MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.)

share|cite|improve this question
What is R here? – Ben Webster Oct 11 '09 at 13:28
@Ben- My guess: R is a commutative ring and the sheaf is on Spec(R). – Anton Geraschenko Oct 11 '09 at 14:23
Your question would certainly benefit from more info – what is R and what type of sheaf cohomology you are considering? – Ilya Nikokoshev Oct 11 '09 at 14:38
Now I agree with ilya, R is probably the real line. – Anton Geraschenko Oct 11 '09 at 14:43
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. – Georges Elencwajg Oct 11 '09 at 18:01
up vote 10 down vote accepted

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.

share|cite|improve this answer
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! – Georges Elencwajg Oct 16 '09 at 19:01
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 – Nicolas Schmidt May 1 '15 at 19:40

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.