Some generalizations of preprojective algebras and their properties

Abstract

In this note we consider a notion of relative Frobenius pairs of commutative rings S/R. To such a pair, we associate an N-graded R -algebra which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of S over R is 4 and R is Noetherian, we prove that the generalized preprojective algebra is itself Noetherian and finite over its center and that it is finitely generated projective in each degree. We also prove that generalized preprojective algebras are of finite global dimension if the rings R and S are regular.