Lemma 38.30.4. Let $A \to C$ be a finite locally free ring map of rank $d$. Let $h \in C$ be an element such that $C_ h$ is étale over $A$. Let $J \subset C$ be an ideal. Set $I = \text{Fit}_0(C/J)$ where we think of $C/J$ as a finite $A$-module. Then $IC_ h = JJ'$ for some ideal $J' \subset C_ h$. If $J$ is finitely generated so are $I$ and $J'$.
Proof. We will use basic properties of Fitting ideals, see More on Algebra, Lemma 15.8.4. Then $IC$ is the Fitting ideal of $C/J \otimes _ A C$. Note that $C \to C \otimes _ A C$, $c \mapsto 1 \otimes c$ has a section (the multiplication map). By assumption $C \to C \otimes _ A C$ is étale at every prime in the image of $\mathop{\mathrm{Spec}}(C_ h)$ under this section. Hence the multiplication map $C \otimes _ A C_ h \to C_ h$ is étale in particular flat, see Algebra, Lemma 10.143.8. Hence there exists a $C_ h$-algebra such that $C \otimes _ A C_ h \cong C_ h \oplus C'$ as $C_ h$-algebras, see Algebra, Lemma 10.143.9. Thus $(C/J) \otimes _ A C_ h \cong (C_ h/J_ h) \oplus C'/I'$ as $C_ h$-modules for some ideal $I' \subset C'$. Hence $IC_ h = JJ'$ with $J' = \text{Fit}_0(C'/I')$ where we view $C'/J'$ as a $C_ h$-module. $\square$
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