The Picard group of the projective line

The degree isomorphism on divisor classes. By [F2] the homomorphism ${\mathrm{deg}}_k$ from the group $\mathrm{CaDiv}({\mathbb{P}}_k^1)\mathrm{/}\mathrm{Prin}({\mathbb{P}}_k^1)$ of Cartier divisor classes to $\mathbb{Z}$ is an isomorphism: it is well defined by [F3], surjective because $m[\mathrm{\infty }]$ has degree $m$, and injective because a degree-zero divisor is principal.

The Cartier-to-Picard dictionary. Since ${\mathbb{P}}_k^1$ is integral, the actual dictionary [F5] induces an isomorphism $\mathrm{CaDiv}({\mathbb{P}}_k^1)\mathrm{/}\mathrm{Prin}({\mathbb{P}}_k^1)\to \mathrm{Pic}({\mathbb{P}}_k^1)$ with inverse $[\mathcal{O}(D)]\mapsto$ the class of $D$. Its compatibility with addition and duals is given by [F5] and the local tensor identifications.

The composite isomorphism. Composing the inverse of the isomorphism of step 1.2 with the degree isomorphism of step 1.1 gives a group isomorphism $\mathrm{Pic}({\mathbb{P}}_k^1)\to \mathbb{Z}$. By the degree-descent supplier [F5], it is given on classes by $[\mathcal{O}(D)]\mapsto {\mathrm{deg}}_{k}D$.

The twists and the degree. Let $d\in \mathbb{Z}$. By [F2], $\mathcal{O}(1)\cong \mathcal{O}([\mathrm{\infty }])=\mathcal{O}(\mathrm{\infty })$, and by the addition and dual isomorphisms of [F5] applied to the multiple $d[\mathrm{\infty }]$ one has $\mathcal{O}(d[\mathrm{\infty }])\cong \mathcal{O}([\mathrm{\infty }]{)}^{\otimes d}\cong \mathcal{O}(1{)}^{\otimes d}$. To compare this with the twisting sheaf, let $U_i=D_{+}(x_i)$ be the standard charts of ${\mathbb{P}}_k^1=\mathrm{Proj}k[x_0,x_1]$; by [F4] the module $S(n{)}_{(x_i)}=x_i^{n}S(0{)}_{(x_i)}$ is free of rank one over $S(0{)}_{(x_i)}$ on the generator $x_i^n$, so $\mathcal{O}(1{)}^{\otimes d}$ is free of rank one on $U_i$ with frame $x_i^d$, and on the overlap the frames satisfy $x_1^d=(x_1\mathrm{/}{x}_0{)}^d{x}_0^d$ with the unit $(x_1\mathrm{/}{x}_0{)}^d$; this is exactly the transition unit of the twist of index $d$ in [F4], so the gluing uniqueness of [F4] gives $\mathcal{O}(1{)}^{\otimes d}\cong \mathcal{O}(d)$ for every $d$ (for $d<0$ duals invert the transition units, and the identification of [F5] provides the dual isomorphism $\mathcal{O}(-D)\cong \mathcal{O}(D{)}^{\vee }$). Hence $\mathcal{O}(d)\cong \mathcal{O}(d[\mathrm{\infty }])$, and the isomorphism of step 2.1 sends $[\mathcal{O}(d)]$ to ${\mathrm{deg}}_k(d[\mathrm{\infty }])=d\cdot [\kappa (\mathrm{\infty }):k]=d$.

Generator and uniqueness. By step 3.1 the class $[\mathcal{O}(1)]$ maps to $1$, so it generates $\mathrm{Pic}({\mathbb{P}}_k^1)$ under the isomorphism of step 2.1, and $d\mapsto [\mathcal{O}(d)]$ is a two-sided inverse $\mathbb{Z}\to \mathrm{Pic}({\mathbb{P}}_k^1)$: it is a group homomorphism because $\mathcal{O}(d)\otimes \mathcal{O}(e)\cong \mathcal{O}(d+e)$ by the same comparison with the addition isomorphisms of [F5], and it is inverse to the isomorphism of step 2.1. Consequently every invertible sheaf $\mathcal{L}$ on ${\mathbb{P}}_k^1$ satisfies $\mathcal{L}\cong \mathcal{O}(d)$ for the integer $d$ determined by $[\mathcal{L}]$, and this $d$ is unique by [F4] (no two distinct twists are isomorphic).

Conclusion and choice accounting. Steps 2.1, 3.1 and 4.1 prove the statement: the degree homomorphism induces an isomorphism $\mathrm{Pic}({\mathbb{P}}_k^1)\to \mathbb{Z}$, $[\mathcal{O}(d)]\mapsto d$, so $[\mathcal{O}(1)]$ is a generator and every invertible sheaf is isomorphic to a unique twist. The Axiom of Choice is inherited through the divisor classification [F2], the twisting-sheaf construction [F4] and the degree-descent theorem [F5], as recorded in [F6]; the remaining arguments multiply finitely many transition units, use the single chart cover of ${\mathbb{P}}_k^1$, and select the integer $d$ determined by a class, so no further choice is made.