The Stacks project

Lemma 66.31.1. 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 $x \in |X|$. The following are equivalent

  1. for some commutative diagram

    \[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]

    where $U$ and $V$ are schemes, $a, b$ are ├ętale, and $u \in U$ mapping to $x$ the module $a^*\mathcal{F}$ is flat at $u$ over $V$,

  2. the stalk $\mathcal{F}_{\overline{x}}$ is flat over the ├ętale local ring $\mathcal{O}_{Y, \overline{y}}$ where $\overline{x}$ is any geometric point lying over $x$ and $\overline{y} = f \circ \overline{x}$.

Proof. During this proof we fix a geometric proof $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ over $x$ and we denote $\overline{y} = f \circ \overline{x}$ its image in $Y$. Given a diagram as in (1) we can find a geometric point $\overline{u} : \mathop{\mathrm{Spec}}(k) \to U$ lying over $u$ with $\overline{x} = a \circ \overline{u}$, see Properties of Spaces, Lemma 65.19.4. Set $\overline{v} = h \circ \overline{u}$ with image $v \in V$. We know that

\[ \mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh} \quad \text{and}\quad \mathcal{O}_{Y, \overline{y}} = \mathcal{O}_{V, v}^{sh} \]

see Properties of Spaces, Lemma 65.22.1. We obtain a commutative diagram

\[ \xymatrix{ \mathcal{O}_{U, u} \ar[r] & \mathcal{O}_{X, \overline{x}} \\ \mathcal{O}_{V, v} \ar[u] \ar[r] & \mathcal{O}_{Y, \overline{y}} \ar[u] } \]

of local rings. Finally, we have

\[ \mathcal{F}_{\overline{x}} = (\varphi ^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} \]

by Properties of Spaces, Lemma 65.29.4. Thus Algebra, Lemma 10.39.9 tells us $(\varphi ^*\mathcal{F})_ u$ is flat over $\mathcal{O}_{V, v}$ if and only if $\mathcal{F}_{\overline{x}}$ is flat over $\mathcal{O}_{V, v}$. Hence the result follows from More on Flatness, Lemma 38.2.5. $\square$


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