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

The relevant theorem supplies, locally on $S=\mathrm{Spec}A$, a bounded complex $K^{\bullet }$ of finite free $A$-modules such thatnaturally for every $A$-module $M$. In particular the fiber cohomology at $s$ is computed by $K^{\bullet }{\otimes }_A\kappa (s)$.

The Euler characteristic of a finite-dimensional complex equals the alternating sum of the dimensions of its terms. Hencewhich is constant wherever one complex works. It is therefore locally constant on $S$, and is constant when $S$ is connected.

Solved by gpt-5.6-sol high.