Permanence properties of $F$-injectivity

Datta, Rankeya; Murayama, Takumi

doi:10.48550/arxiv.1906.11399

Cited by 3 publications

(2 citation statements)

References 47 publications

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“…P is said to descend under faithfully flat maps if for a faithfully flat map of noetherian local rings R → S, S is P implies that R is P . Many singularities and properties of rings descend under faithfully flat maps and we refer the reader to the work of Datta and Murayama [DM20] for an exposition surrounding this topic. We will need that the properties of a ring being normal, Gorenstein, and regular descend along faithfully maps.…”

Section: Lemma 22 Let (R M K) Be An Excellent Local Characteristic Ze...mentioning

confidence: 99%

Vanishing of Tors of absolute integral closures in equicharacteristic zero

Patankar

2024

Trans. Amer. Math. Soc. Ser. B

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We show that R R is regular if T o r i R ( R + , k ) = 0 Tor_{i}^{R}(R^{+},k) = 0 for some i ≥ 1 i\geq 1 assuming further that R R is a N \mathbb {N} -graded ring of dimension 2 2 finitely generated over an algebraically closed equicharacteristic zero field k k . This answers a question of Bhatt, Iyengar, and Ma [Comm. Algebra 47 (2019), pp. 2367–2383]. We use almost mathematics over R + R^{+} to deduce properties of the noetherian ring R R and rational surface singularities. Moreover we observe that R + R^{+} in equicharacteristic zero has a rich module-theoretic structure; it is m m -adically ideal(wise) separated, (weakly) intersection flat, and Ohm-Rush. As an application we show that the hypothesis can be astonishingly vacuous for i ≪ d i m ( R ) i \ll dim(R) . We show that a positive answer to an old question of Aberbach and Hochster [J. Pure Appl. Algebra 122 (1997), pp. 171–184] also answers this question and we use our techniques to study a question of André and Fiorot [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), pp. 81–114] regarding ‘fpqc analogues’ of splinters.

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Section: Lemma 22 Let (R M K) Be An Excellent Local Characteristic Ze...mentioning

confidence: 99%

Vanishing of Tors of absolute integral closures in equicharacteristic zero

Patankar

2024

Trans. Amer. Math. Soc. Ser. B

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“…Moreover, if R is a domain then it is N-1. We briefly indicate the steps needed to prove this result, following the strategy of [DM19] that shows an analogous result for 'Cohen-Macaulay and geometrically F -injective'. The techniques in [DM19] are inspired by arguments of Vélez [Vél95].…”

Section: This Proves (2)mentioning

confidence: 99%

Openness of splinter loci in prime characteristic

Datta¹,

Tucker²

2021

Preprint

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A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and F -compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically étale finite extension. We also show that for a noetherian N-graded ring over a field, the homogeneous maximal ideal detects the splinter property.Corollary 1.0.5. (See Corollary 4.3.9) Let R = ∞ n=0 R n be a noetherian graded ring such that R 0 = k is a field. Let m := n>0 R n be the homogeneous maximal ideal of R. Then R is a splinter if and only if R m is a splinter.

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Torsors over the Rational Double Points in Characteristic $\mathbf{p}$

Liedtke¹,

Martin²,

Matsumoto³

2021

Preprint

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We study torsors under finite group schemes over the punctured spectrum of a singularity x ∈ X in positive characteristic.We show that the Dieudonné module of the (loc,loc)-part Picloc loc,loc X/k of the local Picard sheaf can be described in terms of local Witt vector cohomology, making Picloc loc,loc X/k computable. Together with the class group and the abelianised local étale fundamental group, Picloc loc,loc X/kcompletely describes the finite abelian torsors over X \ {x}.We compute Picloc loc,locfor every rational double point singularity, which complements results of Artin [Ar77] and Lipman [Li69], who determined π ét loc (X) and Cl(X). All three objects turn out to be finite. We extend the Flenner-Mumford criterion for smoothness of a normal surface germ x ∈ X to perfect fields of positive characteristic, generalising work of Esnault and Viehweg: If k is algebraically closed, then X is smooth if and only if Picloc loc,loc X/k , π ét loc (X), and Cl(X) are trivial. Finally, we study the question whether rational double point singularities are quotient singularities by group schemes and if so, whether the group scheme is uniquely determined by the singularity. We give complete answers to both questions, except for some D r n -singularities in characteristic 2. In particular, we will give examples of (F-injective) rational double points that are not quotient singularities.

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Permanence properties of $F$-injectivity

Cited by 3 publications

References 47 publications

Vanishing of Tors of absolute integral closures in equicharacteristic zero

Vanishing of Tors of absolute integral closures in equicharacteristic zero

Openness of splinter loci in prime characteristic

Torsors over the Rational Double Points in Characteristic $\mathbf{p}$

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