The Stacks project

Exercise 111.44.1. Make an example of a field $k$, a curve $X$ over $k$, an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ and a cohomology class $\xi \in H^1(X, \mathcal{L})$ with the following property: for every surjective finite morphism $\pi : Y \to X$ of schemes the element $\xi $ pulls back to a nonzero element of $H^1(Y, \pi ^*\mathcal{L})$. Hint: construct $X$, $k$, $\mathcal{L}$ such that there is a short exact sequence $0 \to \mathcal{L} \to \mathcal{O}_ X \to i_*\mathcal{O}_ Z \to 0$ where $Z \subset X$ is a closed subscheme consisting of more than $1$ closed point. Then look at what happens when you pullback.