Test ideals and base change problems in tight closure theory
HTML articles powered by AMS MathViewer
- by Ian M. Aberbach and Florian Enescu
- Trans. Amer. Math. Soc. 355 (2003), 619-636
- DOI: https://doi.org/10.1090/S0002-9947-02-03162-8
- Published electronically: October 9, 2002
- PDF | Request permission
Abstract:
Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of $F$-rationality under flat base change.References
- Ian M. Aberbach, Tight closure in $F$-rational rings, Nagoya Math. J. 135 (1994), 43–54. MR 1295816, DOI 10.1017/S0027763000004943
- I. M. Aberbach, Some conditions for the equivalence of weak and strong F-regularity, Comm. Algebra 30 (4) (2002), 1635–1651.
- Ian M. Aberbach, Extension of weakly and strongly F-regular rings by flat maps, J. Algebra 241 (2001), no. 2, 799–807. MR 1843326, DOI 10.1006/jabr.2001.8785
- Ian M. Aberbach, Melvin Hochster, and Craig Huneke, Localization of tight closure and modules of finite phantom projective dimension, J. Reine Angew. Math. 434 (1993), 67–114. MR 1195691, DOI 10.1515/crll.1993.434.67
- A. Bravo and K. E. Smith, Behavior of test ideals under smooth and étale homomorphisms, J. Algebra 247 (1) (2002), 78–94.
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Florian Enescu, On the behavior of F-rational rings under flat base change, J. Algebra 233 (2000), no. 2, 543–566. MR 1793916, DOI 10.1006/jabr.2000.8430
- Mitsuyasu Hashimoto, Cohen-Macaulay F-injective homomorphisms, Geometric and combinatorial aspects of commutative algebra (Messina, 1999) Lecture Notes in Pure and Appl. Math., vol. 217, Dekker, New York, 2001, pp. 231–244. MR 1824233
- 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 elements of small order in integral extensions, J. Pure Appl. Algebra 71 (1991), no. 2-3, 233–247. MR 1117636, DOI 10.1016/0022-4049(91)90149-V
- 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 Craig Huneke, Localization and test exponents for tight closure, Michigan Math. J. 48 (2000), 305–329. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786493, DOI 10.1307/mmj/1030132721
- Craig Huneke, Tight closure and strong test ideals, J. Pure Appl. Algebra 122 (1997), no. 3, 243–250. MR 1481089, DOI 10.1016/S0022-4049(97)00053-4
- Ernst Kunz, On Noetherian rings of characteristic $p$, Amer. J. Math. 98 (1976), no. 4, 999–1013. MR 432625, DOI 10.2307/2374038
- S. Loepp and C. Rotthaus, Some results on tight closure and completion, J. Algebra 246 (2001), 859-880.
- 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
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- J. Nishimura, A few examples of local rings, I, preprint, 1994.
- 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
- Juan D. Vélez, Openness of the F-rational locus and smooth base change, J. Algebra 172 (1995), no. 2, 425–453. MR 1322412, DOI 10.1016/S0021-8693(05)80010-9
- A. Vraciu, Strong test ideals, J. Pure Appl. Algebra 167 (2-3) (2002), 361–373.
Bibliographic Information
- Ian M. Aberbach
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 314830
- Email: aberbach@math.missouri.edu
- Florian Enescu
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109; Institute of Mathematics of the Romanian Academy, Bucharest, Romania
- Address at time of publication: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- Email: fenescu@umich.edu
- Received by editor(s): October 30, 2001
- Published electronically: October 9, 2002
- Additional Notes: The first author was partially supported by the NSF and by the University of Missouri Research Board. The second author thanks the University of Michigan for support through the Rackham Predoctoral Fellowship
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 619-636
- MSC (2000): Primary 13A35; Secondary 13B40
- DOI: https://doi.org/10.1090/S0002-9947-02-03162-8
- MathSciNet review: 1932717