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

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!

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.