Koszul hypersurfaces over the exterior algebras

TL;DR

This paper characterizes when quotient algebras of exterior algebras by quadratic forms are Koszul, showing they are Koszul precisely when the quadratic form factors into two linear forms.

Contribution

It provides a complete characterization of Koszul property for hypersurfaces over exterior algebras defined by quadratic forms.

Findings

E/(h) is Koszul iff h is a product of two linear forms

The result characterizes Koszulity in terms of the factorization of quadratic forms

Provides a criterion for Koszulness in exterior algebra hypersurfaces

Abstract

We prove that if $E$ is an exterior algebra over a field, $h$ is a quadratic form, then $E/(h)$ is Koszul if and only if $h$ is a product of two linear forms.