The Stacks project

Lemma 37.6.2. If $f : X \to S$ is a formally unramified 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 at most one dotted arrow making the diagram commute. In other words, in Definition 37.6.1 the condition that $T$ be affine may be dropped.

Proof. This is true because a morphism is determined by its restrictions to affine opens. $\square$

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