A sufficient condition for F-purity

Abstract

Abstract. It is well known that nice conditions on the canonical module of a local ring have a strong impact in the study of strong F -regularity and F -purity. In this note, we prove that if (R, m) is an equidimensional and S 2 local ring that admits a canonical ideal I ∼ = ω R such that R/I is F -pure, then R is F -pure. This greatly generalizes one of the main theorems in [Ene03]. We also provide examples to show that not all Cohen-Macaulay F -pure local rings satisfy the above property. introductionThe purpose of this note is to investigate the condition that R admits a canonical ideal I ∼ = ω R such that R/I is F -pure. This condition was first studied in [Ene03] and also in [Ene12] using pseudocanonical covers. And in [Ene03] it was shown that this implies R is F -pure under the additional hypothesis that R is Cohen-Macaulay and F -injective. Applying some theory of canonical modules for non Cohen-Macaulay rings as well as some recent results in [Sha10] and [Ma12], we are able to drop both the Cohen-Macaulay and F -injective condition: we only need to assume R is equidimensional and S 2 . We also provide examples to show that not all complete F -pure Cohen-Macaulay rings satisfy this condition. In fact, if R is Cohen-Macaulay and F -injective, we show that this property is closely related to whether the natural injective Frobenius action on H d m (R) can be "lifted" to an injective Frobenius action on E R , the injective hull of the residue field of R. And instead of using pseudocanonical covers, our treatment uses the anti-nilpotent condition for modules with Frobenius action introduced in [EH08] and [Sha07].In Section 2 we summarize some results on canonical modules of non Cohen-Macaulay rings. These results are well known to experts. In Section 3 we do a brief review of the notions of F -pure and F -injective rings as well as some of the theory of modules with Frobenius action, and we prove our main result. canonical modules of non Cohen-Macaulay ringsIn this section we summarize some basic properties of canonical modules of non-CohenMacaulay local rings (for those we cannot find references, we give proofs). All these properties are characteristic free. Recall that the canonical module ω R is defined to be a finitely generated R-module satisfying ω∨ denotes the Matlis dual. Throughout this section we only require (R, m) is a Noetherian local ring. We do not need the CohenMacaulay or even excellent condition.

On the interplay between the Frobenius functor and its dual

“…There is a faithfully flat homomorphism R → R[X ] P whose fibres are Gorenstein and so R[X ] P is (S 2 ) and generically Gorenstein. This implies that L is a height one ideal by [19,Proposition 2.4]. Now, we have…”