The following theorem of Serre is well-known:



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> A noetherian scheme $X$ is affine if and only if $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$. (Actually it is enough to have this for $i=1$ and all coherent ideal sheaves.)
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I asked myself whether there is an extension of this theorem to (Artin/Deligne-Mumford) stacks. More precisely:



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> Question: Can one characterize the class of (Artin/Deligne-Mumford) stacks $X$ such that $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$?
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It is certainly not true that affine schemes are here the only examples. For example, take a graded ring $A$ and consider $X = Spec A // \mathbb{G}\_m$ (where the $\mathbb{G}\_m$-action is induced by the grading). The category of quasi-coherent sheaves on $X$ is (by fpqc-descent) equivalent to that of graded $A$-modules and the global sections functor corresponds to taking the zeroth degree of such a graded module. This is clearly exact and thus all higher cohomology groups of all quasi-coherent sheaves vanish. 



Cohomologically trivial stacks
Suppose that $\mathcal X$ is an algebraic stack with finite inertia (for example, a separated Deligne-Mumford stack); then, by a well-known result of Keel and Mori, there exist a moduli space $\pi \colon \mathcal X \to M$. The stack $\mathcal X$ is called *tame* when $\mathrm R^i\pi\_\* F = 0$ for every quasi-coherent sheaf $F$ on $\mathcal X$ and every $i > 0$. From the definition it follows easily that tame stacks with affine moduli spaces have the property you require. In characteristic 0, an algebraic stack with finite diagonal is tame if and only if it is Deligne-Mumford.


There are several different characterizations of tame stacks; see the paper "Tame stacks in positive characteristic" by Dan Abramovich, Martin Olsson and myself. Using the results in that paper, it is not hard to show that a noetherian algebraic stack with finite inertia has the property you want if and only if it is tame with affine moduli space.


[Edit:] here is a proof that if a noetherian algebraic stack $\mathcal X$ with finite inertia has the property you want it is tame with affine moduli space. Let $\mathcal X \to M$ be the moduli space. Let $\mathcal G$ be the residual gerbe over a closed point of $M$; then $\mathcal G$ is closed in $\mathcal X$, so the cohomology of each quasi-coherent sheaf on $\mathcal G$ is trivial. The moduli space of $\mathcal G$ is the spectrum of a field, so $\mathcal G$ is tame. This implies that the automorphism group of an object of $\mathcal G$ is linearly reductive. One of the results in the paper implies that an open neighborhood of $\mathcal G$ in $\mathcal X$ is tame. Since every non-empty closed subset of $M$ contains a closed point of $M$, this implies that these open neighborhoods cover $\mathcal X$, so $\mathcal X$ is tame.


