Lemma 88.27.1 (0GDQ)—The Stacks project

Lemma 88.27.1. In the situation above, let $X_1 \to X$ be a morphism of algebraic spaces with $X_1$ locally Noetherian. Denote $T_1 \subset |X_1|$ the inverse image of $T$ and $U_1 \subset X_1$ the inverse image of $U$. We denote

$\mathcal{C}_{X, T}$ the category whose objects are morphisms of algebraic spaces $f : X' \to X$ which are locally of finite type and such that $U' = f^{-1}U \to U$ is an isomorphism,
$\mathcal{C}_{X_1, T_1}$ the category whose objects are morphisms of algebraic spaces $f_1 : X_1' \to X_1$ which are locally of finite type and such that $f_1^{-1}U_1 \to U_1$ is an isomorphism,
$\mathcal{C}_{X_{/T}}$ the category whose objects are morphisms $g : W \to X_{/T}$ of formal algebraic spaces with $W$ locally Noetherian and $g$ rig-étale,
$\mathcal{C}_{X_{1, /T_1}}$ the category whose objects are morphisms $g_1 : W_1 \to X_{1, /T_1}$ of formal algebraic spaces with $W_1$ locally Noetherian and $g_1$ rig-étale.

Then the diagram

\[ \xymatrix{ \mathcal{C}_{X, T} \ar[d] \ar[r] & \mathcal{C}_{X_{/T}} \ar[d] \\ \mathcal{C}_{X_1, T_1} \ar[r] & \mathcal{C}_{X_{1, /T_1}} } \]

is commutative where the horizonal arrows are given by (88.27.0.1) and the vertical arrows by base change along $X_1 \to X$ and along $X_{1, /T_1} \to X_{/T}$.

Proof. This follows immediately from the fact that the completion functor $(h : Y \to X) \mapsto Y_{/T} = Y_{/|h|^{-1}T}$ on the category of algebraic spaces over $X$ commutes with fibre products. $\square$

The Stacks project

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