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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.

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    $\begingroup$ See the last paragraph of section II.4.12 in Godement, Topologie algébrique et théorie des faisceaux. $\endgroup$
    – A.G
    Commented Jun 23, 2020 at 17:00
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    $\begingroup$ Or Tag 01FF (which implies the result for quasi-compact quasi-separated schemes). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 23, 2020 at 21:08

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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.

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

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Hartshorne Chapter 3 Proposition 2.9

can be apply to this case

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