Construction of Koszul algebras by finite Galois covering

Abstract

It is shown that, the quasi-Koszulities of algebras and modules are Morita invariance. A finite-dimensional $K$-algebra $A$ with an action of $G$ is quasi-Koszul if and only if so is the skew group algebra $A \ast G$, where $G$ is a finite group satisfying $\char K \nmid |G|$. A finite-dimensional $G$-graded $K$-algebra $A$ is quasi-Koszul if and only if so is the smash product $A # G^*$, where $G$ is a finite group satisfying $\char K \nmid |G|$. These results are applied to prove that, if a finite-dimensional connected quiver algebra is Koszul then so are its Galois covering algebras with finite Galois group $G$ satisfying $\char K \nmid |G|$. So one can construct Koszul algebras by finite Galois covering. Moreover, a general construction of Koszul algebras by Galois covering with finite cyclic Galois group is provided. As examples, many Koszul algebras are constructed from exterior algebras and Koszul preprojective algebras by finite Galois covering with either cyclic or noncyclic Galois group.