Let $F$ be a Deligne-Mumford stack that is of finite type, smooth and proper over $\mathrm{Spec~}k$ for a perfect field $k$. Consider $K_m$, the presheaf of $m$-th $K$-groups on $F_{et}$, the etale site of $F$:

$K_m : F_{et} \to Ab$

$(U \to F) \mapsto K_m(U)$

$(f : U \to V) \mapsto (f^{*} : K_m(V) \to K_m(U))$

My question is, what are some simple cases when this is already a sheaf? For example, is it a sheaf when $F = BG$ for a finite group $G$?

## Background

My question is aimed at a computation of motives of DM-stacks. The sheaffification $\mathcal{K}_m = K_m^{++}$ is one way to define the Chow groups of $F$:

$A^m(f) := H^m(F_{et}, \mathcal{K}_m \otimes {\bf Q})$

A twist on this definition leads to a well-behaved theory of motives for DM-stacks described by Toen

### Etale site

Someone might be able to confirm that the cohomology can be computed using the etale site whose objects are etale morphisms from affine schemes, since Laumon and Moret-Bailly show it's equivalent (by the inclusion) to the larger site which contains all etale morphisms from algebraic spaces (Champs algebriques, p.102). This might simplify working with the $K$-groups.

globalinvariant exactly because it's not a sheaf. Consider $K_0$ as a warm-up. If it were a sheaf, it would be zero much too often. – Minhyong Kim Dec 12 '10 at 2:56spectraaren't there ? – Zoran Skoda Dec 25 '10 at 20:34