# The annihilator numbers of $S/I$

Let $S=K[x_{1},x_{2},...,x_{n}]$ and $I$ be a strongly stable ideal of $S$. Compute the annihilator numbers of $S/I$ with respect to the almost regular sequence $x_{n},x_{n-1},...,x_{1}$. (Herzog and Hibi, Monomial Ideals, Exercise 4.8.)

Note that a sequence $x_{1},x_{2},...,x_{n}$ is called almost regular if the length of $(0:_{M/{({x_{1},x_{2},...,x_{i-1})M}}}x_{i})$ is finite and the ideal $I$ is strongly stable if $x_{i}(u/x_{j})\in I$ for all monomials $u$ belong to $I$ and all $i<j$ such that $x_{j}$ divides $u$. We denote by $A_{i-1}(y;M)$ the graded module $0:_{M/{({y_{1},y_{2},...,y_{i-1})M}}}$ $y_{i}$ and call the numbers $a_{ij}(y;M)=\dim_{K} A_{i}(y;M)_{j}$ if $i < n$ and $\beta_{0j}(M)$ if $i=n$ the annihilator numbers of $M$ with respect to the sequence $y$.

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