  Our purpose in this work is multifold. First, we provide general criteria for
the finiteness of the projective and injective dimensions of a finite module
$M$ over a (commutative) Noetherian ring $R$. Second, in the other direction,
we investigate the impact of the finiteness of certain homological dimensions
of $M$ if $R$ is local, mainly when $R$ is Cohen-Macaulay and with a partial
focus on duals. Along the way, we produce various freeness criteria for
modules. Finally, we give applications, including characterizations of when $R$
is Gorenstein (and other ring-theoretic properties as well, sometimes in the
prime characteristic setting), particularly by means of its anticanonical
module, and in addition we address special cases of some long-standing
conjectures; for instance, we confirm the 1985 conjecture of Vasconcelos on
normal modules in case the module of differentials is almost Cohen-Macaulay.
