The structure of F-pure rings

By Ian M. Aberbach, Florian Enescu. 2003

76 citations recorded by the source corpus.

Abstract.For a reduced F-finite ring R of characteristic p>0 and q=pe one can write where Mq has no free direct summands over R. We investigate the structure of F-finite, F-pure rings R by studying how the numbers aq grow with respect to q. This growth is quantified by the splitting dimension and the splitting ratios of R which we study in detail. We also prove the existence of a special prime ideal (R) of R, called the splitting prime, that has the property that R/(R) is strongly F-regular. We show that this ideal captures significant information with regard to the F-purity of R.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ce83ff28acfbee7002f43e9f989df3d4b735a948

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

F-signature of pairs and the asymptotic behavior of Frobenius splittings (2011, methodology): “pression in (a) takes only integer values. 17 Remark 4.6. The F-splitting dimension of a local ring Ris simply the F-splitting dimension of (R,CR) and was ﬁrst introduced by I. Aberbach and F. Enescu [AE05] using the characterization in (b). The equality of (b) and (c) in this case gives a positive ”
Observations on the F-signature of local rings of characteristic p (2006, background): “E=k. A similar formula with essentially the same eect can be found in [ AE1 ].”
A survey of test ideals (2011, background): “ar very naturally. For example, suppose that Ris a ring and CR is the complete algebra of maps on R. Suppose that I⊆Ris a CR-compatible ideal (such as the test ideal τ(R) or the splitting prime8 P of [AE05].). One can then restrict each element of CRto R/I. This yields an algebra of maps DR/I = CR|R”