Last time we looked at what the infinitesimal lifting property was and that a particularly nice class of schemes always satisfies it. A related concept is that of an infinitesimal extension of a scheme. Suppose is a scheme of finite type over and . A pair is an infinitesimal extension of by if is a sheaf of ideals satisfying and as schemes .

Given , as an example we always have the trivial infinitesimal extension by , which is where the structure sheaf on is given by where multiplication is defined by . Now notice that the way that is a sheaf of ideals given this new structure sheaf, is . Take and see what happens . Thus and clearly , so this is really an infinitesimal extension.

Let’s go back to our nice simple case from last time: Let Spec be nonsingular. Then the claim is that any infinitesimal extension by a coherent sheaf is isomorphic to the trivial one.

This immediately converts to an algebra problem. Let be the -module such that . Suppose we have an infinitesimal extension . Then is an ideal of with , , and as an -module.

This gives us an exact sequence . Consider the isomorphism , by last time there exists a lift . This is a retraction and hence the sequence splits giving (we do need to check that the multiplication is the right one, but it is by computation). Taking the sheaves associated to this, we see that , the trivial extension.

It’s good to notice that by what we did yesterday that these lifts are not actually unique. They are actually “parametrized” in some sense by where is the tangent sheaf. So it is reasonable to guess that plays a role here. Since we’ve classified the infinitesimal extensions of non-singular affine varieties, and non-singular varieties are locally of that form, let’s extend the result.

If is a non-singular variety over , and , then there is a one-to-one correspondence between the set of infinitesimal extensions of by up to isomorphism and .

Take a finite affine cover of , . Then we have a natural iso , so we’ll think in terms of ech cocycles.

If is an infinitesimal extension, then on each affine it must look trivial, so . So . Any two liftings differ by a section of . This is exactly saying that the pairwise intersections satisfy the cocycle condition and hence gives us an element of .

This basically reverses for the converse. Given an element of , then it satisfies the cocycle condition and hence patches by differences in the right thing to give an infinitesimal extension.

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What are and ?

Depending on what you know, this could be a rather hard thing to answer. I’ll just throw some words out there. is the structure sheaf (sheaf of regular functions), and is the sheaf of relative differentials.