Lemma 20.20.2. Let $X$ be an irreducible topological space. Then $H^ p(X, \underline{A}) = 0$ for all $p > 0$ and any abelian group $A$.

**Proof.**
Recall that $\underline{A}$ is the constant sheaf as defined in Sheaves, Definition 6.7.4. It is clear that for any nonempty open $U \subset X$ we have $\underline{A}(U) = A$ as $X$ is irreducible (and hence $U$ is connected). We will show that the higher Čech cohomology groups $\check{H}^ p(\mathcal{U}, \underline{A})$ are zero for any open covering $\mathcal{U} : U = \bigcup _{i\in I} U_ i$ of an open $U \subset X$. Then the lemma will follow from Lemma 20.11.8.

Recall that the value of an abelian sheaf on the empty open set is $0$. Hence we may clearly assume $U_ i \not= \emptyset $ for all $i \in I$. In this case we see that $U_ i \cap U_{i'} \not= \emptyset $ for all $i, i' \in I$. Hence we see that the Čech complex is simply the complex

We have to see this has trivial higher cohomology groups. We can see this for example because this is the Čech complex for the covering of a $1$-point space and Čech cohomology agrees with cohomology on such a space. (You can also directly verify it by writing an explicit homotopy.) $\square$

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