Resumen de Some results on the subadditivity condition of syzygies

Abed Abedelfatah

• 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).