The Stacks project

Exercise 111.65.2 (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. morphisms from a scheme $X$ to the affine scheme $\mathop{\mathrm{Spec}}(A)$,

2. cohomology of a quasi-coherent module $\mathcal{F}$ on an affine scheme $X$,

3. the Picard group of $\mathbf{P}^1_ k$ where $k$ is a field,

4. the dimensions of fibres of a flat proper morphism $X \to S$ for $S$ Noetherian,

5. $\mathbf{G}_ m$-equivariant modules on a scheme $S$, and

6. Bezout's theorem on intersections (restrict to a special case if you like).