Deformation Theory for Vector Bundles

Deformation Theory for Vector Bundles Nitin Nitsure 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. The original version appeared in the vol- ume ‘Moduli spaces and vector bundles’ LMS lecture note series 359 (2009) in honour of Peter Newstead. Contents 1 Introduction : Basic examples ................................. 1 2 General theory .......................................... 4 3 Calculations for basic examples ................................ 17 1 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. For any A in Art k , let h A : Art k → Sets be the deformation functor defined by taking h A (B)= Hom k-alg (A, B). Recall the well-known Yoneda lemma, which asserts that there is a natural bijection Hom(h A ,F ) → F (A) under which a natural transformation α : h A → F is identified with the element α(id A ) ∈ F (A). To simplify notation, given any natural transformation α : h A → F , we denote again by α ∈ F (A) the element α(id A ) ∈ F (A). Any element α ∈ F (A) will be called a family parametrised by A (the reason for this nomenclature will be clear from the examples). Given f : B → A and β ∈ F (B), we denote the family F (f )β ∈ F (A) simply by β | A , when f is understood. Let Art k denote the category of complete local noetherian k-algebras with residue field k as objects and all k-algebra homomorphisms as morphisms. Given any R

Deformation Theory for Vector Bundles

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