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
f_*\mathcal{F} is locally projective on S, and
\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
Comments (0)