Theorem 7.1 in chapter 2 of Hartshorne's text says that invertible sheaves on a scheme $X$ together with its given global generating sections correspond to morphisms from $X$ to $\mathbb{P}_A^n$ (here $A$ is some fixed ring). I wonder if given any invertible sheaf $\mathcal{L}$ on a scheme $X$, there always exists a finite set of global sections in $\mathcal{L}(X)$ that generate $\mathcal{L}$?
[Math] Does an invertible sheaf always have global generating sections
algebraic-geometry
Best Answer
It is not true for general schemes $X$ that the global sections $\mathcal L(X)$ of an invertible sheaf $\mathcal L$ on $X$ generate the stalks of $\mathcal L$ because it may happen that $\mathcal L$ has no non-zero section at all: this is the case for Serre's twisting sheaves $\mathcal O_{\mathbb P^n}(r)$ on $\mathbb P^n$ as soon as $r$ is negative.
(Jyrki's example in his comment is the case $n=1, r=-2$)
However your question has an affirmative answer if $X$ is affine: Serre (him again!) has proved that in the affine case any coherent sheaf $\mathcal F$ on $X$ has its stalks generated by the global sections $\mathcal F(X)$ of $\mathcal F$. [This result is known as Serre's Theorem A]
Since any affine scheme is quasi-compact, even finitely many global sections suffice to generate all the stalks of $\mathcal F$.