The Stacks project

Lemma 75.16.3. Let $S$ be a scheme. Let $f : X \to Y$ be a formally étale morphism of algebraic spaces over $S$. Then given any solid commutative diagram

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

where $T \subset T'$ is a first order thickening of algebraic spaces over $Y$ there exists exactly one dotted arrow making the diagram commute. In other words, in Definition 75.16.1 the condition that $T$ be affine may be dropped.

Proof. Let $U' \to T'$ be a surjective étale morphism where $U' = \coprod U'_ i$ is a disjoint union of affine schemes. Let $U_ i = T \times _{T'} U'_ i$. Then we get morphisms $a'_ i : U'_ i \to X$ such that $a'_ i|_{U_ i}$ equals the composition $U_ i \to T \to X$. By uniqueness (see Lemma 75.14.3) we see that $a'_ i$ and $a'_ j$ agree on the fibre product $U'_ i \times _{T'} U'_ j$. Hence $\coprod a'_ i : U' \to X$ descends to give a unique morphism $a' : T' \to X$. $\square$


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