In algebraic geometry, we have the following famous base change theorem [Hartshorne III Theorem 12.11]:
Let $f:X\to Y$ be a projective morphism of noetherian schemes, and let $\mathcal F$ be a coherent sheaf on $X$, flat over $X$. Let $y$ be a point of $Y$. Then if the natural map $$\varphi^i(y): R^i f_*(\mathcal F)\otimes k(y)\to H^i(X_y,\mathcal F_y)$$ is surjecitve, then it is an isomorphism.
On the other hand, in the category of topogical spaces, we also have a base change theorem as above [Cohomology of sheaves lversen, Birger Theorem 1.4 P315 ]
Let $f:X\to Y$ denote a continuous proper map between compact spaces. For $y\in Y$ we have a natural isomorphism $$(R^i f_*\mathcal F)_y\cong H^i(f^{-1}(y),\mathcal F); i\in\mathbb Z$$ as $\mathcal F$ varies through the category of sheaves on $X$.
In the theorem of base change of topological spaces, we have a very general result: we do not ask for any conditions for the sheaf $\mathcal F$, but we can always get an isomorphism. However, in the theorem of base change of schemes, even we suppose that the sheaf $\mathcal F$ is flat, we do not have an isomorphism in the general cases. So I am a litte confused. I think that the second theorem is more general, and the first theorem is just a special case of the second one. Why do we just get a weaker result?
PS. In the book of lversen, he suppose that the topological spaces are Hausdorff. But we can consider the category of complex analytic spaces in which we also have a similar theorem as the first one.