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. $\square$

Comment #7837 by Manu on

If $\underline{A}(U) = A$ for all non-empty $U$, then doesn't that mean $\underline{A}$ is flabby and hence acyclic? That makes proof much simpler. Or am I missing something?

Comment #8061 by on

A bit sad to remove this argument, but what you say may work better for most people. Thanks! See changes here.

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