2008/08/11 by Rodney Y. Sharp, Sharp, Rodney Y.
Mathematics · #13A35 #13D45 #13E05 #13E10 #13H10 (Primary) 13J10 (Secondary) #16S36 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #msc:13A35 #msc:13D45 #msc:13E05 #msc:13E10 #msc:13H10 #msc:13J10 #msc:16S36
paper · pdf · doi:10.48550/arxiv.0808.1483
This is to appear in the Journal of Algebra
arxiv created 2008/08/11 · openalex publication_date 2008/08/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 = 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 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.