  Gorenstein homological dimensions are refinements of the classical
homological dimensions, and finiteness singles out modules with amenable
properties reflecting those of modules over Gorenstein rings. As opposed to
their classical counterparts, these dimensions do not immediately come with
practical and robust criteria for finiteness, not even over commutative
noetherian local rings. In this paper we enlarge the class of rings known to
admit good criteria for finiteness of Gorenstein dimensions: It now includes,
for instance, the rings encountered in commutative algebraic geometry and, in
the non-commutative realm, $k$--algebras with a dualizing complex.
