The finite type of modules of bounded projective dimension and Serre's conditions

Let $R$ be a commutative Noetherian ring. For a natural number $k$, 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 $\left(S_k\right)$. 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 $\left(C_{k+1}\right)$.