Linear resolutions and Gr\"{o}bner basis of Hankel determinantal ideals

TL;DR

This paper investigates Hankel determinantal ideals, demonstrating their algebraic properties such as quadratic Gr"obner bases, Koszulness, and linear resolutions for products of these ideals, advancing understanding of their algebraic structure.

Contribution

It establishes that the multi-Rees algebra of Hankel determinantal ideals has a quadratic Gr"obner basis and proves linear resolutions for all products of these ideals, revealing their algebraic regularity.

Findings

Multi-Rees algebra is quadratic Gr"obner basis and Koszul.

Products of ideals have linear resolutions.

Generators of product ideals form a Gr"obner basis.

Abstract

In this paper, we study the family of determinantal ideals of "close" cuts of Hankel matrices, say $ \mathcal{f} $. We show that the multi-Rees algebra of ideals in $ \mathcal{f} $ is defined by a quadratic Gr\"{o}bner basis, it is Koszul, normal Cohen-Macaulay domain and it has a nice Sagbi basis. As a consequence of Koszulness, we prove that every product $ I_1\ldots I_l $ of ideals of $ \mathcal{f} $ has linear resolution. Moreover, we show that natural generators of every product $ I_1\ldots I_l $ form a Gr\"{o}bner basis.