The Stacks project

Lemma 85.11.1. With notation as above. The morphism $a : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ is flat if and only if $a_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ is flat for $n \geq 0$.

Proof. Since $g_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O})$ is flat, we see that if $a$ is flat, then $a_ n = a \circ g_ n$ is flat as a composition. Conversely, suppose that $a_ n$ is flat for all $n$. We have to check that $\mathcal{O}$ is flat as a sheaf of $a^{-1}\mathcal{O}_\mathcal {D}$-modules. Let $\mathcal{F} \to \mathcal{G}$ be an injective map of $a^{-1}\mathcal{O}_\mathcal {D}$-modules. We have to show that

\[ \mathcal{F} \otimes _{a^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O} \to \mathcal{G} \otimes _{a^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O} \]

is injective. We can check this on $\mathcal{C}_ n$, i.e., after applying $g_ n^{-1}$. Since $g_ n^* = g_ n^{-1}$ because $g_ n^{-1}\mathcal{O} = \mathcal{O}_ n$ we obtain

\[ g_ n^{-1}\mathcal{F} \otimes _{g_ n^{-1}a^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O}_ n \to g_ n^{-1}\mathcal{G} \otimes _{g_ n^{-1}a^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O}_ n \]

which is injective because $g_ n^{-1}a^{-1}\mathcal{O}_\mathcal {D} = a_ n^{-1}\mathcal{O}_\mathcal {D}$ and we assume $a_ n$ was flat. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 85.11: Cohomology and augmentations of ringed simplicial sites

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Your email address will not be published.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 0DH3. Beware of the difference between the letter 'O' and the digit '0'.