Algebra extensions and derived-discrete algebras

Abstract

Let $\ddot{I}•:A\to B$ be an algebra extension. We prove that if $\ddot{I}•$ is split, the derived-discreteness of $A$ implies the derived-discreteness of $B$; if $\ddot{I}•$ is separable and the right $A$-module $B$ is projective, the converse holds. We prove an analogous statement for piecewise hereditary algebras.