On the subadditivity condition of edge ideal

Abed Abedelfatah

doi:10.1007/s10801-024-01363-6

On the subadditivity condition of edge ideal

Abed Abedelfatah

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

Abstract

Let S=K[x₁,…,x_n], where K is a field, and t_i(S/I) denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i, where I is an edge ideal. In this paper, we prove that if t_b(S/I)≥⌈3b2⌉ for some b≥0, then the subadditivity condition t_a+b(S/I)≤t_a(S/I)+t_b(S/I) holds for all a≥0. In addition, we prove that t_a+4(S/I)≤t_a(S/I)+t₄(S/I) for all a≥0 (the case b=0,1,2,3 is known). We conclude that if the projective dimension of S/I is at most 9, then I satisfies the subadditivity condition.

Original language	English
Pages (from-to)	1061-1069
Number of pages	9
Journal	Journal of Algebraic Combinatorics
Volume	60
Issue number	4
DOIs	https://doi.org/10.1007/s10801-024-01363-6
State	Published - 1 Dec 2024
Externally published	Yes

Keywords

Betti numbers
Edge ideal
Monomial ideal
Simplicial complex
Subadditivity condition

ASJC Scopus subject areas

Algebra and Number Theory
Discrete Mathematics and Combinatorics

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10.1007/s10801-024-01363-6

Cite this

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title = "On the subadditivity condition of edge ideal",

abstract = "Let S=K[x1,…,xn], where K is a field, and ti(S/I) denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i, where I is an edge ideal. In this paper, we prove that if tb(S/I)≥⌈3b2⌉ for some b≥0, then the subadditivity condition ta+b(S/I)≤ta(S/I)+tb(S/I) holds for all a≥0. In addition, we prove that ta+4(S/I)≤ta(S/I)+t4(S/I) for all a≥0 (the case b=0,1,2,3 is known). We conclude that if the projective dimension of S/I is at most 9, then I satisfies the subadditivity condition.",

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AU - Abedelfatah, Abed

N1 - Publisher Copyright: © The Author(s) 2024.

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Y1 - 2024/12/1

N2 - Let S=K[x1,…,xn], where K is a field, and ti(S/I) denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i, where I is an edge ideal. In this paper, we prove that if tb(S/I)≥⌈3b2⌉ for some b≥0, then the subadditivity condition ta+b(S/I)≤ta(S/I)+tb(S/I) holds for all a≥0. In addition, we prove that ta+4(S/I)≤ta(S/I)+t4(S/I) for all a≥0 (the case b=0,1,2,3 is known). We conclude that if the projective dimension of S/I is at most 9, then I satisfies the subadditivity condition.

AB - Let S=K[x1,…,xn], where K is a field, and ti(S/I) denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i, where I is an edge ideal. In this paper, we prove that if tb(S/I)≥⌈3b2⌉ for some b≥0, then the subadditivity condition ta+b(S/I)≤ta(S/I)+tb(S/I) holds for all a≥0. In addition, we prove that ta+4(S/I)≤ta(S/I)+t4(S/I) for all a≥0 (the case b=0,1,2,3 is known). We conclude that if the projective dimension of S/I is at most 9, then I satisfies the subadditivity condition.

KW - Betti numbers

KW - Edge ideal

KW - Monomial ideal

KW - Simplicial complex

KW - Subadditivity condition

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DO - 10.1007/s10801-024-01363-6

M3 - Article

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SN - 0925-9899

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JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 4

ER -

On the subadditivity condition of edge ideal

Abstract

Keywords

ASJC Scopus subject areas

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