The Stacks project

27.21 Locally projective modules

A consequence of the work done in the algebra chapter is that it makes sense to define a locally projective module as follows.

Definition 27.21.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. We say $\mathcal{F}$ is locally projective if for every affine open $U \subset X$ the $\mathcal{O}_ X(U)$-module $\mathcal{F}(U)$ is projective.

Lemma 27.21.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent

  1. $\mathcal{F}$ is locally projective, and

  2. there exists an affine open covering $X = \bigcup U_ i$ such that the $\mathcal{O}_ X(U_ i)$-module $\mathcal{F}(U_ i)$ is projective for every $i$.

In particular, if $X = \mathop{\mathrm{Spec}}(A)$ and $\mathcal{F} = \widetilde{M}$ then $\mathcal{F}$ is locally projective if and only if $M$ is a projective $A$-module.

Proof. First, note that if $M$ is a projective $A$-module and $A \to B$ is a ring map, then $M \otimes _ A B$ is a projective $B$-module, see Algebra, Lemma 10.93.1. Hence if $U$ is an affine open such that $\mathcal{F}(U)$ is a projective $\mathcal{O}_ X(U)$-module, then the standard open $D(f)$ is an affine open such that $\mathcal{F}(D(f))$ is a projective $\mathcal{O}_ X(D(f))$-module for all $f \in \mathcal{O}_ X(U)$. Assume (2) holds. Let $U \subset X$ be an arbitrary affine open. We can find an open covering $U = \bigcup _{j = 1, \ldots , m} D(f_ j)$ by finitely many standard opens $D(f_ j)$ such that for each $j$ the open $D(f_ j)$ is a standard open of some $U_ i$, see Schemes, Lemma 25.11.5. Hence, if we set $A = \mathcal{O}_ X(U)$ and if $M$ is an $A$-module such that $\mathcal{F}|_ U$ corresponds to $M$, then we see that $M_{f_ j}$ is a projective $A_{f_ j}$-module. It follows that $A \to B = \prod A_{f_ j}$ is a faithfully flat ring map such that $M \times _ A B$ is a projective $B$-module. Hence $M$ is projective by Algebra, Theorem 10.94.5. $\square$

Lemma 27.21.3. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. If $\mathcal{G}$ is locally projective on $Y$, then $f^*\mathcal{G}$ is locally projective on $X$.

Proof. See Algebra, Lemma 10.93.1. $\square$


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