Definition 38.5.5. Let $S$, $X$, $\mathcal{F}$, $x$, $s$ be as in Definition 38.5.2. Consider a complete dévissage $(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k, z_ k, y_ k)_{k = 1, \ldots , n}$ of $\mathcal{F}/X/S$ at $x$. Let us define a standard shrinking of this situation to be given by standard opens $S' \subset S$, $X' \subset X$, $Z'_ k \subset Z_ k$, and $Y'_ k \subset Y_ k$ such that $s_ k \in S'$, $x_ k \in X'$, $z_ k \in Z'$, and $y_ k \in Y'$ and such that
\[ (Z'_ k, Y'_ k, i'_ k, \pi '_ k, \mathcal{G}'_ k, \alpha '_ k, z_ k, y_ k)_{k = 1, \ldots , n} \]
is a one step dévissage of $\mathcal{F}'/X'/S'$ at $x$ where $\mathcal{G}'_ k = \mathcal{G}_ k|_{Z'_ k}$ and $\mathcal{F}' = \mathcal{F}|_{X'}$.
Comments (0)