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 $KH\_n$".



Gersten for homotopy invariant K-theory of non-singular varieties.
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.


