Lemma 76.14.3 (05ZY)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.14: Formally unramified morphisms
Lemma 76.14.3 (cite)

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Lemma 76.14.3. Let $S$ be a scheme. If $f : X \to Y$ is a formally unramified morphism of algebraic spaces over $S$ , then given any solid commutative diagram

$\xymatrix{ X \ar[d]_ f & T \ar[d]^ i \ar[l] \\ Y & T' \ar[l] \ar@{-->}[lu] }$

where $T \subset T'$ is a first order thickening of algebraic spaces over $S$ there exists at most one dotted arrow making the diagram commute. In other words, in Definition 76.14.1 the condition that $T$ be an affine scheme may be dropped.

Proof. This is true because there exists a surjective étale morphism $U' \to T'$ where $U'$ is a disjoint union of affine schemes (see Properties of Spaces, Lemma 66.6.1) and a morphism $T' \to X$ is determined by its restriction to $U'$ . $\square$

Comments (2)

Comment #8790 by Niven on February 02, 2024 at 14:44

Should the $S$ in the diagram be a $Y$ instead?

Comment #9292 by Stacks project on May 22, 2024 at 13:10

Thanks and fixed here.

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