Derived representation type and field extensions
Volume 168 / 2022
Colloquium Mathematicum 168 (2022), 105-117
MSC: Primary 16G10; Secondary 16E35.
DOI: 10.4064/cm8376-3-2021
Published online: 6 October 2021
Abstract
Let $A$ be a finite-dimensional algebra over a field $k$. We define $A$ to be $\mathbf C$-dichotomic if it has the dichotomy property of the representation type on the category of certain bounded complexes of projective $A$-modules. If $k$ admits a finite separable field extension $K/k$ such that $K$ is algebraically closed (the real number field for example), we prove that $A$ is $\mathbf C$-dichotomic. As a consequence, the second derived Brauer–Thrall type theorem holds for $A$, i.e., $A$ is either derived-discrete or strongly derived-unbounded.
