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The Stacks project

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

Should the in the diagram be a instead?


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