Two-affine projective line and its twists

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and write $k[t]$ and $k[u]$ for the polynomial rings in one variable over $k$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution). The two-affine projective line ${\mathbb{P}}_k^1$ is the scheme obtained by gluing the two affine schemes $\mathrm{Spec}k[t]$ and $\mathrm{Spec}k[u]$ along their basic opens $D(t)$ and $D(u)$, which are affine open subschemes isomorphic to $\mathrm{Spec}k[t,t^{-1}]$ and $\mathrm{Spec}k[u,u^{-1}]$ (The spectrum of a principal localisation is the distinguished open D(f), Sections and restrictions on distinguished opens of an affine scheme, Affine open subschemes), identified through the mutually inverse ring isomorphisms $k[t,t^{-1}]\to k[u,u^{-1}]$, $t\mapsto u^{-1}$, and $k[u,u^{-1}]\to k[t,t^{-1}]$, $u\mapsto t^{-1}$ (Gluing affine schemes along compatible open isomorphisms). Its two charts $U_0\cong \mathrm{Spec}k[t]$ and $U_{\mathrm{\infty }}\cong \mathrm{Spec}k[u]$ are open subschemes covering ${\mathbb{P}}_k^1$, and their overlap $W=U_0\cap U_{\mathrm{\infty }}\cong \mathrm{Spec}k[t,t^{-1}]$ carries the two coordinate functions, mutually inverse units with $tu=1$.

Here the distinguished-open identification is an isomorphism of schemes, not only of spaces. For a ring $A$ and $f\in A$, the localization-spectrum map $\mathrm{Spec}(A_f)\to D_A(f)$ is a homeomorphism (The spectrum of a principal localisation is the distinguished open D(f)). On the basis open $D_{A_f}(g\mathrm{/}{f}^m)$ it maps to $D_A(fg)$; the structure-sheaf section rings on these opens are respectively $(A_f{)}_{g\mathrm{/}{f}^m}$ and $A_{fg}$, canonically isomorphic by the universal property of localization, and the identifications commute with further restrictions (Sections and restrictions on distinguished opens of an affine scheme, The localization construction extends to the structure sheaf on Spec A, Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$). Thus the homeomorphism identifies the restricted structure sheaves and is a local-ringed-space isomorphism. For $A=k[t]$, $f=t$, the localization $A_f$ is $k[t,t^{-1}]$ by the same universal property; similarly $k[u{]}_u=k[u,u^{-1}]$. These are the scheme chart identifications used in the gluing above.

For every $n\in \mathbb{Z}$, let $\mathcal{O}(n)$ be the sheaf of ${\mathcal{O}}_{{\mathbb{P}}_k^1}$-modules glued from the structure sheaves ${\mathcal{O}}_{U_0}$ and ${\mathcal{O}}_{U_{\mathrm{\infty }}}$ with frames $e_0=1$ on $U_0$ and $e_{\mathrm{\infty }}=1$ on $U_{\mathrm{\infty }}$, related on $W$ by $e_{\mathrm{\infty }}=t^n{e}_0,\text{equivalently}{e}_0=t^{-n}{e}_{\mathrm{\infty }},$ through the transition isomorphism given by multiplication by the unit $t^{-n}$ of $k[t,t^{-1}]$ (Compatible local sheaves glue uniquely up to unique isomorphism, A gluing datum for sheaves on an open cover). Each $\mathcal{O}(n)$ is an invertible sheaf: it is free of rank one on each chart with the displayed frame. In particular $\mathcal{O}(0)$ is the structure sheaf ${\mathcal{O}}_{{\mathbb{P}}_k^1}$, and on the overlap the sections are written $k[t,t^{-1}]e_0$, so that $a(t)e_0=t^{-n}a(t)e_{\mathrm{\infty }}=u^{n}a(u^{-1})e_{\mathrm{\infty }}$.