Non-zero sheaf cohomology - MathOverflow

Non-zero sheaf cohomology

2009-10-11T12:38:53Z

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 ?

(Here cohomology means derived functor cohomology as in, say, Hartshorne or EGA. Anyway this cohomology coincides with Cech cohomology since R is paracompact.)

Answer by Ilya Nikokoshev for Non-zero sheaf cohomology

2009-10-11T14:37:28Z

Since now we know that R in your question refers to real line equipped with standard topology, sheaf cohomology will always have H^i(F) = 0 for i>1 — depending on how you define sheaf cohomology this is a theorem of different difficulty.

Answer by Sam Derbyshire for Non-zero sheaf cohomology

2009-10-16T02:14:32Z

The result you're after is in Hartshorne:

Theorem 2.7 (Grothendieck)
Let X be a noetherian topological space of dimension n.
Then Hⁱ(X,F) = 0 for all i > n and all sheaves of abelian groups F.

Answer by Abdó Roig-Maranges for Non-zero sheaf cohomology

2009-10-16T17:48:06Z

The sheaf cohomology Hⁱ(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 "Sheaves on manifolds" proposition III.3.2.2.