First-Order Deformations

September 26, 2010 by hilbertthm90 | 5 Comments

Today we actually get to some deformations. Let ${X_0}$ be a scheme of finite type over ${k}$ . First, we’ll be working with the “ring of dual numbers” a lot, so we’ll just define it to be ${R=k[\epsilon]/(\epsilon^2)}$ . Let’s recall a few useful properties first.

To give a map in ${k}$ -schemes: Spec ${R\rightarrow X_0}$ is equivalent specifying a ${k}$ -point and an element of the Zariski tangent space at that point.

Flatness is another important concept. An ${R}$ -module, ${M}$ , is flat if and only if the map ${M/\epsilon M\rightarrow M}$ by multiplication by ${\epsilon}$ is an injective. The proof is quite straightforward: Consider the exact sequence ${0\rightarrow k\stackrel{\epsilon}{\rightarrow} R\rightarrow k \rightarrow 0}$ . If ${M}$ flat, tensor this with ${M}$ and you get ${0\rightarrow M/\epsilon M\stackrel{\epsilon}{\rightarrow} M\rightarrow M/\epsilon M\rightarrow 0}$ , and hence injectivity. If the map is injective, then ${Tor^1(M,k)=0}$ , so ${M}$ is flat.

Let’s move on now that those are out of the way. What exactly should these deformations be? Let’s say we have a nice family of schemes. This means that there is a map ${f:X\rightarrow T}$ , and nice in our case means flat. ${T}$ parametrizes this family, since the fiber over any point gives a scheme ${X_t}$ . There is a special fiber ${X_0}$ , and deformations of ${X_0}$ are the schemes that occur in this family in a neighborhood of ${X_0}$ . (Recall, this is still a “touchy-feely” idea of what a deformation is, don’t take this to be the definition).

For instance, you could parametrize some curves over ${\mathbb{A}^1_k}$ by ${Spec\left(k[x,y,t]/(xy-t)\right)\rightarrow Spec(k[t])}$ . We could think about this family as hyperbolas $X_t=\{(x,y): xy=t\}$ . As $t$ approaches 0, the hyperbolas degenerate into the coordinate axes. Deformations in this family of the special fiber ${X_0}$ are all irreducible, yet the special fiber is reducible.

Now for what we really care about in this post. A first-order deformation of ${X_0}$ is a scheme ${X'}$ , flat over ${R}$ such that ${X'\otimes_R k\simeq X_0}$ . Note the terminology comes from the fact that the family is over Spec ${R}$ which remembers “tangent” information. A second-order deformation would keep track of information via a family over Spec ${k[\epsilon]/(\epsilon^3)}$ , etc (anyone see a completion happening in the near future?).

A first-order deformation ${X'}$ is something that completes the fiber diagram:

${\begin{matrix} X_0 & \rightarrow & X'\\ \downarrow & & \downarrow f\\ Spec k & \rightarrow & Spec R \end{matrix}}$

where ${f}$ is flat. Maybe a notation will be useful later: Def( ${X_0/k}$ ).

Let’s use the last couple of posts to classify the first-order deformations of a non-singular scheme over an algebraically closed field. The claim is that Def( ${X/k}$ ) (where these are up to isomorphism) is in bijective correspondence with ${H^1(X, \mathcal{T}_X)}$ . So by the last post it is enough to show bijective correspondence with the infinitesimal extensions of ${X}$ by ${\mathcal{O}_X}$ .

The proof is just adapting what we said at the start of the post. If ${X'\in Def(X/k)}$ , then by flatness we can tensor with ${0\rightarrow k\rightarrow R\rightarrow k\rightarrow 0}$ and it remains exact: ${0\rightarrow \mathcal{O}_X\rightarrow \mathcal{O}_{X'}\rightarrow \mathcal{O}_X}$ . Thus ${X'}$ is an infinitesimal extension of ${X}$ by ${\mathcal{O}_X}$ . Conversely, any such extension is flat and hence a first-order deformation.

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Author: hilbertthm90

I am a mathematics graduate student fascinated in how all my interests fit together.

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First-Order Deformations

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