# Sheaf cohomology commutes with colimits of sheaves

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.

• See the last paragraph of section II.4.12 in Godement, Topologie algébrique et théorie des faisceaux.
– A.G
Jun 23 '20 at 17:00
• Or Tag 01FF (which implies the result for quasi-compact quasi-separated schemes). Jun 23 '20 at 21:08

Let $$E'$$ be a coherent topos. For every integer $$q$$, the functor $$H^q(E', -)$$ commutes with filtered inductive limits of abelian sheaves.