The Stacks project

Lemma 37.8.2. If $f : X \to S$ is a formally ├ętale morphism, 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 schemes over $S$ there exists exactly one dotted arrow making the diagram commute. In other words, in Definition 37.8.1 the condition that $T$ be affine may be dropped.

Proof. Let $T' = \bigcup T'_ i$ be an affine open covering, and let $T_ i = T \cap T'_ i$. Then we get morphisms $a'_ i : T'_ i \to X$ fitting into the diagram. By uniqueness we see that $a'_ i$ and $a'_ j$ agree on any affine open subscheme of $T'_ i \cap T'_ j$. Hence $a'_ i$ and $a'_ j$ agree on $T'_ i \cap T'_ j$. Thus we see that the morphisms $a'_ i$ glue to a global morphism $a' : T' \to X$. The uniqueness of $a'$ we have seen in Lemma 37.6.2. $\square$

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