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 \(a+b\le \mathrm {proj\,dim}(S/I)\).

让\（S = K [X_1，\ ldots，x_n] \） ，其中ķ是一个字段，和\（t_i \）表示在最小分级自由的最大移位小号渐变代数的-分辨率小号/我在程度我。在本文中，我们证明：

• 如果I是S的单项式理想并且\（a \ ge b-1 \ ge 0 \）是使得\（a + b \ le \ mathrm {proj \，dim}（S / I）\）的整数，则

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

• 如果\（I = I _ {\ Delta} \）其中\（\ Delta \）是一个简单复数，则\（\ dim（\ Delta）<t_a-a \）或\（\ dim（\ Delta）<t_b -b \），然后

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

• 如果我是\（m_1，\ ldots，m_r \）最小生成的单项式理想，这样\（\ frac {{{{\，\ mathrm {lcm} \，}}（m_1，\ ldots，m_r）} { {{\，\ mathrm {lcm} \，}}（m_1，\ ldots，\ widehat {m} _i，\ ldots，m_r）} \ notin K \）对于所有i，其中\（\ widehat {m} _i \）装置，其\（M_I \）被省略，然后\（T_ {A + b} \文件T_A + T_B \）对所有\（A，b \ GE 0 \）与\（A + b \文件\ mathrm {proj \，dim}（S / I）\）。