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Community Bot
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I'm not sure what you mean by "coherent sheaf", as that term is usually only used in the presence of something like a complex structure. But by this answerthis answer, the cohomology of any sheaf vanishes in degrees above $n$ on any topological $n$-manifold. Essentially, it can be shown that any open cover of an $n$-manifold admits a refinement for which all $(n+2)$-fold intersections are empty, so the Cech cohomology automatically vanishes above $n$.

I'm not sure what you mean by "coherent sheaf", as that term is usually only used in the presence of something like a complex structure. But by this answer, the cohomology of any sheaf vanishes in degrees above $n$ on any topological $n$-manifold. Essentially, it can be shown that any open cover of an $n$-manifold admits a refinement for which all $(n+2)$-fold intersections are empty, so the Cech cohomology automatically vanishes above $n$.

I'm not sure what you mean by "coherent sheaf", as that term is usually only used in the presence of something like a complex structure. But by this answer, the cohomology of any sheaf vanishes in degrees above $n$ on any topological $n$-manifold. Essentially, it can be shown that any open cover of an $n$-manifold admits a refinement for which all $(n+2)$-fold intersections are empty, so the Cech cohomology automatically vanishes above $n$.

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Eric Wofsey
Eric Wofsey
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I'm not sure what you mean by "coherent sheaf", as that term is usually only used in the presence of something like a complex structure. But by this answer, the cohomology of any sheaf vanishes in degrees above $n$ on any topological $n$-manifold. Essentially, it can be shown that any open cover of an $n$-manifold admits a refinement for which all $(n+2)$-fold intersections are empty, so the Cech cohomology automatically vanishes above $n$.