Remark 38.6.5. Let $A \to B$ be a finite type ring map and let $N$ be a finite $B$-module. Let $\mathfrak q$ be a prime of $B$ lying over the prime $\mathfrak r$ of $A$. Set $X = \mathop{\mathrm{Spec}}(B)$, $S = \mathop{\mathrm{Spec}}(A)$ and $\mathcal{F} = \widetilde{N}$ on $X$. Let $x$ be the point corresponding to $\mathfrak q$ and let $s \in S$ be the point corresponding to $\mathfrak p$. Then

1. if there exists a complete dévissage of $\mathcal{F}/X/S$ over $s$ then there exists a complete dévissage of $N/B/A$ over $\mathfrak p$, and

2. there exists a complete dévissage of $\mathcal{F}/X/S$ at $x$ if and only if there exists a complete dévissage of $N/B/A$ at $\mathfrak q$.

There is just a small twist in that we omitted the condition on the relative dimension in the formulation of “a complete dévissage of $N/B/A$ over $\mathfrak p$” which is why the implication in (1) only goes in one direction. The notion of a complete dévissage at $\mathfrak q$ does have this condition built in. In any case we will only use that existence for $\mathcal{F}/X/S$ implies the existence for $N/B/A$.

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