The relevant theorem supplies, locally on $S=\operatorname{Spec}A$, a bounded complex $K^\bullet$ of finite free $A$-modules such that
$$
H^p(X,\mathcal F\otimes_AM)\cong H^p(K^\bullet\otimes_AM)
$$
naturally 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. Hence
$$
\chi(X_s,\mathcal F_s)=\sum_i(-1)^i\operatorname{rank}K^i,
$$
which 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.


Solution

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

 Solution

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