On the subadditivity condition of edge ideal

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2024年10月8日

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Abed Abedelfatah

摘要

Let $S=K[x_1,\ldots ,x_n]$ $S = K [x_{1}, \dots, x_{n}]$ , where K is a field, and $t_i(S/I)$ $t_{i} (S / I)$ 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)\ge \lceil \frac{3b}{2} \rceil $ $t_{b} (S / I) \geq ⌈ \frac{3 b}{2} ⌉$ for some $b\ge 0$ $b \geq 0$ , then the subadditivity condition $t_{a+b}(S/I)\le t_a(S/I)+t_b(S/I)$ $t_{a + b} (S / I) \leq t_{a} (S / I) + t_{b} (S / I)$ holds for all $a\ge 0$ $a \geq 0$ . In addition, we prove that $t_{a+4}(S/I)\le t_a(S/I)+t_4(S/I)$ $t_{a + 4} (S / I) \leq t_{a} (S / I) + t_{4} (S / I)$ for all $a\ge 0$ $a \geq 0$ (the case $b=0,1,2,3$ $b = 0, 1, 2, 3$ is known). We conclude that if the projective dimension of S/I is at most 9, then I satisfies the subadditivity condition.