Multi-Rees algebras of strongly stable ideals

Selvi Kara; Kuei Nuan Lin; Gabriel Sosa Castillo

doi:10.1007/s13348-022-00385-2

Multi-Rees algebras of strongly stable ideals

Selvi Kara
, Kuei Nuan Lin
, Gabriel Sosa Castillo

Penn State Greater Allegheny

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

1 Scopus citations

Abstract

We prove that the multi-Rees algebra R(I₁⊕ ⋯ ⊕ I_r) of a collection of strongly stable ideals I₁, … , I_r is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of R(I₁⊕ ⋯ ⊕ I_r) based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of R(I₁⊕ I₂) where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.

Original language	English (US)
Pages (from-to)	213-246
Number of pages	34
Journal	Collectanea Mathematica
Volume	75
Issue number	1
DOIs	https://doi.org/10.1007/s13348-022-00385-2
State	Published - Jan 2024

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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title = "Multi-Rees algebras of strongly stable ideals",

abstract = "We prove that the multi-Rees algebra R(I1⊕ ⋯ ⊕ Ir) of a collection of strongly stable ideals I1, … , Ir is of fiber type. In particular, we provide a Gr{\"o}bner basis for its defining ideal as a union of a Gr{\"o}bner basis for its special fiber and binomial syzygies. We also study the Koszulness of R(I1⊕ ⋯ ⊕ Ir) based on parameters associated to the collection. Furthermore, we establish a quadratic Gr{\"o}bner basis of the defining ideal of R(I1⊕ I2) where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.",

author = "Selvi Kara and Lin, \{Kuei Nuan\} and Castillo, \{Gabriel Sosa\}",

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T1 - Multi-Rees algebras of strongly stable ideals

AU - Kara, Selvi

AU - Lin, Kuei Nuan

AU - Castillo, Gabriel Sosa

PY - 2024/1

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N2 - We prove that the multi-Rees algebra R(I1⊕ ⋯ ⊕ Ir) of a collection of strongly stable ideals I1, … , Ir is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of R(I1⊕ ⋯ ⊕ Ir) based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of R(I1⊕ I2) where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.

AB - We prove that the multi-Rees algebra R(I1⊕ ⋯ ⊕ Ir) of a collection of strongly stable ideals I1, … , Ir is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of R(I1⊕ ⋯ ⊕ Ir) based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of R(I1⊕ I2) where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.

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Multi-Rees algebras of strongly stable ideals

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