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

Let k be a field. Is there a realization functor

$DM_{gm}(k,\mathbb{Z}/n)^{op} \to D^b_c(k, \mathbb{Z}/n)$

from category of motives to category of complexes of étale sheaves of $\mathbb{Z}/n$ modules with bounded constructible cohomology sheaves?

share|cite|improve this question
What do you mean by realization? With the little explanation you offer, I feel tempted to answer: the trivial functor. – Fernando Muro Feb 6 '13 at 16:21

A similar functor, but restricted to the effective part of $DM_{gm}$, is established in Voevodsky's paper on $DM_{gm}$. Actually it is an equivalence (if $n$ is prime to the characteristic of $k$). Also look at Ayoub's paper on etale realization.

share|cite|improve this answer
Why only effective? The target category seems to me to admit Tate twists. – Will Sawin Mar 20 '13 at 15:25
For this, you need to consider the étale version of effective motives with $\mathbb{Z}/n\mathbb{Z}$-coefficients ($n$ prime to $char(k)$). Then the equivalence with the derived category of Galois representations with $\mathbb{Z}/n\mathbb{Z}$-coefficients (which is reformulation of Suslin-Voevodsky's rigidity theorem) imply that the Tate twist is already invertible in the triangulated category of effective étale motives with $\mathbb{Z}/n\mathbb{Z}$-coefficients (so that the adjective 'effective' may be dropped after all). But there is no need to restrict to geometric motives though. – Denis-Charles Cisinski Apr 6 '13 at 23:44

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.