Point varieties and point-exactness of Koszul algebras

TL;DR

This paper introduces the point-exact condition for Koszul algebras, linking it to the (G1) condition, and demonstrates its preservation under certain algebraic operations, with applications to skew polynomial algebras.

Contribution

It defines the point-exact condition for Koszul algebras and shows its stability under quotients and specific algebraic constructions, advancing understanding of algebraic properties.

Findings

Point-exact condition characterizes (G1) for Koszul algebras.

The (G1) condition and point-exactness are preserved under taking quotients by regular normal elements.

Skew polynomial algebras satisfy the point-exact condition.

Abstract

In this paper, we introduce the point-exact condition for a Koszul algebra $A$, which is useful for characterizing the (G1) condition of $A$ in the sense of Mori. Let $B = A/(f)$, where $f \in A_2$ is a regular normal element. We show that if $A$ satisfies the (G1) condition and is point-exact up to degree $\ell \geq 2$, then $B$ also satisfies the (G1) condition and is point-exact up to degree $\ell$. Moreover, we show that skew polynomial algebras satisfy the point-exact condition.