Lemma 20.26.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. 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 $\mathcal{O}_ X$-modules. If two out of three of $\mathcal{K}_ i^\bullet $ are K-flat, so is the third.

**Proof.**
By Modules, Lemma 17.17.7 for every complex $\mathcal{L}^\bullet $ we obtain a short exact sequence

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

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

## 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Your email address will not be published.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.

## Comments (0)