F-rationality of Rees algebras

Abstract

RESULTSLet R be a noetherian ring and I an R-ideal. The Rees algebra of R with respect to I is R R (I) := ⊕ n≥0 I n ; the extended Rees algebra of R with respect to I is R ′ R (I) := ⊕ n∈Z I n , where, for n ≤ 0 I n := R.Several authors have studied the singularities of Rees algebras: E. Hyry [Hyr99] for rational singularties in characteristic zero; A. K. Singh [Sin00] in prime characteristic for strong Fregularity and F-purity; and N. Hara, K.-i. in prime characteristic for F-rationality and F-regularity respectively.In this article, we study the F-rationality of R R (I) and R ′ R (I). Our primary aim is to understand some questions of N. Hara, K.-i. Watanabe and K.-i. Yoshida [HWY02a, Section 1], which ask for necessary and sufficient conditions for Rees algebras to be F-rational. We list our results below, postponing definitions to Section 2. Theorem 1.1. Let (R, m) be an excellent local domain of prime characteristic and I an m-primary R-ideal. Then the following are equivalent:This settles [HWY02a, Conjecture 4.1] which asserted the conclusion of the above theorem. That the F-rationality of R R (I) implies the F-rationality of R ′ R (I) is [HWY02a, Theorem 4.2], but we give a different proof, which follows directly from some observations on the tight closure of zero in the local cohomology modules of R R (I) and of R ′ R (I) and on the F-rationality of Proj R R (I) that we discuss in Section 3. As applications of the results of Section 3, we get a sufficient (but not necessary) condition for the F-rationality of R given the F-rationality of R R (I) (Proposition 4.3) and recover [HWY02a, Theorem 3.1] about the F-rationality of Rees algebras of integrally closed ideals in two-dimensional F-rational rings (Theorem 4.5).Our next result partially answers [HWY02a, Question 3.7], which asked whether the result holds (for R R (I)), without any restriction on the dimension. Since our proof uses the principalization result of V. Cossart and O. Piltant [CP08, Proposition 4.2], we put some conditions on R.Theorem 1.2. Let R be a three-dimensional finite-type domain over a field of prime characteristic and m a maximal ideal. Assume that R is a rational singularity. Let I be an m-primary ideal. Let S be a graded R-algebra with R R (I) ⊆ S ⊆ R R (I). Suppose that Proj S is F-rational. Then the ring ⊕ n≥0 S Nn is F-rational for every integer N ≫ 0. This paper arose from trying to understand whether the results of Hyry [Hyr99] (who, in characteristic zero, relate the rationality of a Rees algebra to that of the corresponding blow-up) have counterparts for F-rationality.2010 Mathematics Subject Classification. Primary: 13A30, 13A35.In Section 2, we give the definitions, some known results needed in our proofs and some preliminary lemmas. The subsequent sections contain the proofs of the above theorems.Acknowledgements. We thank the referee for his/her comments.

Rees algebras and generalized depth‐like conditions in prime characteristic

“…This can be seen as a complementary result to other work examining F ‐singularities for Rees algebras, specifically F ‐rational singularities [6, 13], F ‐regular singularities [7], or F ‐pure singularities [3].…”

The theory of F-rational signature

“…We explored a connection with Rees algebras in Corollary 3.18, but it is likely that one can say more. For example, it was conjectured in [HWY02] and proved in [KK21] that for an m-primary ideal I the extended Rees algebra R[It, t −1 ] is F-rational if and only if R and the Rees algebra R[It] are F-rational. It is desirable to give a connection in terms of F-rational signature akin to Corollary 3.18.…”