The Stacks project

Lemma 21.17.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $0 \to \mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \mathcal{K}_3^\bullet \to 0$ be a short exact sequence of complexes such that the terms of $\mathcal{K}_3^\bullet $ are flat $\mathcal{O}$-modules. If two out of three of $\mathcal{K}_ i^\bullet $ are K-flat, so is the third.

Proof. By Modules on Sites, Lemma 18.28.9 for every complex $\mathcal{L}^\bullet $ we obtain a short exact sequence

\[ 0 \to \text{Tot}(\mathcal{L}^\bullet \otimes _\mathcal {O} \mathcal{K}_1^\bullet ) \to \text{Tot}(\mathcal{L}^\bullet \otimes _\mathcal {O} \mathcal{K}_1^\bullet ) \to \text{Tot}(\mathcal{L}^\bullet \otimes _\mathcal {O} \mathcal{K}_1^\bullet ) \to 0 \]

of complexes. Hence the lemma follows from the long exact sequence of cohomology sheaves and the definition of K-flat complexes. $\square$


Comments (1)

Comment #9529 by nkym on

In the proof the last two should be and


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Your email address will not be published.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 0G7B. Beware of the difference between the letter 'O' and the digit '0'.