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Trevor Malcolm
science forum beginner
Joined: 15 May 2005
Posts: 4
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 Posted: Mon Feb 07, 2005 11:17 pm Post subject:
Vanishing Sheaf Cohomology
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Hartshorne proves in his book that if X is a noetherian topological
space of dimension n, then H^i(X,F)=0 for any abelian sheaf F on X and
i>n. Is this result true without the noetherian hypothesis? In analogy
with differential geometry and singular/de Rham cohomology I would think
it's true, but who knows.
Thanks
TM |
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Jannick Asmus
science forum Guru
Joined: 25 Mar 2005
Posts: 312
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| Posted: Tue Feb 08, 2005 8:19 am Post subject:
Re: Vanishing Sheaf Cohomology
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On 08.02.2005 01:17, Trevor Malcolm wrote:
| Quote: | Hartshorne proves in his book that if X is a noetherian topological
space of dimension n, then H^i(X,F)=0 for any abelian sheaf F on X and
i>n. Is this result true without the noetherian hypothesis? In analogy
with differential geometry and singular/de Rham cohomology I would think
it's true, but who knows.
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Have a look at the historical note following the proof of the Vanishing
Theorem of Grothendieck in Hartshorne's book. There are standard
references of how this result in algebraic geometry generalizes to
comparing assertions of cohomology theories on complex analytic spaces
and paracompact spaces.
I recommend Grothendieck's famous Tohoku article if you are interested
in the original version of the vanishing theorem.
J. |
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