A Short Description of the Theory of Deformations of Imbedded Schemes and Morphisms

Deformation theory is often an important tool in dealing with a variety of questions related to moduli spaces. We have discussed the basic idea before: we study families of objects over Artinian rings restricting to a given object over Spec k, and this tells us, for instance, the dimension of the tangent space of a moduli space, or whether it is smooth at a point. We begin by developing this general picture a bit further. We will restrict to the situation of being over a field. If one is interested in mixed-characteristic situations, all these ideas to categories of Artin Λ-algebras, where Λ is a fixed complete Noetherian (such as a Witt vector ring), but the setup gets a bit wordier in this case. Suppose we are given a functor F : Schk → Set, and an element η0 ∈ F (Spec k). We can define a deformation problem (functor) associated to η0 and F as the functor Fη0 : Art(k) → Set associating to each A ∈ Art(k) the set of objects η ∈ F (SpecA) restricting to η0 ∈ F (Spec k). Here Art(k) denotes the category of Artin local k-algebras with residue field k, so we are given a canonical map Spec k → SpecA. If F is not representable (for instance, if it is a moduli functor parametrizing objects with automorphisms), this will not necessarily produce a well-behaved deformation functor. As in our earlier discussion of deformations of smooth varieties, we would want to functor to include a choice of morphism η0 → η, which doesn’t make sense in the general setting of functors, and points towards the usefulness of stacks. However, if F is represented by a locally Noetherian scheme X, and η0 corresponds to x ∈ X with k = k(x), then Fη0 has several nice properties: • The tangent space Fη0 (k[ǫ]/(ǫ )) is the tangent space of X at x. • If X is locally of finite type over Spec k, it is smooth over Spec k at x if and only if Fη0 is formally smooth: that is, for each A ։ A ′ in Art(k), we have Fη0(A) ։ Fη0(A ). • It is pro-representable (meaning it is representable, but we have to enlarge from the category of Artinian rings to the category of complete Noetherian rings in order to find a representing object), and in fact is represented by Spec ÔX,x.