Nagoya Mathematical Journal

A dual to tight closure theory

Neil Epstein and Karl Schwede

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Abstract

We introduce an operation on modules over an $F$-finite ring of characteristic $p$. We call this operation tight interior. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to phantom homology. Using our dual notion of persistence, we obtain new and interesting transformation rules for tight interiors, and so in particular for the test ideal. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory with Blickle’s notion of Cartier modules, and in the process we prove new existence results for Blickle’s test submodule. Finally, we apply the theory we developed to the study of test ideals in nonnormal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases.

Article information

Source
Nagoya Math. J. Volume 213 (2014), 41-75.

Dates
First available in Project Euclid: 31 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1383240639

Digital Object Identifier
doi:10.1215/00277630-2376749

Mathematical Reviews number (MathSciNet)
MR3290685

Zentralblatt MATH identifier
1315.13016

Subjects
Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]
Secondary: 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 13B40: Étale and flat extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25] 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14F18: Multiplier ideals

Citation

Epstein, Neil; Schwede, Karl. A dual to tight closure theory. Nagoya Math. J. 213 (2014), 41--75. doi:10.1215/00277630-2376749. https://projecteuclid.org/euclid.nmj/1383240639.


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References

  • [Bl] M. Blickle, Test ideals via algebras of $p^{-e}$-linear maps, J. Algebraic Geom. 22 (2013), 49–83.
  • [BlB] M. Blickle and G. Böckle, Cartier modules: Finiteness results, J. Reine Angew. Math. 661 (2011), 85–123.
  • [Bo] N. Bourbaki, Commutative Algebra, Chapters 1–7, reprint of the 1989English translation, Elem. Math. (Berlin), Springer, Berlin, 1998.
  • [BSm] A. Bravo and K. E. Smith, Behavior of test ideals under smooth and étale homomorphisms, J. Algebra 247 (2002), 78–94.
  • [BM] H. Brenner and P. Monsky, Tight closure does not commute with localization, Ann. of Math. (2) 171 (2010), 571–588.
  • [BrS] M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math. 60, Cambridge University Press, Cambridge, 1998.
  • [BHe] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993.
  • [G] O. Gabber, “Notes on some $t$-structures” in Geometric Aspects of Dwork Theory, Vols. 1, 2 (Padova, 2001), Walter de Gruyter, Berlin, 2004, 711–734.
  • [HT] N. Hara and S. Takagi, On a generalization of test ideals, Nagoya Math. J. 175 (2004), 59–74.
  • [Ha] R. Hartshorne, Residues and Duality, with an appendix by P. Deligne, Lecture Notes in Math. 20, Springer, Berlin, 1966.
  • [HaS] R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic $p$, Ann. of Math. (2) 105 (1977), 45–79.
  • [He] J. Herzog, Ringe der Charakteristik $p$ und Frobeniusfunktoren, Math. Z. 140 (1974), 67–78.
  • [HH1] M. Hochster and C. Huneke, “Tight closure and strong $F$-regularity” in Colloque en l’honneur de Pierre Samuel (Orsay, 1987), Mém. Soc. Math. Fr. (N.S.) 38 (1989), 119–133.
  • [HH2] M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31–116.
  • [HH3] Melvin Hochster and Craig Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62.
  • [HH4] Melvin Hochster and Craig Huneke, Tight closure in equal characteristic zero, preprint, http://www.lsa.umich.edu/math/~hochster/tcz.ps (accessed 26 September 2013).
  • [H] C. Huneke, Tight Closure and Its Applications, with an appendix by M. Hochster, CBMS Reg. Conf. Ser. Math. 88, Amer. Math. Soc., Providence, 1996.
  • [HS] C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge University Press, Cambridge, 2006.
  • [L] G. Lyubeznik, $F$-modules: Applications to local cohomology and $D$-modules in characteristic $p>0$, J. Reine Angew. Math. 491 (1997), 65–130.
  • [LS1] G. Lyubeznik and K. E. Smith, Strong and weak $F$-regularity are equivalent for graded rings, Amer. J. Math. 121 (1999), 1279–1290.
  • [LS2] G. 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.
  • [M] D. J. McCulloch, Tight closure and base change, Ph.D. dissertation, University of Michigan, Ann Arbor, Michigan, 1997.
  • [PS] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale: Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 42 (1973), 47–119.
  • [S1] K. Schwede, $F$-adjunction, Algebra Number Theory 3 (2009), 907–950.
  • [S2] Karl Schwede, Test ideals in non-$\mathbb{{Q}}$-Gorenstein rings, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5925–5941.
  • [ST] K. Schwede and K. Tucker, On the behavior of test ideals under finite morphisms, preprint, arXiv:1003.4333v3 [math.AG].
  • [Sm1] K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), 41–60.
  • [Sm2] Karen E. Smith, Test ideals in local rings, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3453–3472.
  • [Sm3] Karen E. Smith, The multiplier ideal is a universal test ideal, Comm. Algebra 28 (2000), 5915–5929.
  • [Sm4] Karen E. Smith, Tight closure commutes with localization in binomial rings, Proc. Amer. Math. Soc. 129 (2001), 667–669.
  • [T] W. N. Traves, Differential operators on monomial rings, J. Pure Appl. Algebra 136 (1999), 183–197.
  • [V] J. C. Vassilev, Test ideals in quotients of $F$-finite regular local rings, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4041–4051.