The Stacks project

Exercise 111.63.3. Let $k$ be a field. Let $X$ be a scheme over $k$. Assume $X = X_1 \cup X_2$ is an open covering with $X_1$, $X_2$ both isomorphic to $\mathbf{P}^1_ k$ and $X_1 \cap X_2$ isomorphic to $\mathbf{A}^1_ k$. (Such a scheme exists, for example you can take $\mathbf{P}^1_ k$ with $\infty $ doubled.) Show that $\dim _ k H^1(X, \mathcal{O}_ X)$ is infinite.