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Stanley depth of monomial ideals with small number of generators. (English) Zbl 1185.13027

Summary: For a monomial ideal \(I \subset S = K[x _{1}\dots ,x _{n }]\), we show that \(\text{sdepth}(S/I) \geq n - g(I)\), where \(g(I)\) is the number of the minimal monomial generators of \(I\). If \(I =\nu I^{\prime}\), where \(\nu \in S\) is a monomial, then we see that \(\text{sdepth}(S/I) = \text{sdepth}(S/I^{\prime})\). We prove that if \(I\) is a monomial ideal \(I \subset S\) minimally generated by three monomials, then \(I\) and \(S/I\) satisfy the Stanley conjecture. Given a saturated monomial ideal \(I \subset K[x _{1},x _{2},x _{3}]\) we show that \(\text{sdepth}(I) = 2\). As a consequence, \(\text{sdepth}(I) \geq \text{sdepth}(K[x _{1},x _{2},x _{3}]/I) +1\) for any monomial ideal in \(I \subset K[x _{1},x _{2},x _{3}]\).

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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References:

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