Derived Representation Type and Field Extensions
Jie Li, Chao Zhang
TL;DR
The paper extends the second derived Brauer-Thrall dichotomy to algebras over arbitrary infinite fields by introducing C-dichotomic algebras, defined via the derived representation type of complexes of projectives. It proves that C-dichotomy is preserved under finite separable extensions to an algebraically closed field K, and leverages the known dichotomy over algebraically closed fields to conclude that A is either derived-discrete or strongly derived-unbounded when K/k is finite separable with K algebraically closed. Consequently, the second derived Brauer-Thrall type theorem holds for such A, with R as a concrete example. The results unify derived representation type behavior across ground fields and illuminate the role of field extensions in derived category dichotomies.
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 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.