Lemma 88.6.2. Let I be an ideal of a Noetherian ring A. Let C^ h be the henselization of a finite type A-algebra C with respect to the ideal IC. Let J \subset C^ h be an ideal. Then there exists a finite type A-algebra B such that B^\wedge \cong (C^ h/J)^\wedge .
Proof. By More on Algebra, Lemma 15.12.4 the ring C^ h is Noetherian. Say J = (g_1, \ldots , g_ m). The ring C^ h is a filtered colimit of étale C algebras C' such that C/IC \to C'/IC' is an isomorphism (see proof of More on Algebra, Lemma 15.12.1). Pick an C' such that g_1, \ldots , g_ m are the images of g'_1, \ldots , g'_ m \in C'. Setting B = C'/(g'_1, \ldots , g'_ m) we get a finite type A-algebra. Of course (C, IC) and C', IC') have the same henselizations and the same completions. It follows easily from this that B^\wedge = (C^ h/J)^\wedge . \square
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