Generic Initial Ideals of Artinian Ideals Having Lefschetz Properties or The Strong Stanley Property

Details Generic Initial Ideals of Artinian Ideals Having Lefschetz Properties or The Strong Stanley Property Generic Initial Ideals of Artinian Ideals Having Lefschetz Properties or The Strong Stanley Property

2006-10-24

arxiv.org/abs/math/0610733v1

Mathematics

Commutative Algebra

Scientific paper

For a standard Artinian $k$-algebra $A=R/I$, we give equivalent conditions for $A$ to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of the generic initial ideal $\mathrm{gin}(I)$ of $I$ under the reverse lexicographic order. Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of $\mathrm{gin}(I)$ are determined just by the the Hilbert function of $I$ if $A$ has the weak Lefschetz property. Furthermore, for the case that $A$ is a standard Artinian $k$-algebra of codimension 3, we show that every graded Betti numbers of $\mathrm{gin}(I)$ are determined by the graded Betti numbers of $I$ if $A$ has the weak Lefschetz property. And if $A$ has the strong Lefschetz (resp. Stanley) property, then we show that the minimal system of generators of $\mathrm{gin}(I)$ is determined by the graded Betti numbers (resp. by the Hilbert function) of $I$.

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