Lemma 70.4.2 (07SH)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 70: Limits of Algebraic Spaces
Section 70.4: Limits of algebraic spaces
Lemma 70.4.2 (cite)

Lemma 70.4.2. Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_ i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$ with affine transition maps. Let $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$ . Let $0 \in I$ . Suppose that $T \to X_0$ is a morphism of algebraic spaces. Then

$T \times _{X_0} X = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} T \times _{X_0} X_ i$

as algebraic spaces over $S$ .

Proof. The limit $X$ is an algebraic space by Lemma 70.4.1. The equality is formal, see Categories, Lemma 4.14.10. $\square$

The Stacks project

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