Example 20.20.5. 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 20.20.4 for $d = 0$ but not the conclusion.

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