The Stacks project

Lemma 38.3.4. Assumptions and notation as in Lemma 38.3.2. The following are equivalent

  1. $\mathcal{F}$ is flat over $S$ in a neighbourhood of $x$,

  2. $\mathcal{G}$ is flat over $S'$ in a neighbourhood of $z'$, and

  3. $\pi _*\mathcal{G}$ is flat over $S'$ in a neighbourhood of $y'$.

The following are equivalent also

  1. $\mathcal{F}_ x$ is flat over $\mathcal{O}_{S, s}$,

  2. $\mathcal{G}_{z'}$ is flat over $\mathcal{O}_{S', s'}$, and

  3. $(\pi _*\mathcal{G})_{y'}$ is flat over $\mathcal{O}_{S', s'}$.

Proof. To prove the equivalence of (1), (2), and (3) we also consider: (4) $g^*\mathcal{F}$ is flat over $S$ in a neighbourhood of $x'$. We will use Lemma 38.2.3 to equate flatness over $S$ and $S'$ without further mention. The ├ętale morphism $g$ is flat and open, see Morphisms, Lemma 29.36.13. Hence for any open neighbourhood $U' \subset X'$ of $x'$, the image $g(U')$ is an open neighbourhood of $x$ and the map $U' \to g(U')$ is surjective and flat. Thus (4) $\Leftrightarrow $ (1) by Morphisms, Lemma 29.25.13. Note that

\[ \Gamma (X', g^*\mathcal{F}) = \Gamma (Z', \mathcal{G}) = \Gamma (Y', \pi _*\mathcal{G}) \]

Hence the flatness of $g^*\mathcal{F}$, $\mathcal{G}$ and $\pi _*\mathcal{G}$ over $S'$ are all equivalent (this uses that $X'$, $Z'$, $Y'$, and $S'$ are all affine). Some omitted topological arguments (compare More on Morphisms, Lemma 37.47.4) regarding affine neighbourhoods now show that (4) $\Leftrightarrow $ (2) $\Leftrightarrow $ (3).

To prove the equivalence of (a), (b), (c) consider the commutative diagram of local ring maps

\[ \xymatrix{ \mathcal{O}_{X', x'} \ar[r]_\iota & \mathcal{O}_{Z', z'} & \mathcal{O}_{Y', y'} \ar[l]^\alpha & \mathcal{O}_{S', s'} \ar[l]^\beta \\ \mathcal{O}_{X, x} \ar[u]^\gamma & & & \mathcal{O}_{S, s} \ar[lll]_\varphi \ar[u]_\epsilon } \]

We will use Lemma 38.2.4 to equate flatness over $\mathcal{O}_{S, s}$ and $\mathcal{O}_{S', s'}$ without further mention. The map $\gamma $ is faithfully flat. Hence $\mathcal{F}_ x$ is flat over $\mathcal{O}_{S, s}$ if and only if $g^*\mathcal{F}_{x'}$ is flat over $\mathcal{O}_{S', s'}$, see Algebra, Lemma 10.39.9. As $\mathcal{O}_{S', s'}$-modules the modules $g^*\mathcal{F}_{x'}$, $\mathcal{G}_{z'}$, and $\pi _*\mathcal{G}_{y'}$ are all isomorphic, see More on Morphisms, Lemma 37.47.4. This finishes the proof. $\square$

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