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
$$
\dim H^p(K^\bullet\otimes\kappa(s))
=\dim K^p-\operatorname{rank}d^{p-1}_s-\operatorname{rank}d^p_s,
$$
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.

Solution

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

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{wiki=Semicontinuity_theorem}

In a proper flat family with a coherent sheaf, the fiber dimension $s\mapsto\dim_{\kappa(s)}H^p(X_s,\mathcal F_s)$ is upper semicontinuous, and the fiberwise Euler characteristic is locally constant.


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 Incoming links: Semicontinuity theorem for coherent cohomology