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).

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

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

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

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

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

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