# Differentials I

Setup: alg. closed. and is an algebraic k-scheme: meaning is quasi compact and of finite type. Note there is a bijection

closed pts of X

where I’ve abbreviated as . The scheme D is a one pt. space so given , there is a backwards map of local rings . The map is k-linear and so its clear what the map does to elements outside of , the map being a local homomorphism means it restricts to give , as there is a factorization

Also there is a canonical isomorphism of k-vector spaces: sending , thus the map gives us a closed point and a module homomorphism . It can be checked there is a bijection

and now thickenings and the sheaf of differentials

## Thickenings

First the commutative algebra version. Let A be a ring an I and ideal such that any element of I is nilpotent. For example consider the maximal ideal m in . Every element of m is nilpotent (because it has some as a factor) but note that is nonzero for every . In any case, for as above note that any prime of A necessarily contains I, hence the surjection gives a homeomorphism on the associated map of spectra.

Now moving to the category of schemes, a thickening is a closed immersion which is bijective. Equivalently, a thickening is a closed immersion defined a qcoh sheaf of ideals I s.t. every local section of I is nilpotent. Think of T’ as a slightly fatter version of T (thicker), topologically the same but scheme theoretically a little bigger. A nilpotent thickening is one such that the defining ideal is nilpotent. If , then defines an nth order thickening. Finally, given a (locally) closed immersion , it is defined by a sheaf of ideals so defines another (locally) closed immersion and there is a factorization .

Here’s an example. Let be a qcoh sheaf on X. Set and define algebra structure on :

This makes into a qcho sheaf of algebras, so I can construct with a given affines morphism to X. Using the identity map on X, and sheaf map which is just projection to , and the universal property of relative spec I get a morphism . Note in the algebra structure makes into a sheaf of ideals. In fact gives a map of sheaves , and by construction, the kernel of this sheaf maps is the sheaf of ideals defined by , and also , in summary

is a closed immersion which is actually a 1st order thickening.

## Sheaf of Differentials

Say is a morphism of schemes. Then is at least a locally closed immersion. Let by its nth order thickening, so there is a factorization of the diagonal

without bothering to worry about pullbacks or pushforwards, there is a map of sheaves . When n = 1, the kernel of this sheaf map (pulled back to a sheaf on X) is the sheaf of relative differentials denoted :

This is looks much nicer in the affines case: and . First, the diagonal corresponds to a ring homomorphism . It is surjective with kernel , so we can write . So

becomes

and so . Now restricting the two projections , I get the diagram

So on sheaves there are maps . For set , and let be the composition

It turns out that is a derivation, in fact a pretty special one. In the affine case is the map

Here is a proposition for the affines case:

Prop: Let , and be the map above. Then

- is A-linear and annihilates the image of A in B.
- is generated as a left module by

——————————————-

pf: 1 is a simple check, and 2 is a calculation. Here is the proof of 3. A general element can be expressed as . That maps to zero means so

QED.

Cor. For the composition

- is A linear and annihilates the image of A in B
- is generated as a B module by

——————————————–

QED.

Note: In contrast, Hartshorne begins by just defining to be the module generated by the formal symbols modulo all the derivation relations.

## About this entry

You’re currently reading “Differentials I,” an entry on Math Meandering

- Published:
- June 16, 2009 / 11:10 pm

- Category:
- Ogus Excerpts, Ogus/Hartshorne/Miranda

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