# Relating properties of homological dimension two algebras

Let Q be a connected, non-ADE quiver. The preprojective algebra of Q is well-behaved in the following sense: it is 2-Calabi–Yau, a noncommutative complete intersection (NCCI), and prime. If further Q is extended ADE then the preprojective algebra of Q is a noncommutative crepant resolution (NCCR) over its center, which is isomorphic to functions on the corresponding du Val singularity. In this talk, I will explain joint work with Travis Schedler which proves these properties for a multiplicative analogue of the preprojective algebra, defined by Crawley-Boevey and Shaw, in the case Q contains a cycle. Current work in progress aims to prove this for general Q. The technique involves defining a new notion, the strong free product property (SFPP), which implies these notions. One then proves the SFPP using multiple applications of Bergman’s Diamond Lemma for ring theory. Applications to topology and geometry include computations of certain Chekanov–Eliashberg dg-algebras / wrapped Fukaya categories following Etgü–Lekili, and a description of the formal local structure of quiver varieties.