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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?

edited Feb 9 at 13:02

quid
13.3k●33●54

asked Feb 6 at 13:49

Grilo
105●5

What do you mean by realization? With the little explanation you offer, I feel tempted to answer: the trivial functor. – Fernando Muro Feb 6 at 16:21

Jakob · Answer 1 · 2013-02-20 13:48:20Z UTC

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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.

answered Feb 20 at 13:48

Jakob
314●7

	Why only effective? The target category seems to me to admit Tate twists. – Will Sawin Mar 20 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 at 23:44

Realisation functor

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