The Stacks project

Lemma 65.30.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over the point $x \in |X|$. Let $\overline{y} = f \circ \overline{x}$. The following are equivalent

  1. $f$ is flat at $x$, and

  2. the map on ├ętale local rings $\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}$ is flat.

Proof. Choose a 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 $x$. 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 64.19.4. 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 64.22.1. 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 with flat horizontal arrows. We have to show that the left vertical arrow is flat if and only if the right vertical arrow is. Algebra, Lemma 10.39.9 tells us $\mathcal{O}_{U, u}$ is flat over $\mathcal{O}_{V, v}$ if and only if $\mathcal{O}_{X, \overline{x}}$ is flat over $\mathcal{O}_{V, v}$. Hence the result follows from More on Flatness, Lemma 38.2.5. $\square$


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