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The Stacks project

Lemma 38.5.4. Let S, X, \mathcal{F}, x, s be as in Definition 38.5.2. Let (S', s') \to (S, s) be a morphism of pointed schemes which induces an isomorphism \kappa (s) = \kappa (s'). Let (Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k, z_ k, y_ k)_{k = 1, \ldots , n} be a complete dévissage of \mathcal{F}/X/S at x. Let (Z'_ k, Y'_ k, i'_ k, \pi '_ k, \mathcal{G}'_ k, \alpha '_ k)_{k = 1, \ldots , n} be as constructed in Lemma 38.5.3 and let x' \in X' (resp. z'_ k \in Z', y'_ k \in Y') be the unique point mapping to both x \in X (resp. z_ k \in Z_ k, y_ k \in Y_ k) and s' \in S'. If S' is affine, then (Z'_ k, Y'_ k, i'_ k, \pi '_ k, \mathcal{G}'_ k, \alpha '_ k, z'_ k, y'_ k)_{k = 1, \ldots , n} is a complete dévissage of \mathcal{F}'/X'/S' at x'.

Proof. Combine Lemma 38.5.3 and Lemma 38.4.5. \square


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