Test ideals and base change problems in tight closure theory

Authors:
Ian M. Aberbach and Florian Enescu

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

Published electronically:
October 9, 2002

MathSciNet review:
1932717

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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 -rationality under flat base change.

**[1]**I. M. Aberbach,*Tight closure in**-rational rings*, Nagoya Math. J.**135**(1994), 43-54. MR**95g:13020****[2]**I. M. Aberbach,*Some conditions for the equivalence of weak and strong F-regularity*, Comm. Algebra**30 (4)**(2002), 1635-1651.**[3]**I. M. Aberbach,*Extension of weakly and strongly**-regular rings by flat maps*, J. Algebra**241 (2)**(2001), 799-807.MR**2002f:13008****[4]**I. M. Aberbach, M. Hochster, and C. Huneke,*Localization of tight closure and modules of finite phantom projective dimension*, J. Reine Angew. Math. (Crelle's Journal)**434**(1993), 67-114. MR**94h:13005****[5]**A. Bravo and K. E. Smith,*Behavior of test ideals under smooth and étale homomorphisms*, J. Algebra**247 (1)**(2002), 78-94.**[6]**W. Bruns and J. Herzog,*Cohen-Macaulay rings*, Cambridge University Press, Cambridge, 1998. MR**95h:13020****[7]**F. Enescu,*On the behavior of F-rational rings under flat base change*, J. Algebra**233**(2000), 543-566. MR**2001j:13007****[8]**M. Hashimoto,*Cohen-Macaulay F-injective homomorphisms*, Geometric and combinatorial aspects of commutative algebra (Messina, 1999), Lecture Notes in Pure and Appl. Math., 217, Marcel Dekker, New York, 2001, pp. 231-244. MR**2002d:13007****[9]**M. Hochster and C. Huneke,*Tight closure, invariant theory,and the Briançon-Skoda theorem*, J. Amer. Math. Soc.**3**(1990), 31-116. MR**91g:13010****[10]**M. Hochster and C. Huneke,*Tight closure and elements of small order in integral extensions*, J. Pure Appl. Algebra**71**(1991), 233-247. MR**92i:13002****[11]**M. Hochster and C. Huneke,*-regularity, test elements, and smooth base change*, Trans. Amer. Math. Soc.**346**(1994), 1-62. MR**95d:13007****[12]**M. Hochster and C. Huneke,*Localization and test exponents for tight closure (Dedicated to William Fulton on the occasion of his 60th birthday)*, Michigan Math. J.**48**(2000), 305-329. MR**2002a:13001****[13]**C. Huneke,*Tight closure and strong test ideals*, J. Pure Appl. Algebra**122**(1997), 243-250. MR**98g:13003****[14]**E. Kunz,*On Noetherian rings of characteristic*, Amer. J. Math. (1976), 999-1013. MR**55:5612****[15]**S. Loepp and C. Rotthaus,*Some results on tight closure and completion*, J. Algebra**246**(2001), 859-880.**[16]**G. Lyubeznik and K. E. Smith,*On the commutation of the test ideal with localization and completion*, Trans. Amer. Math. Soc.**353**(2001), 3149-3180. MR**2002f:13010****[17]**H. Matsumura,*Commutative ring theory*, Cambridge University Press, Cambridge, 1986. MR**88h:13001****[18]**J. Nishimura,*A few examples of local rings, I*, preprint, 1994.**[19]**K. E. Smith,*Test ideals in local rings*, Trans. Amer. Math. Soc.**347**(1995), 3453-3472. MR**96c:13008****[20]**J. Vélez,*Openness of the F-rational locus and smooth base change*, J. Algebra**172(2)**(1995), 425-453. MR**96g:13003****[21]**A. Vraciu,*Strong test ideals*, J. Pure Appl. Algebra**167 (2-3)**(2002), 361-373.

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Additional Information

**Ian M. Aberbach**

Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211

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

DOI:
https://doi.org/10.1090/S0002-9947-02-03162-8

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

Article copyright:
© Copyright 2002
American Mathematical Society