980) [MR571055 (81j:14030)] · Zbl 0412.14024
[9] Hara N.: Geometric interpretation of tight closure and test ideals. Trans. Am. Math. Soc. 353(5), 1885–1906 (2001) (electronic) [MR1813597 (2001m:13009)] · Zbl 0976.13003
[10] Hara N.: A characteristic p analog of multiplier ideals and applications. Comm. Algebra 33(10), 3375–3388 (2005) [MR2175438 (2006f:13006)] · Zbl 1090.13003
[11] Hara N., Takagi S.: On a generalization of test ideals. Nagoya Math. J. 175, 59–74 (2004) [MR2085311 (2005g:13009)] · Zbl 1094.13004
[12] Hara N., Watanabe K.-I.: F-regular and F-pure rings versus log terminal and log canonical singularities. J. Algebraic Geom. 11(2), 363–392 (2002) [MR1874118 (2002k:13009)] · Zbl 1013.13004
[13] Hara N., Yoshida K.-I.: A generalization of tight closure and multiplier ideals, Trans. Am. Math. Soc. 355(8), 3143–3174 (2003) (electronic) [MR1974679 (2004i:13003)] · Zbl 1028.13003
[14] Hartshorne, R.: Generalized divisors on Gorenstein schemes. In: Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), vol. 8, pp. 287–339 (1994) [MR1291023 (95k:14008)] · Zbl 0826.14005
[15] Hironaka H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I. Ann. Math. (2) 79, 109–203 (1964) [MR0199184 (33 #7333)] · Zbl 0122.38603
[16] Hironaka H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. II. Ann. Math. (2) 79, 205–326 (1964) [MR0199184 (33 #7333)] · Zbl 0122.38603
[17] Hochster, M.: Foundations of tight closure theory, lecture notes from a course taught on the University of Michigan Fall 2007. Available online at http://www.math.lsa.umich.edu/\(\sim\)hochster/711F07/711.html
[18] Hochster M., Huneke C.: Tight closure, invariant theory, and the Briançon-Skoda theorem. J. Am. Math. Soc. 3(1), 31–116 (1990) [MR1017784 (91g:13010)] · Zbl 0701.13002
[19] Hochster M., Huneke C.: F-regularity, test elements, and smooth base change. Trans. Am. Math. Soc. 346(1), 1–62 (1994) [MR1273534 (95d:13007)] · Zbl 0844.13002
[20] Hochster, M., Huneke, C.: Tight closure in equal characteristic zero (2006, preprint)
[21] Hochster M., Roberts J.L.: The purity of the Frobenius and local cohomology. Advances Math. 21(2), 117–172 (1976) [MR0417172 (54 #5230)] · Zbl 0348.13007
[22] Huneke, C., Swanson, I.: Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, vol.336. Cambridge University Press, Cambridge (2006) [MR2266432] · Zbl 1117.13001
[23] Kawamata Y.: Subadjunction of log canonical divisors. II. Am. J. Math. 120(5), 893–899 (1998) [MR1646046 (2000d:14020)] · Zbl 0919.14003
[24] Kollár J., Shepherd-Barron N.I.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988) [MR922803 (88m:14022)] · Zbl 0642.14008
[25] Kollár, J.: Singularities of pairs, Algebraic geometry–Santa Cruz 1995. In: Proc. Sympos. Pure Math., vol. 62, pp. 221–287. Amer. Math. Soc., Providence (1997) [MR1492525 (99m:14033)]
[26] Kollár, J., 14 coauthors: Flips and abundance for algebraic threefolds. Société Mathématique de France, Paris (1992) [Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992). MR1225842 (94f:14013)]
[27] Kollár, J., Mori, S.: Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998) [With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. MR1658959 (2000b:14018)] · Zbl 0926.14003
[28] Kovács, S., Schwede, K., Smith, K.: Cohen-Macaulay semi-log canonical singularities are Du Bois arXiv:0801.1541
[29] Lauritzen N., Raben-Pedersen U., Thomsen J.F.: Global F-regularity of Schubert varieties with applications to \({\fancyscript {D}}\) -modules. J. Am. Math. Soc. 19(2), 345–355 (2006) (electronic) [MR2188129 (2006h:14005)] · Zbl 1098.14038
[30] Lazarsfeld, R.: 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, Berlin (2004) [Positivity for vector bundles, and multiplier ideals. MR2095472 (2005k:14001b)]
[31] Lyubeznik G., Smith K.E.: On the commutation of the test ideal with localization and completion, Trans. Am. Math. Soc. 353(8), 3149–3180 (2001) (electronic) [MR1828602 (2002f:13010)] · Zbl 0977.13002
[32] Mehta V.B., Ramanathan A.: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. Math. (2) 122(1), 27–40 (1985) [MR799251 (86k:14038)] · Zbl 0601.14043
[33] Schwede K.: Generalized test ideals, sharp F-purity, and sharp test elements. Math. Res. Lett. 15(6), 1251–1261 (2008) (MR2470398) · Zbl 1185.13010
[34] Schwede K.E.: F-injective singularities are Du Bois. Am. J. Math. 131(2), 445–473 (2009) · Zbl 1164.14001
[35] Sharp R.Y.: Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure. Trans. Am. Math. Soc. 359(9), 4237–4258 (2007) (electronic) [MR2309183 (2008b:13006)] · Zbl 1130.13002
[36] Smith K.E.: The D-module structure of F-split rings. Math. Res. Lett. 2(4), 377–386 (1995) [MR1355702 (96j:13024)] · Zbl 0848.13010
[37] Smith K.E.: F-rational rings have rational singularities. Am. J. Math. 119(1), 159–180 (1997) [MR1428062 (97k:13004)] · Zbl 0910.13004
[38] Smith K.E.: The multiplier ideal is a universal test ideal. Comm. Algebra 28(12), 5915–5929 (2000) [Special issue in honor of Robin Hartshorne. MR1808611 (2002d:13008)] · Zbl 0979.13007
[39] Takagi S.: F-singularities of pairs and inversion of adjunction of arbitrary codimension. Invent. Math. 157(1), 123–146 (2004) (MR2135186) · Zbl 1121.13008
[40] Takagi S.: An interpretation of multiplier ideals via tight closure. J. Algebraic Geom. 13(2), 393–415 (2004) [MR2047704 (2005c:13002)] · Zbl 1080.14004
[41] Takagi S.: Formulas for multiplier ideals on singular varieties. Am. J. Math. 128(6), 1345–1362 (2006) [MR2275023 (2007i:14006)] · Zbl 1109.14005
[42] Takagi S.: A characteristic p analogue of plt singularities and adjoint ideals. Math. Z. 259(2), 321–341 (2008) [MR2390084 (2009b:13004)] · Zbl 1143.13007
[43] Traverso C.: Seminormality and Picard group. Ann. Scuola Norm. Sup. Pisa 24(3), 585–595 (1970) [MR0277542 (43 #3275)] · Zbl 0205.50501
[44] Vassilev J.C.: Test ideals in quotients of F-finite regular local rings. Trans. Am. Math. Soc. 350(10), 4041–4051 (1998) [MR1458336 (98m:13009)] · Zbl 0913.13005
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