About q-approximations and q-hulls over a Noetherian ring,some refinements of the Auslander-Bridger theory

We revisit the notion of q-approximations for a module over a Noetherian ring, originally due to Auslander and Bridger without a name and rediscovered later by Evans and Griffith. We introduce a somewhat symmetric notion of q-hull and provide existence theorems for both q-approximations and q-hulls. The main feature here are existence theorems and characterizations of minimal such ones when the ring is local. The q-hulls being close to the morphism obtained by Auslander and Bridger in their “approximation theorem”, we also obtain for the latter a minimal statement in the case when the ring is local.