Derived Brauer–Thrall Type Theorem I for Artin Algebras

Abstract

We define the global cohomological range for artin algebras, and define the derived bounded algebras to be the algebras with finite global cohomological range, then we prove the first Brauer–Thrall type theorem for bounded derived categories of artin algebras, i.e., derived bounded algebras are precisely the derived finite algebras. Moreover, the main theorem establishes that the derived bounded artin algebras are just piecewise hereditary algebras of Dynkin type, and can be also characterized as those artin algebras with derived dimension zero, which can be regarded as a generalization of the results of Han–Zhang [11, Theorem 1] and Chen–Ye–Zhang [4, Theorem] in the context of finite-dimensional algebras over algebraically closed fields, respectively.