Let $\mathcal{M}\_g$ be the moduli stack of smooth genus $g$ curves. Let $F$ be a coherent sheaf on $\mathcal{M}\_g$. 


Is $H^i(\mathcal{M}\_g,F)$ always finite dimensional? For example, $F=(f\_\*\Omega^1\_{\mathcal{C}\_{g}/\mathcal{M}\_g})^\vee$, where $f\colon\mathcal{C}\_g\to \mathcal{M}\_g$ is the universal curve?


(In the case $F=\mathcal{O}\_{M\_g}$. We know $H^0(M\_g,F)=\mathbb{C}$, see GTM187, "Moduli of curves", p45b)



Is $H^i(\mathcal{M}_g,F)$ necessarily finite dimensional for a coherent sheaf $F$?
Let $X$ be a separated Deligne-Mumford stack of finite type over a field of characteristic 0. Let $\pi\colon X \to M$ be the moduli space; assume that $M$ is quasi-projective. I claim that if $\mathrm{H}^i(X, F)$ is finite-dimensional for all locally free sheaves on $X$, then $X$ is proper, or, equivalently, $M$ is projective.


In fact, the pushforward $\pi\_\*$ is exact on quasi-coherent sheaves and $\pi\_\*\mathcal{O}\_{X} = \mathcal{O}\_{M}$; hence for any locally free sheaf $F$ on $M$ we have $\mathrm{H}^i(M, F) = \mathrm{H}^i(X, \pi^\*F)$; so it is enough to assume that $X = M$ is a quasi-projective variety. I claim that in this case we have that $\mathrm{H}^i(X, F)$ is finite dimensional for any coherent sheaf $F$ on $X$.


I prove this by descending induction on $i$, starting from the fact that $\mathrm{H}^i(X, F) = 0$ if $i > \dim X$. If $F$ is a coherent sheaf, there exists an exact sequence
$$
0 \longrightarrow K \longrightarrow L \longrightarrow F \longrightarrow 0
$$
where $L$ is locally free; hence for all $i$ we have an exact sequence
$$
\mathrm{H}^i(X, L) \longrightarrow \mathrm{H}^i(X, F) \longrightarrow \mathrm{H}^{i+1}(X, K)
$$
which proves what we want.


Now, if $X$ is not projective, it contains a closed affine curve $C$, and then $\mathrm{H}^i(X, \mathcal{O}\_C)$ is infinite-dimensional. This completes the proof.


