Combinatorial interpretations of some Boij-Söderberg decompositions

Uwe Nagel; Stephen Sturgeon

doi:10.1016/j.jalgebra.2013.01.027

Combinatorial interpretations of some Boij-Söderberg decompositions

Uwe Nagel, Stephen Sturgeon

Mathematics

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules.

Original language	English
Pages (from-to)	54-72
Number of pages	19
Journal	Journal of Algebra
Volume	381
DOIs	https://doi.org/10.1016/j.jalgebra.2013.01.027
State	Published - May 1 2013

Bibliographical note

Funding Information:
✩ This work was partially supported by a grant from the Simons Foundation (#208869 to Uwe Nagel) and by the National Security Agency under Grant No. H98230-12-1-0247. The second author also would like to thank MSRI for organizing and funding the summer workshop on commutative algebra in 2011 as well as Daniel Erman for his inspiring lectures. * Corresponding author. E-mail addresses: [email protected] (U. Nagel), [email protected] (S. Sturgeon).

Keywords

Boij-Söderberg theory
Ferrers hypergraph
Gorenstein ring
Linear resolution
O-sequence
Quasi-Gorenstein module

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2013.01.027

Cite this

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abstract = "Boij-S{\"o}derberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules.",

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author = "Uwe Nagel and Stephen Sturgeon",

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AU - Nagel, Uwe

AU - Sturgeon, Stephen

N1 - Funding Information: ✩ This work was partially supported by a grant from the Simons Foundation (#208869 to Uwe Nagel) and by the National Security Agency under Grant No. H98230-12-1-0247. The second author also would like to thank MSRI for organizing and funding the summer workshop on commutative algebra in 2011 as well as Daniel Erman for his inspiring lectures. * Corresponding author. E-mail addresses: [email protected] (U. Nagel), [email protected] (S. Sturgeon).

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N2 - Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules.

AB - Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules.

KW - Boij-Söderberg theory

KW - Ferrers hypergraph

KW - Gorenstein ring

KW - Linear resolution

KW - O-sequence

KW - Quasi-Gorenstein module

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EP - 72

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Combinatorial interpretations of some Boij-Söderberg decompositions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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