Article ID Journal Published Year Pages File Type
4587424 Journal of Algebra 2009 17 Pages PDF
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=τ0⊂τ1⊂⋯⊂τt=R of radical ideals of R such that, for each i=0,…,t−1, the reduced local ring R/τi is F-pure and its test ideal (has positive height and) is exactly τi+1/τi. 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.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory