Tight closure commutes with localization in binomial rings

Smith, Karen

doi:10.1090/S0002-9939-00-05626-4

Tight closure commutes with localization in binomial rings
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by Karen E. Smith PDF

Proc. Amer. Math. Soc. 129 (2001), 667-669 Request permission

Abstract:

It is proved that tight closure commutes with localization in any domain which has a module finite extension in which tight closure is known to commute with localization. It follows that tight closure commutes with localization in binomial rings, in particular in semigroup or toric rings.

References

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
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Aldo Conca, Hilbert-Kunz function of monomial ideals and binomial hypersurfaces, Manuscripta Math. 90 (1996), no. 3, 287–300. MR 1397658, DOI 10.1007/BF02568307
David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), no. 1, 1–45. MR 1394747, DOI 10.1215/S0012-7094-96-08401-X
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
Craig Huneke, Tight closure and its applications, CBMS Regional Conference Series in Mathematics, vol. 88, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With an appendix by Melvin Hochster. MR 1377268, DOI 10.1016/0167-4889(95)00136-0
Mordechai Katzman, The complexity of Frobenius powers of ideals, J. Algebra 203 (1998), no. 1, 211–225. MR 1620654, DOI 10.1006/jabr.1997.7246
McCulloch, D., Tight closure and base change, thesis, University of Michigan (1997).
Smith, K.E. and Swanson, I., Linear growth of primary components of monomial ideals, Communications in Algebra 25 (1997), 3071–3079.
Traves, W., Differential operators on monomial rings, Jour. Pure and Applied Algebra 136 (1999), 183–197.