Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 4 / b / Solution

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.