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. Since $X$ is irreducible, any nonempty open $U$ is irreducible and a fortiori connected. Hence for $U \subset X$ nonempty open we have $\underline{A}(U) = A$. We have $\underline{A}(\emptyset ) = 0$. Thus $\underline{A}$ is a flasque abelian sheaf on $X$. The vanishing follows from Lemma 20.12.3.
$\square$

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