MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $KH_n$".

share|cite|improve this question
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
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.