Alexander duality for the alternative polarizations of strongly stable ideals

Abstract

We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal $I\subset \mathbb{k}[x_1,\dots ,x_n]$ with $\mathrm{deg}(\mathsf{m})\le d$ for all $\mathsf{m}\in G(I)$, its dual $I^{\ast }\subset \mathbb{k}[y_1,\dots ,y_d]$ is a strongly stable ideal with $\mathrm{deg}(\mathsf{m})\le n$ for all $\mathsf{m}\in G(I^{\ast })$. This duality has been constructed by Fl$ø$ystad et al. in a different manner, so we emphasis applications here. For example, we will describe the Hilbert serieses of the local cohomologies $H_{\mathfrak{m}}^i(S/I)$ using the irreducible decomposition of $I$ (through the Betti numbers of $I^{\ast }$).

21 pages. We clarified the proofs especially in Section 3 and added Section 6