5 - Deformation Theory for Vector Bundles

Summary

Abstract

These expository notes give an introduction to the elements of deformation theory which is meant for graduate students interested in the theory of vector bundles and their moduli.

Introduction : Basic examples

For simplicity, we will work over a fixed base field k which may be assumed to be algebraically closed. All schemes and all morphisms between them will be assumed to be over the base k, unless otherwise indicated. In this section we introduce four examples which are of basic importance in deformation theory, with special emphasis on vector bundles.

Basic example 1: Deformations of a point on a scheme

We begin by setting up some notation. Let Art_k be the category of all artin local k-algebras, with residue field k. In other words, the objects of Art_k are local k-algebras with residue field k which are finite-dimensional as k-vector spaces, and morphisms are all k-algebra homomorphisms. Note that k is both an initial and a final object of Art_k. By a deformation functor we will mean a covariant functor F : Art_k → Sets for which F(k) is a singleton point. As k is an initial object of Art_k, this condition means that we can as well regard F to be a functor to the category of pointed sets.