Graded annihilators and tight closure test ideals

TL;DR

This paper explores the relationship between the structure of the injective envelope of a module over a Noetherian local ring of prime characteristic and the properties of tight closure test ideals, extending known results to more general rings.

Contribution

It establishes that certain module structures imply the existence of tight closure test elements and relates test ideals to special ideals, generalizing previous results beyond F-finite rings.

Findings

If the injective envelope has a torsion-free Frobenius module structure, then the ring has a test element and is F-pure.

The paper constructs a chain of radical ideals with specific properties related to F-purity and test ideals.

Provides an analogue of Cowden Vassilev's result for complete, not necessarily F-finite rings.

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.