The Stacks project

Lemma 20.10.3. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be a covering. Denote $j_{i_0\ldots i_ p} : U_{i_0 \ldots i_ p} \to X$ the open immersion. Consider the chain complex $K(\mathcal{U})_\bullet $ of presheaves of $\mathcal{O}_ X$-modules

\[ \ldots \to \bigoplus _{i_0i_1i_2} (j_{i_0i_1i_2})_{p!}\mathcal{O}_{U_{i_0i_1i_2}} \to \bigoplus _{i_0i_1} (j_{i_0i_1})_{p!}\mathcal{O}_{U_{i_0i_1}} \to \bigoplus _{i_0} (j_{i_0})_{p!}\mathcal{O}_{U_{i_0}} \to 0 \to \ldots \]

where the last nonzero term is placed in degree $0$ and where the map

\[ (j_{i_0\ldots i_{p + 1}})_{p!}\mathcal{O}_{U_{i_0\ldots i_{p + 1}}} \longrightarrow (j_{i_0\ldots \hat i_ j \ldots i_{p + 1}})_{p!} \mathcal{O}_{U_{i_0\ldots \hat i_ j \ldots i_{p + 1}}} \]

is given by $(-1)^ j$ times the canonical map. Then there is an isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K(\mathcal{U})_\bullet , \mathcal{F}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \]

functorial in $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{PMod}(\mathcal{O}_ X))$.

Proof. We saw in the discussion just above the lemma that

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}( (j_{i_0\ldots i_ p})_{p!}\mathcal{O}_{U_{i_0\ldots i_ p}}, \mathcal{F}) = \mathcal{F}(U_{i_0\ldots i_ p}). \]

Hence we see that it is indeed the case that the direct sum

\[ \bigoplus \nolimits _{i_0 \ldots i_ p} (j_{i_0 \ldots i_ p})_{p!}\mathcal{O}_{U_{i_0 \ldots i_ p}} \]

represents the functor

\[ \mathcal{F} \longmapsto \prod \nolimits _{i_0\ldots i_ p} \mathcal{F}(U_{i_0\ldots i_ p}). \]

Hence by Categories, Yoneda Lemma 4.3.5 we see that there is a complex $K(\mathcal{U})_\bullet $ with terms as given. It is a simple matter to see that the maps are as given in the lemma. $\square$

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Your email address will not be published.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 01EL. Beware of the difference between the letter 'O' and the digit '0'.