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The cleanness of (symbolic) powers of Stanley-Reisner ideals

Somayeh BandariAli Soleyman Jahan — 2017

Czechoslovak Mathematical Journal

Let Δ be a pure simplicial complex on the vertex set [ n ] = { 1 , ... , n } and I Δ its Stanley-Reisner ideal in the polynomial ring S = K [ x 1 , ... , x n ] . We show that Δ is a matroid (complete intersection) if and only if S / I Δ ( m ) ( S / I Δ m ) is clean for all m and this is equivalent to saying that S / I Δ ( m ) ( S / I Δ m , respectively) is Cohen-Macaulay for all m . By this result, we show that there exists a monomial ideal I with (pretty) cleanness property while S / I m or S / I ( m ) is not (pretty) clean for all integer m 3 . If dim ( Δ ) = 1 , we also prove that S / I Δ ( 2 ) ( S / I Δ 2 ) is clean if and only if S / I Δ ( 2 ) ( S / I Δ 2 ,...

Pretty cleanness and filter-regular sequences

Somayeh BandariKamran Divaani-AazarAli Soleyman Jahan — 2014

Czechoslovak Mathematical Journal

Let K be a field and S = K [ x 1 , ... , x n ] . Let I be a monomial ideal of S and u 1 , ... , u r be monomials in S . We prove that if u 1 , ... , u r form a filter-regular sequence on S / I , then S / I is pretty clean if and only if S / ( I , u 1 , ... , u r ) is pretty clean. Also, we show that if u 1 , ... , u r form a filter-regular sequence on S / I , then Stanley’s conjecture is true for S / I if and only if it is true for S / ( I , u 1 , ... , u r ) . Finally, we prove that if u 1 , ... , u r is a minimal set of generators for I which form either a d -sequence, proper sequence or strong s -sequence (with respect to the reverse lexicographic...

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