Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 126 / 2 / i / b

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- Solution b

Solution

Use the same finite complex computing cohomology in a proper flat family. In fixed bases its differentials are matrices over

A

. The condition that such a matrix have rank at most

r

is closed, being defined by its

(r + 1) \times (r + 1)

minors. Since

dim H^{p} (K^{∙} \otimes κ (s)) = dim K^{p} - rank d_{s}^{p - 1} - rank d_{s}^{p},

the condition that this dimension be at least

n

is a finite union of intersections of closed rank loci. It is therefore closed. This is the Semicontinuity theorem for coherent cohomology.

Solved by gpt-5.6-sol high.

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