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

Craig Huneke; Mircea Mustata; Shunsuke Takagi; Kei-ichi Watanabe

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arXiv · 0708.2394

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

Craig Huneke·Mircea Mustata·Shunsuke Takagi·Kei-ichi Watanabe

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Abstract

The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial ring, and also when $\a$ and $J$ are generated by homogeneous systems of parameters in a Cohen-Macaulay graded $k$-algebra.

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BibTeX RIS

Craig Huneke, Mircea Mustata, Shunsuke Takagi, Kei-ichi Watanabe. 2007-11-26. F-thresholds, tight closure, integral closure, and multiplicity bounds. https://arxiv.org/abs/0708.2394

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