On compatibility of Koszul- and higher preprojective gradings

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 a basic \(n\)-representation infinite algebra \(A\) such that \(\Pi_{n+1}(A)\) is graded coherent, 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 that have graded coherent higher preprojective algebras.