Non-zero sheaf cohomology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:39:21Z http://mathoverflow.net/feeds/question/274 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/274/non-zero-sheaf-cohomology Non-zero sheaf cohomology Georges Elencwajg 2009-10-11T12:38:53Z 2009-10-16T17:48:06Z <p>Let R denote the real line with its usual topology. Does there exist a sheaf F of abelian groups on R whose second cohomology group H^2(R,F) is non-zero? What about H^j(R,F) for integers j>=2 ?</p> <p>(Here cohomology means derived functor cohomology as in, say, Hartshorne or EGA. Anyway this cohomology coincides with Cech cohomology since R is paracompact.)</p> http://mathoverflow.net/questions/274/non-zero-sheaf-cohomology/282#282 Answer by Ilya Nikokoshev for Non-zero sheaf cohomology Ilya Nikokoshev 2009-10-11T14:37:28Z 2009-10-12T17:58:35Z <p>Since now we know that <code>R</code> in your question refers to real line equipped with standard topology, sheaf cohomology will always have <code>H^i(F) = 0</code> for <code>i&gt;1</code> &mdash; depending on how you define sheaf cohomology this is a theorem of different difficulty.</p> http://mathoverflow.net/questions/274/non-zero-sheaf-cohomology/702#702 Answer by Sam Derbyshire for Non-zero sheaf cohomology Sam Derbyshire 2009-10-16T02:14:32Z 2009-10-16T02:14:32Z <p>The result you're after is in Hartshorne:</p> <p><strong>Theorem 2.7</strong> (Grothendieck) <br /> Let <em>X</em> be a noetherian topological space of dimension <em>n</em>. <br /> Then H<sup><em>i</em></sup>(<em>X</em>,F) = 0 for all <em>i</em> > <em>n</em> and all sheaves of abelian groups F.</p> http://mathoverflow.net/questions/274/non-zero-sheaf-cohomology/770#770 Answer by AbdÃ³ Roig-Maranges for Non-zero sheaf cohomology AbdÃ³ Roig-Maranges 2009-10-16T17:48:06Z 2009-10-16T17:48:06Z <p>The sheaf cohomology H<sup>i</sup>(X,F) of a (topological) manifold X of dimension n vanishes for i > n. This is a topological version of Grothendieck's vanishing theorem above. You can find this result in Kashiwara-Schapira's "<a href="http://books.google.com/books?id=qfWcUSQRsX4C&amp;lpg=PA475&amp;ots=Dk-Z-JgqhP&amp;dq=kashiwara%20schapira%20sheaves%20on%20manifolds&amp;pg=PP1#v=onepage&amp;q=&amp;f=false" rel="nofollow">Sheaves on manifolds</a>" proposition III.3.2.2.</p>