$F$-injective rings and $F$-stable primes

Abstract:

The notion of stability of the highest local cohomology module with respect to the Frobenius functor originates in the work of R. Hartshorne and R. Speiser. R. Fedder and K.-i. Watanabe examined this concept for isolated singularities by relating it to $F$-rationality. The purpose of this note is to study what happens in the case of non-isolated singularities and to show how this stability concept encapsulates a few of the subtleties of tight closure theory. Our study can be seen as a generalization of the work by Fedder and Watanabe. We introduce two new ring invariants, the $F$-stability number and the set of $F$-stable primes. We associate to every ideal $I$ generated by a system of parameters and $x \in I^\ast - I$ an ideal of multipliers denoted $I(x)$ and obtain a family of ideals $Z_{I,R}$. The set $\operatorname {Max}(Z_{I,R})$ is independent of $I$ and consists of finitely many prime ideals. It also equals $\operatorname {Max} \{P| P$ prime ideal such that $R_{P}$ is $F$-stable$\}$. The maximal height of such primes defines the $F$-stability number.