Alexander duality for the alternative polarizations of strongly stable ideals

TL;DR

This paper introduces an Alexander duality for strongly stable ideals, linking their properties to dual ideals and enabling the computation of local cohomology Hilbert series through irreducible decompositions.

Contribution

It defines a new form of Alexander duality for strongly stable ideals and explores its applications, including Hilbert series calculations via dual Betti numbers.

Findings

Established duality between strongly stable ideals and their duals.

Connected Hilbert series of local cohomologies to irreducible decompositions.

Provided methods to compute invariants using duality.

Abstract

We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal $I \subset \Bbbk[x_1, \ldots, x_n]$ with ${\rm deg}(\mathsf{m}) \le d$ for all $\mathsf{m} \in G(I)$, its dual $I^* \subset \Bbbk[y_1, \ldots, y_d]$ is a strongly stable ideal with ${\rm deg}(\mathsf{m}) \le n$ for all $\mathsf{m} \in G(I^*)$. This duality has been constructed by Fl$\o$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 $I^*$).