Local cohomology, arrangements of subspaces and monomial ideals

Montaner, Josep Àlvarez; López, R.; Armengou, Santiago Zarzuela

doi:10.1016/s0001-8708(02)00050-6

Cited by 32 publications

(20 citation statements)

References 9 publications

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“…It is also clear that any sum of the I k s gives another prime ideal of the same kind, so the poset P is made up by prime ideals of this form, in particular all of them are cohomologically complete intersection ideals; so, the assumptions of Theorem 4.17 are satisfied in this case. This example is the one already studied in detail in [5].…”

Section: Lemma 415mentioning

confidence: 96%

“…These extension problems were studied in [5] for the spectral sequence where A = K[x 1 , • • • , x d ] is the polynomial ring over a field K and I is a squarefree monomial ideal. Namely, in the subcategory of Z d -graded modules introduced by Yanagawa [64] under the notion of straight modules, these extension problems are nontrivial and are described by the multiplication by the variables x i .…”

Section: Mayer-vietoris Spectral Sequence Of Local Cohomology Modulesmentioning

confidence: 99%

“…It turns out that the results by Hochster (respectively, Terai) and Gräbe (respectively, Mustaţ ǎ) are equivalent, by means of the duality established by Miller [40,Corollary 6.7]. It is also worth noting that, in the last decade, several generalizations of Hochster's decomposition have appeared in the literature, being [62, Theorem 1], [19,Theorem 5.1], [10, Theorems 1.1 and 1.3], and [28, Theorem 4.5 and Lemma 4.6] the most relevant for the purposes of this paper (see also [38, Another different approach was adopted by Àlvarez Montaner et al in [5]; indeed, building upon a Mayer-Vietoris spectral sequence (1) (where A is a polynomial ring with coefficients on a field, I is the defining ideal of an arrangement of linear varieties, I = I 1 ∩ . .…”

Section: Introductionmentioning

confidence: 99%

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On Some Local Cohomology Spectral Sequences

Montaner

Boix

Zarzuela

2018

International Mathematics Research Notices

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We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The 1st type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain by applying a family of functors to a single module. For the 2nd type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their 2nd page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules of Stanley–Reisner rings given by Hochster.

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Section: Lemma 415mentioning

confidence: 96%

Section: Mayer-vietoris Spectral Sequence Of Local Cohomology Modulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Some Local Cohomology Spectral Sequences

Montaner

Boix

Zarzuela

2018

International Mathematics Research Notices

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“…This structure has been described by Terai [33] and Mustaţȃ [31] in terms of some simplicial complexes associated with the monomial ideal J . The approach considered in [7] gives an interpretation in terms of the components appearing in the minimal primary decomposition of J which will be more convenient for our purposes.…”

Section: Bass Numbers Of Local Cohomology Modulesmentioning

confidence: 99%

Bass numbers of local cohomology of cover ideals of graphs

Montaner

Sohrabi

2020

J Algebr Comb

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We develop splitting techniques to study the Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally, we are able to compute all the Bass numbers and the shape of the injective resolution of local cohomology modules by considering the connected components of all the induced subgraphs. Indeed, our method gives us a very simple criterion for the vanishing of these local cohomology modules in terms of the number of connected components of the induced subgraphs.

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“…We note that [AGZ,Corollary 2.2], using [Mu,Theorem 3.3], shows that when K has characteristic zero and when R = K[x 1 , . .…”

Section: Lyubeznik Characteristic Of Stanley-reisner Ringsmentioning

confidence: 99%

Generalized Lyubeznik numbers

Núñez-Betancourt¹,

Witt²

2014

Nagoya Mathematical Journal

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Abstract. Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers, but that captures finer information. These generalized Lyubeznik numbers are defined as lengths of certain iterated local cohomology modules in a category of D-modules, and in order to define them, we develop the theory of a functor Lyubeznik utilized in proving that his original invariants are well defined. In particular, this functor gives an equivalence of categories with a category of D-modules. These new invariants are indicators of F -regularity and F -rationality in characteristic p > 0, and have close connections with characteristic cycle multiplicities in characteristic zero. We compute the generalized Lyubeznik numbers associated to monomial ideals using interpretations as lengths in a category of straight modules, as well as provide examples of these invariants associated to certain determinantal ideals.

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Local cohomology, arrangements of subspaces and monomial ideals

Cited by 32 publications

References 9 publications

On Some Local Cohomology Spectral Sequences

On Some Local Cohomology Spectral Sequences

Bass numbers of local cohomology of cover ideals of graphs

Generalized Lyubeznik numbers

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