The Stacks project

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. $\square$

Comments (2)

Comment #7837 by Manu on

If for all non-empty , then doesn't that mean 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.

There are also:

  • 2 comment(s) on Section 20.20: Vanishing on Noetherian topological spaces

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