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Let $k$ be a field and $DM_k$ denote the triangulated category of geometric motives with $ \mathbb{Q}$ coeffients over $k$. Recall that there exists a motive functor $M: Var_k\rightarrow DM_k$, which yields an fully faithful embedding of tensor $ \mathbb{Q}$-categories $CHM_k\rightarrow DM_k$, where $CHM_k$ is athe category of Chow motives with $\mathbb{Q}$-coefficients over $k$.

For $\ell$ prime to the characteristic of $k$ one has the $\ell$-adic realisation functor $r_{\ell}:DM_k\rightarrow D^b(Vec_{\mathbb{Q}\ell})$. As I understand it, if $X$ is smooth projective variety over $k$ then the cohomology of $r_{\ell}(M(X))$ is just the $\ell$-adic cohomology of $X$.

Now the conjectural motivic $t$-structure on $DM_k$ has the property that the realisation functor $r_{\ell}$ is $t$-exact.Thus those objects in the heart of this $t$-structure (the conjectural category of mixed motives $MM_k$) have realisation with trivial cohomology outside of degree zero.

Here is what is confusing me: for a general smooth projective variety its $\ell$-adic cohomology is not always concentrated in degree zero. Thus for such $X$, the motive $M(X)$ is not in the category of mixed motives. This can't be correct as the category of mixed motives should contain the category of pure motives. Where am I going wrong?

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  • $\begingroup$ In the second paragraph, I think you should note that the complex in $D^b(Vec_{Q_\ell})$ is one with chain groups equal to the $\ell$-adic cohomology and chain maps zero, so you can just take the components rather than actually taking the cohomology. $\endgroup$
    – David Corwin
    Sep 20, 2012 at 20:36
  • $\begingroup$ To pinpoint the confusion implicit in Marc Hoyois's answer. When taking the derived motive $M(X)\in DM(k,\mathbb Q)$ you have the "natural" degrees for each piece of the motivic complex, but when taking the (full) abelian motive $H(X)\in MM(k,\mathbb Q)$ (i pick notation "$H$" to avoid confusion) you gather the abelian summands cut out of $M(X)$ by the t-structure $\tau$ shifting each of them appropriately. So $H(X)=\sum_i H_i(M(X))=\sum_i H_0^\tau(M(X)[i])=\sum_i\tau_{\leq 0}\tau_{\geq 0}M(X)[i]$. In other words you cram all the abelian summands in "degree 0". $\endgroup$
    – plm
    Jul 9 at 22:26

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The image of the inclusion $CHM_k\hookrightarrow DM_k$ is indeed not contained in the heart $M_k$ of the motivic t-structure.

$CHM_k$ does map to the heart, but via a different functor, namely, the projection $CHM_k\to NM_k$ to numerical motives, followed by the inclusion $NM_k\hookrightarrow M_k$ ($NM_k$ should be the subcategory of semi-simple objects in $M_k$). This functor $CHM_k\to M_k$ should be equivalent to the composition of the inclusion $CHM_k\hookrightarrow DM_k$ followed by the functor $DM_k \to M_k$ which sends $M$ to $\bigoplus_{i\in \mathbb{Z}} H^i(M)$.

I would draw a commutative square here if I knew how, but you can see it in Yves André's book "Une introduction aux motifs", 21.1.5.

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  • $\begingroup$ Dear Mark, would you know why passing from 𝐷𝑀 to 𝑀𝑀 corresponds on the semisimple objects to passing from 𝐶𝐻𝑀 to 𝑁𝑀, that is, to quotienting by the ⊗-ideal generated by numerical equivalences of correspondences. What would be the analog for étale modules over Spec 𝑘 or Hodge structures, and their respective derived categories ? PS: Also, are you sure that the sum of the $H^i(M)$s is direct in general, as you wrote ? $\endgroup$
    – plm
    Jul 9 at 22:27
  • $\begingroup$ So i read further about Chow groups and motives: i found that Beilinson conjectured a spectral sequence between Ext groups of motives (the conjectural abelian motivic summands) and motivic cohomology groups ( $Hom_{DM}(\mathbb 1,M(X)(r)[p+q]$) ), in particular the Chow groups. Thus i think we can conjecturally understand rational equivalence as "numerical equivalences of all orders", and the process of passing from Chow to numerical motives (ie passing to the heart) as forgetting the "higher numerical equivalences" (requiring only "1st-order numerical equivalence"). $\endgroup$
    – plm
    Jul 10 at 22:28

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