## 38.6 Translation into algebra

It may be useful to spell out algebraically what it means to have a complete dévissage. We introduce the following notion (which is not that useful so we give it an impossibly long name).

Definition 38.6.1. Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$ lying over the prime $\mathfrak p$ of $R$. A *elementary étale localization of the ring map $R \to S$ at $\mathfrak q$* is given by a commutative diagram of rings and accompanying primes

\[ \xymatrix{ S \ar[r] & S' \\ R \ar[u] \ar[r] & R' \ar[u] } \quad \quad \xymatrix{ \mathfrak q \ar@{-}[r] & \mathfrak q' \\ \mathfrak p \ar@{-}[u] \ar@{-}[r] & \mathfrak p' \ar@{-}[u] } \]

such that $R \to R'$ and $S \to S'$ are étale ring maps and $\kappa (\mathfrak p) = \kappa (\mathfrak p')$ and $\kappa (\mathfrak q) = \kappa (\mathfrak q')$.

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

\[ \xymatrix{ & A_1 & & A_2 & & ... & & A_ n \\ S \ar[ru] & & B_1 \ar[lu] \ar[ru] & & ... \ar[lu] \ar[ru] & & ... \ar[lu] \ar[ru] & & B_ n \ar[lu] } \]

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

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

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

$B_ i \to A_ i$ is finite,

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

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

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

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

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

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 $\mathfrak r$.

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

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

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

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

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

Lemma 38.6.6. Let $R \to S$ be a finite type ring map. Let $M$ be a finite $S$-module. Let $\mathfrak q$ be a prime ideal of $S$. There exists an elementary étale localization $R' \to S', \mathfrak q', \mathfrak p'$ of the ring map $R \to S$ at $\mathfrak q$ such that there exists a complete dévissage of $(M \otimes _ S S')/S'/R'$ at $\mathfrak q'$.

**Proof.**
This is a reformulation of Proposition 38.5.7 via Remark 38.6.5
$\square$

## Comments (0)