The structure of F-pure rings

Aberbach, Ian M.; Enescu, Florian

doi:10.1007/s00209-005-0776-y

Cited by 53 publications

(43 citation statements)

References 16 publications

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“…Note that we do not need to assume that the ambient ring R is strongly F-regular (the characteristic p analogue of Kawamata log terminal). The following result generalizes [1,Theorem 4.7] to triples (R, , a • ), to the non-local case, and to the situation where R is not (necessarily) the quotient of a regular ring. Corollary 7.8 [1,Theorem 4.7] Suppose that (R, , a • ) is a triple and P is a center of sharp F-purity for (R, , a • ) that is maximal (as an ideal) among the centers of F-purity for (R, , a • ) with respect to ideal containment.…”

Section: Remark 77mentioning

confidence: 84%

“…In particular, we show that a center of F-purity of maximal height (as an ideal) automatically cuts out a strongly F-regular scheme; compare with Kawamata's subadjunction theorem [23]. In fact, in a reduced F-finite F-pure local ring, the center of F-purity of maximal height is the splitting prime as defined by Aberbach and Enescu [1]. We also prove the following theorem:…”

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confidence: 96%

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Centers of F-purity

Schwede

2009

Math. Z.

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In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair $(R, \Delta)$. We call these analogues centers of $F$-purity. We prove positive characteristic analogues of subadjunction-like results, prove new stronger subadjunction-like results, and in some cases, lift these new results to characteristic zero. Using a generalization of centers of $F$-purity which we call uniformly $F$-compatible ideals, we give a characterization of the test ideal (which unifies several previous characterizations). Finally, in the case that $\Delta = 0$, we show that uniformly $F$-compatible ideals coincide with the annihilators of the $\mathcal{F}(E_R(k))$-submodules of $E_R(k)$ as defined by Smith and Lyubeznik.Comment: Typos corrected. To appear in Math.

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Section: Remark 77mentioning

confidence: 84%

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confidence: 96%

Centers of F-purity

Schwede

2009

Math. Z.

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“…In [AL], I. Aberbach and G. Leuschke define the s-dimension of (R, m), denoted by sdim(R), to be the largest integer i such that lim sup e→∞ # e R, R /q α(R)+i > 0 in case R is F -finite. Recently, I. Aberbach and F. Enescu showed results concerning sdim(R) in [AE1]. We would like to remark that the notion may just as well be defined as the largest integer i such that lim sup e→∞ (#( e R)/q i ) > 0 for any Noetherian local ring of characteristic p. The results in this section may be used to analyze the behavior of s-dimension under localization and flat local extension.…”

Section: Equalities Hold Inmentioning

confidence: 93%

Observations on the F-signature of local rings of characteristic p

Yao

2006

Journal of Algebra

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Let (R, m, k) be a d-dimensional Noetherian reduced local ring of prime characteristic p such that R 1/p e are finite over R for all e ∈ N (i.e. R is F -finite). Consider the sequence {a e /q α(R)+d } ∞ e=0 , in which α(R) = log p [k : k p ], q = p e , and a e is the maximal rank of free R-modules appearing as direct summands of R-module R 1/q . Denote by s − (R) and s + (R) the liminf and limsup, respectively, of the above sequence as e → ∞. If s − (R) = s + (R), then the limit, denoted by s(R), is called the F -signature of R. It turns out that the F -signature can be defined in a way that is independent of the module finite property of R 1/q over R. We show that: (1) If s + (R) 1 − 1/(d!p d ), then R is regular; (2) If R is excellent such that R P is Gorenstein for every P ∈ Spec(R) \ {m}, then s(R) exists;(3) If (R, m) → (S, n) is a local flat ring homomorphism, then s ± (R) s ± (S) and, if furthermore S/mS is Gorenstein, s ± (S) s ± (R)s(S/mS).

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“…To show the converse, we use the splitting prime. For the definition and the behavior, see [1]. Let p be the splitting prime.…”

Section: Local Casementioning

confidence: 99%

F-signature of graded Gorenstein rings

Sannai

Watanabe

2011

Journal of Pure and Applied Algebra

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a b s t r a c tFor a commutative ring R, the F -signature was defined by Huneke and Leuschke [Math. Ann. 324 (2) (2002) [391][392][393][394][395][396][397][398][399][400][401][402][403][404]. It is an invariant that measures the order of the rank of the free direct summand of R (e) . Here, R (e) is R itself, regarded as an R-module through e-times Frobenius action F e .In this paper, we show a connection of the F -signature of a graded ring with other invariants. More precisely, for a graded F -finite Gorenstein ring R of dimension d, we give an inequality among the F -signature s(R), a-invariant a(R) and Poincaré polynomial P(R, t).Moreover, we show that R (e) has only one free direct summand for any e, if and only if R is F -pure and a(R) = 0. This gives a characterization of such rings.

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The structure of F-pure rings

Cited by 53 publications

References 16 publications

Centers of F-purity

Centers of F-purity

Observations on the F-signature of local rings of characteristic p

F-signature of graded Gorenstein rings

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