The Stacks project

Definition 38.6.2. Let $R \to S$ be a finite type ring map. Let $\mathfrak r$ be a prime of $R$. Let $N$ be a finite $S$-module. A complete dévissage of $N/S/R$ over $\mathfrak r$ is given by $R$-algebra maps

finite $A_ i$-modules $M_ i$ and $B_ i$-module maps $\alpha _ i : B_ i^{\oplus r_ i} \to M_ i$ such that

1. $S \to A_1$ is surjective and of finite presentation,

2. $B_ i \to A_{i + 1}$ is surjective and of finite presentation,

3. $B_ i \to A_ i$ is finite,

4. $R \to B_ i$ is smooth with geometrically irreducible fibres,

5. $N \cong M_1$ as $S$-modules,

6. $\mathop{\mathrm{Coker}}(\alpha _ i) \cong M_{i + 1}$ as $B_ i$-modules,

7. $\alpha _ i : \kappa (\mathfrak p_ i)^{\oplus r_ i} \to M_ i \otimes _{B_ i} \kappa (\mathfrak p_ i)$ is an isomorphism where $\mathfrak p_ i = \mathfrak rB_ i$, and

8. $\mathop{\mathrm{Coker}}(\alpha _ n) = 0$.

In this situation we say that $(A_ i, B_ i, M_ i, \alpha _ i)_{i = 1, \ldots , n}$ is a complete dévissage of $N/S/R$ over $\mathfrak r$.