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A variant Same question for $\check{C}$ech cohomology: is it true that $$\check{H}^*(X,\prod \limits_{i \in I} \mathcal F_i)=\prod \limits_{i \in I} \check{H}^*(X,\mathcal F_i) \;? … $$ (Of course, $\check{C}$ech cohomology often coincides with derived functor cohomology but still the question should be considered independently) A prayer Godement's book Topologie algébrique et théorie …

Question: Is the top singular cohomology group $H^n(M,\mathbb Z)$ zero? …

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

Actually a more general result is true: If $X$ is a complex space (maybe not smooth) and if $H^1(X,\mathcal I)$ is zero for all coherent sheaves of ideals $\mathcal I\subset \mathcal O_X$, then $X$ is …

) and other cohomologies. 1) If $X$ is locally contractible then the cohomology of a constant sheaf coincides with singular cohomology. … Then the cohomology of $\F$ is already calculated by the Cech cohomology OF THE COVERING $(U_i)$: no need to pass to the inductive limit on all covers. …

Here is a completely elementary example which shows that group cohomology is not empty verbiage, but can solve a problem ("parametrization of rational circle") whose statement has nothing to do with cohomology … So we have obtained from group cohomology the well-known parametrization for the rational points of the unit circle $x^2+y^2=1$ $$x=\frac{u^2-v^2}{u^2+v^2}, \quad y=\frac{2uv}{u^2+v^2}$$. …

Although we clearly all have more or less the same answers, here is how I like to organize things. I) Let $\mathcal F$ be a sheaf of abelian groups on the topological space $X$. It is said to be sof …