Strong F-regularity and boundedness questions in tight closure.

Abstract

Let R be a commutative Noetherian integral domain of prime characteristic p and let $R\sp+$ denote the integral closure of R in an algebraic closure of its fraction field. We examine the question of whether tight closure commutes with localization, i.e., whether ($I\sp{\*})\sb{P}$ = ($I\sb{P})\sp{\*}$ for I any ideal of R and P a prime ideal. Three different approaches to this problem are: to show that weak F-regularity is strong F-regularity, to prove the uniform annihilation of the lowest local cohomology of a certain family of modules, or to show $I\sp{\*}$ = $IR\sp+\ \cap\ R.$. We show that if R is an F-finite weakly F-regular domain with a canonical ideal J such that there exists an n with $J\sp{(n)}$ locally principal except at finitely many closed points, then R is strongly F-regular. We also develop other canonical ideal conditions for a weakly F-regular ring to be F-regular when localized at primes of height two or less. Finally we extend the known class of modules for which $N\sbsp{M}{\*}$ = $N\sbsp{M}{+}$ to those finitely generated R-modules, $N \subseteq M$, such that ${M\over N}\ \otimes R\sp+$ has a finite resolution by finitely generated projective $R\sp+$-modules.