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On vanishing patterns in $j$-strands of edge ideals
Commutative Algebra · Submitted 28 Oct 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Next, we apply our result on strand connectivity to establish the subadditivity conjecture for edge ideals, $t_{a+b}\leq t_a+t_b$, in case $b=2,3$ (the case $b=1$ is known).
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The abstract, sentence by sentence
We consider two problems regarding vanishing patterns in the Betti table of edge ideals $I$ in polynomial algebra $S$. First, we show that the $j$-strand is connected if $j=3$ (for $j=2$ this is easy and known), and give examples where the $j$-strand is not connected for any $j>3$. Next, we apply our result on strand connectivity to establish the subadditivity conjecture for edge ideals, $t_{a+b}\leq t_a+t_b$, in case $b=2,3$ (the case $b=1$ is known). Here $t_i$ stands for the maximal shifts in the minimal free $S$-resolution of $S/I$
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider two problems regarding vanishing patterns in the Betti table of edge ideals $I$ in polynomial algebra $S$. | background 0.56 | 0.44 | 0.04 | unclear 0.65 |
| s1 | First, we show that the $j$-strand is connected if $j=3$ (for $j=2$ this is easy and known), and give examples where the $j$-strand is not connected for any $j>3$. | result 0.94 | 0.94 | 0.43 | asserted 1.00 |
| s2 | Next, we apply our result on strand connectivity to establish the subadditivity conjecture for edge ideals, $t_{a+b}\leq t_a+t_b$, in case $b=2,3$ (the case $b=1$ is known). | result 0.73 | 0.95 | 0.44 | asserted 1.00 |
| s3 | Here $t_i$ stands for the maximal shifts in the minimal free $S$-resolution of $S/I$ | background 0.93 | 0.11 | 0.03 | asserted 0.78 |
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Original abstract
We consider two problems regarding vanishing patterns in the Betti table of edge ideals $I$ in polynomial algebra $S$. First, we show that the $j$-strand is connected if $j=3$ (for $j=2$ this is easy and known), and give examples where the $j$-strand is not connected for any $j>3$. Next, we apply our result on strand connectivity to establish the subadditivity conjecture for edge ideals, $t_{a+b}\leq t_a+t_b$, in case $b=2,3$ (the case $b=1$ is known). Here $t_i$ stands for the maximal shifts in the minimal free $S$-resolution of $S/I$
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1 arXiv paper cite this in 1 sentence
Papers citing it
Cites it as background 1 paper
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These include some algebras of krull dimension at most 1 (Eisenbud, Huneke and Ulrich [ref] ), when $a=1$ (Fernández-Ramos and Gimenez [ref] if $I$ is generated by degree 2 monomials, Herzog and Srinivasan [ref] when $b$ is the projective dimension of $S/I$ or when $I$ is any monomial ideal), in certain homological degrees in the case of Gorenstein algebras (El Khoury and Srinivasan [ref] ) and the case $a=1,2,3$ for a monomial ideal generated in degree 2 (Abedelfatah and Nevo [ref] ).
Lattice Complements and the Subadditivity of Syzygies of Simplicial Forests · Introduction · 0.90
What it cites · 1 sentence without a confident label
Cites as background 3 papers
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Given a graded ideal $I$ in $S$ let $t_i$ denote the maximal shifts in the minimal graded free $S$-resolution of $S/I$, namely [equation] The subadditivity relation [equation] was proved under certain conditions on $I$ or for certain values of $a$ and $b$, e.g. in [ref] , and is conjectured to hold under other conditions on $I$ for all values $a$ and $b$ [ref] .
A note on the subadditivity problem for maximal shifts in free resolutions · Introduction · 0.86
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Given a graded ideal $I$ in $S$ let $t_i$ denote the maximal shifts in the minimal graded free $S$-resolution of $S/I$, namely [equation] The subadditivity relation [equation] was proved under certain conditions on $I$ or for certain values of $a$ and $b$, e.g. in [ref] , and is conjectured to hold under other conditions on $I$ for all values $a$ and $b$ [ref] .
A note on the subadditivity of Syzygies · Introduction · 0.86
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For other recent results on other vanishing patterns, see e.g. [ref] .
An alternative algorithm for computing the Betti table of a monomial ideal · Introduction · 0.93
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