  The numerical invariants (global) cohomological length, (global)
cohomological width, and (global) cohomological range of complexes (algebras)
are introduced. Cohomological range leads to the concepts of derived bounded
algebras and strongly derived unbounded algebras naturally. The first and
second Brauer-Thrall type theorems for the bounded derived category of a
finite-dimensional algebra over an algebraically closed field are obtained. The
first Brauer-Thrall type theorem says that derived bounded algebras are just
derived finite algebras. The second Brauer-Thrall type theorem says that an
algebra is either derived discrete or strongly derived unbounded, but not both.
Moreover, piecewise hereditary algebras and derived discrete algebras are
characterized as the algebras of finite global cohomological width and finite
global cohomological length respectively.
