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The Stacks project

Lemma 38.12.2. Let R \to S be a ring map of finite presentation. Let N be a finitely presented S-module flat over R. Let \mathfrak r \subset R be a prime ideal. Assume there exists a complete dévissage of N/S/R over \mathfrak r. Then there exists an f \in R, f \not\in \mathfrak r such that

N_ f \cong B_1^{\oplus r_1} \oplus \ldots \oplus B_ n^{\oplus r_ n}

as R-modules where each B_ i is a smooth R_ f-algebra with geometrically irreducible fibres. Moreover, N_ f is projective as an R_ f-module.

Proof. Let (A_ i, B_ i, M_ i, \alpha _ i)_{i = 1, \ldots , n} be the given complete dévissage. We prove the lemma by induction on n. Note that the assertions of the lemma are entirely about the structure of N as an R-module. Hence we may replace N by M_1, and we may think of M_1 as a B_1-module. See Remark 38.6.3 in order to see why M_1 is of finite presentation as a B_1-module. By Lemma 38.12.1 we may, after replacing R by R_ f for some f \in R, f \not\in \mathfrak r, assume the map \alpha _1 : B_1^{\oplus r_1} \to M_1 is R-universally injective. Since M_1 and B_1^{\oplus r_1} are R-flat and finitely presented as B_1-modules we see that \mathop{\mathrm{Coker}}(\alpha _1) is R-flat (Algebra, Lemma 10.82.7) and finitely presented as a B_1-module. Note that (A_ i, B_ i, M_ i, \alpha _ i)_{i = 2, \ldots , n} is a complete dévissage of \mathop{\mathrm{Coker}}(\alpha _1). Hence the induction hypothesis implies that, after replacing R by R_ f for some f \in R, f \not\in \mathfrak r, we may assume that \mathop{\mathrm{Coker}}(\alpha _1) has a decomposition as in the lemma and is projective. In particular M_1 = B_1^{\oplus r_1} \oplus \mathop{\mathrm{Coker}}(\alpha _1). This proves the statement regarding the decomposition. The statement on projectivity follows as B_1 is projective as an R-module by Lemma 38.9.3. \square


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