The Stacks project

Remark 86.2.3 (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 (86.2.0.1) for $(A_ i, I_ i)$. There is a base change functor

86.2.3.1
\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 (86.2.0.2) for $(A_ i, I_ i)$. If $I_2$ is finitely generated, then there is a base change functor

86.2.3.2
\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 10.95.3). If both $I_1$ and $I_2$ are finitely generated, then the two base change functors agree via the functors (86.2.0.3) which are equivalences by Lemma 86.2.1.


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