  Consider the obvious functor from the unbounded derived category of all
finitely generated modules over a left noetherian ring $R$ to the unbounded
derived category of all modules. We answer the natural question whether this
functor defines an equivalence onto the full subcategory of complexes with
finitely generated cohomology modules in two special cases. If $R$ is a
quasi-Frobenius ring of infinite global dimension, then this functor is not
full. If $R$ has finite left global dimension, this functor is an equivalence.
We also prove variants of the latter assertion for left coherent rings, for
noetherian schemes and for locally noetherian Grothendieck categories.
