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Ivorra defined a tensor triangulated functor from Voevodsky's triangulated category of motives to the derived category of complexes of etale sheaves of $\mathbb{Z}/n$ modules with bounded cohomology sheaves: \begin{eqnarray*} DM_{gm}(k, \mathbb{Z}/n)^{op} &\to& D_c^b(k, \mathbb{Z}/n)\\ M(X) &\mapsto& R\pi_*(\mathbb{Z}/n)_X \end{eqnarray*}

What I wonder is that if this is compatible with the gysin triangle defined by Voevodsky

\begin{eqnarray*} M(X - Z) \to M(X) \to M(Z)(c)[2c] \to M(X - Z)[1] \end{eqnarray*}

meaning if it leads to a commutative diagram of two localization sequences (I am sorry I don't know how to draw a diagram in mathoverflow).

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Yes, it is.:) Gysin triangles (along with orientable cohomology theories) were studied in detail in several papers of Deglise. For example, have a look at section 4 of http://perso.ens-lyon.fr/frederic.deglise/docs/2013/ssp.pdf or apply the (more advanced) methods of http://arxiv.org/abs/1305.5361.

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  • $\begingroup$ Thanks for the answer. But I am still confused. Deglise's papers shows the naturality of the Gysin triangle (Prop. 2.3 in arxiv.org/pdf/0804.2415.pdf). What I want to learn is that if I construct a natural map from Motivic cohomology (or homology) to etale cohomology using above realization, is the two localization sequences of the pair $(X,Z)$, where $Z$ smooth closed subscheme of $X$ of pure codimension, in both motivic (co)homology and etale cohomology fit in a commutative diagram. $\endgroup$
    – Grilo
    Commented Dec 10, 2014 at 14:36
  • $\begingroup$ This is definitely true for the paper your cite. Yet I believe that the statement you want can be extracted from other papers of Deglise. The alternative author is Ayoub (but I have never read much of him). $\endgroup$
    – Mikhail Bondarko
    Commented Dec 10, 2014 at 16:05

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