A section of an invertible sheaf has a canonical zero subscheme

Since $L$ is locally free of rank one, choose an affine trivializing open cover $U_i=\mathrm{Spec}{A}_i$ with generator $e_i$. By [F1] and [F2], there is a unique $f_i\in A_i$ such that $s{\mathrm{\mid }}_{U_i}=f_i{e}_i$. Put $Z_i=\mathrm{Spec}(A_i\mathrm{/}(f_i))$ and let $q_i:Z_i\to U_i$ be the quotient morphism. Its image is $V(f_i)$ by [F4], and it is a homeomorphism onto that image. On each distinguished open $D(g)\subseteq U_i$, [F3] identifies the restricted quotient with the quotient map $A_{i,g}\to A_{i,g}\mathrm{/}(f_i)$; thus the map of structure sheaves is surjective locally. Hence $q_i$ is a closed immersion by [F7]. If $f_i$ is a unit then $A_i\mathrm{/}(f_i)=0$ and $Z_i$ is empty; if $f_i=0$ then $Z_i=U_i$.

On $U_i\cap U_j$, the two generators differ by an invertible function: $e_i=u_{ij}{e}_j$. Comparing the expressions for the same restricted section gives $f_j=u_{ij}{f}_i$, so the generated ideals agree. Refine each overlap by affine opens $W=\mathrm{Spec}R$. On such a $W$, both local zero schemes restrict to $\mathrm{Spec}(R\mathrm{/}(f_i{\mathrm{\mid }}_W))$ and $\mathrm{Spec}(R\mathrm{/}(f_j{\mathrm{\mid }}_W))$; equality of the ideals gives the canonical identity-on-$R$ isomorphism. On distinguished opens of $W$, [F3] identifies the restrictions with the corresponding localized quotients.

These overlap isomorphisms are compatible when further restricted: each acts on residue classes by the identity on functions from ${\mathcal{O}}_X$. They therefore satisfy the identity and cocycle conditions, including on triple overlaps. By [F5] the $Z_i$ glue to a scheme $Z$, with each $Z_i$ an open subscheme of $Z$. Their maps $q_i$ to the open subsets $U_i\subseteq X$ agree on overlaps, so [F6] glues them to a unique morphism $q:Z\to X$.

The restriction $q^{-1}(U_i)\to U_i$ is $q_i$, a closed immersion by step 1.1. The $U_i$ cover $X$, so [F8] implies that $q$ is a closed immersion. For any other affine trivializing cover, refine both covers by affine opens. On each such open the two equations differ by a unit, hence define the same quotient ring and the same morphism to $X$. The uniqueness in [F5] and [F6] then gives a unique isomorphism over $X$ between the two constructions.

No radical or regularity condition entered the construction: it quotients by $(f_i)$, even when $f_i$ is a zero divisor or nilpotent. For example, on $X=\mathrm{Spec}(k[ϵ]\mathrm{/}({ϵ}^3))$ with $L={\mathcal{O}}_X$ and $s={ϵ}^2$, the zero scheme is $\mathrm{Spec}(k[ϵ]\mathrm{/}({ϵ}^2))$, which is still nonreduced. Thus the construction retains precisely the nilpotents not killed by the section equation.