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arXiv · 2411.13283

On compatibility of Koszul- and higher preprojective gradings

Darius Dramburg·Mads Hustad Sandøy
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Abstract

We investigate compatibility of gradings for an almost Koszul or Koszul algebra $R$ that is also the higher preprojective algebra $Π_{n+1}(A)$ of an $n$-hereditary algebra $A$. For an $n$-representation finite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ 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 $Π_{n+1}(A)$ can be endowed with a Koszul grading. From this we deduce that a higher preprojective grading of an (almost) Koszul algebra $R = Π_{n+1}(A)$ 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.

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Darius Dramburg, Mads Hustad Sandøy. 2025-10-16. On compatibility of Koszul- and higher preprojective gradings. https://arxiv.org/abs/2411.13283

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