The Stacks project

Proposition 38.19.2. Let $f : X \to S$ be an affine, finitely presented morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation, flat over $S$. Then the following are equivalent

  1. $f_*\mathcal{F}$ is locally projective on $S$, and

  2. $\mathcal{F}$ is pure relative to $S$.

In particular, given a ring map $A \to B$ of finite presentation and a finitely presented $B$-module $N$ flat over $A$ we have: $N$ is projective as an $A$-module if and only if $\widetilde{N}$ on $\mathop{\mathrm{Spec}}(B)$ is pure relative to $\mathop{\mathrm{Spec}}(A)$.

Proof. The implication (1) $\Rightarrow $ (2) is Lemma 38.17.4. Assume $\mathcal{F}$ is pure relative to $S$. Note that by Lemma 38.18.3 this implies $\mathcal{F}$ remains pure after any base change. By Descent, Lemma 35.7.7 it suffices to prove $f_*\mathcal{F}$ is fpqc locally projective on $S$. Pick $s \in S$. We will prove that the restriction of $f_*\mathcal{F}$ to an étale neighbourhood of $s$ is locally projective. Namely, by Lemma 38.12.5, after replacing $S$ by an affine elementary étale neighbourhood of $s$, we may assume there exists a diagram

\[ \xymatrix{ X \ar[dr] & & X' \ar[ll]^ g \ar[ld] \\ & S & } \]

of schemes affine and of finite presentation over $S$, where $g$ is étale, $X_ s \subset g(X')$, and with $\Gamma (X', g^*\mathcal{F})$ a projective $\Gamma (S, \mathcal{O}_ S)$-module. Note that in this case $g^*\mathcal{F}$ is universally pure over $S$, see Lemma 38.17.4. Hence by Lemma 38.18.2 we see that the open $g(X')$ contains the points of $\text{Ass}_{X/S}(\mathcal{F})$ lying over $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$. Set

\[ E = \{ t \in S \mid \text{Ass}_{X_ t}(\mathcal{F}_ t) \subset g(X') \} . \]

By More on Morphisms, Lemma 37.25.5 $E$ is a constructible subset of $S$. We have seen that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \subset E$. By Morphisms, Lemma 29.22.4 we see that $E$ contains an open neighbourhood of $s$. Hence after replacing $S$ by an affine neighbourhood of $s$ we may assume that $\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')$. By Lemma 38.7.4 this means that

\[ \Gamma (X, \mathcal{F}) \longrightarrow \Gamma (X', g^*\mathcal{F}) \]

is $\Gamma (S, \mathcal{O}_ S)$-universally injective. By Algebra, Lemma 10.89.7 we conclude that $\Gamma (X, \mathcal{F})$ is Mittag-Leffler as an $\Gamma (S, \mathcal{O}_ S)$-module. Since $\Gamma (X, \mathcal{F})$ is countably generated and flat as a $\Gamma (S, \mathcal{O}_ S)$-module, we conclude it is projective by Algebra, Lemma 10.93.1. $\square$

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