发表机构
University of Utah; Nihon University（犹他大学; 日本大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究正规分次环的 $b$-不变量，证明有限阿贝尔群不变环的 $b$-不变量非正，构造了负 $b$-不变量的非强 $F$-正则环，否证了相关猜想。
AI 中文摘要
$b$-不变量是 Okuma、Watanabe 和 Yoshida 最近对正规分次环定义的一个不变量。我们研究了有限群的不变环的 $b$-不变量，证明了在阿贝尔群情形下它非正，但在其他情形下则不一定；更一般地，我们给出了有限群不变环的反典范模的描述。转向无限群，我们记录了行列式环族的反典范模和 $b$-不变量。我们还构造了特征零的、具有孤立奇点和负 $b$-不变量的标准分次 Cohen-Macaulay 环，这些环不是强 $F$-正则型的，从而否证了 Okuma-Watanabe-Yoshida 的一个猜想。
英文摘要
The $b$-invariant of a normal graded ring was defined recently by Okuma, Watanabe, and Yoshida. We study this for invariant rings of finite groups, proving that it is nonpositive in the case of abelian groups, but not necessarily otherwise; more generally, we give a description of the anticanonical module for invariant rings of finite groups. Turning to infinite groups, we record the anticanonical module and the $b$-invariant for families of determinantal rings. We also construct standard-graded Cohen-Macaulay rings of characteristic zero, with isolated singularities and negative $b$-invariant, that are not of strongly $F$-regular type, thereby disproving a conjecture of Okuma-Watanabe-Yoshida.
Comments18 pages; comments welcome
