Let $X$ be a scheme (I don't think any other conditions are needed on $X$) and $\mathcal{L}$ an invertible sheaf on $X$. I've read somewhere that the map of sheaves $$\varphi:\mathcal{O}_X \rightarrow \mathcal{L}$$ corresponds to a global section $s \in \Gamma(X,\mathcal{L})$. Is this a basic fact about invertible sheaves, or sheaves in general? How is this correspondence constructed?
Also, since $\mathcal{L}$ is a line bundle, then $\varphi$ is an isomorphism if and only if $s$ is nowhere vanishing. Why is this true?
Any references (particularly in Hartshorne) would be appreciated, I'm trying to understand the invertible sheaves better.
Best Answer
Consider the morphism $O_X \rightarrow L$ that sends $a \in O_X(U)$ to $a s|_U \in L(U)$. This is a morphism of $O_X$-modules as if $V \subseteq U$, then $(a s|_U)|_V = a|_V s|_V$. Note that $1 \in O_X(X)$ is sent to $s$
Conversely, given a morphism $f : O_X \rightarrow L$, let $s = f(1) \in L(X)$. Then, for open sets $U \subseteq X$ and $a \in O_X(U)$, $f(a) = a * f(1|_U) = a s|_U$.
These two constructions are inverses. Note that I never used that $L$ is invertible, so this holds for any sheaf of modules on any ringed space.