2013/01/29 by Rodney Y. Sharp, Sharp, Rodney Y.
Mathematics · #13A35 #13E05 #13E10 #13H05 #13J10 #16S36 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13A35 #msc:13E05 #msc:13E10 #msc:13H05 #msc:13J10 #msc:16S36
paper · pdf · doi:10.48550/arxiv.1301.6890
This has been accepted for publication in the Journal of Pure and Applied Algebra. arXiv admin note: text overlap with arXiv:1108.1660
arxiv created 2013/01/29 · arxiv updated 2013/01/30
Let R be a (commutative Noetherian) local ring of prime characteristic that is F-pure. This paper studies a certain finite set \mathcal I of radical ideals of R that is naturally defined by the injective envelope of the simple R-module. This set \mathcal I contains 0 and R, and is closed under taking primary components. For a multiplicatively closed subset S of R, the concept of tight closure with respect to S, or S-tight closure, is discussed, together with associated concepts of S-test element and S-test ideal. It is shown that an ideal of R belongs to \mathcal I if and only if it is the S'-test ideal of R for some multiplicatively closed subset S' of R. When R is complete, \mathcal I is also `closed under taking test ideals', in the following sense: for each proper ideal C in \mathcal I, it turns out that R/C is again F-pure, and if J and K are the unique ideals of R that contain C and are such that J/C is the (tight closure) test ideal of R/C and K/C is the big test ideal of R/C, then both J and K belong to \mathcal I. The paper ends with several examples.