actions of the absolute Galois group and the motivic Galois group on étale cohomology

Question

Let $K$ be a field of characteristic $0$; let $\ell$ be any prime; and let $\mathrm{Mot}(K, \mathbb{Q}_{\ell})$ be a Tannakian category of motives over $K$ with coefficients in $\mathbb{Q}_{\ell}$. So, we may assume some conjectures which imply that our category of motives (defined with respect to any equivalence relation on cycles) is Tannakian, or we may apply the construction of Ives André to get the category of "motivated" motives, which is known to be Tannakian. We may choose the étale cohomology realization functor taking $\mathrm{Mot}(F, \mathbb{Q}_{\ell})$ to the category of graded $\mathbb{Q}_{\ell}$-vector spaces as our fiber functor. Then we have the motivic Galois group $G_{\mathrm{mot}, F}$, which is an algebraic group over $\mathbb{Q}_{\ell}$ yielding an equivalence between $\mathrm{Mot}(K, \mathbb{Q}_{\ell})$ and the category of graded $G_{\mathrm{mot}, K}$-representations. In particular, for any smooth projective variety $X$ over $K$, the group $G_{\mathrm{mot}, K}$ acts on each piece $H_{\mathrm{ét}}^i(X_{\bar{K}}, \mathbb{Q}_{\ell})$ of the étale cohomology.

Meanwhile, the absolute Galois group $\Gamma_K$ acts on each $H_{\mathrm{ét}}^i(X_{\bar{K}}, \mathbb{Q}_{\ell})$ as well. Now $G_{\mathrm{mot}, K}$ and $\Gamma_K$ are related by a quotient map induced by the embedding of the category of Artin motives into $\mathrm{Mot}(K, \mathbb{Q}_{\ell})$. I believe this quotient map admits a section $\Gamma_K \hookrightarrow G_{\mathrm{mot}, K}$ induced by the functor from $\mathrm{Mot}(K, \mathbb{Q}_{\ell})$ to Artin motives given by applying $\pi_0$ (or the $0$th cohomology).

My question is, how are the two actions on étale cohomology compatible? I believe that the action of the absolute Galois group factors as the composition of the section $\Gamma_K \hookrightarrow G_{\mathrm{mot}, K}$ with the action of the motivic Galois group, but how do I prove it? Is it explicitly proven somewhere in the literature? Is there some aspect of my construction of the question which is fuzzy? I would appreciate any imput.

Answer 1

The absolute Galois group is a quotient of the motivic Galois group. The $\ell$-adic cohomology defines a section to the quotient map only on $\mathbb{Q}_{\ell}$-points. This section gives the action of the Galois group on the $\ell$-adic cohomology (by definition). Such things are discussed in Saavedra Rivano's book and in Section 6 of the article of Deligne and Milne on Tannakian categories in LNM 900.

Answer 2

Compatibility with $\pi_0$ (if I understand what you mean by that correctly) follows from general nonsense from a commutative triangle between the functor taking an algebraic variety to its motive, $\pi_0$ taking a motive to a representation, and etale cohomology taking a variety to its Galois representation.

Indeed, the definition of a morphism $G_1 \to G_2$ associated to a functor from $G_2$-reps to $G_1$-reps is it is precisely the unique morphism (up to conjugation) where the $G_1$-action on a $G_2$-rep pulled back by this morphism is isomorphic to the $G_1$-action on the functor applied to the $G_2$-rep.

This commutative triangle should be part of the definition of the conjectural theory of motives and its etale realization functor.