The Stacks project

76.5 Flat finitely presented modules

This is the analogue of More on Flatness, Section 38.12.

Proposition 76.5.1. 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

  1. $f$ is locally of finite presentation,

  2. $\mathcal{F}$ is of finite presentation, and

  3. $\mathcal{F}$ 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 ├ętale 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 immmediately reduces to the case of schemes, which is More on Flatness, Proposition 38.12.4. $\square$

Lemma 76.5.2. 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

  1. $f$ is of finite presentation,

  2. $\mathcal{F}$ is of finite presentation, and

  3. $\mathcal{F}$ 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 ├ętale, 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 immmediately reduces to the case of schemes, which is More on Flatness, Lemma 38.12.5. $\square$


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