Chern classes of vector bundles on a stack

Question

Is there literature about chern classes of vector bundles on DM-stacks? I had a look at a lot of different papers about intersection theory on stacks and related stuff and this seems to be known, but I couldn't find a good reference on this topic. The strongest statement I could find was "there are chern classes in the Chow-cohomology satisfying all the usual properties".

A particular thing I am interested in, is the projection formula, i.e. given a proper morphism of DM-stacks $p:M\longrightarrow N$, a vectorbundle $E$ on $N$ and a cycle $\alpha\in A_*(M)_\mathbb{Q}$ is it true that $p_*(c_i(p^*E)\cap\alpha)=c_i(E)\cap p_*\alpha$? I guess this is one of the usual properties, but does it hold for any proper morphism, or only representable ones?

What I did was trying to check the definition of the Chow-cohomology from Vitsoli's paper on Intersection theory on stacks for chern classes, but I got already stuck at the compatibility with the Gysin homomorphism, because of my poor knowledge about normal bundles and stacks... so I gave up trying to prove it myself.

Can anyone help me out with a hint to more detailed literature, or some hints how the proof of the projection formula works? I would also be happy with some precise statement under which circumstances the projection formula holds. Thank you in advance.

Answer 1

The formula holds for arbitrary proper morphisms. If $X \to Y$ is a proper morphism of DM stacks, where $X$ has finite inertia (the hypotheses in my paper are more stringent, but the theory has been refined since then), there exists a finite map $V \to X$, where $V$ is an algebraic space, and the pushforward from the rational Chow group of $V$ to that of $X$ is surjective. This allows to reduce the non-representable case to the representable one.

About the more general question, I am reasonably certain that the construction of Chern classes in Fulton's book goes through in the formalism in my old paper without any major problem, but it's been a while since I wrote it, so it's hard to be completely positive.