Let $X$ be a scheme and $ \mathscr{L}$ be a line bundle on $X$. In a few proofs I have seen the scheme 
$$ L = \mathscr{S}{\rm pec} \oplus\_{n \in \mathbb{Z}} \mathscr{L}^{\otimes n} \to X$$
pop up. Does the scheme $ L $ have a name? How should I be thinking about $L$?



How should I think about this scheme constructed from a line bundle
Denote by $\mathcal{A}$ the $\mathbb{Z}$-graded sheaf of $\mathcal{O}\_X$-algebras, $$ \mathcal{A} = \oplus\_{n\in \mathbb{Z}} \mathcal{L}^{\otimes n}. $$
There is a natural isomorphism of invertible $\mathcal{A}$-modules,
$$\phi: \mathcal{A}\otimes\_{\mathcal{O}\_X}\mathcal{L} \to \mathcal{A}.$$
As a map of graded modules, this is homogeneous of degree $1$, i.e., the degree $0$ part, $\mathcal{A}\_0\otimes\_{\mathcal{O}\_X}\mathcal{L} = \mathcal{L}$ gets mapped via the identity map to the degree $1$ part $\mathcal{A}\_1=\mathcal{L}$. There is a unique extension of this map, $\phi\_0=\text{Id}\_{\mathcal{L}}$, to an isomorphism of $\mathcal{A}$-modules.


Denote by $\pi:L\to X$ the natural projection. Associated to this isomorphism $\phi$, there is an isomorphism of $\mathcal{O}\_L$-modules, $\widetilde{\phi}:\pi^\*\mathcal{L}\to \mathcal{O}\_L$. This isomorphism is universal: for every $X$-scheme, $f:Y\to X$, and for every isomorphism $\psi:f^\*\mathcal{L}\to \mathcal{O}\_Y$ of $\mathcal{O}\_Y$-modules, there is a unique morphism of $X$-schemes, $g:Y\to L$, such that $g^\*\widetilde{\phi}$ equals $\psi$. 


Although $\widetilde{\phi}$ is universal, of course it is not unique. In particular, on the product $\mathbb{G}\_m\times L$, there is a modification of $\text{pr}\_L^\*\widetilde{\phi}$ that scales this by the universal invertible element in $\mathcal{O}\_{\mathbb{G}\_m}$. In particular, this modification of $\widetilde{\phi}$ gives rise to a morphism of $X$-schemes, $$ m\_L :\mathbb{G}\_m\times L \to L. $$ Of course this is an action of $\mathbb{G}\_m$ on $L$ over $X$. The corresponding $\mathbb{Z}$-grading of $\pi\_\*\mathcal{L}$ is the evident grading of $\mathcal{A}$. In particular, since $\mathcal{A}$ is isomorphic to $\mathcal{O}\_X[t,t^{-1}]$ as graded sheaves of algebras locally on $X$, the action $m\_L$ makes $L$ into a $\mathbb{G}\_m$-torsor. This is the "$\mathbb{G}\_m$-torsor naturally associated to $\mathcal{L}$". 


Of course, as is always the case, about half of the mathematical word chooses the opposite convention. But that is no problem: whenever somebody talks about **THE** torsor associated to an invertible sheaf, just ask them for the universal property that their space satisfies, and then use that to calibrate with your own convention.


