Cohen-Macaulay approximation in fibred categories

We extend the Auslander-Buchweitz axioms and prove Cohen-Macaulay approximation results for fibred categories. We show that these axioms apply for the fibred category of pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. In particular such a pair admits an approximation with a flat family of maximal Cohen-Macaulay modules and a hull with a flat family of modules with finite injective dimension. The existence of minimal approximations and hulls in the local, flat case implies extension of upper semi-continuous invariants. As an example of MCM approximation we define a relative version of Auslander's fundamental module. (C) 2012 Elsevier Inc. All rights reserved.