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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