Some results on the subadditivity condition of syzygies

Abstract

Let $S=K[x_1,\ldots ,x_n]$, 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 $a\ge b-1\ge 0$ are integers such that $a+b\le \mathrm {proj\,dim}(S/I)$, then

  $$\begin{aligned} t_{a+b}\le t_a+t_1+t_2+\cdots +t_b-\frac{b(b-1)}{2}. \end{aligned}$$

• If $I=I_{\Delta }$ where $\Delta$ is a simplicial complex such that $\dim (\Delta )< t_a-a$ or $\dim (\Delta )< t_b-b$, then

  $$\begin{aligned} t_{a+b}\le t_a+t_b. \end{aligned}$$

• If I is a monomial ideal that minimally generated by $m_1,\ldots ,m_r$ such that $\frac{{{\,\mathrm{lcm}\,}}(m_1,\ldots ,m_r)}{{{\,\mathrm{lcm}\,}}(m_1,\ldots ,\widehat{m}_i,\ldots ,m_r)}\notin K$ for all i, where $\widehat{m}_i$ means that $m_i$ is omitted, then $t_{a+b}\le t_a+t_b$ for all $a,b\ge 0$ with $a+b\le \mathrm {proj\,dim}(S/I)$.