If X is an algebraic scheme, K\_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and there exists a natural map from the k-th Chow group A\_k(X) to the k-th graded part Gr\_k K\_0(X), just by mapping [V] to [O\_V]. This is example 15.1.5 in Fulton's book.


This even becomes an isomorphism after tensoring with Q! That's Corollary 18.3.2 in Fulton's book.


All of the definitions surely make sense for DM-stacks. We've got Chow-groups and K\_0 can be graded in the same way. I'm not sure whether the natural map above actually passes to rational equivalence, but I'll just assume this to be true. 


Here's the question: Is the map still an isomorphism?



k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks?
Maybe you might have a look at Toën's paper:
"On motives for Deligne-Mumford stacks", IMRN No. 17 (2000), 909-928.
He defines Chow groups of a Deligne-Mumford stack X *with coefficients in the characters* of X, and proves that the corresponding graded ring of Chow groups is isomorphic to K\_0 (see the Remark following def. 3.3 of loc. cit.).


