# Differentials III

This is just a few examples of the stuff in the previous two posts.

### Affine Stuff

Set up:

- ring and
- , with affine morphism

Claim: . Recall, for ,

Its not hard to see that is generated by so is generated by in particular

Hence for an module,

### Deformation Stuff

Keeping the same notation recall, from the first post on differentials, that is a first order thickening of X. In particular, there is a diagram

So the map is a deformation of the identity. The above diagram has a corresponding sheaf diagram:

By the previous discussion about deformations, if is any derivation, then there is another commutative sheaf diagram

So this gives another deformation of the identity. As is nonempty (because it contains ) its actually a torsor so its isomorphic to :

(1)

In the case for some B-module , then the previous affine calculation shows

### Example with VE

Let a scheme and qcoh on , set . Let be the map to the base. Recall there is a universal arrow which is the same as which is the same as

(2)

The point here is to show

Notice by (2) and the universal map there is natural map , to show its an isomorphism it suffices (by Yoneda) to show

for every module M. So, I compute

The second isomorphism uses (1), the next isomorphism (I’m speculating) is justified as follows: a deformation of the identity corresponds to a sheaf map and a commutative sheaf diagram

Commutativity of the diagram means that where , and , the map is the correspondence in the second to last isomorphism. The last isomorphism comes from the universality of (2).

There is also a statement about , namely

Thm: Let qcoh on , , the ideal of the diagonal , then there is a natural isomorphism

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H is some hyperplane that makes the thm make sense. But I didn’t take good notes, so no proof.

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- Published:
- June 19, 2009 / 2:48 pm

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