Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F\_{\alpha})\_{\alpha \in I}$ a direct system of $O\_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof of the fact that colimits of sheaves commute with sheaf cohomology, ie that for all $i\ge0$ the canonical morphism of $R$-modules


$$\varinjlim\_i H^{i}(X, F\_{\alpha}) \to H^i(X, \varinjlim\_{\alpha} F\_{\alpha}) $$


induced by $F\_{\alpha} \to \varinjlim\_{\alpha} F\_{\alpha}$, applying the naturality of cohomology functor and the universal property of colimits. That's of course a well known fact used by people involved in research on algebraic geometry literally by reflex but I nowhere found a well explained proof of this isomorphism.



Sheaf cohomology commutes with colimits of sheaves
I think the canonical reference (true for any coherent topos, so for every qcqs scheme, as Remy mentions) is SGA 4 II, Expose VI, Corollaire 5.2.



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