Lecture 16: Differentials

We covered 15.1–15.7 of Gathmann's notes.

• Derivations (only implicit in Gathmann's notes). Module of Kähler differentials. Corresponding sheaf.
• Kähler differentials of polynomial ring / affine space.
• Kähler differentials of quotient ring / affine scheme. Duals of fibers are tangent spaces.
• 2nd construction of sheaf of Kähler differentials using the ideal of the diagonal.
• Proposition 15.6: A variety is smooth <=> sheaf of Kähler differentials is locally free.

Suggested exercises: There are no exercises in this section. In class we proved that there were no relations in Example 15.3(a). You could try to prove that there are no further relations needed in Example 15.3(b). One way to approach this is to show that the quotient satisfies the universal property. It also follows from the more general "Second fundamental exact sequence" of Kähler differentials (which also generalizes Prop 15.10).