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

@article{Huneke2007FThresholdsTC,
  title={F-Thresholds, tight closure, integral closure, and multiplicity bounds},
  author={Craig Huneke and Mircea Mustaţǎ and Shunsuke Takagi and Kei-ichi Watanabe},
  journal={Michigan Mathematical Journal},
  year={2007},
  volume={57},
  pages={463-483}
}
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… 
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    TURKISH JOURNAL OF MATHEMATICS
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In a recent paper, De Stefani and N\'{u}\~{n}ez-Betancourt proved that for a standard-graded $F$-pure $k$-algebra $R$, its diagonal $F$-threshold $c(R)$ is always at least $-a(R)$, where $a(R)$ is
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