Graded annihilators and tight closure test ideals

Abstract

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$. The main purposes of this paper are to show that if the injective envelope $E$ of the simple $R$-module has a structure as a torsion-free left module over the Frobenius skew polynomial ring over $R$, then $R$ has a tight closure test element (for modules) and is $F$-pure, and to relate the test ideal of $R$ to the smallest '$E$-special' ideal of $R$ of positive height. A byproduct is an analogue of a result of Janet Cowden Vassilev: she showed, in the case where $R$ is an $F$-pure homomorphic image of an $F$-finite regular local ring, that there exists a strictly ascending chain $0={\tau }_0\subset {\tau }_1\subset ...\subset {\tau }_t=R$ of radical ideals of $R$ such that, for each $i=0,...,t-1$, the reduced local ring $R/{\tau }_i$ is $F$-pure and its test ideal (has positive height and) is exactly ${\tau }_{i+1}/{\tau }_i$. This paper presents an analogous result in the case where $R$ is complete (but not necessarily $F$-finite) and $E$ has a structure as a torsion-free left module over the Frobenius skew polynomial ring. Whereas Cowden Vassilev's results were based on R. Fedder's criterion for $F$-purity, the arguments in this paper are based on the author's work on graded annihilators of left modules over the Frobenius skew polynomial ring.

This is to appear in the Journal of Algebra

References in corpus (1)

• Parameter test ideals of Cohen Macaulay rings

Cited by in corpus (1)

• Right and Left Modules over the Frobenius Skew Polynomial Ring in the F-Finite Case