# Proposition 3.93 of Harris-Morrison (rational classes on Deligne-Mumford moduli stack vs. rational classes on Deligne-Mumford moduli space)

Proposition 3.93 of Harris-Morrison's "Moduli of Curves" describes the relationship between rational divisor classes on the Deligne-Mumford moduli stack and rational divisor classes on the Deligne-Mumford moduli space.

Does this proposition generalize to Chow classes, K-theory classes, cohomology classes?

I assume the answer is "yes" -- is there a reference or an easy proof?

For cohomology, I don't know a reference off the top of my head, but the argument is not difficult, here is a sketch. If $\mathcal X$ is a Deligne-Mumford stack (with finite inertia) and $\pi: \mathcal X \to \mathbf X$ is its moduli space, you need to show that $\mathrm R^i \pi_* \mathbb Q_{\mathcal X}$ (for the classical case) is zero fr $i > 0$, and concides with $\mathbb Q_{\mathbf X}$ for $i = 0$. This is an étale local problem on $\mathbb X$, so we may assume that $\mathcal X$ is of the form $[U/G]$, where $G$ is a finite group acting on a scheme $U$. Then it is a standard fact that the rational cohomology of $[U/G]$, which is the $G$-equivariant cohomology of $U$, coincides with the $G$-invariants in the classical cohomology of $U$, and this makes the result clear. The argument for étale cohomology is similar.
On the other hand, this is most definitely false for K-theory. For example, if $G$ is a finite group acting a point, the K-theory ring of the quotient stack $[\mathrm{pt}/G]$ is the representation ring of $G$, which is in general very far from being trivial. The connection between the K-theory of the stack and the K-theory of the space is complicated, and is given in its general form by Toën's Riemann-Roch.
• @ Kevin: just in case you want some references for the proof of $R^i\pi_*Q_l=0$ for $i\ne 0$ rather than Angelo's direct proof (which is excellent), you may follow the references cited in the proof of 7.3.2 in arxiv.org/PS_cache/arxiv/pdf/1008/1008.3689v1.pdf (this part doesn't use the assumption that X is proper smooth, hence applies in general). – shenghao Feb 15 '11 at 9:11