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Suppose ${\mathbb{P}}_k^n$ had a cover by $n$ affine opens. Since projective space is separated, every finite intersection in this cover is affine. The acyclic cover theorem would therefore compute the cohomology of every quasi-coherent sheaf by its Čech complex. A cover with only $n$ members has no degree-$n$ cochains, so it would implyBut top cohomology of projective space givesa contradiction. Hence no such affine cover exists.

Solved by gpt-5.6-sol high.

For the cover $\mathcal{U}=\{X_1,X_2\}$ of the affine plane with doubled origin, the overlap is the punctured affine plane $U$. Since $H^0(U,{\mathcal{O}}_U)=k[x,y]$, the Čech complex beginsThis map is surjective, and the normalized complex has no terms in degrees at least two. Consequently

The Mayer-Vietoris sequence for sheaf cohomology also gives $H^1(Y,{\mathcal{O}}_Y)=0$, but its next part givesPart b with $f=1$ shows that the group on the right is infinite-dimensional. Thus

This does not contradict the acyclic cover theorem. Although $X_1$ and $X_2$ are affine, their intersection $U$ is not acyclic: it has nonzero first structure-sheaf cohomology. Equivalently, this affine cover does not satisfy the theorem's hypotheses; the doubled-origin plane is not a semi-separated scheme.

Solved by gpt-5.6-sol high.