Deformation theory

Definition

Let $k$ be a field and let $X$ be a scheme over $k$. Let $\mathrm{Art}_k$ be the category of local Artinian $k$-algebras $A$ with residue field $k$. For $A \in \mathrm{Art}_k$, a deformation of $X$ over $A$ is a pair $(\mathcal{X}, \phi)$ where:

1. $\mathcal{X}$ is a scheme with a flat morphism $\mathcal{X} \to \operatorname{Spec} A$;
2. $\phi \colon \mathcal{X} \times_{\operatorname{Spec} A} \operatorname{Spec} k \to X$ is an isomorphism of $k$-schemes, where the left side is a fiber product.

Two deformations $(\mathcal{X}, \phi)$ and $(\mathcal{X}', \phi')$ are isomorphic if there is an isomorphism $\mathcal{X} \to \mathcal{X}'$ over $A$ compatible with $\phi$ and $\phi'$. The deformation functor of $X$ is $\operatorname{Def}_X \colon \mathrm{Art}_k \to \mathrm{Set}$, sending $A$ to the set of isomorphism classes of deformations of $X$ over $A$. A first-order deformation is a deformation over the dual numbers $k[\varepsilon] = k[\varepsilon]/(\varepsilon^2)$. Deformation theory is the study of $\operatorname{Def}_X$ and of the analogous functors for other objects, such as sheaves, morphisms and subschemes.