Today we actually get to some deformations. Let be a scheme of finite type over . First, we’ll be working with the “ring of dual numbers” a lot, so we’ll just define it to be . Let’s recall a few useful properties first.

To give a map in -schemes: Spec is equivalent specifying a -point and an element of the Zariski tangent space at that point.

Flatness is another important concept. An -module, , is flat if and only if the map by multiplication by is an injective. The proof is quite straightforward: Consider the exact sequence . If flat, tensor this with and you get , and hence injectivity. If the map is injective, then , so 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 , and nice in our case means flat. parametrizes this family, since the fiber over any point gives a scheme . There is a special fiber , and deformations of are the schemes that occur in this family in a neighborhood of . (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 by . We could think about this family as hyperbolas . As approaches 0, the hyperbolas degenerate into the coordinate axes. Deformations in this family of the special fiber are all irreducible, yet the special fiber is reducible.

Now for what we really care about in this post. A first-order deformation of is a scheme , flat over such that . Note the terminology comes from the fact that the family is over Spec which remembers “tangent” information. A second-order deformation would keep track of information via a family over Spec , etc (anyone see a completion happening in the near future?).

A first-order deformation is something that completes the fiber diagram:

where is flat. Maybe a notation will be useful later: Def().

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() (where these are up to isomorphism) is in bijective correspondence with . So by the last post it is enough to show bijective correspondence with the infinitesimal extensions of by .

The proof is just adapting what we said at the start of the post. If , then by flatness we can tensor with and it remains exact: . Thus is an infinitesimal extension of by . Conversely, any such extension is flat and hence a first-order deformation.