$F$-深度与$F$-幂零环：推广与应用
$F$-depth and $F$-nilpotent rings: generalizations and applications
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中文总结 AI 辅助
本文综述了素特征环中Frobenius检验指数的历史、与局部上同调上Frobenius作用幂零性的联系，并推广至较少幂零性的情形。
中文摘要 AI 辅助
在素特征环中，计算理想的Frobenius闭包是一个困难的问题。对于给定的理想，其Frobenius检验指数为执行该计算提供了一种有价值的均匀性度量。因此，寻找理想的Frobenius检验指数的统一上界是可取的。即使在低维的良好环中，Brenner表明考虑所有理想的集合通常是毫无希望的，但对于Cohen-Macaulay环，Katzman-Sharp证明了参数理想类存在Frobenius检验指数的统一上界。随后控制Frobenius检验指数的努力通常涉及研究Frobenius作用在局部上同调上的行为以及该作用的幂零程度。本综述文章的目标是考察Frobenius检验指数问题的历史及其与由Frobenius作用在局部上同调上定义的局部环奇点类型的关系。我们还探讨了这些先前结果在假设Frobenius作用中较少幂零性的情况下的若干推广。
英文摘要
Computation of the Frobenius closure of ideals in rings of prime characteristic is a difficult problem. For a given ideal, its Frobenius test exponent provides a valuable degree of uniformity in performing the calculation. Hence, it is desirable to find uniform upper bounds on the Frobenius test exponent of ideals. Even in nice rings of low dimension, Brenner showed considering the collection of all ideals is generally hopeless, but for Cohen-Macaulay rings, Katzman-Sharp there are uniform upper bounds on the Frobenius test exponent for the class of parameter ideals. Subsequent efforts in controlling the Frobenius test exponents have typically involved studying the Frobenius action on local cohomology and the degree to which this action is nilpotent. The goal of this survey article is to examine the history of the Frobenius test exponent problem and its relationships to singularity types for local rings defined in terms of the Frobenius action on local cohomology. We also explore several generalizations of these prior results to a setting where less nilpotence in the Frobenius action is assumed.
