TITLE: What is the line bundle on $\mathbb{P}^1_\mathbb{C}$ whose transition function is $e^z$
QUESTION [5 upvotes]: Let $U_0,U_\infty$ be the two affine patches of $\mathbb{P}^1_\mathbb{C}$, neighborhoods of "0" and "$\infty$" respectively. Let $L$ be the line bundle on $\mathbb{P}^1_\mathbb{C}$ constructed by gluing the trivial bundles over $U_0$ and $U_\infty$ via the function $e^z$, which is a nowhere vanishing holomorphic function on $U_0\cap U_\infty$.
I've never taken complex geometry (my background is in algebraic geometry, and $e^z$ isn't an algebraic function), so my question is - does $L$ exist in the algebraic world? Which one is it? (for which $n\in\mathbb{Z}$ is $L\cong\mathcal{O}(n)$?). If it isn't any of them, what is the "right statement" to say that it doesn't exist? For example, is $e^z$ somehow not in $\mathcal{O}_{\mathbb{P}^1}(U_0\cap U_\infty)$?

REPLY [2 votes]: Since $e^z$ extends as a non-zero function over $U_0$, you can change basis on $U_0$ so that your gluing function is now just $1$.  This makes it clear that you have the trivial bundle.
(Incidentally, by GAGA, any holomorphic line bundle on $\mathbb P^1$ has to be algebraic, so what you wrote down had to be $\mathscr O(n)$ for some $n$.)
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