On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group

Well-definedness of the map. Let $D$ be a Cartier divisor with local-equation datum $\{(U_i,f_i)\}$; by [F3] the subsheaf ${\mathcal{O}}_X(D)\subseteq {\mathcal{K}}_X$ is well defined and independent of the chosen datum, and by [F4] it is invertible, so its isomorphism class $[{\mathcal{O}}_X(D)]$ lies in $\mathrm{Pic}(X)$ by [F6]. This assigns to every $D\in \mathrm{CaDiv}(X)$ a well-defined element ${\phi }_X(D)$ of $\mathrm{Pic}(X)$.

Homomorphism. For Cartier divisors $D,E$ the canonical isomorphism ${\mathcal{O}}_X(D+E)\cong {\mathcal{O}}_X(D){\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(E)$ of [F5] gives ${\phi }_X(D+E)=[{\mathcal{O}}_X(D+E)]=[{\mathcal{O}}_X(D)][{\mathcal{O}}_X(E)]={\phi }_X(D){\phi }_X(E)$ by the product rule in [F6], and ${\mathcal{O}}_X(0)={\mathcal{O}}_X$ by [F3] gives ${\phi }_X(0)=[{\mathcal{O}}_X]$, the identity of $\mathrm{Pic}(X)$; hence ${\phi }_X$ is a homomorphism of abelian groups.

Principal divisors have trivial class. Let $f\in \mathrm{\Gamma }(X,{\mathcal{K}}_X^{\times })$ and put $D={\mathrm{div}}_C(f)=q_X(f)$; by [F2] the divisor $D$ is represented by the single global equation $f$ on $X$, so [F3] gives ${\mathcal{O}}_X(D)=f^{-1}{\mathcal{O}}_X\subseteq {\mathcal{K}}_X$. Multiplication by $f^{-1}$ is an isomorphism ${\mathcal{O}}_X\to {\mathcal{O}}_X(D)$, $a\mapsto a{f}^{-1}$, of ${\mathcal{O}}_X$-modules, with inverse given by multiplication by $f$, so $[{\mathcal{O}}_X(D)]=[{\mathcal{O}}_X]$ and ${\phi }_X(D)=0$ by [F6]; thus ${\mathrm{Prin}}_C(X)\subseteq \ker {\phi }_X$ by [F1] and [F2].

An isomorphism of the divisor sheaf with ${\mathcal{O}}_X$. Conversely let $D$ be a Cartier divisor with ${\phi }_X(D)=[{\mathcal{O}}_X]$, choose an isomorphism $u:{\mathcal{O}}_X(D)\to {\mathcal{O}}_X$ of ${\mathcal{O}}_X$-modules, and fix a local-equation datum $\{(U_i,f_i){\}}_{i\in I}$ for $D$, so that $f_i\in {\mathcal{K}}_X(U_i{)}^{\times }$ and $f_i\mathrm{/}{f}_j\in {\mathcal{O}}_X(U_i\cap U_j{)}^{\times }$ for all $i,j$ by [F1]. By [F4] the sheaf ${\mathcal{O}}_X(D){\mathrm{\mid }}_{U_i}$ is freely generated by $f_i^{-1}$; the chosen isomorphism $u$ is a single selection from the nonempty set of isomorphisms, so no choice principle is used.

A rational section exists. Now assume that $X$ is integral, and let $\mathcal{L}$ be an invertible ${\mathcal{O}}_X$-module. By [F9] the scheme $X$ is nonempty, reduced and irreducible, hence has a generic point $\eta$; by [F8] the sheaf $K_X(\mathcal{L})$ is the constant sheaf with value the stalk ${\mathcal{L}}_{\eta }$, a one-dimensional vector space over the field $K(X)$, which is nonzero, so the set of nonzero elements of ${\mathcal{L}}_{\eta }$ is nonempty. Choose such an element $s$; it is a global section of $K_X(\mathcal{L})$, that is, a rational section of $\mathcal{L}$, and this is a single selection from a nonempty set, not a choice principle.

