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Here is the question:

if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective?

If $X$ is smooth, this is known for the Zariski sheaf associated to $KH_n$. I am wondering if anyone knows off the top of their head how essential smoothness is to the proof.

Edit: I originally mistakenly stated that $KH$ is a presheaf with transfers; its not (see page 105 of "Cohomological theory of presheaves" - Voeovdsky and "Triangulated categories of motives over a field" 3.1.11) however, its Zariski sheafification does have transfers on smooth schemes after work by Deglise.

Edit 2: (after the comment by Cisinski): I originally was missing "for the Zariski sheaf associated to $KH_n$".

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This map is not injective, even if $X$ is regular: it follows from the exactness of the Gersten complex that this map is injective for regular schemes iff $KH_n$ is separated for the Zariski topology (when restricted to regular schemes); it is easy to produce counter-examples to the latter assertion (because vector bundles are locally free...). In the case of singular schemes, we might have $KH_n(X)\neq 0$ for $n<0$ while $KH_n(\kappa(x))=0$ for $n<0$ (because fields are regular), which gives another source of counter-examples. –  Denis-Charles Cisinski Nov 2 '11 at 20:47

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up vote 2 down vote accepted

Your question is an interesting one. But smoothness is essential to Quillen's proof of Gersten's conjecture (in his paper, Higher Algebraic K-theory : I), which, I'm assuming, you intend to use to deduce the consequence you state.

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