Lemma 20.26.11. Let $(X, \mathcal{O}_ X)$ be a ringed space. For any complex $\mathcal{G}^\bullet $ there exists a $K$-flat complex $\mathcal{K}^\bullet $ and a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet $. Moreover, each $\mathcal{K}^ n$ is a flat $\mathcal{O}_ X$-module.

**Proof.**
Choose a diagram as in Lemma 20.26.10. Each complex $\mathcal{K}_ n^\bullet $ is a bounded above complex of flat modules, see Modules, Lemma 17.16.5. Hence $\mathcal{K}_ n^\bullet $ is K-flat by Lemma 20.26.8. The induced map $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism by construction. Thus $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet $ is K-flat by Lemma 20.26.9. Property (3) of Lemma 20.26.10 shows that $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^ m$ is a direct sum of flat modules and hence flat which proves the final assertion.
$\square$

## 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Your email address will not be published.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)