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 Auslanderʼs fundamental module.