Question

This, of course, requires some heavier theorems in Cech cohomology: If $X$ is a separated scheme that's covered by $d$ affine opens and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then $H^p(X,\mathcal{F}) = 0$ for all $p \geq d$. As a corollary: if $X$ is a quasi-projective scheme over a Noetherian ring $A$, and $\mathcal{F}$ is a quasi-coherent sheaf, then $H^p(X,\mathcal{F}) = 0$ for all $p > d$ where $d = \dim(X)$ (note that $X$ is covered by $d+1$ affines).