The Stacks project

Lemma 75.23.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite presentation. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module. Let $x \in |X|$ with image $y \in |Y|$. If $\mathcal{F}$ is flat at $x$ over $Y$, then the following are equivalent

  1. $(\mathcal{F}_{\overline{y}})_{\overline{x}}$ is a flat $\mathcal{O}_{X_{\overline{y}}, \overline{x}}$-module,

  2. $(\mathcal{F}_{\overline{y}})_{\overline{x}}$ is a free $\mathcal{O}_{X_{\overline{y}}, \overline{x}}$-module,

  3. $\mathcal{F}_{\overline{y}}$ is finite free in an étale neighbourhood of $\overline{x}$ in $X_{\overline{y}}$, and

  4. $\mathcal{F}$ is finite free in an étale neighbourhood of $x$ in $X$.

Here $\overline{x}$ is a geometric point of $X$ lying over $x$ and $\overline{y} = f \circ \overline{x}$.

Proof. Pick a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are étale such that there is a point $u \in U$ mapping to $x$. Let $v \in V$ be the image of $u$. Applying Lemma 75.23.1 to $\text{id} : X \to X$ over $Y$ we see that (1) translates into the condition “$\mathcal{F}|_{U_ v}$ is flat over $U_ v$ at $u$”. In other words, (1) is equivalent to $(\mathcal{F}|_{U_ v})_ u$ being a flat $\mathcal{O}_{U_ v, u}$-module. By the case of schemes (More on Morphisms, Lemma 37.16.7), we find that this implies that $\mathcal{F}|_ U$ is finite free in an open neighbourhood of $u$. In this way we see that (1) implies (4). The implications (4) $\Rightarrow $ (3) and (2) $\Rightarrow $ (1) are immediate. For the implication (3) $\Rightarrow $ (2) use the description of local rings and stalks in Properties of Spaces, Lemmas 65.22.1 and 65.29.4. $\square$

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