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The Stacks project

Lemma 88.4.5. Let A_1 \to A_2 be a map of Noetherian rings. Let I_ i \subset A_ i be an ideal such that V(I_1A_2) = V(I_2). Let B_1 be in (88.2.0.2) for (A_1, I_1). Let B_2 be the base change of B_1 as in Remark 88.2.3. If B_1 is rig-smooth over (A_1, I_1), then B_2 is rig-smooth over (A_2, I_2).

Proof. Follows from Lemma 88.4.4 and Definition 88.4.1 and the fact that I_2^ c is contained in I_1A_2 for some c \geq 0 as A_2 is Noetherian. \square


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