## 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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