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Pretty cleanness and filter-regular sequences

Let $K$ be a field and $S=K[x_1,\ldots , x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,\ldots , u_r$ be monomials in $S$. We prove that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then $S/I$ is pretty clean if and only if $S/(I,u_1,\ldots , u_r)$ is pretty clean. Also, we show that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then Stanley's conjecture is t…