  For a ring $A$, we consider the question whether every bounded above cochain
complex of injective $A$-modules which is acyclic is null-homotopic. We show
that if $A$ is left and right noetherian and has a dualizing complex, then this
implies that the finitistic dimension of $A$ is finite. In the appendix,
Nakamura and Thompson show that the opposite holds over any ring. Our results
give several new necessary and sufficient conditions for a ring to have finite
finitistic dimension in a very general setting. Applications include a
generalization of a recent result of Rickard about relations between unbounded
derived categories and finitistic dimension, as well as several new
characterizations of noetherian rings which satisfy the Gorenstein symmetry
conjecture.
