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?
Best Answer
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 ยง3.3.11.1. 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.