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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 graded ideal which has linear quotients with respect to a homogeneous system of generators $f_{1},...,f_{m}$ with deg$(f_{1})\leq...$ deg$(f_{m})$. Prove that

$\beta_{i,i+j}(I)=\sum\limits_{k=1,deg(f_{k})=j}^{m}$ $\left( \begin{array}{c} r_{k}\\ i \end{array} \right)$

where $r_{k}$ is the cardinality of the minimal system of linear forms which generate $(f_{1},...,f_{k-1}):f_{k}$.

The case when deg$f_{1}=$deg$f_{2}=...=$deg$f_{m}$ may be proved easily because in this case $I$ has linear resolution.

When degrees are not equal, how to prove? We know from Theorem 8.2.15 in this book that $I$ is componentwise linear.

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  • $\begingroup$ Is there a reason for asking two questions on Exercise 8.8 in Monomial Ideals by Herzog and Hibi, rather than putting them together in one post? $\endgroup$ – Stefan Kohl Jan 12 '14 at 16:48
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    $\begingroup$ @Stefan Kohl No reason. Just not to be very long in one post. If you think the same reasoning can lead to the solution of both then please give me some hints on either of them. I will be highly grateful. $\endgroup$ – user118827 Jan 13 '14 at 3:46

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