Why are line bundles called principal $\mathbb G_m$-bundle?

Question

We see the standard statement that $H^1(X,F)$ claasifies principal $F$-sheaves (say $F$ is a sheaf of groups) on $X$. By definition, these are sheaves $G$ of sets with an $F$-action such that they have a trivialising neighbourhood $(U_\alpha)_\alpha$ on which the sheaf $G$-considered as an $F$-set is isomorphic to $F$ acting on itself by left translation.

We next see the claim that $H^1(X,O_X^\times)$ is the isomorphsim classes of line bundles on $X$. But by the above paragraph, this cohomology gives us principal $O_X^\times$-sheaves. But a line bundle is locally isomorphic to $O_X$. How are these two same, as $O_X^\times$ does $\textit{not}$ act freely and transitively on $O_X$?

Merci en avance!

Answer 1

There is an equivalence of categories between the groupoid of line bundles on $X$ and the groupoid of principal $\mathbb{G}_{\mathrm{m}}$-bundles which works as follows.

Given a principal $\mathbb{G}_{\mathrm{m}}$-bundle $P$ one defines a line bundle by $$ L := P \times^{\mathbb{G}_{\mathrm{m}}} \mathbb{A}^1 = (P \times \mathbb{A}^1) / \mathbb{G}_{\mathrm{m}}, $$ where $\mathbb{G}_{\mathrm{m}}$ acts on $\mathbb{A}^1$ in the standard way.

Conversely, given a line bundle $L$ one defines a principal $\mathbb{G}_{\mathrm{m}}$-bundle as $$ P := L \setminus 0_X, $$ the complement of the zero section.