Graded annihilators and tight closure test ideals

Abstract

Let R be a commutative Noetherian local ring of prime characteristic p, with maximal ideal m . The main purposes of this paper are to show that if the injective envelope E of R / m has a structure as an x-torsion-free left module over the Frobenius skew polynomial ring over R (in the indeterminate x), 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 = τ 0 ⊂ τ 1 ⊂ ⋯ ⊂ τ t = R of radical ideals of R such that, for each i = 0 , … , t − 1 , the reduced local ring R / τ i is F-pure and its test ideal (has positive height and) is exactly τ i + 1 / τ 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 an x-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.