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.)



Non-zero sheaf cohomology
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](http://books.google.com/books?id=qfWcUSQRsX4C&lpg=PA475&ots=Dk-Z-JgqhP&dq=kashiwara%20schapira%20sheaves%20on%20manifolds&pg=PP1#v=onepage&q=&f=false)" 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”](http://books.google.com/books?id=qfWcUSQRsX4C&lpg=PA475&ots=Dk-Z-JgqhP&dq=kashiwara%20schapira%20sheaves%20on%20manifolds&pg=PP1#v=onepage&q=&f=false)* by Christian Houzel. (English)
Grundlehren der Mathematischen Wissenschaften, 292. Berlin etc.: Springer-Verlag, pp. x+512 (1990), [MR1074006](https://mathscinet.ams.org/mathscinet-getitem?mr=1074006), [Zbl 0709.18001](https://zbmath.org/0709.18001).


