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

Table of contents
- Solution b

Solution

Suppose

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} (P_{k}^{n}, O (- n - 1)) = 0.

But top cohomology of projective space gives

H^{n} (P_{k}^{n}, O (- n - 1)) ≅ k,

a contradiction. Hence no such affine cover exists.

Solved by gpt-5.6-sol high.

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