Definition 38.6.4. Let $R \to S$ be a finite type ring map. Let $\mathfrak q$ be a prime of $S$ lying over the prime $\mathfrak r$ of $R$. Let $N$ be a finite $S$-module. A complete dévissage of $N/S/R$ at $\mathfrak q$ is given by a complete dévissage $(A_ i, B_ i, M_ i, \alpha _ i)_{i = 1, \ldots , n}$ of $N/S/R$ over $\mathfrak r$ and prime ideals $\mathfrak q_ i \subset B_ i$ lying over $\mathfrak r$ such that

1. $\kappa (\mathfrak r) \subset \kappa (\mathfrak q_ i)$ is purely transcendental,

2. there is a unique prime $\mathfrak q'_ i \subset A_ i$ lying over $\mathfrak q_ i \subset B_ i$,

3. $\mathfrak q = \mathfrak q'_1 \cap S$ and $\mathfrak q_ i = \mathfrak q'_{i + 1} \cap A_ i$,

4. $R \to B_ i$ has relative dimension $\dim _{\mathfrak q_ i}(\text{Supp}(M_ i \otimes _ R \kappa (\mathfrak r)))$.

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