Some results on the subadditivity condition of syzygies

Abed Abedelfatah

doi:10.1007/s13348-020-00312-3

Some results on the subadditivity condition of syzygies

Abed Abedelfatah

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

2 Scopus citations

Abstract

Let S= K[x₁, … , x_n] , where K is a field, and t_i denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i. In this paper, we prove:If I is a monomial ideal of S and a≥ b- 1 ≥ 0 are integers such that a+b≤projdim(S/I), then (Formula presented.)If I= I_Δ where Δ is a simplicial complex such that dim (Δ) < t_a- a or dim (Δ) < t_b- b, then (Formula presented.)If I is a monomial ideal that minimally generated by m₁, … , m_r such that lcm(m1,…,mr)lcm(m1,…,m^i,…,mr)∉K for all i, where m^ _i means that m_i is omitted, then t_a₊_b≤ t_a+ t_b for all a, b≥ 0 with a+b≤projdim(S/I).

Original language	English
Pages (from-to)	173-179
Number of pages	7
Journal	Collectanea Mathematica
Volume	73
Issue number	2
DOIs	https://doi.org/10.1007/s13348-020-00312-3
State	Published - 1 May 2022
Externally published	Yes

Keywords

Betti numbers
Monomial ideal
Simplicial complex
Subadditivity condition

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1007/s13348-020-00312-3

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title = "Some results on the subadditivity condition of syzygies",

abstract = "Let S= K[x1, … , xn] , where K is a field, and ti denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i. In this paper, we prove:If I is a monomial ideal of S and a≥ b- 1 ≥ 0 are integers such that a+b≤projdim(S/I), then (Formula presented.)If I= IΔ where Δ is a simplicial complex such that dim (Δ) < ta- a or dim (Δ) < tb- b, then (Formula presented.)If I is a monomial ideal that minimally generated by m1, … , mr such that lcm(m1,…,mr)lcm(m1,…,m\textasciicircum{}i,…,mr)∉K for all i, where m\textasciicircum{} i means that mi is omitted, then ta+b≤ ta+ tb for all a, b≥ 0 with a+b≤projdim(S/I).",

keywords = "Betti numbers, Monomial ideal, Simplicial complex, Subadditivity condition",

author = "Abed Abedelfatah",

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AU - Abedelfatah, Abed

PY - 2022/5/1

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N2 - Let S= K[x1, … , xn] , where K is a field, and ti denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i. In this paper, we prove:If I is a monomial ideal of S and a≥ b- 1 ≥ 0 are integers such that a+b≤projdim(S/I), then (Formula presented.)If I= IΔ where Δ is a simplicial complex such that dim (Δ) < ta- a or dim (Δ) < tb- b, then (Formula presented.)If I is a monomial ideal that minimally generated by m1, … , mr such that lcm(m1,…,mr)lcm(m1,…,m^i,…,mr)∉K for all i, where m^ i means that mi is omitted, then ta+b≤ ta+ tb for all a, b≥ 0 with a+b≤projdim(S/I).

AB - Let S= K[x1, … , xn] , where K is a field, and ti denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i. In this paper, we prove:If I is a monomial ideal of S and a≥ b- 1 ≥ 0 are integers such that a+b≤projdim(S/I), then (Formula presented.)If I= IΔ where Δ is a simplicial complex such that dim (Δ) < ta- a or dim (Δ) < tb- b, then (Formula presented.)If I is a monomial ideal that minimally generated by m1, … , mr such that lcm(m1,…,mr)lcm(m1,…,m^i,…,mr)∉K for all i, where m^ i means that mi is omitted, then ta+b≤ ta+ tb for all a, b≥ 0 with a+b≤projdim(S/I).

KW - Betti numbers

KW - Monomial ideal

KW - Simplicial complex

KW - Subadditivity condition

UR - https://www.scopus.com/pages/publications/85099031046

U2 - 10.1007/s13348-020-00312-3

DO - 10.1007/s13348-020-00312-3

M3 - Article

AN - SCOPUS:85099031046

SN - 0010-0757

VL - 73

SP - 173

EP - 179

JO - Collectanea Mathematica

JF - Collectanea Mathematica

IS - 2

ER -

Some results on the subadditivity condition of syzygies

Abstract

Keywords

ASJC Scopus subject areas

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