Effective Cartier divisors are closed subschemes cut out by regular equations

Affine setup. Let $D$ be effective with datum $\{(U_i,f_i)\}$ as in [F1]. Refine the cover by affine opens $U_i=\mathrm{Spec}{A}_i$ and write $f_i\in A_i$. For every overlap $W_{ij}=U_i\cap U_j$, the Cartier datum gives $f_i\mathrm{/}{f}_j\in {\mathcal{O}}_X(W_{ij}{)}^{\times }$, so $f_i{\mathcal{O}}_{W_{ij}}=f_j{\mathcal{O}}_{W_{ij}}$. No affineness of $W_{ij}$ is needed.

The converse datum. Now let $Z\hookrightarrow X$ be locally cut out by nonzerodivisors. Cover $X$ by affine opens $U_i=\mathrm{Spec}{A}_i$ with $Z\cap U_i=\mathrm{Spec}(A_i\mathrm{/}{f}_i{A}_i)$ for nonzerodivisors $f_i\in A_i$. Then $I_Z{\mathrm{\mid }}_{U_i}=f_i{\mathcal{O}}_{U_i}$ by [F5], and the sections $f_i$ are regular: a nonzerodivisor of $A_i$ remains a nonzerodivisor after localisation at every prime, so its germ at each point of $U_i$ is a nonzerodivisor.

The glued ideal sheaf. Let ${\mathcal{I}}_D$ be the subsheaf of ${\mathcal{O}}_X$ whose sections over an open $V$ are the $s\in {\mathcal{O}}_X(V)$ with $s{\mathrm{\mid }}_{V\cap U_i}\in f_i{\mathcal{O}}_X(V\cap U_i)$ for every $i$. This is a subsheaf with ideal values, hence an ideal sheaf, and ${\mathcal{I}}_D{\mathrm{\mid }}_{U_i}=f_i{\mathcal{O}}_{U_i}$: on $U_i$ every $s$ lies in $f_i{\mathcal{O}}_{U_i}$ by the condition $i$, while the conditions for $j$ add nothing because $f_j{\mathcal{O}}_{U_j}$ and $f_i{\mathcal{O}}_{U_i}$ agree on $U_i\cap U_j$ by step 1.1.

The affine pieces. For each $i$ let $Z_i=\mathrm{Spec}(A_i\mathrm{/}{f}_i{A}_i)$ and let $j_i:Z_i\to U_i$ be the closed immersion induced by the quotient map $A_i\to A_i\mathrm{/}{f}_i{A}_i$; by [F5] the ideal of $j_i$ is $f_i{A}_i$, and its structure sheaf is the quotient. On the overlap $W_{ij}$, the restrictions of the ideals generated by $f_i$ and $f_j$ are equal by step 1.1. Their quotient sheaves therefore define the same closed subscheme of $W_{ij}$, giving canonical overlap isomorphisms $Z_i{\times }_{U_i}{W}_{ij}\cong Z_j{\times }_{U_j}{W}_{ij}$. These isomorphisms satisfy the cocycle conditions because they are induced by equality of the restricted ideal sheaves.

Invertibility. For every $i$, multiplication $m_i:{\mathcal{O}}_{U_i}\to {\mathcal{I}}_D{\mathrm{\mid }}_{U_i}$, $a\mapsto f_{i}a$, is an isomorphism: it is injective because $f_i$ is regular by [F1], and it is surjective because every section of $f_i{\mathcal{O}}_{U_i}$ is of the form $f_{i}a$ by definition of the principal ideal sheaf. Hence ${\mathcal{I}}_D$ is invertible.

Gluing. The affine schemes $Z_i$ with the open subschemes $Z_i{\times }_{U_i}{W}_{ij}$ and the canonical identifications supplied by step 2.2 satisfy the identity and cocycle conditions, so by [F7] they glue to a scheme $Z_D$ which is covered by open subschemes identified with the $Z_i$, with overlaps identified with the common closed subschemes of step 2.2. The local morphisms $j_i:Z_i\to U_i\subseteq X$ agree on these overlaps, so they glue to a morphism $i_D:Z_D\to X$: continuous maps that agree on an open cover glue topologically, and the structure-sheaf maps ${\mathcal{O}}_X{\mathrm{\mid }}_{U_i}\to (j_i{)}_{\ast }{\mathcal{O}}_{Z_i}$ agree on overlaps and glue by the sheaf axiom.

Independence of the datum. If $\{(V_k,g_k)\}$ is another local-equation datum for $D$, [F10] gives a common refinement on which the equations differ by regular units. They therefore generate the same ideal sheaf there, so the ideals glued in step 2.1 agree and define the same closed subscheme.

The glued morphism is a closed immersion with ideal ${\mathcal{I}}_D$. The restriction of $i_D$ over $U_i$ is the closed immersion $j_i$ (Affine schemes and their coordinate rings), so $i_D$ is a closed immersion by [F6]. Its ideal sheaf is ${\mathcal{I}}_D$: on $U_i$ the kernel of ${\mathcal{O}}_{U_i}\to (j_i{)}_{\ast }{\mathcal{O}}_{Z_i}$ is $f_i{\mathcal{O}}_{U_i}$ by [F5], and ${\mathcal{I}}_D{\mathrm{\mid }}_{U_i}=f_i{\mathcal{O}}_{U_i}$ by step 2.1; these local identifications agree on overlaps by step 2.2 and glue. In particular $I_D={\mathcal{I}}_D$ is locally generated by the local equations $f_i$ and is invertible by step 3.1.

The datum is Cartier. For each pair $i,j$, the two ideals $f_i{\mathcal{O}}_{W_{ij}}=I_Z{\mathrm{\mid }}_{W_{ij}}=f_j{\mathcal{O}}_{W_{ij}}$ agree on the overlap, so $f_i=u_{ij}{f}_j$ and $f_j=v_{ij}{f}_i$ for sections $u_{ij},v_{ij}\in {\mathcal{O}}_X(W_{ij})$; substituting gives $u_{ij}{v}_{ij}{f}_j=f_j$, and since $f_j$ is regular on $W_{ij}$ (step 1.2) one gets $u_{ij}{v}_{ij}=1$, so $u_{ij}$ is a unit. Hence $\{(U_i,f_i)\}$ is a local-equation datum with regular equations and unit ratios, i.e. an effective Cartier divisor $D_Z$, and its ideal sheaf is ${\mathcal{I}}_{D_Z}=I_Z$ by the construction of steps 2.1 and 4.1.

Conclusion. Part 1 is steps 1.1, 2.1–4.1 and 3.3: every effective Cartier divisor determines a closed immersion $i_D:Z_D\hookrightarrow X$ whose ideal sheaf is locally generated by its equations and is invertible, independently of the chosen datum. Part 2 is steps 1.2 and 5.1: a closed subscheme locally cut out by nonzerodivisors gives an effective Cartier divisor $D_Z$ with $I_{D_Z}=I_Z$, so the closed subscheme cut out by $D_Z$ is $Z$ itself.