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)

 
 

 

Test ideals in quotients
of $F$-finite regular local rings


Author: Janet Cowden Vassilev
Journal: Trans. Amer. Math. Soc. 350 (1998), 4041-4051
MSC (1991): Primary 13A35
DOI: https://doi.org/10.1090/S0002-9947-98-02128-X
MathSciNet review: 1458336
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $S$ be an $F$-finite regular local ring and $I$ an ideal contained in $S$. Let $R=S/I$. Fedder proved that $R$ is $F$-pure if and only if $(I^{[p]}:I)\break \nsubseteq \mathfrak{m}^{[p]}$. We have noted a new proof for his criterion, along with showing that $(I^{[q]}:I) \subseteq (\tau ^{[q]}:\tau )$, where $\tau $ is the pullback of the test ideal for $R$. Combining the the $F$-purity criterion and the above result we see that if $R=S/I$ is $F$-pure then $R/\tau $ is also $F$-pure. In fact, we can form a filtration of $R$, $I \subseteq \tau = \tau _{0} \subseteq \tau _{1} \subseteq \ldots \subseteq \tau _{i} \subseteq \ldots $ that stabilizes such that each $R/\tau _{i}$ is $F$-pure and its test ideal is $\tau _{i+1}$. To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let $R=T/I$, where $T$ is either a polynomial or a power series ring and $I= P_{1} \cap \ldots \cap P_{n}$ is generated by monomials and the $R/P_{i}$ are regular. Set $J = \Sigma (P_{1} \cap \ldots \cap \hat {P_{i}} \cap \ldots \cap P_{n})$. Then $J=\tau =\tau _{par}$.


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

  • [BH] Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993. MR 95h:13020
  • [Ei] Eisenbud, D., Commutative Algebra: With a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995. MR 97a:13001
  • [Fe] Fedder, R., $F$-purity and rational singularity, Trans. Amer. Math. Soc 278 (1983), 461-480. MR 84h:13031
  • [FW] Fedder, R. and Watanabe, K., A characterization of F-regularity in terms of F-purity, Commutative Algebra, MSRI Publications, Springer-Verlag, 1989, pp. 227-245. MR 91k:13009
  • [Gl] Glassbrenner, D, Strong $F$-regularity in images of regular rings, Proc. Amer. Math. Soc. 124 (1996), 345-353. MR 96d:13004
  • [GW] Goto, S, Watanabe, K, The structure of one-dimensional $F$-pure rings, J. of Alg. 49 (1977), 415-421. MR 56:11989
  • [Ha] Hara, N., A characterization of rational singularities in terms of injectivity of Frobenius maps, preprint.
  • [Ho] Hochster, M., Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463-488. MR 57:3111
  • [HH1] Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 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
  • [Hu] Huneke, C., Hilbert functions and symbolic powers, Mich. Math. J. 34 (1987), 293-318. MR 89b:13037
  • [HS] Huneke, C. and Smith, K., Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Matth. 484 (1997), 127-152. CMP 97:09
  • [Ku1] Kunz, E., Characterization of regular local rings for characteristic $p$, Amer. J. Math 91 (1969), 772-784. MR 40:5609
  • [Ku2] Kunz, E., On Noetherian rings of characteristic $p$, Am. J. Math. 98 (1976), 999-1013. MR 55:5612
  • [S1] Smith, K., Test ideals in local rings, Trans. Amer. Math. Soc. 347 (1995), 3453-3472. MR 96c:13008
  • [S2] Smith, K., The $D$-module structure of $F$-split rings, Math Research Letters 2 (1995), 377-386.

Similar Articles

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

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


Additional Information

Janet Cowden Vassilev
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90024
Address at time of publication: Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, Virginia 23284
Email: jcvassil@saturn.vcu.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02128-X
Keywords: Tight closure, test element, $F$-finite, $F$-pure
Received by editor(s): November 4, 1996
Additional Notes: I would like to express my appreciation to Purdue University for hosting me during the time that I completed these results. I also thank Craig Huneke for many helpful conversations.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society