Why are infinitesimal deformation defined to be over Artin rings?

I am wondering what is the motivation for defining infinitesimal deformations to be over the spectrum of Artinian rings, i.e rings that have a finite number of prime maximal ideals.

I have been trying to understand the connection between deformations and the functorial description of moduli, i.e as as a functor from, let us say schemes, to sets. However, this is is becoming difficult because I don't understand the motivation in defining (infinitesimal) deformations to be over Artinian rings

I found "Elementary deformation theory" in FGA explained very helpful in learning some basic deformation theory.

In the following whenever I say local artinian ring I really mean local artinian $$k$$-algebra.

The basic motivation comes from the fact that tangent space of a scheme $$X$$ at a point $$x\in X$$ can be described as the vector space consisting of maps $$\operatorname{Spec} k[t]/(t^2) \rightarrow X$$ sending the underlying closed point to $$x$$. Now if $$X$$ is some moduli space parametrising certain objects $$S$$ , then a map from $$\operatorname{Spec} k[t]/(t^2)\rightarrow X$$ is equivalent to giving a first order deformation $$S$$ over $$\operatorname{Spec}k[t]/(t^2)$$.

In general the local behaviour of $$x\in X$$ can be understood via the completion $$\hat{\mathcal{O}_{X,x}}$$ of the local ring $$\mathcal{O}_{X,x}$$. For example $$\mathcal{O}_{X,x}$$ is regular iff $$\hat{\mathcal{O}_{X,x}}$$ is regular and $$\dim_k \mathcal{O}_{X,x} = \dim_k \mathcal{O}_{X,x}$$. Now $$\hat{\mathcal{O}_{X,x}}$$ is something which can be understood by understanding maps from local artinian rings to $$X$$ mapping the underlying closed point to $$x\in X$$. In particular, the deformation functor $$D: (Loc/k) \rightarrow Sets$$ sending local artinian rings $$A$$ to the set of maps $$\operatorname{Spec} A \rightarrow X$$ satisfying the above is representable by $$\hat{\mathcal{O}_{X,x}}$$. So understanding deformations of objects over local artinian rings is equivalent to understanding this functor, which is equivalent to understanding $$\hat{\mathcal{O}_{X,x}}$$.

A concrete example of how all this is used to understand the moduli space is via constructing a tangent-obstruction theory. The tangent-obstruction theory for a deformation functor $$D$$ (a functor from $$(Loc/k) \rightarrow Sets$$) consists of two vector spaces $$V_1$$ (tangent space) and $$V_2$$ (obstruction space) such that

(1) whenever you have a small extension $$0\rightarrow M \rightarrow B \rightarrow A \rightarrow 0$$ i.e. $$A,B$$ are local artinian rings, and $$\mathfrak{m}_B \cdot M = 0$$, there exists an exact sequence of sets $$V_1\otimes_k M \rightarrow D(B) \rightarrow D(A) \stackrel{ob}{\rightarrow} V_2\otimes_k M$$

where exactness at $$D(A)$$ means that if $$ob(x) = 0$$ for some $$x\in D(A)$$ then there exists $$x'\in D(B)$$ mapping to $$x\in D(A)$$ (i.e. if the obstruction vanishes then you can lift your deformation), and that exactness at $$D(B)$$ means that $$V_1\otimes_k M$$ acts transitively on the fibers.

(2) If $$M=k$$, then $$D(A)$$ forms an affine space under $$V_1\otimes_k M$$.

It is a fact that if $$R$$ represents $$D$$, then $$\dim_k V_1 - \dim_k V_2 \le \dim R \le \dim_k V_1$$. As above if your moduli problem is representable by a scheme $$X$$, then the deformation functor above $$D$$ is representable by $$\hat{\mathcal{O}_{X,x}}$$. It turns out that while you might not know what your scheme looks like (e.g. Hilbert scheme), it is quite often not too hard to construct $$V_1,V_2$$. So this allows you to deduce some information about the moduli space at $$x\in X$$. For example, the proof of the existence of rational curves passing through any $$x\in X$$, when $$X$$ is smooth projective Fano (which can be found in e.g. Debarre's "higher dimensional algebraic geometry") relies on knowing that some moduli space has positive dimension at some point, which is done by understanding its tangent obstruction theory to give a lower bound.

• @Ioch Typo: deBarre -> Debarre. – Ariyan Javanpeykar Oct 11 '18 at 10:33
• @AriyanJavanpeykar fixed. Thanks! – loch Oct 11 '18 at 10:34