# Products of ideals of linear forms in quadric hypersurfaces

## TL;DR

This paper extends the known result that products of ideals of linear forms have linear resolutions from polynomial rings to quadric hypersurfaces, using a new approach based on approximation systems.

## Contribution

It introduces a flexible version of Derksen and Sidman's approximation systems to prove linear resolutions for products of ideals of linear forms in quadric hypersurfaces.

## Key findings

- Products of ideals of linear forms in quadric hypersurfaces have linear resolutions.
- Develops a new method based on approximation systems for this class of problems.
- Extends previous results from polynomial rings to more general hypersurfaces.

## Abstract

Conca and Herzog proved that any product of ideals of linear forms in a polynomial ring has a linear resolution. The goal of this paper is to establish the same result for any quadric hypersurface. The main tool we develop and use is a flexible version of Derksen and Sidman's approximation systems.