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The Stacks project

Remark 88.2.3 (Base change). Let \varphi : A_1 \to A_2 be a ring map and let I_ i \subset A_ i be ideals such that \varphi (I_1^ c) \subset I_2 for some c \geq 1. This induces ring maps A_{1, cn} = A_1/I_1^{cn} \to A_2/I_2^ n = A_{2, n} for all n \geq 1. Let \mathcal{C}_ i be the category (88.2.0.1) for (A_ i, I_ i). There is a base change functor

88.2.3.1
\begin{equation} \label{restricted-equation-base-change-systems} \mathcal{C}_1 \longrightarrow \mathcal{C}_2,\quad (B_ n) \longmapsto (B_{cn} \otimes _{A_{1, cn}} A_{2, n}) \end{equation}

Let \mathcal{C}_ i' be the category (88.2.0.2) for (A_ i, I_ i). If I_2 is finitely generated, then there is a base change functor

88.2.3.2
\begin{equation} \label{restricted-equation-base-change-complete} \mathcal{C}_1' \longrightarrow \mathcal{C}_2',\quad B \longmapsto (B \otimes _{A_1} A_2)^\wedge \end{equation}

because in this case the completion is complete (Algebra, Lemma 10.96.3). If both I_1 and I_2 are finitely generated, then the two base change functors agree via the functors (88.2.0.3) which are equivalences by Lemma 88.2.1.


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