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 ΠR(S) 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 ΠR(S) is itself noetherian and finite over its center and that each ΠR(S)d is finitely generated projective. We also prove that ΠR(S) is of finite global dimension if R and S are regular.