Versal deformations of pairs and Cohen-Macaulay approximation

TL;DR

This paper develops a deformation theory for pairs of algebra and module with isolated singularities, using André-Quillen cohomology, and explores Cohen-Macaulay approximation's impact on versal deformation spaces.

Contribution

It establishes the existence of versal deformations for algebra-module pairs with singularities and introduces a cohomological framework including a Kodaira-Spencer class.

Findings

Existence of versal henselian deformations for pairs with isolated singularities

A long exact sequence relating algebra, module, and pair cohomology

Cohen-Macaulay approximation induces maps between versal base spaces

Abstract

For a pair (algebra, module) with equidimensional and isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an Andr\'e-Quillen cohomology for pairs is a central ingredient in the Artin theory used. In particular we give a long exact sequence relating the algebra cohomology and the module cohomology with the cohomology of the pair and define a Kodaira-Spencer class for pairs. Cohen-Macaulay approximation induces maps between versal base spaces for pairs and cohomology conditions imply properties like smoothness, isomorphism and linear section.