# Differentials II

Recall, given a (locally) closed immersion , it comes from a sheaf of ideals, and we can use this sheaf of ideals to construct thickenings of . Doing this in the case of for a Y-scheme X (with map ), I get a particular sheaf of -modules, called the sheaf of differentials on X (I’m neglecting to put pushforwards or pullbacks into the notation):

In the previous post, it was shown that the projection give rise to a derivation . More generally, for a given sheaf of modules , I can ask for derivations (i.e. linear and satisfies Leibniz rule):

Now given an module homomorphism , I get a derivation into E simply by first applying , i.e. . So there is a natural map

This map is an isomorphism in the affine case. Its also an isomorphism when it comes to deformations…

## Deformations

This whole discussion happens in for some base scheme . Suppose I have and a thickening . Intuitively, a deformation of is “thickening” or “stretching” of its domain to , i.e. I want such that is a factorization of :

Note:

- is a closed immersion so defined by an ideal , then is a sheaf of modules. A
- s are topologically the same, a deformation is determined by a map of sheaves, i.e. by a commutative diagram:

- When is a first order thickening there is an action given by where is a sheaf maps that fits into a commutative diagram as above.

For the last bullet point, its clear that is linear, and that modulo i.e. that fits into a commutative diagram as above. So it just remains to show is a ring homomorphism. Now

Recall is what gives its module structure. On the other hand

and the last term is zero for a first order thickening.

I think this is the first time I’ve had a need to mention torsor. Intuitively, for a group G, a G-torsor is a set S with an action of G such that for any two elements a,b of S there is a unique element of G taking a to b. In fact any G-torsor is isomorphic to G, but there is no preferred isomorphism. Ogus’ definiton a G pseudo torsor is a G-set such that is bijective for every . A G torsor is a G pseudo torsor that is nonempty. I think the point of defining G pseudo torsor is because in some case you might want to define a set of ‘would be’ torsors, but in certain cases you might get the emptyset, for example in our case it turns out the above action makes into a torsor, but in some cases some maps might not have any deformations. In any case, here is a theorem:

Thm: Given and a first order thickening then is a pseudo torsor.

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I’m not going to go through the proof, but here’s an outline. What needs to be shown is that

sending is an bijective. One way to define a free action is to say the above map is injective. So check the action is free to get injectivity. The difficult part is surjectivity. Given the task is to find a derivation such that . The given a map basically you play with this to get a map of short exact sequences:

into

More specifically there should be some scheme diagram that gives the latter two vertical arrows, and the universal property of kernel should then give a sheaf map . Then precomposing should give the desired derivation, this uses something like factorizes as . In effect this constructs an inverse that factorizes through :

As I’ve outlined is bijective, the first map above is injective, and the latter in surjective. Also it shouldn’t be hard to see that is injective, this gives

Cor. With the same assumption as the thm, the natural map

is bijective. QED

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Next, maybe some examples.

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- June 18, 2009 / 10:23 pm

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