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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…