77.5 Flat finitely presented modules
This is the analogue of More on Flatness, Section 38.12.
Proposition 77.5.1. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent sheaf on X. Let x \in |X| with image y \in |Y|. Assume that
f is locally of finite presentation,
\mathcal{F} is of finite presentation, and
\mathcal{F} is flat at x over Y.
Then there exists a commutative diagram of pointed schemes
\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^ g \ar[d] \\ (Y, y) & (Y', y') \ar[l] }
whose horizontal arrows are étale such that X', Y' are affine and such that \Gamma (X', g^*\mathcal{F}) is a projective \Gamma (Y', \mathcal{O}_{Y'})-module.
Proof.
As formulated this proposition immediately reduces to the case of schemes, which is More on Flatness, Proposition 38.12.4.
\square
Lemma 77.5.2. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent sheaf on X. Let y \in |Y|. Set F = f^{-1}(\{ y\} ) \subset |X|. Assume that
f is of finite presentation,
\mathcal{F} is of finite presentation, and
\mathcal{F} is flat over Y at all x \in F.
Then there exists a commutative diagram of algebraic spaces
\xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ Y & Y' \ar[l]_ h }
such that h and g are étale, there is a point y' \in |Y'| mapping to y, we have F \subset g(|X'|), the algebraic spaces X', Y' are affine, and \Gamma (X', g^*\mathcal{F}) is a projective \Gamma (Y', \mathcal{O}_{Y'})-module.
Proof.
As formulated this lemma immediately reduces to the case of schemes, which is More on Flatness, Lemma 38.12.5.
\square
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