Let $(X,O_{X})$ be a scheme and $L$ be an invertible sheaf on it. Denote by $\Gamma(X,L)$ the global sections of $L$. Can it happen that $\Gamma(X,L)$ is not finitely generated as an $\Gamma (X,O_{X})$ module?
If $X$ is affine then $\Gamma(X,L)$ is finitely generated as we will have a finite cover where $L$ will be finitely generated and one can prove global sections will be finitely generated as well. Perhaps one needs to look at non quasi compact schemes for an example.
Sure. For each $n$, let $X_n$ be a scheme which has an invertible sheaf $L_n$ such that $\Gamma(X_n,L_n)$ cannot be generated by fewer than $n$ elements as a $\Gamma(X_n,O_{Y_n})$-module (for instance, $X_n=\mathbb{P}^1$ and $L_n=O(n)$). Let $X$ be the disjoint union $\coprod X_n$ and let $L$ be the invertible sheaf on $X$ whose restriction to $X_n$ is $L_n$ for each $n$. If there were finitely many elements of $\Gamma(X,L)$ that generated it as a $\Gamma(X,O_X)$-modules, then their restrictions would generate $\Gamma(X_n,L_n)$ for each $n$ (note that the restriction map is surjective since $\Gamma(X,L)$ is just the product $\prod_n\Gamma(X_n,L_n)$). By our choice of $X_n$ and $L_n$, this is impossible.
