Let $X$ be a finite CW complex. Is there an integer $N,$ such that $H^i(X,F)=0$ for all $i>N$ and all abelian sheaves $F$ on $X?$ The cohomology is defined to be the derived functor of the global section.
You can take $N=\dim X$, according to Proposition 3.1.5 in Dimca, [Alexandru. Sheaves in topology. Universitext. Springer-Verlag, Berlin, 2004. xvi+236 pp. MR2050072]