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Exercise 8.8 in Monomial Ideals by Herzog and Hibi:

Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either $\deg(u_{i})<\deg(u_{j})$ or $\deg u_{i}=\deg u_{j}$ and $u_{j}<_{\text{ lex}}u_{i}$. Show that $I$ has linear quotients with respect to $u_{1},...,u_{m}$ and compute the graded Betti numbers of $I$ in terms of its minimal monomial generators.

We know that a stable monomial ideal $I$ generated in one degree say $d$ has linear quotients with respect to $u_{1}>_{lex}u_{2}>_{lex}...>_{lex}u_{m}$. More precisely, $(u_{1},...,u_{i-1}):u_{i}=(x_{1},...,x_{\max(u_{i})-1})$. Moreover, $I$ has linear resolution and $\beta_{i}(I)=\sum\limits_{u\in G(I)}$ $\left( \begin{array}{c} \max(u)-1\\ i \end{array} \right)$.

What happens in the case when $I$ is not necessarily generated in one degree? Would someone please help or give some hint about the exercise?

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