The Stacks project

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10.82 Descent for finite projective modules

In this section we give an elementary proof of the fact that the property of being a finite projective module descends along faithfully flat ring maps. The proof does not apply when we drop the finiteness condition. However, the method is indicative of the one we shall use to prove descent for the property of being a countably generated projective module—see the comments at the end of this section.

Lemma 10.82.1. Let $M$ be an $R$-module. Then $M$ is finite projective if and only if $M$ is finitely presented and flat.

Proof. This is part of Lemma 10.77.2. However, at this point we can give a more elegant proof of the implication (1) $\Rightarrow $ (2) of that lemma as follows. If $M$ is finitely presented and flat, then take a surjection $R^ n \to M$. By Lemma 10.80.3 applied to $P = M$, the map $R^ n \to M$ admits a section. So $M$ is a direct summand of a free module and hence projective. $\square$

Here are some properties of modules that descend.

Lemma 10.82.2. Let $R \to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. Then

  1. if the $S$-module $M \otimes _ R S$ is of finite type, then $M$ is of finite type,

  2. if the $S$-module $M \otimes _ R S$ is of finite presentation, then $M$ is of finite presentation,

  3. if the $S$-module $M \otimes _ R S$ is flat, then $M$ is flat, and

  4. add more here as needed.

Proof. Assume $M \otimes _ R S$ is of finite type. Let $y_1, \ldots , y_ m$ be generators of $M \otimes _ R S$ over $S$. Write $y_ j = \sum x_ i \otimes f_ i$ for some $x_1, \ldots , x_ n \in M$. Then we see that the map $\varphi : R^{\oplus n} \to M$ has the property that $\varphi \otimes \text{id}_ S : S^{\oplus n} \to M \otimes _ R S$ is surjective. Since $R \to S$ is faithfully flat we see that $\varphi $ is surjective, and $M$ is finitely generated.

Assume $M \otimes _ R S$ is of finite presentation. By (1) we see that $M$ is of finite type. Choose a surjection $R^{\oplus n} \to M$ and denote $K$ the kernel. As $R \to S$ is flat we see that $K \otimes _ R S$ is the kernel of the base change $S^{\oplus n} \to M \otimes _ R S$. As $M \otimes _ R S$ is of finite presentation we conclude that $K \otimes _ R S$ is of finite type. Hence by (1) we see that $K$ is of finite type and hence $M$ is of finite presentation.

Part (3) is Lemma 10.38.8. $\square$

Proposition 10.82.3. Let $R \to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. If the $S$-module $M \otimes _ R S$ is finite projective, then $M$ is finite projective.

The next few sections are about removing the finiteness assumption by using dévissage to reduce to the countably generated case. In the countably generated case, the strategy is to find a characterization of countably generated projective modules analogous to Lemma 10.82.1, and then to prove directly that this characterization descends. We do this by introducing the notion of a Mittag-Leffler module and proving that if a module $M$ is countably generated, then it is projective if and only if it is flat and Mittag-Leffler (Theorem 10.92.3). When $M$ is finitely generated, this statement reduces to Lemma 10.82.1 (since, according to Example 10.90.1 (1), a finitely generated module is Mittag-Leffler if and only if it is finitely presented).

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