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 37.43.4) 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 37.43.4. This finishes the proof. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 057U. Beware of the difference between the letter 'O' and the digit '0'.