The Stacks project

Lemma 86.22.2. Assume we have

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

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

  3. a morphism $f : X' \to X$ locally of finite type and ├ętale over $X \setminus T$,

  4. a morphism $h : Y \to X$,

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

Then there exists an ├ętale 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 $\alpha = a_{/T} \circ b_{/T}^{-1}$.

Proof. The notation using the subscript ${}_{/T}$ in the statement refers to the construction which to a morphism of schemes $g : V \to X$ associates the morphism $g_{/T} : V_{/g^{-1}T} \to X_{/T}$ of formal algebraic spaces; it is a functor from the category of schemes over $X$ to the category of formal algebraic spaces over $X_{/T}$, see Section 86.20. Having said this, the lemma is just a reformulation of Lemma 86.8.7. $\square$


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