Lemma 86.22.2. Assume we have

Noetherian affine schemes $X$, $X'$, and $Y$,

a closed subset $T \subset |X|$,

a morphism $f : X' \to X$ locally of finite type and étale over $X \setminus T$,

a morphism $h : Y \to X$,

a morphism $\alpha : Y_{/T} \to X'_{/T}$ over $X_{/T}$ (see proof for notation).

Then there exists an étale morphism $b : Y' \to Y$ of affine schemes which induces an isomorphism $b_{/T} : Y'_{/T} \to Y_{/T}$ and a morphism $a : Y' \to X'$ over $X$ such that $\alpha = a_{/T} \circ b_{/T}^{-1}$.

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