The weak Lefschetz property for \(\mathfrak{m}\)-full ideals and componentwise linear ideals. (English) Zbl 1285.13026

Harima, Tadahito; Watanabe, Junzo

Ill. J. Math. 56, No. 3, 957-966 (2012).

This paper is based on two results of A. Wiebe [Commun. Algebra 32, No. 12, 4601–4611 (2004; Zbl 1089.13500)] and A. Conca et al. [Math. Z. 265, No. 3, 715–734 (2010; Zbl 1236.13009)]. Wiebe proved a 3-statement equivalence for a componentwise linear ideal to have the weak Lefschetz property in terms of the graded Betti numbers; while Conca et al. showed that componentwise linear ideals are \(\mathfrak m\)-full. Therefore, the idea of the authors to study whether Wiebe’s equivalence holds for \(\mathfrak m\)-full ideals naturally appears.
Firstly they give the equivalence of the three original statements of Wiebe for a \(\mathfrak m\)-full ideal \(I\) with the additional assumption that \(I + (x) / (x)\) is \(\mathfrak m\)-full where \(x\) is a general linear form. Secondly they show that if the generic initial ideal with respect to the graded reverse lexicographic order is stable, then the ideal is componentwise linear if and only if it is completely \(\mathfrak m\)-full.

Reviewer: Adrian Popescu (Kaiserslautern)

MSC:

13E10	Commutative Artinian rings and modules, finite-dimensional algebras
13A02	Graded rings
13F20	Polynomial rings and ideals; rings of integer-valued polynomials

Keywords:

weak Lefschetz property; \(\mathfrak{m}\)-full; componentwise linear

Citations:

Zbl 1089.13500; Zbl 1236.13009

References:

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