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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.

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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.

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