Simple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">D</mml:mi></mml:math>-module components of local cohomology modules

Hartshorne, Robin; Polini, Claudia

doi:10.1016/j.jalgebra.2018.09.005

Cited by 4 publications

(4 citation statements)

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“…[18], [15]). This paper continues the line of research under the same theme: we prove F-module analogues of results obtained by Hartshorne and Polini in [7] and by the authors in [16] for D-modules over formal power series or polynomial rings. The theory of (F-finite) F-modules will be reviewed in the next section.…”

Section: Introductionsupporting

confidence: 76%

“…where k is a field of characteristic zero, and M is a holonomic D(R, k)-module, then dim k Hom D (M, E) is equal to the maximal integer t for which there exists a Dlinear surjection M → E t . This statement is part of [7,Corollary 5.2]. An easier "dual" statement is the following [16,Lemma 2.3]: dim k Hom D (R, M ) is equal to the maximal integer t for which there exists a D-linear injection R t → M .…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…Theorem 1.2. [7,Theorems 4.8,6.4] Let X ⊆ P n k be a smooth, irreducible projective scheme of codimension c < n, where k is an algebraically closed field of characteristic zero. Let I ⊆ R = k[x 0 , .…”

Section: Introductionmentioning

confidence: 99%

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A prime-characteristic analogue of a theorem of Hartshorne-Polini

Switala

Zhang

2020

Journal of Pure and Applied Algebra

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Let R be an F -finite Noetherian regular ring containing an algebraically closed field k of positive characteristic, and let M be an F-finite F-module over R in the sense of Lyubeznik (for example, any local cohomology module of R). We prove that the Fp-dimension of the space of F-module morphisms M → E(R/m) (where m is any maximal ideal of R and E(R/m) is the R-injective hull of R/m) is equal to the k-dimension of the Frobenius stable part of HomR(M, E(R/m)). This is a positive-characteristic analogue of a recent result of Hartshorne and Polini for holonomic D-modules in characteristic zero. We use this result to calculate the F-module length of certain local cohomology modules associated with projective schemes.2010 Mathematics Subject Classification. Primary 13A35; secondary 13A02, 13D45.

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Section: Introductionsupporting

confidence: 76%

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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A prime-characteristic analogue of a theorem of Hartshorne-Polini

Switala

Zhang

2020

Journal of Pure and Applied Algebra

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“…We proceed with part (b). Since here we consider local cohomology supported in the cone over a smooth projective variety, there are several results available in this direction [6,7,28,30,40]. Proof.…”

Section: Local Cohomology Of Smentioning

confidence: 99%

Local cohomology on a subexceptional series of representations

Lörincz¹,

Weyman²

2023

Annales De l'Institut Fourier

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We consider a series of four subexceptional representations coming from the third line of the Freudenthal-Tits magic square; using Bourbaki notation, these are representations (G , X) corresponding to (C 3 , ω 3 ), (A 5 , ω 3 ), (D 6 , ω 5 ), and (E 7 , ω 6 ). In each case X has five G = G × C-orbits, displaying some uniform behavior, e.g. their dimensions or defining ideals. In this paper, we determine some further invariants and analyze their uniformity within the series. We describe the category of G-equivariant coherent D X -modules as the category of representations of a quiver. We construct explicitly the simple equivariant D-modules and describe their G-structures. We determine the D-module structure of local cohomology modules supported in orbit closures, and calculate intersection cohomology groups and Lyubeznik numbers. While our results for (A 5 , ω 3 ), (D 6 , ω 5 ), (E 7 , ω 6 ) are still completely uniform, the case (C 3 , ω 3 ) displays a surprisingly different behavior, for which we give two explanations: the middle orbit is not simply-connected, and its closure is not Gorenstein.Résumé. -Nous considérons une série de quatre représentations sous-exceptionnelles venant de la troisième ligne du carré magique de Freudenthal-Tits: (G , X) = (C 3 , ω 3 ), (A 5 , ω 3 ), (D 6 , ω 5 ), ou (E 7 , ω 6 ), en utilisant la notation de Bourbaki. Dans chaque cas, X a cinq G = G × C-orbites, qui se révèlent avoir un comportement uniforme, avec par exemple leurs dimensions ou leurs idéaux définissants. Dans cet article, nous obtenons plus d'invariants et nous étudions leur uniformité dans cette série des représentations. Nous décrivons la catégorie des D X -modules cohérentes G-équivariantes, et décrivons leurs G-structures. Nous déterminons, pour les modules de cohomologie locale avec support dans des clôtures d'orbites, leur structure comme D-module, et calculons des groupes de cohomologie d'intersection et des nombres de Lyubeznik. Alors que nos résultats pour (A 5 , ω 3 ), (D 6 , ω 5 ), (E 7 , ω 6 ) sont uniformes, le cas (C 3 , ω 3 ) fait apparaître un comportement différent et unique, pour lequel nous donnons deux explications : l'orbite moyenne n'est pas simplement connexe, et sa clôture n'est pas Gorenstein.

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Local Cohomology—An Invitation

Walther

Zhang

2021

Commutative Algebra

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Simple D-module components of local cohomology modules

Cited by 4 publications

References 13 publications

A prime-characteristic analogue of a theorem of Hartshorne-Polini

A prime-characteristic analogue of a theorem of Hartshorne-Polini

Local cohomology on a subexceptional series of representations

Local Cohomology—An Invitation

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