Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the commutation of the test ideal with localization and completion


Authors: Gennady Lyubeznik and Karen E. Smith
Journal: Trans. Amer. Math. Soc. 353 (2001), 3149-3180
MSC (1991): Primary 13A35; Secondary 13C99
DOI: https://doi.org/10.1090/S0002-9947-01-02643-5
Published electronically: January 18, 2001
MathSciNet review: 1828602
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

Let $R$ be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of $R$ commutes with localization and, if $R$ is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull $E$ of the residue field of every local ring of $R$ is equal to the finitistic tight closure of zero in $E$. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each $R$-module is introduced and studied. This theory gives rise to an ideal of $R$ which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of $R$ in general, and shown to equal the test ideal under the hypothesis that $0_E^*=0_E^{fg*}$in every local ring of $R$.


References [Enhancements On Off] (What's this?)

  • [AM] Aberbach, I., and MacCrimmon, B., Some results on test ideals, Proc. Edinburgh Math. Soc. (2) 42 (1999), 541-549. CMP 2000:04
  • [A1] Aoyama, Y., On the depth and the projective dimension of the canonical module, Japan. J. Math.(N.S.) 6 (1980), 61-66. MR 82h:13007
  • [A2] Aoyama, Y., Some basic results on canonical modules, J. Math. Kyoto Univ. 23 (1983), 85-94. MR 84i:13015
  • [C] Vassilev, J., Test ideals in quotients of F-finite regular local rings, Trans. Amer. Math. Soc. 350 (1998), 4041-4051. MR 98m:13009
  • [G] Glassbrenner, D., Strong F-regularity in images of regular rings, Proceedings of AMS 124 (1996), 345-353. MR 96d:13004
  • [Ha] Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York, 1993. MR 57:3116 (1st ed.)
  • [HH1] Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briancon-Skoda theorem, JAMS 3 (1990), 31-116. MR 91g:13010
  • [HH2] Hochster, M. and Huneke, C., Tight closure and strong F-regularity, Mémoires de la Société Mathématique de France 38 (1989), 119-133. MR 91i:13025
  • [HH3] Hochster, M. and Huneke, C., F-regularity, test elements and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 1-62. MR 95d:13007
  • [HH4] Hochster, M. and Huneke, C., Applications of the Existence of Big Cohen-Macaulay Algebras, Advances in Math 113 (1995), 45-117. MR 96d:13014
  • [HH5] Hochster, M. and Huneke, C., Indecomposable canonical modules and connectedness, in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra, eds. Heinzer, Huneke, Sally, Contemporary Mathematics 159 (1994), 197-208. MR 95e:13014
  • [Ku] Kunz, E., On Noetherian rings of characteristic $p$, Amer. J. Math. 97 (1975), 791-813. MR 55:5612
  • [L] Lyubeznik, G., F-modules: Applications to local cohomology and D-modules in characteristic $p > 0$, J. f. d. reine u. angew. Math. 491 (1997), 65-130. MR 99c:13005
  • [LS] Lyubeznik, G. and Smith, K., Strong and weak $F$-regularity are equivalent for graded rings, American Journal of Mathematics 121 (1999), 1279-1290. CMP 2000:05
  • [Mac] MacCrimmon, B., Strong $F$-regularity and boundedness questions in tight closure, University of Michigan, thesis (1996).
  • [Mats] Matsumura, H., Commutative Ring Theory, Cambridge University Press, Cambridge, 1989. MR 90i:13001
  • [Na] Nagata, M., Local Rings, Interscience Publishers (a division of John Wiley and Sons), New York, 1962. MR 27:5790
  • [P1] Popescu, D., General Néron desingularization, Nagoya Math. J. 100 (1985), 97-126. MR 87f:13019
  • [P2] Popescu, D., General Néron desingularization and approxiation, Nagoya Math. J. 104 (1986), 85-115. MR 88a:14007
  • [R] Radu, N., Une classe d'anneaux noetheriens, Rev. Roumaine Math. Pures Appl. 37 (1992), 79-82. MR 93g:13014
  • [S1] Smith, Karen E., Tight closure of parameter ideals and F-rationality, University of Michigan, thesis (1993).
  • [S2] Smith, Karen E., Tight closure of parameter ideals, Invent. Math. 115 (1994), 41-60. MR 94k:13006
  • [S3] Smith, Karen E., Test ideals in local rings, Trans. Amer. Math. Soc. 347 (9) (1995), 3453-3472. MR 96c:13008
  • [S4] Smith, Karen E., F-rational rings have rational singularities, Amer. Jour. Math. 119 (1) (1997), 159-180. MR 97k:13004
  • [Sw] Swan, R., Néron-Popescu desingularization, Algebra and Geometry (Taipei, 1995), Lect. Algebra Geom., vol. 2, International Press, Cambridge, MA, 135-192. MR 2000h:13006
  • [W] Williams, L., Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, Jour. of Alg. 172 (1995), 721-743. MR 96f:13003

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13A35, 13C99

Retrieve articles in all journals with MSC (1991): 13A35, 13C99


Additional Information

Gennady Lyubeznik
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: gennady@math.umn.edu

Karen E. Smith
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: kesmith@math.lsa.umich.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02643-5
Keywords: Tight closure, test ideal, localisation, Frobenius action
Received by editor(s): January 4, 1999
Received by editor(s) in revised form: July 12, 1999, and March 25, 2000
Published electronically: January 18, 2001
Additional Notes: Both authors were supported by the National Science Foundation
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society