Is it true that if $X$ is a compact space and $\{F_i\}_{i \in I}$ an inductive system of sheaves of $k$-modules on $X$, then the functor of global sections commutes with colimits?
I mean, I am asking if it is true that in such hypotheses
$$ \varinjlim \Gamma(X; F_i) \simeq \Gamma(X; \varinjlim F_i)$$
I know that filtrant inductive limits are exact, i.e. commute with finite limits and colimits.
Also, I can write $\varinjlim \Gamma(X; F_i) = \varinjlim \Gamma(X; \text{for}(F_i)) $, where "for" stands for the forgetful functor. But I can't still move the colimit inside $\Gamma (X; -)$ (can do only if it was finite). Then I think I should use compactness hypothesis but I am a bit confused because on one hand I need some finiteness condition on the index category $I$, on the other I have finiteness condition on open coverings of the topological space.
How to continue? Thank you very much.
Quasi-compactness of $X$ only gives injectivity of the canonical map $$\varinjlim_i \Gamma(X,F_i) \to \Gamma(X,\varinjlim_i F_i).$$ It is bijective if the transition maps $F_i \to F_j$ are injective. Another sufficient condition for bijectivity is the following: Every open covering of $X$ has a refinement of the form $(U_i)_{i \in I}$ where $I$ is finite and the intersections $U_i \cap U_j$ are quasi-compact. A reference is Stacks Project, Tag 0738. If you do not know what a site is, just replace it there by "topological space". An important example where the assumption applies is that of a quasi-compact quasi-separated scheme.