F-Thresholds, tight closure, integral closure, and multiplicity bounds

By Craig Huneke, M. Mustaţă, S. Takagi, Kei-ichi Watanabe. 2007

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This is a joint work with Craig Huneke, Mircea Mustaţă and Kei-ichi Watanabe. Let R be a Noetherian ring of prime characteristic p and denote by R◦ the set of elements of R that are not contained in any minimal prime ideal. The tight closure I∗ of an ideal I ⊆ R is defined to be the ideal of R consisting of all elements x ∈ R for which there exists c ∈ R◦ such that cx ∈ I [q] for all large q = p, where I [q] is the ideal generated by the q powers of all elements of I. The ring R is called F-rational if J∗ = J for every ideal J ⊆ R generated by parameters. Let a be a fixed proper ideal of R such that a ∩ R◦ = ∅. To each ideal J of R such that a ⊆ √ J , we associate an F-threshold as follows. For every q = p, let

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/fe56c866ea3ed132bceb035471a4a5acba9e84b7