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?


It does generalize to Chow classes and to (classical or étale) rational cohomology. For Chow classes this is part of the basics of intersection theory of stacks, for example, in my paper Intersection theory on algebraic stacks and on their moduli spaces.

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.

[Edit] I guess I was really answering a different question, on whether there is an isomorphism between Chow groups, and so on, of the stack, and the corresponding groups of the moduli space. The Proposition in Harris-Mumford is actually more refined. The answer is still positive for Chow classes; for cohomology I don't know how to make sense out of it, because I don't know how to interpret the order of the stabilizer of the generic point.

  • $\begingroup$ @ 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). $\endgroup$ – shenghao Feb 15 '11 at 9:11

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