The Stacks project

Lemma 60.18.4. Let $\mathcal{C}$ be a category endowed with the chaotic topology. Let $X$ be an object of $\mathcal{C}$ such that every object of $\mathcal{C}$ has a morphism towards $X$. Assume that $\mathcal{C}$ has products of pairs. Then for every abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ the total cohomology $R\Gamma (\mathcal{C}, \mathcal{F})$ is represented by the complex

\[ \mathcal{F}(X) \to \mathcal{F}(X \times X) \to \mathcal{F}(X \times X \times X) \to \ldots \]

associated to the cosimplicial abelian group $[n] \mapsto \mathcal{F}(X^ n)$.

Proof. Note that $H^ q(X^ p, \mathcal{F}) = 0$ for all $q > 0$ as any presheaf is a sheaf on $\mathcal{C}$. The assumption on $X$ is that $h_ X \to *$ is surjective. Using that $H^ q(X, \mathcal{F}) = H^ q(h_ X, \mathcal{F})$ and $H^ q(\mathcal{C}, \mathcal{F}) = H^ q(*, \mathcal{F})$ we see that our statement is a special case of Cohomology on Sites, Lemma 21.13.2. $\square$

Comments (6)

Comment #3625 by shanbei on

second line of this proof:

It should be ""?

Comment #3655 by BB on

The statement of the lemma assumes that C has products, but the proof only uses finite non-empty products. In potential applications, I think one does not want to assume C has a final object (as implied by the existence of products, at least under some conventions). So I think it might be good to replace "has products" with "has finite non-empty products".

Comment #5445 by Hao on

In the proof, "sheaves are presheaves" is somehow confusing. Maybe "any presheaf is a sheaf" is better.

What is the definition of "total complex" in the statement?

Comment #5668 by on

THanks and fixed here. The terminology "total cohomology" is just a way of referring to .

There are also:

  • 4 comment(s) on Section 60.18: General remarks on cohomology

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Your email address will not be published.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 07JM. Beware of the difference between the letter 'O' and the digit '0'.