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 $x'$.

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