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Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero? What about $H^{j}\left(\mathbb{R},F\right)$ for integers $j\ge 2$ ?

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

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  • $\begingroup$ What is R here? $\endgroup$
    – Ben Webster
    Commented Oct 11, 2009 at 13:28
  • $\begingroup$ @Ben- My guess: R is a commutative ring and the sheaf is on Spec(R). $\endgroup$ Commented Oct 11, 2009 at 14:23
  • $\begingroup$ Your question would certainly benefit from more info – what is R and what type of sheaf cohomology you are considering? $\endgroup$ Commented Oct 11, 2009 at 14:38
  • $\begingroup$ Now I agree with ilya, R is probably the real line. $\endgroup$ Commented Oct 11, 2009 at 14:43
  • $\begingroup$ Yes, R is the real line with its usual topology. Cohomology is derived functor cohomology for the functor "Global Sections" (the cohomology used in Hartshorne or EGA, defined via injective resolutions) . It coincides with Cech cohomology since R is paracompact. @ilya n. Could you please explain how the vanishing of cohomology in dimensions 2 and larger is a simple consequence of the definition of Cech cohomology ? Of course I do not assume that my sheaf is constant (in which case even the first cohomology group would vanish). Thanks to all for your interest. $\endgroup$ Commented Oct 11, 2009 at 18:01

2 Answers 2

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The sheaf cohomology ${H}^i(X,F)$ 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.

Reference

Masaki Kashiwara, Pierre Schapira, [Houzel, Christian] Sheaves on manifolds. With a short history “Les débuts de la théorie des faisceaux” by Christian Houzel. (English) Grundlehren der Mathematischen Wissenschaften, 292. Berlin etc.: Springer-Verlag, pp. x+512 (1990), MR1074006, Zbl 0709.18001.

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  • $\begingroup$ Thank you,Abdo', you have given my question a definitive answer. Congratulations on your erudition: I have the pleasant feeling I will read more from you on this site! $\endgroup$ Commented Oct 16, 2009 at 19:01
  • $\begingroup$ This holds true in much more generality actually: One has $H^i(X,F) = 0$ for $i > n$ where $n$ is the Lebesgue covering dimension of $X$, for every paracompact Hausdorff space $X$. For a proof see the book of Godement, II.5.12 $\endgroup$ Commented May 1, 2015 at 19:40
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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.

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