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