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Almost complete intersections and Stanley’s conjecture. (English) Zbl 1297.13024

Let \(\mathbb{K}\) be a field and \(S=\mathbb{K}[x_1,\dots,x_n]\) be the polynomial ring in \(n\) variables over the field \(\mathbb{K}\). Let \(M\) be a nonzero finitely generated \(\mathbb{Z}^n\)-graded \(S\)-module. Let \(u\in M\) be a homogeneous element and \(Z\subseteq \{x_1,\dots,x_n\}\). The \(\mathbb{K}\)-subspace \(u\mathbb{K}[Z]\) generated by all elements \(uv\) with \(v\in\mathbb{K}[Z]\) is called a Stanley space of dimension \(|Z|\), if it is a free \(\mathbb{K}[\mathbb{Z}]\)-module. Here, as usual, \(|Z|\) denotes the number of elements of \(Z\). A (vector space) decomposition \(\mathcal{D}\) of \(M\) as a finite direct sum of Stanley spaces is called a Stanley decomposition of \(M\). The minimum dimension of a Stanley space in \(\mathcal{D}\) is called Stanley depth of \(\mathcal{D}\) and is denoted by \(\text{sdepth}(\mathcal {D})\). The quantity \[ \text{sdepth}(M):=\max\big\{\text{sdepth}(\mathcal{D})\mid \mathcal{D}\;\text{is\;a\;Stanley\;decomposition\;of}\;M\big\} \] is called Stanley depth of \(M\). Stanley conjectured that \[ \text{depth}(M) \leq \text{sdepth}(M) \] for all \(\mathbb{Z}^n\)-graded \(S\)-modules \(M\).
For every monomial ideal \(I\), the cardinality of the set of minimal monomial generators of \(I\) is denoted by \(\mu(I)\). The ideal \(I\) is said to be almost complete intersection if \(\mu(I)=\text{ht}(I)+1\). In the paper under review, the authors prove that \(S/I\) satisfies Stanley’s conjecture if either:
1) \(I\) is almost complete intersection,
2) \(I\) can be generated by less than four monomials; or
3) \(I\) is the Stanley-Reisner ideal of a locally complete intersection simplicial complex.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
05E40 Combinatorial aspects of commutative algebra
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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