A note on dimension of triangulated categories

Abstract:

In this note we study the behavior of the dimension of the perfect derived category $\operatorname {Perf}(A)$ of a dg-algebra $A$ over a field $k$ under a base field extension $K/k$. In particular, we show that the dimension of a perfect derived category is invariant under a separable algebraic extension $K/k$. As an application we prove the following statement: Let $A$ be a self-injective algebra over a perfect field $k$. If the dimension of the stable category $\underline {\textrm {mod}}A$ is $0$, then $A$ is of finite representation type. This theorem is proved by M. Yoshiwaki in the case when $k$ is an algebraically closed field. Our proof depends on his result.