I now want to talk about some of the material in Hartshorne, II.8. First, we need some preliminaries from commutative algebra.

Let be a commutative ring, an -algebra, and a -module. Then an **-derivation** of in is a linear map satisfying for and

The set of all such derivations forms a -module . If we regard this as a set, clearly, we have a contravariant functor

because if is a homomorphism of -algebras, we can pull back a derivation.

Before proceeding, I should say something about the canonical example. Let be a smooth manifold and the local ring (of germs of smooth functions) at . Then becomes an -module if the germ acts by multiplication by . More precisely, we have an exact sequence

for the maximal ideal of functions vanishing at , and this is the way is an -module.

Anyway, an -derivation is just a tangent vector at .

Now back to the algebraic theory. It turns out that the functor is representable. In other words, for each -algebra , there is a -module such that

the isomorphism being functorial. In addition, there must be a “universal” derivation (corresponding to the identity in the above functorial isomorphism), that any derivation factors through.

The construction of is straightforward. We define it as the -module generated by symbols , modulo the relations for , , and . It is now clear that we have a functorial isomorphism as above. Now, is called the module of **Kahler differentials** of over . (more…)