Regularity of canonical and deficiency modules for monomial ideals

Kummini, Manoj; Murai, Shinji

doi:10.2140/pjm.2011.249.377

Cited by 2 publications

(1 citation statement)

References 8 publications

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“…Remark 5.3. In [20], it is established that for a monomial ideal I (not necessarily square-free), the deficiency modules of R = S/I satisfies the same regularity bounds as in Propositions 3.1 and 4.1, so our main results above applied for this situation as well.…”

Section: Applications and Examplesmentioning

confidence: 79%

Regularity, singularities and $h$-vector of graded algebras

Dao,

Ma,

Varbaro

2019

Preprint

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Let R be a standard graded algebra over a field. We investigate how the singularities of Spec R or Proj R affect the h-vector of R, which is the coefficients of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if R satisfies Serre's condition (S r ) and have reasonable singularities (Du Bois on the punctured spectrum or F -pure), then h 0 , . . . , h r ≥ 0. Furthermore the multiplicity of R is at leastWe also prove that equality in many cases forces R to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain Ext modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and F -pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others.

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Section: Applications and Examplesmentioning

confidence: 79%

Regularity, singularities and $h$-vector of graded algebras

Dao,

Ma,

Varbaro

2019

Preprint

View full text Add to dashboard Cite

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Regularity, singularities and ℎ-vector of graded algebras

Dao,

Ma,

Varbaro

2024

Trans. Amer. Math. Soc.

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Let R R be a standard graded algebra over a field. We investigate how the singularities of Spec ⁡ R \operatorname {Spec} R or Proj ⁡ R \operatorname {Proj} R affect the h h -vector of R R , which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if R R satisfies Serre’s condition ( S r ) (S_r) and has reasonable singularities (Du Bois on the punctured spectrum or F F -pure), then h 0 h_0 , …, h r ≥ 0 h_r\geq 0 . Furthermore the multiplicity of R R is at least h 0 + h 1 + ⋯ + h r − 1 h_0+h_1+\dots +h_{r-1} . We also prove that equality in many cases forces R R to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain Ext \operatorname {Ext} modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and F F -pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others.

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Regularity of canonical and deficiency modules for monomial ideals

Abstract: We show that the Castelnuovo-Mumford regularity of the canonical or a deficiency module of the quotient of a polynomial ring by a monomial ideal is bounded by its dimension.

Cited by 2 publications

References 8 publications

Regularity, singularities and $h$-vector of graded algebras

Regularity, singularities and $h$-vector of graded algebras

Regularity, singularities and ℎ-vector of graded algebras

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