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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-126/2/i/b/solution