On compatibility of Koszul- and higher preprojective gradings

TL;DR

This paper explores the compatibility of gradings in higher preprojective algebras derived from $n$-hereditary algebras, establishing conditions under which these algebras are Koszul and how gradings relate to cuts of existing gradings.

Contribution

It proves that certain gradings on higher preprojective algebras imply the original algebra is Koszul and characterizes these gradings as cuts of known gradings, extending understanding of algebra compatibility.

Findings

Koszul property is necessary for compatible gradings in $n$-representation finite and infinite cases.

Higher preprojective gradings are isomorphic to cuts of (almost) Koszul gradings.

$n$-APR tilting preserves Koszul property in $n$-representation infinite algebras.

Abstract

We investigate compatibility of gradings for an almost Koszul or Koszul algebra $R$ that is also the higher preprojective algebra $\Pi_{n+1}(A)$ of an $n$-hereditary algebra $A$. For an $n$-representation finite algebra $A$, we show that $A$ must be Koszul if $\Pi_{n+1}(A)$ can be endowed with an almost Koszul grading. For an acyclic basic $n$-representation infinite algebra $A$, we show that $A$ must be Koszul if $\Pi_{n+1}(A)$ can be endowed with a Koszul grading. From this we deduce that a higher preprojective grading of an (almost) Koszul algebra $R = \Pi_{n+1}(A)$ is, in both cases, isomorphic to a cut of the (almost) Koszul grading. Up to a further assumption on the tops of the degree $0$ subalgebras for the different gradings, we also show a similar result without the basic assumption in the $n$-representation infinite case. As an application, we show that $n$-APR tilting preserves the property of being Koszul for $n$-representation infinite algebras.