The transported generators are units. Fix $i$ and put $e_i:=u(f_i^{-1})\in {\mathcal{O}}_X(U_i)$; then the composite of the generator isomorphism $a\mapsto a{f}_i^{-1}$ of [F4] with $u$ is the endomorphism $a\mapsto a{e}_i$ of ${\mathcal{O}}_{U_i}$, and it is an isomorphism of ${\mathcal{O}}_{U_i}$-modules. Its surjectivity gives an element $b\in {\mathcal{O}}_X(U_i)$ with $b{e}_i=1$, so $e_i\in {\mathcal{O}}_X(U_i{)}^{\times }$ with inverse $b$; in particular $f_i{e}_i\in {\mathcal{K}}_X(U_i{)}^{\times }$, because $f_i$ is a unit of ${\mathcal{K}}_X(U_i)$ by [F1] and the unit $e_i$ maps to a unit of ${\mathcal{K}}_X(U_i)$ under the ring homomorphism ${\mathcal{O}}_X(U_i)\to {\mathcal{K}}_X(U_i)$ of [F11].

Surjectivity. By [F7] the rational section $s$ has a well-defined Cartier divisor $D:={\mathrm{div}}_C(s)$ on the integral scheme $X$, and there is a canonical isomorphism ${\mathcal{O}}_X(D)\to \mathcal{L}$, so $[\mathcal{L}]=[{\mathcal{O}}_X(D)]={\phi }_X(D)$ in $\mathrm{Pic}(X)$ by [F6] and step 1.1. As $\mathcal{L}$ was an arbitrary invertible ${\mathcal{O}}_X$-module, the map ${\phi }_X$ is surjective.

Gluing the global equation. For each $i$ put $g_i:=f_i{e}_i\in {\mathcal{K}}_X(U_i{)}^{\times }$, a unit by step 2.1. On $U_i\cap U_j$ one has $f_i^{-1}=(f_j\mathrm{/}{f}_i)f_j^{-1}$ with $f_j\mathrm{/}{f}_i\in {\mathcal{O}}_X(U_i\cap U_j{)}^{\times }$ by [F1], and the ${\mathcal{O}}_X$-linearity of $u$ gives $e_i=u(f_i^{-1})=(f_j\mathrm{/}{f}_i)u(f_j^{-1})=(f_j\mathrm{/}{f}_i)e_j$, so $g_i=f_i{e}_i=f_j{e}_j=g_j$; by [F12] the $g_i$ glue to a unique global section $f\in \mathrm{\Gamma }(X,{\mathcal{K}}_X)$ with $f{\mathrm{\mid }}_{U_i}=g_i$. The local inverses $g_i^{-1}$ agree on the overlaps as well, because they are the inverses of the equal restrictions $g_i{\mathrm{\mid }}_{U_i\cap U_j}=g_j{\mathrm{\mid }}_{U_i\cap U_j}$, so they glue to an inverse of $f$ and $f\in \mathrm{\Gamma }(X,{\mathcal{K}}_X^{\times })$.

The divisor is principal. On each $U_i$ one has $f_i=f{\mathrm{\mid }}_{U_i}{e}_i^{-1}$ with $e_i^{-1}\in {\mathcal{O}}_X(U_i{)}^{\times }$ by step 2.1, so the local equations $f_i$ of $D$ differ from the restrictions of the global meromorphic unit $f$ by units of ${\mathcal{O}}_X$, and by the local-equation description of [F1] the divisor $D$ is represented by the single global equation $f$; thus $D=q_X(f)={\mathrm{div}}_C(f)$ is principal by [F2]. Hence $\ker {\phi }_X\subseteq {\mathrm{Prin}}_C(X)$.

The kernel. By step 1.3 every principal Cartier divisor lies in the kernel of ${\phi }_X$, and by step 4.1 every divisor in the kernel is principal; since ${\mathrm{Prin}}_C(X)$ is a subgroup of $\mathrm{CaDiv}(X)$ by [F1] and [F2], the kernel of ${\phi }_X$ is exactly ${\mathrm{Prin}}_C(X)$.

The induced isomorphism. By step 1.2 the map ${\phi }_X$ is a group homomorphism, by step 5.1 its kernel is ${\mathrm{Prin}}_C(X)$, and by step 2.2 its image is all of $\mathrm{Pic}(X)$ when $X$ is integral; the first isomorphism theorem [F10] therefore identifies $\mathrm{CaDiv}(X)\mathrm{/}{\mathrm{Prin}}_C(X)$ with $\mathrm{Pic}(X)$ through the map induced by ${\phi }_X$.