Betti numbers of symmetric shifted ideals

Jennifer Biermann; Hernán de Alba; Federico Galetto; Satoshi Murai; Uwe Nagel; Augustine O'Keefe; Tim Römer; Alexandra Seceleanu

doi:10.1016/j.jalgebra.2020.04.037

Betti numbers of symmetric shifted ideals

Jennifer Biermann, Hernán de Alba, Federico Galetto, Satoshi Murai, Uwe Nagel, Augustine O'Keefe, Tim Römer, Alexandra Seceleanu

Mathematics

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

11 Scopus citations

Abstract

We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.

Original language	English
Pages (from-to)	312-342
Number of pages	31
Journal	Journal of Algebra
Volume	560
DOIs	https://doi.org/10.1016/j.jalgebra.2020.04.037
State	Published - Oct 15 2020

Bibliographical note

Funding Information:
The research of the fourth author is partially supported by KAKENHI 16K05102 . The fifth author was partially supported by Simons Foundation grant # 317096 . The last author was supported by NSF grant DMS–1601024 and EPSCoR award OIA–1557417 .

Funding Information:
The research of the fourth author is partially supported by KAKENHI 16K05102. The fifth author was partially supported by Simons Foundation grant #317096. The last author was supported by NSF grant DMS?1601024 and EPSCoR award OIA?1557417.

Publisher Copyright:
© 2020 Elsevier Inc.

Keywords

Betti numbers
Equivariant resolution
Linear quotients
Shifted ideal
Star configuration
Symbolic power

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2020.04.037

Cite this

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title = "Betti numbers of symmetric shifted ideals",

abstract = "We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.",

keywords = "Betti numbers, Equivariant resolution, Linear quotients, Shifted ideal, Star configuration, Symbolic power",

author = "Jennifer Biermann and {de Alba}, Hern{\'a}n and Federico Galetto and Satoshi Murai and Uwe Nagel and Augustine O'Keefe and Tim R{\"o}mer and Alexandra Seceleanu",

note = "Funding Information: The research of the fourth author is partially supported by KAKENHI 16K05102 . The fifth author was partially supported by Simons Foundation grant # 317096 . The last author was supported by NSF grant DMS–1601024 and EPSCoR award OIA–1557417 . Funding Information: The research of the fourth author is partially supported by KAKENHI 16K05102. The fifth author was partially supported by Simons Foundation grant #317096. The last author was supported by NSF grant DMS?1601024 and EPSCoR award OIA?1557417. Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2020",

month = oct,

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doi = "10.1016/j.jalgebra.2020.04.037",

language = "English",

volume = "560",

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T1 - Betti numbers of symmetric shifted ideals

AU - Biermann, Jennifer

AU - de Alba, Hernán

AU - Galetto, Federico

AU - Murai, Satoshi

AU - Nagel, Uwe

AU - O'Keefe, Augustine

AU - Römer, Tim

AU - Seceleanu, Alexandra

N1 - Funding Information: The research of the fourth author is partially supported by KAKENHI 16K05102 . The fifth author was partially supported by Simons Foundation grant # 317096 . The last author was supported by NSF grant DMS–1601024 and EPSCoR award OIA–1557417 . Funding Information: The research of the fourth author is partially supported by KAKENHI 16K05102. The fifth author was partially supported by Simons Foundation grant #317096. The last author was supported by NSF grant DMS?1601024 and EPSCoR award OIA?1557417. Publisher Copyright: © 2020 Elsevier Inc.

PY - 2020/10/15

Y1 - 2020/10/15

N2 - We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.

AB - We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.

KW - Betti numbers

KW - Equivariant resolution

KW - Linear quotients

KW - Shifted ideal

KW - Star configuration

KW - Symbolic power

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U2 - 10.1016/j.jalgebra.2020.04.037

DO - 10.1016/j.jalgebra.2020.04.037

M3 - Article

AN - SCOPUS:85085758583

VL - 560

SP - 312

EP - 342

ER -

Betti numbers of symmetric shifted ideals

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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