The Stacks project

Lemma 21.44.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Given a solid diagram of complexes of $\mathcal{O}_ U$-modules

\[ \xymatrix{ \mathcal{E}^\bullet \ar@{..>}[dr] \ar[r]_\alpha & \mathcal{F}^\bullet \\ & \mathcal{G}^\bullet \ar[u]_ f } \]

with $\mathcal{E}^\bullet $ strictly perfect, $\mathcal{E}^ j = 0$ for $j < a$ and $H^ j(f)$ an isomorphism for $j > a$ and surjective for $j = a$, then there exists a covering $\{ U_ i \to U\} $ and for each $i$ a dotted arrow over $U_ i$ making the diagram commute up to homotopy.

Proof. Our assumptions on $f$ imply the cone $C(f)^\bullet $ has vanishing cohomology sheaves in degrees $\geq a$. Hence Lemma 21.44.6 guarantees there is a covering $\{ U_ i \to U\} $ such that the composition $\mathcal{E}^\bullet \to \mathcal{F}^\bullet \to C(f)^\bullet $ is homotopic to zero over $U_ i$. Since

\[ \mathcal{G}^\bullet \to \mathcal{F}^\bullet \to C(f)^\bullet \to \mathcal{G}^\bullet [1] \]

restricts to a distinguished triangle in $K(\mathcal{O}_{U_ i})$ we see that we can lift $\alpha |_{U_ i}$ up to homotopy to a map $\alpha _ i : \mathcal{E}^\bullet |_{U_ i} \to \mathcal{G}^\bullet |_{U_ i}$ as desired. $\square$


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Your email address will not be published.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 08FQ. Beware of the difference between the letter 'O' and the digit '0'.