Linear resolutions of powers and products

TL;DR

This paper explores families of homogeneous ideals in polynomial rings where all products have linear resolutions, linking this property to primary decompositions and properties of multi-Rees algebras, using Gr"obner bases and Sagbi deformation.

Contribution

It provides new examples of ideal families with universally linear resolutions for their products, connecting algebraic properties with homological and combinatorial structures.

Findings

Polymatroidal ideals have linear resolutions for all products.

Ideals generated by linear forms exhibit this property.

Borel fixed ideals of maximal minors also satisfy the condition.

Abstract

The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see, this condition is strongly correlated to good primary decompositions of the products and good homological and arithmetical properties of the associated multi-Rees algebras. The following families will be discussed in detail: polymatroidal ideals, ideals generated by linear forms and Borel fixed ideals of maximal minors. The main tools are Gr\"obner bases and Sagbi deformation.