The Stacks project

Lemma 75.14.3. 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

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

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 75.14.1 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 65.6.1) and a morphism $T' \to X$ is determined by its restriction to $U'$. $\square$

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