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

Let be a commutative Noetherian ring. For a natural number , we prove that the class of modules of projective dimension bounded by is of finite type if and only if satisfies Serre's condition . 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 ‐dimensional version of the Govorov–Lazard theorem holds if and only if satisfies the ‘almost’ Serre condition .