Geometric interpretation of tight closure and test ideals

Hara, Nobuo

doi:10.1090/S0002-9947-01-02695-2

Geometric interpretation of tight closure and test ideals
HTML articles powered by AMS MathViewer

by Nobuo Hara PDF

Trans. Amer. Math. Soc. 353 (2001), 1885-1906 Request permission

Abstract:

We study tight closure and test ideals in rings of characteristic $p \gg 0$ using resolution of singularities. The notions of $F$-rational and $F$-regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider classes of singularities. The test ideal is the annihilator of the tight closure relations and plays a crucial role in the tight closure theory. It is proved that, in a normal $\mathbb Q$-Gorenstein ring of characteristic $p \gg 0$, the test ideal is equal to so-called the multiplier ideal, which is an important ideal in algebraic geometry. This is proved in more general form, and to do this we study the behavior of the test ideal and the tight closure of the zero submodule in certain local cohomology modules under cyclic covering. We reinterpret the results also for graded rings.

References

I. Aberbach and B. MacCrimmon, Some results on test ideals, Proc. Edinburgh Math. Soc. (2) 42 (1999), 541–549.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Michel Demazure, Anneaux gradués normaux, Introduction à la théorie des singularités, II, Travaux en Cours, vol. 37, Hermann, Paris, 1988, pp. 35–68 (French). MR 1074589
Lawrence Ein, Multiplier ideals, vanishing theorems and applications, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 203–219. MR 1492524, DOI 10.1090/s0894-0347-97-00223-3
Montserrat Teixidor i Bigas and Loring W. Tu, Theta divisors for vector bundles, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990) Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 327–342. MR 1188206, DOI 10.1090/conm/136/1188206
Richard Fedder, A Frobenius characterization of rational singularity in $2$-dimensional graded rings, Trans. Amer. Math. Soc. 340 (1993), no. 2, 655–668. MR 1116312, DOI 10.1090/S0002-9947-1993-1116312-3
Richard Fedder and Keiichi Watanabe, A characterization of $F$-regularity in terms of $F$-purity, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 227–245. MR 1015520, DOI 10.1007/978-1-4612-3660-3_{1}1
Hans Grauert and Oswald Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292 (German). MR 302938, DOI 10.1007/BF01403182
Nobuo Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), no. 5, 981–996. MR 1646049
Nobuo Hara and Kei-ichi Watanabe, The injectivity of Frobenius acting on cohomology and local cohomology modules, Manuscripta Math. 90 (1996), no. 3, 301–315. MR 1397659, DOI 10.1007/BF02568308
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
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, Tight closure and strong $F$-regularity, Mém. Soc. Math. France (N.S.) 38 (1989), 119–133. Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR 1044348
Melvin Hochster and Craig Huneke, Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geom. 3 (1994), no. 4, 599–670. MR 1297848
Melvin Hochster and Joel L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976), no. 2, 117–172. MR 417172, DOI 10.1016/0001-8708(76)90073-6
Craig Huneke, Tight closure and its applications, CBMS Regional Conference Series in Mathematics, vol. 88, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With an appendix by Melvin Hochster. MR 1377268, DOI 10.1016/0167-4889(95)00136-0
Craig Huneke and Karen E. Smith, Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math. 484 (1997), 127–152. MR 1437301
Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
Joseph Lipman and Bernard Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97–116. MR 600418
G. Lyubeznik and K. E. Smith, Strong and weak F-regularity are equivalent for graded rings, Amer. J. Math. 121 (1999), 1279–1290.
—, On the commutation of the test ideal with localization and completion, Trans. Amer. Math. Soc. (to appear)
B. MacCrimmon, Weak F-regularity is strong F-regularity for rings with isolated non-$\mathbb {Q}$-Gorenstein points, Trans. Amer. Math. Soc. (to appear).
V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian J. Math. 1 (1997), no. 2, 249–271. MR 1491985, DOI 10.4310/AJM.1997.v1.n2.a4
Alan Michael Nadel, Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 19, 7299–7300. MR 1015491, DOI 10.1073/pnas.86.19.7299
K. Nakakoshi and K.-i. Watanabe, in preparation.
Karen E. Smith, Test ideals in local rings, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3453–3472. MR 1311917, DOI 10.1090/S0002-9947-1995-1311917-0
Karen E. Smith, $F$-rational rings have rational singularities, Amer. J. Math. 119 (1997), no. 1, 159–180. MR 1428062
—, The multiplier ideal is a universal test ideal, preprint.
Masataka Tomari and Keiichi Watanabe, Normal $Z_r$-graded rings and normal cyclic covers, Manuscripta Math. 76 (1992), no. 3-4, 325–340. MR 1185023, DOI 10.1007/BF02567764
Keiichi Watanabe, Rational singularities with $k^{\ast }$-action, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 339–351. MR 686954
Keiichi Watanabe, $F$-regular and $F$-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), no. 2-3, 341–350. MR 1117644, DOI 10.1016/0022-4049(91)90157-W
—, F-regular and F-pure rings vs. log-terminal and log-canonical singularities, preprint.
Lori J. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), no. 3, 721–743. MR 1324179, DOI 10.1006/jabr.1995.1067