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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?

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2 Answers

up vote 7 down vote accepted

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 § 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.

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This also applies to algebraic stacks, when their cohomology is defined as usual for the etale site (but I'm not sure if this is the correct definition). –  Martin Brandenburg Feb 5 '13 at 9:27
Since etale maps are open, it seems to suffice that every etale map to the given Deligne--Mumford stack has quasi-compact source. This should be implied by the stack being noetherian. Thanks. –  Lennart Meier Feb 6 '13 at 3:38
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I am not sure if you are only really interested in properly stacky things, but it is perhaps worth pointing out that the result you mentioned from Hartshorne is true in significantly greater generality.

For any quasi-compact quasi-separated scheme $X$ (in fact for any spectral space $X$, or for something even slightly weaker) and any filtered system $(\mathcal{F}_\alpha)$ of sheaves of abelian groups on $X$ there is an isomorphism $$\mathrm{colim}\; H^i(X, \mathcal{F}_\alpha) \cong H^i(X, \mathrm{colim}\; \mathcal{F}_\alpha)$$ for all $i \geq 0$.

I have a feeling this is well known (as usual to those who well know it), but at least one place this is written down is in a paper of Kempf, more specifically in Theorem 8.

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Kempf's paper is very useful and clearly written. I would recommend it to anyone interested on cohomology of schemes. –  Leo Alonso Feb 5 '13 at 9:58
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