Derived Representation Type and Field Extensions

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 complexes of projective $A$-modules. $\mathbf{C}$-dichotomy implies the dichotomy properties of representation type on the levels of homotopy category and derived category. 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.