Cohomological dimension, Lyubeznik numbers, and connectedness in mixed characteristic

Hernández, Daniel J.; Núñez-Betancourt, Luis; Pérez, Felipe; Witt, Emily E.

doi:10.1016/j.jalgebra.2018.07.019

Cited by 6 publications

(6 citation statements)

References 34 publications

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“…Assume that the punctured spectrum of S/I is disconnected. Let n denote the maximal ideal of S, so that there exist ideals I 1 and I 2 of S that are not n-primary for which rad (I 1 ∩ I 2 ) = rad I and rad (I 1 + I 2 ) = n. Consider the Mayer-Vietoris sequence, It should be mentioned that the part of the proof in the above paragraph, is same to that of [HNBPW18], Theorem 3.8, but for the sake of completeness we keep it here.…”

Section: The Main Resultsmentioning

confidence: 99%

“…In [HNBPW18], the SVT has been extended to complete unramified regular local ring of mixed characteristic: Let R be a d-dimensional complete unramified regular local ring of mixed characteristic, whose residue field is separably closed. Let J be an ideal of R for which dim(R/p) ≥ 3 for every minimal prime p of J.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we extend the result of the SVT partially to complete ramified regular local rings only for the extended ideals. In proving the main result, we reduce the conditions given on ramified case to the unramified case and then we apply the result of [HNBPW18], Theorem 3.8. The main result is the following (see Corollary 1): Let (R, m) be a d-dimensional complete unramified regular local ring of mixed characteristic, with a separably closed residue field and {J α } be a countable collection ideals of it.…”

Section: Introductionmentioning

confidence: 99%

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A note on the second vanishing theorem

Bhattacharyya

2020

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Let R be a regular local ring R of dimension d and H d−1 J (R) be the local cohomology module supported at the ideal J. The second vanishing theorem for the local cohomology or the SVT is well known in regular local rings containing field and for complete unramified regular local ring of mixed characteristic. It states that under certain conditions on J (see below for detail), H d−1 J (R) vanishes if and only if the punctured spectrum R/J is connected. In this paper, we extend the result of SVT partially to complete ramified regular local ring only for the extended ideals. When punctured spectrum is not connected but it has t number of connected components, we show that the Matlis duals of these local cohmology modules are free modules with t − 1 generators. They also support the Conjecture 1 and the result of Corollary 1.2 of [LY18]. Thus, we extends the result of Corollary 1.2 of [LY18] in unramified and ramified situation for certain local cohomology modules.2010 Mathematics Subject Classification. 13D45.

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Section: The Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A note on the second vanishing theorem

Bhattacharyya

2020

Preprint

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“…On the other hand, the Hartshorne-Lichtenbaum vanishing theorem [BS13, 8.2.1] says that if R is a complete local domain, then H n I (R) vanishes if and only if dim(R/I) ≥ 1; in other words, this result characterizes when cd(I) ≤ n − 1. Nowadays, one also knows necessary and sufficient conditions to guarantee cd(I) ≤ n − 2 and cd(I) ≤ n − 3; the interested reader can consult [DT16,HNBPW18] and the references given therein for additional information.…”

Section: Introductionmentioning

confidence: 99%

Certain endomorphism rings of local cohomology modules and Lyubeznik numbers

Boix¹,

Eghbali²

2022

Preprint

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The goal of this paper is twofold; on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne-Lichtenbaum Vanishing Theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules. On the other hand, building upon previous work by Àlvarez Montaner, we exhibit the shape of Lyubeznik tables of the so-called partially sequentially Cohen-Macaulay rings as introduced by Sbarra and Strazzanti.

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“…A special case of Theorem 1.3, when dim(R/a) ≥ 3 and R/a is equidimensional, can be found in [HNnBPW18].…”

Section: Introductionmentioning

confidence: 99%

The Second Vanishing Theorem for Local Cohomology Modules

Zhang¹

2021

Preprint

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We prove the Second Vanishing Theorem for local cohomology modules of an unramified regular local ring in its full generality and provide a new proof of the Second Vanishing Theorem in prime characteristic p. As an application of our vanishing theorem for unramified regular local rings, we extend our topological characterization of the highest Lyubeznik number of an equi-characteristic local ring to the setting of mixed characteristic. An upper bound of local cohomological dimension in mixed characteristic is also obtained by partially extending Lyubeznik's vanishing theorem in prime characteristic p to mixed characteristic.

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Cohomological dimension, Lyubeznik numbers, and connectedness in mixed characteristic

Cited by 6 publications

References 34 publications

A note on the second vanishing theorem

A note on the second vanishing theorem

Certain endomorphism rings of local cohomology modules and Lyubeznik numbers

The Second Vanishing Theorem for Local Cohomology Modules

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