# When is the K-theory presheaf a sheaf?

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.

-
In some sense, $K$-theory is a global invariant 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 2:56
Right, but there are some descent properties for presheaves of K-theory spectra aren't there ? – Zoran Skoda Dec 25 '10 at 20:34
Zoran: Yes, Zariski (or better, Nisnevich) but not etale in general. – Dustin Clausen Dec 25 '10 at 21:17

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.

-
Thanks for resolving the question in the case of Milnor's K-theory. Unfortunately, my question was about the K-theory defined for schemes by the $Q$ construction. I think it agrees for $K_0$ and $K_1$, but not for the higher K-groups. – expz Jan 25 '11 at 16:31
For a field, it agrees for K_0, K_1, and K_2. – Benjamin Antieau Jan 31 '11 at 5:25