I’ve basically recuperated from my test and I’m trying to get back into the AG frame of mind. I have about 5 posts half written, so I’m going to actually try to finish this one and start up a nice little series. I’m taking a class on deformation theory this quarter (which hasn’t actually started yet), so this series will review some of the very, very small amount of deformation theory scattered throughout the exercises of Hartshorne.
Let’s start with some basics on the infinitesimal lifting property. First assume an algebraically closed field and a finitely generated -algebra with Spec a nonsingular variety (over ). Suppose is exact with a -algebra and an ideal with . Then satisfies the infinitesimal lifting property: whenever there is a -algebra hom , there is a lift making the obvious diagram commute.
First note that if are two such lifts then, is a -derivation of into . A quick subtlety is that a “-derivation” is an -module map that is a derivation and evaluates to zero on . So we need to understand how is an -module. But , so it is a -module, which in turn is an -module (via and which will be used). The reason is that is a lift of the zero map since and both lift . Since the sequence is exact and lands in the kernel, it is in the image of the one before it, i.e. .
Evaluates to 0 on : Since , . Thus .
Since is a universal object, we can consider . Conversely, given any , we can compose with the universal map to get is a -derivation. Compose this with the inclusion , call this . Since composing again with gives , is a lift of and hence is a lift of (note we’ve only guaranteed -linear so far, not algebra hom). Finally let’s check it preserves multiplication:
Now let for which for some . So we get another exact sequence . We now check that there is a map such that the square commutes and this induces an -linear map .
Note a map out of is completely determined by where the go. Since surjective, choose any such that . Extend this to get . By definition makes the square commute. Chasing around exactness, we get that if , then considering gives . Thus restricting gives . Since we have , so this descends to a map . It is clearly -linear.
Let Spec and Spec . The sheaf of ideals defines as a subscheme. Then by nonsingularity (Theorem 8.17 of Hartshorne) we have an exact sequence . Take global sections of this sequence to get the exact sequence ( vanishes by Serre).
Now apply the functor to get the exact sequence . Exactness on the right is due to being locally free and hence projective, so vanishes.
That surjectivity is exactly what we needed to say a lift exists. Take as constructed before. Then choose that maps to it. Compose with the universal map and inclusion to get a derivation (we’ll just relabel this ). Set . Since if we have it descends to a map which is the desired lift.
Next time we’ll move on to infinitesimal extensions and rephrase what we just did in those terms.