First some indication that it really is a difficult problem: Both Vistoli and Gillet in their classics on intersection theory on stacks remark that their should be a Riemann-Roch theorem for proper representable morphisms, but that they are not able to prove it. I think thats more than enough evidence.

The two existing proofs by Toen and Joshua both involve using not the naive Chow ring, but a modified version. This makes both proofs quite heavy on K-Theory, and I don't really get them.

So what makes the proof using the naive Chow-Ring so difficult? If I remember correctly from reading Fulton-Langs "Riemann-Roch Algebra", the basic technique is to factor a morphism as a regular imedding followed by a projection. The cases of a regular imbeddings and projections are treated by a hands-on methods. Here are some reasons I can think of why this might not work for stacks:

- Its hard to find such a factorization.
- There's a problem with identifying the Chow-Ring of a stack with the K-Group equipped with the gamma-filtration.
- The factorization exists, but the hands-on part is too difficult.
- Maybe the K-Group doesn't have a lamda-ring structure?

alreadyclassics on stacks? $\endgroup$ – Mariano Suárez-Álvarez Oct 5 '10 at 14:50