The Stacks project

Example 20.21.5. Let $X = \mathbf{N}$ endowed with the topology whose opens are $\emptyset $, $X$, and $U_ n = \{ i \mid i \leq n\} $ for $n \geq 1$. An abelian sheaf $\mathcal{F}$ on $X$ is the same as an inverse system of abelian groups $A_ n = \mathcal{F}(U_ n)$ and $\Gamma (X, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits A_ n$. Since the inverse limit functor is not an exact functor on the category of inverse systems, we see that there is an abelian sheaf with nonzero $H^1$. Finally, the reader can check that $H^ p(X, j_!\mathbf{Z}_ U) = 0$, $p \geq 1$ if $j : U = U_ n \to X$ is the inclusion. Thus we see that $X$ is an example of a space satisfying conditions (2), (3), and (4) of Lemma 20.21.4 for $d = 0$ but not the conclusion.


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Your email address will not be published.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 0BX0. Beware of the difference between the letter 'O' and the digit '0'.