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

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