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The Stacks project

Lemma 38.4.5. Let S, X, \mathcal{F}, x, s be as in Definition 38.4.2. Let (Z, Y, i, \pi , \mathcal{G}, z, y) be a one step dévissage of \mathcal{F}/X/S at x. Let (S', s') \to (S, s) be a morphism of pointed schemes which induces an isomorphism \kappa (s) = \kappa (s'). Let (Z', Y', i', \pi ', \mathcal{G}') be as constructed in Lemma 38.4.4 and let x' \in X' (resp. z' \in Z', y' \in Y') be the unique point mapping to both x \in X (resp. z \in Z, y \in Y) and s' \in S'. If S' is affine, then (Z', Y', i', \pi ', \mathcal{G}', z', y') is a one step dévissage of \mathcal{F}'/X'/S' at x'.

Proof. By Lemma 38.4.4 (Z', Y', i', \pi ', \mathcal{G}') is a one step dévissage of \mathcal{F}'/X'/S' over s'. Properties (1) – (4) of Definition 38.4.2 hold for (Z', Y', i', \pi ', \mathcal{G}', z', y') as the assumption that \kappa (s) = \kappa (s') insures that the fibres X'_{s'}, Z'_{s'}, and Y'_{s'} are isomorphic to X_ s, Z_ s, and Y_ s. \square


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