When is the K-theory presheaf a sheaf?

Question

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.

Answer 1

In general, these presheaves are not sheaves, even on the etale sites of fields. As an easy example, $K_2(\mathbb{C})$ is non-torsion divisible, but $K_2(\mathbb{R})$ has a $2$-torsion element given in symbols by $(-1,-1)$ in Milnor K-theory. But, $K_2(\mathbb{R})$, if $K_2$ were a sheaf, would be the $\mathbb{Z}/2$-fixed points of $K_2(\mathbb{C})$. This cannot happen in this example.

Using the fact that $K_{2i}$ of an algebraically closed field is a non-torsion uniquely divisible group, I imagine one can construct counter-examples for any even K-group.

I would imagine that odd K-groups are also not sheaves.

However, for finite fields, the situation might be different, by Quillen's computation. There, it looks as if the K-groups might be sheaves.

For details on $K_2$ and Milnor $K$-theory, look up Matsumoto's Theorem. For other K-groups of algebraically closed fields, see Suslin's paper On the K-theory of algebraically closed fields.

In general, the place to start thinking about the etale site and algebraic K-theory would be Thomason's paper Algebraic K-theory and etale cohomology.