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
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
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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