Maps to projective space equal generating line-bundle data

Data attached to a morphism. Let $\phi :X\to {\mathbb{P}}_S^n$ be an $S$-morphism. Then $L_{\phi }:={\phi }^{\ast }\mathcal{O}(1)$ is an invertible ${\mathcal{O}}_X$-module and the pullbacks $s_i:={\phi }^{\ast }{x}_i\in \mathrm{\Gamma }(X,L_{\phi })$, $i=0,\dots ,n$, generate $L_{\phi }$: on ${\phi }^{-1}(U_i)$ the section $s_i$ is the pullback of the frame $x_i{\mathrm{\mid }}_{U_i}=e_i$, hence a frame there, and the sets ${\phi }^{-1}(U_i)$ cover $X$.

The universal property in the reverse direction. Conversely, given an invertible $L$ and generating sections $s_0,\dots ,s_n$, [F2] supplies an $S$-morphism $\mathrm{\Phi }(L;s):X\to {\mathbb{P}}_S^n$ with ${\mathrm{\Phi }}^{\ast }{x}_i$ corresponding to $s_i$ under an isomorphism ${\mathrm{\Phi }}^{\ast }\mathcal{O}(1)\cong L$ and with ${\mathrm{\Phi }}^{-1}(D_{+}(x_i))=X_{s_i}$. The construction depends only on the isomorphism class of $(L;s_i)$: an isomorphism $\alpha :L\to L^{\mathrm{\prime }}$ with $\alpha (s_i)=s_i^{\mathrm{\prime }}$ transports a trivialisation of ${\mathrm{\Phi }}^{\ast }\mathcal{O}(1)$ by $L$ into one by $L^{\mathrm{\prime }}$.

The case $n=0$. Here ${\mathbb{P}}_S^0\cong S$ by [F4], so the left side is the singleton $\{$structure morphism $X\to S\}$. On the right side, a pair $(L;s_0)$ with $s_0$ generating $L$ has $s_0$ a global frame: the evaluation map ${\mathcal{O}}_X\to L$ is an isomorphism. Mapping $(L;s_0)$ to the isomorphism class of the trivialisation it defines identifies all such pairs with the single class of $({\mathcal{O}}_X;1)$, so both sides are singletons; the unique morphism $X\to {\mathbb{P}}_S^0$ corresponds to the unit section of ${\mathcal{O}}_X$.

The two constructions are inverse: data-to-morphism-to-data. Start with data $(L;s_0,\dots ,s_n)$ and let $\phi =\mathrm{\Phi }(L;s)$. Then the data attached to $\phi$ in step 1.1 are $({\phi }^{\ast }\mathcal{O}(1);{\phi }^{\ast }{x}_0,\dots ,{\phi }^{\ast }{x}_n)$, which by [F2] is isomorphic to $(L;s_0,\dots ,s_n)$ under the very isomorphism used to define $\mathrm{\Phi }$; hence the composite data $\mapsto$ morphism $\mapsto$ data is the identity on isomorphism classes.

The two constructions are inverse: morphism-to-data-to-morphism. Let $\phi :X\to {\mathbb{P}}_S^n$ be an $S$-morphism and let $(L_{\phi };s_i={\phi }^{\ast }{x}_i)$ be its data as in step 1.1. Let $\psi =\mathrm{\Phi }(L_{\phi };s)$ be the morphism supplied by step 1.2. Then ${\psi }^{-1}(D_{+}(x_i))=X_{s_i}={\phi }^{-1}(D_{+}(x_i))$ and on $X_{s_i}$ the chart coordinates agree: $x_j^{(i)}\circ \psi =s_j\mathrm{/}{s}_i={\phi }^{\ast }(x_j)\mathrm{/}{\phi }^{\ast }(x_i)=x_j^{(i)}\circ \phi$ by [F2] and the definitions. Since the open sets $X_{s_i}$ cover $X$, [F3] gives $\psi =\phi$.

Naturality. For a morphism $g:X^{\mathrm{\prime }}\to X$ the data of $\phi \circ g$ are the pullbacks along $g$ of the data of $\phi$, and $\mathrm{\Phi }$ is compatible with this operation because the universal property [F2] is: the morphism associated to the pulled-back data is $\mathrm{\Phi }(L;s)\circ g$, by uniqueness in [F2]. The same uniqueness gives compatibility with base change $S^{\mathrm{\prime }}\to S$.

Conclusion. Step 1.1 attaches generating line-bundle data to every morphism, step 1.2 produces a morphism from data, steps 2.1 and 2.2 show the two operations are mutually inverse on isomorphism classes, step 2.3 gives naturality, and step 1.3 covers $n=0$. The Axiom of Choice [A1] is inherited from the projective-space and sheaf constructions; no choice is made here. [A1, step 1.2, step 2.1, step 2.2, step 2.3, step 1.3] \qed