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
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