It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}\_i$ the canonical map


$$ \bigoplus\_i H^\*(X; \mathcal{F}\_i) \to H^\*(X; \bigoplus\_i \mathcal{F}\_i) $$


is an isomorphism.


My line of interest is whether there are other situations where these kinds of things are true. One example came to my mind:


First observe that for a finite group $G$ and a sequence of $G$-modules $M\_i$ the canonical map


$$ \bigoplus\_i H^\*(G; M\_i) \to H^\*(X; \bigoplus\_i M\_i) $$


is an isomorphism. Indeed, the bar resolution of $\mathbb{Z}$ over $\mathbb{Z}[G]$ is finitely generated in every degree. Thus, homming out ouf this resolution commutes with direct sums, as does homology. 


Now let $X$ be a scheme with an action by a finite group $G$. Dividing by $G$, we get a map $\pi: X \to X//G$. For every quasi-coherent sheaf $\mathcal{F}$, we have a spectral sequence


$$ H^p(G; H^q(X; \pi^\*\mathcal{F})) \Rightarrow H^{p+q}(X//G; \mathcal{F}) $$


(at least if $X//G$ is flat over $\mathbb{Z}$). 


It easily follow that under these conditions (and $X$ noetherian) for every collection of quasi-coherent sheaves $\mathcal{F}\_i$ on $X//G$, we have an isomorphism


$$ \bigoplus\_i H^\*(X//G; \mathcal{F}\_i) \to H^\*(X//G; \bigoplus\_i \mathcal{F}\_i) $$


Having these examples is not totally satisfactory since there might be a more general theorem with a uniform proof. So, my question is:



> 
> What are reasonably general conditions on a Deligne--Mumford stack such that arbitrary direct sums of quasi-coherent sheaves commute with cohomology?
> 
> 
> 



When does sheaf cohomology commute with arbitrary direct sums?
A Grothendieck topology is called *noetherian* if every object is quasi-compact (which is defined as usual). On such a topology, sheaf cohomology commutes with filtered colimits (and in particular with arbitrary direct sums). A proof can be found in Tamme's *Intoduction to Etale cohomology*, Theorem §3.3.11.1. For the etale topology on the spectrum of a field this shows that Galois cohomology commutes with arbitrary direct sums. For the Zariski topology on a scheme we recover the well-known result cited in Hartshorne. An even more general statement can be found in the Stacks project, [Lemma 19.16.2](http://stacks.math.columbia.edu/tag/0739).


