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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