Cohen-Macaulay approximation in fibred categories

TL;DR

This paper extends Cohen-Macaulay approximation theory to fibred categories, providing new results on existence, properties, and deformation theory of Cohen-Macaulay modules in a fibred setting, with applications to algebraic geometry.

Contribution

It generalizes Cohen-Macaulay approximation axioms to fibred categories and develops deformation theory for pairs of algebra and module, including cohomological tools and versal families.

Findings

Existence of flat families of maximal Cohen-Macaulay modules

Extension of upper semi-continuous invariants

Development of deformation theory for algebra-module pairs

Abstract

We extend the Auslander-Buchweitz axioms and prove Cohen-Macaulay approximation results for fibred categories. Then 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. In the second part we study the induced maps of deformation functors and deduce properties like injectivity, isomorphism and smoothness under general, mainly cohomological conditions on the module. We also provide deformation theory for pairs (algebra, module), e.g. a cohomology for such pairs, a long exact sequence linking this cohomology to the Andr\'e-Quillen cohomology of the algebra and the Ext cohomology of the module, Kodaira-Spencer classes and maps including a secondary Kodaira-Spencer class, and existence of a versal family for pairs with isolated singularity.