摘要翻译：
理想$\a$相对于理想$j$的F阈值$C^j(\a)$是通过比较$\a$的幂与$j$的Frobenius幂而得到的正特征不变量。我们证明了在温和的假设下，我们可以用F-阈值检测参数理想的整闭包或紧闭包中的包容。当$\a$和$J$是零维理想，并且$J$是由一个参数系统生成时，我们根据多重性$e(\a)$和$e(J)$构造了一个有界$c^J(\a)$的猜想。证明了当$J$是多项式环中的单理想时，以及当$a$和$J$是由Cohen-Macaulay分次$K$-代数中的齐次参数系统生成时的猜想。
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英文标题：
《F-thresholds, tight closure, integral closure, and multiplicity bounds》
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作者：
Craig Huneke, Mircea Mustata, Shunsuke Takagi and Kei-ichi Watanabe
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Commutative Algebra        交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  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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PDF链接：
https://arxiv.org/pdf/0708.2394