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I'm trying to solve Exercise III.2.7(a) in Hartshorne's Algebraic Geometry. And what I got is ${ H^1(S^1, \mathbb{Z}) = 0 }$ instead of expected ${ H^1(S^1, \mathbb{Z}) = \mathbb{Z} }$. My reasoning was the following:

  1. $S^1$ is irreducible and $\mathbb{Z}$ is a constant sheaf, so by Exercise II.1.16(a), $\mathbb{Z}$ is flasque.

  2. Applying Proposition III.2.5 we get ${ H^i(S^1, \mathbb{Z}) = 0 }$ for all ${ i > 0 }$. In particular, ${ H^1(S^1, \mathbb{Z}) = 0 }$.

Please help me find the mistake.

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$S^1$ is given its usual topology in that exercise, which means it's not irreducible. For instance, $S^1\cap ([-1,0]\times[-1,1])$ and $S^1\cap ([0,1]\times[-1,1])$ are two proper closed subsets whose union is all of $S^1$.

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