The Stacks project

Exercise 111.63.2 (Theorems). Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. cohomology of abelian sheaves on a Noetherian topological space $X$ of dimension $d$,

2. sheaf of differentials $\Omega ^1_{X/k}$ of a smooth variety over a field $k$,

3. dualizing sheaf $\omega _ X$ of a smooth projective variety $X$ over the field $k$,

4. a smooth proper genus $0$ curve over an algebraically closed field $k$, and

5. the genus of a plane curve of degree $d$.