Effective Cartier divisors on a regular scheme have no embedded associated points

Local structure of the divisor. Fix $z\in Z$ and choose a local-equation neighbourhood $U=\mathrm{Spec}A$ of $z$ on which $Z$ is cut out by a regular element $t\in A$, so that $Z\cap U=\mathrm{Spec}A\mathrm{/}(t)$ and ${\mathcal{O}}_{Z,z}={\mathcal{O}}_{X,z}\mathrm{/}(t_z)$; here $z\in Z$ means that the germ $t_z$ lies in the maximal ideal ${\mathfrak{m}}_z$ of ${\mathcal{O}}_{X,z}$, and $t_z$ is a nonzerodivisor on ${\mathcal{O}}_{X,z}$ because regularity of the section is a stalk condition.

A nonzero local ring at a point of $Z$. The ring ${\mathcal{O}}_{X,z}$ is nonzero with maximal ideal ${\mathfrak{m}}_z$, and ${\mathcal{O}}_{Z,z}$ is its nonzero quotient by the proper ideal $t_z{\mathcal{O}}_{X,z}$, because $t_z\in {\mathfrak{m}}_z$.

(a). The regular local ring ${\mathcal{O}}_{X,z}$ is Cohen--Macaulay by [F2], and $t_z\in {\mathfrak{m}}_z$ is a nonzerodivisor; applying [F3] to $A={\mathcal{O}}_{X,z}$ and $f=t_z$ shows that ${\mathcal{O}}_{Z,z}$ is nonzero Cohen--Macaulay of dimension $\dim {\mathcal{O}}_{X,z}-1$.

(b), first claim. For each $z\in Z$ the local ring ${\mathcal{O}}_{Z,z}$ is a nonzero Cohen--Macaulay local ring by step 2.1, so [F4] applied to $M={\mathcal{O}}_{Z,z}$ over $R={\mathcal{O}}_{Z,z}$ shows that every associated prime of ${\mathcal{O}}_{Z,z}$ is minimal in ${\mathrm{Supp}}_R(M)=\mathrm{Spec}{\mathcal{O}}_{Z,z}$.

(b), associated primes lie over generic points. Let $U=\mathrm{Spec}A$ be an affine chart as in step 1.1 with $C=A\mathrm{/}(t)\mathrm{\ne }0$, so $\mathrm{Spec}C$ is an open subscheme of $Z$. If $\mathfrak{p}\in {\mathrm{Ass}}_C(C)$, then localizing at $\mathfrak{p}$ and applying [F5] gives $\mathfrak{p}{C}_{\mathfrak{p}}\in {\mathrm{Ass}}_{C_{\mathfrak{p}}}(C_{\mathfrak{p}})$; the local ring $C_{\mathfrak{p}}={\mathcal{O}}_{Z,\mathfrak{p}}$ is Cohen--Macaulay by step 2.1, so by step 3.1 the prime $\mathfrak{p}{C}_{\mathfrak{p}}$ is minimal in $\mathrm{Spec}{C}_{\mathfrak{p}}$, which means that $\mathfrak{p}$ is a minimal prime of $C$. Hence every associated prime of $C$ is a generic point of an irreducible component of $Z\cap U$.

(b), detection. Let $c\in C$ be the image of a section and suppose $c\mathrm{/}1=0$ in $C_{\mathfrak{q}}$ for every minimal prime $\mathfrak{q}$ of $C$. By step 4.1 every associated prime of $C$ is minimal, so $c$ vanishes at every associated prime of $C$, and the injectivity of [F6] for $R=C$ gives $c=0$.

Conclusion. Part (a) is step 2.1 and the first claim of (b) is step 3.1. For the second claim of (b), cover $Z$ by affine charts $U=\mathrm{Spec}A$ as in step 1.1; on each chart the restriction of $s$ vanishes at the generic points of all irreducible components of $Z\cap U$, which are the minimal primes of the coordinate ring $C$, so step 5.1 shows that this restriction is zero. As the restrictions to an open cover vanish, $s=0$.