Loading web-font TeX/Math/Italic

The Stacks project

Definition 64.18.8. If X is a separated scheme of finite type over an algebraically closed field k and \mathcal{F} = \left\{ \mathcal{F}_ n\right\} _{n\geq 1} is a \mathbf{Z}_\ell -sheaf on X, then we define

H^ i(X, \mathcal{F}) := \mathop{\mathrm{lim}}\nolimits _ n H^ i(X, \mathcal{F}_ n) \quad \text{and}\quad H_ c^ i(X, \mathcal{F}) := \mathop{\mathrm{lim}}\nolimits _ n H_ c^ i(X, \mathcal{F}_ n).

If \mathcal{F} = \mathcal{F}'\otimes \mathbf{Q}_\ell for a \mathbf{Z}_\ell -sheaf \mathcal{F}' then we set

H_ c^ i(X , \mathcal{F}) := H_ c^ i(X, \mathcal{F}')\otimes _{\mathbf{Z}_\ell }\mathbf{Q}_\ell .

We call these the \ell -adic cohomology of X with coefficients \mathcal{F}.


Comments (2)

Comment #75 by Keenan Kidwell on

I couldn't find the definition of for a sheaf on the \'{e}tale site of anywhere else in the chapter. Did I just miss it?

Comment #82 by on

Fixed by adding a definition in Remark 78.2. But of course this needs a lot more work. Thanks.

There are also:

  • 2 comment(s) on Section 64.18: On l-adic sheaves

Your email address will not be published.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.