Resolution of ideals associated to subspace arrangements

Conca, Aldo; Tsakiris, Manolis C.

doi:10.2140/ant.2022.16.1121

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Resolution of ideals associated to subspace arrangements

Aldo Conca and Manolis C. Tsakiris

Vol. 16 (2022), No. 5, 1121–1140

DOI: 10.2140/ant.2022.16.1121

Abstract

Let $I_{1}, \dots, I_{n}$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_{1} \dots I_{n}$ . We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_{P}$ , where $P$ is a certain poset ideal associated to the underlying subspace arrangement.

Keywords

subspace arrangements, free resolutions

Mathematical Subject Classification

Primary: 13D02

Milestones

Received: 23 November 2020

Revised: 8 April 2021

Accepted: 24 July 2021

Published: 16 August 2022

Authors

Aldo Conca
Dipartimento di Matematica Università di Genova Genova Italy

Manolis C. Tsakiris
Dipartimento di Matematica Università di Genova Genova Italy	Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Vol. 16, No. 5, 2022