  Let $R$ be a commutative noetherian ring. We prove that the class of modules
of projective dimension bounded by $k$ is of finite type if and only if $R$
satisfies Serre's condition $(S_k)$. In particular, this answers positively a
question of Bazzoni and Herbera in the specific setting of a Gorenstein ring.
Applying similar techniques, we also show that the $k$-dimensional version of
the Govorov-Lazard Theorem holds if and only if $R$ satisfies the "almost"
Serre condition $(C_{k+1})$.
