The Stacks project

Lemma 38.10.1. Let $(R, \mathfrak m)$ be a local ring. Let $R \to S$ be a finitely presented flat ring map with geometrically integral fibres. Write $\mathfrak p = \mathfrak mS$. Let $\mathfrak q \subset S$ be a prime ideal lying over $\mathfrak m$. Let $N$ be a finite $S$-module. There exist $r \geq 0$ and an $S$-module map

\[ \alpha : S^{\oplus r} \longrightarrow N \]

such that $\alpha : \kappa (\mathfrak p)^{\oplus r} \to N \otimes _ S \kappa (\mathfrak p)$ is an isomorphism. For any such $\alpha $ the following are equivalent:

  1. $N_{\mathfrak q}$ is $R$-flat,

  2. $\alpha $ is $R$-universally injective and $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat,

  3. $\alpha $ is injective and $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat,

  4. $\alpha _{\mathfrak p}$ is an isomorphism and $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat, and

  5. $\alpha _{\mathfrak q}$ is injective and $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat.

Proof. To obtain $\alpha $ set $r = \dim _{\kappa (\mathfrak p)} N \otimes _ S \kappa (\mathfrak p)$ and pick $x_1, \ldots , x_ r \in N$ which form a basis of $N \otimes _ S \kappa (\mathfrak p)$. Define $\alpha (s_1, \ldots , s_ r) = \sum s_ i x_ i$. This proves the existence.

Fix an $\alpha $. The most interesting implication is (1) $\Rightarrow $ (2) which we prove first. Assume (1). Because $S/\mathfrak mS$ is a domain with fraction field $\kappa (\mathfrak p)$ we see that $(S/\mathfrak mS)^{\oplus r} \to N_{\mathfrak p}/\mathfrak mN_{\mathfrak p} = N \otimes _ S \kappa (\mathfrak p)$ is injective. Hence by Lemmas 38.7.5 and 38.9.3. the map $S^{\oplus r} \to N_{\mathfrak p}$ is $R$-universally injective. It follows that $S^{\oplus r} \to N$ is $R$-universally injective, see Algebra, Lemma 10.82.10. Then also the localization $\alpha _{\mathfrak q}$ is $R$-universally injective, see Algebra, Lemma 10.82.13. We conclude that $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat by Algebra, Lemma 10.82.7.

The implication (2) $\Rightarrow $ (3) is immediate. If (3) holds, then $\alpha _{\mathfrak p}$ is injective as a localization of an injective module map. By Nakayama's lemma (Algebra, Lemma 10.20.1) $\alpha _{\mathfrak p}$ is surjective too. Hence (3) $\Rightarrow $ (4). If (4) holds, then $\alpha _{\mathfrak p}$ is an isomorphism, so $\alpha $ is injective as $S_{\mathfrak q} \to S_{\mathfrak p}$ is injective. Namely, elements of $S \setminus \mathfrak p$ are nonzerodivisors on $S$ by a combination of Lemmas 38.7.6 and 38.9.3. Hence (4) $\Rightarrow $ (5). Finally, if (5) holds, then $N_{\mathfrak q}$ is $R$-flat as an extension of flat modules, see Algebra, Lemma 10.39.13. Hence (5) $\Rightarrow $ (1) and the proof is finished. $\square$

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