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The Stacks project

Lemma 38.10.2. Let (R, \mathfrak m) be a local ring. Let R \to S be a ring map of finite presentation. Let N be a finite S-module. Let \mathfrak q be a prime of S lying over \mathfrak m. Assume that N_{\mathfrak q} is flat over R, and assume there exists a complete dévissage of N/S/R at \mathfrak q. Then N is a finitely presented S-module, free as an R-module, and there exists an isomorphism

N \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-algebra with geometrically irreducible fibres.

Proof. Let (A_ i, B_ i, M_ i, \alpha _ i, \mathfrak q_ i)_{i = 1, \ldots , n} be the given complete dévissage. We prove the lemma by induction on n. Note that N is finitely presented as an S-module if and only if M_1 is finitely presented as an B_1-module, see Remark 38.6.3. Note that N_{\mathfrak q} \cong (M_1)_{\mathfrak q_1} as R-modules because (a) N_{\mathfrak q} \cong (M_1)_{\mathfrak q'_1} where \mathfrak q'_1 is the unique prime in A_1 lying over \mathfrak q_1 and (b) (A_1)_{\mathfrak q'_1} = (A_1)_{\mathfrak q_1} by Algebra, Lemma 10.41.11, so (c) (M_1)_{\mathfrak q'_1} \cong (M_1)_{\mathfrak q_1}. Hence (M_1)_{\mathfrak q_1} is a flat R-module. Thus we may replace (S, N) by (B_1, M_1) in order to prove the lemma. By Lemma 38.10.1 the map \alpha _1 : B_1^{\oplus r_1} \to M_1 is R-universally injective and \mathop{\mathrm{Coker}}(\alpha _1)_{\mathfrak q} is R-flat. Note that (A_ i, B_ i, M_ i, \alpha _ i, \mathfrak q_ i)_{i = 2, \ldots , n} is a complete dévissage of \mathop{\mathrm{Coker}}(\alpha _1)/B_1/R at \mathfrak q_1. Hence the induction hypothesis implies that \mathop{\mathrm{Coker}}(\alpha _1) is finitely presented as a B_1-module, free as an R-module, and has a decomposition as in the lemma. This implies that M_1 is finitely presented as a B_1-module, see Algebra, Lemma 10.5.3. It further implies that M_1 \cong B_1^{\oplus r_1} \oplus \mathop{\mathrm{Coker}}(\alpha _1) as R-modules, hence a decomposition as in the lemma. Finally, B_1 is projective as an R-module by Lemma 38.9.3 hence free as an R-module by Algebra, Theorem 10.85.4. This finishes the proof. \square


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