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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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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. –  David Corwin Sep 20 '12 at 20:36

1 Answer 1

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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