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 imply
$$
H^n(\mathbb P_k^n,\mathcal O(-n-1))=0.
$$
But <top cohomology of projective space> gives
$$
H^n(\mathbb P_k^n,\mathcal O(-n-1))\cong k,
$$
a contradiction. Hence no such affine cover exists.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-113/4/b/solution