Basic deformation theory of smooth formal schemes

TL;DR

This paper develops a deformation theory for smooth formal schemes, analyzing obstructions and classifications of liftings via Ext groups, providing foundational results for understanding their deformation behavior.

Contribution

It introduces a comprehensive deformation framework for smooth formal schemes, detailing obstruction classes and classification of liftings through Ext groups.

Findings

Obstruction to global lifting lies in Ext^1 group.

Classification of liftings is governed by Ext^1 group.

Existence of smooth liftings characterized by vanishing Ext^2 group.

Abstract

We provide the main results of a deformation theory of smooth formal schemes. First we deal with the case of global lifting of smooth morphisms. We prove that the obstruction to the existence of a global lifting lies in a Ext^1 group. Then we study uniqueness and existence of lifting of smooth formal schemes. The set of isomorphism classes of smooth liftings is classified by a Ext^1 group and there exists an obstruction in a Ext^2 group whose vanishing characterizes the existence of smooth liftings.