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Consider the category $\mathsf{Chow}_{\mathbb{Q}}$ of rational pure effective chow $k$-motives. The full subcategory of Artin motives (generated by (X,p,0), X smooth projective zero-dimensional, let $k$ with $char(k)=0$ for simplicity) is equivalent to the category of Galois representations (over $\mathbb{Q}$).

Is it true that the full subcategory of Artin-Tate motives (we add Tate motive $\mathbb{Q}(-1)$ and its powers) is equivalent to the category of graded Galois representations ?

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  • $\begingroup$ Yes; this is true.:) $\endgroup$
    – Mikhail Bondarko
    Commented Jul 30, 2018 at 9:43
  • $\begingroup$ @MikhailBondarko Thank you. Now, I understand that my question was a bit misleading, since in general Tate motive is a bit different notion than Lefschetz motive. May be this subcategory of Chow motives is better to call Artin-Lefschetz motives. But I have a bit different question now: Is it true that K_0(Artin-Lefschetz) embedded in K_0(Chow_Q) ? $\endgroup$
    – Q. Q.
    Commented Aug 22, 2018 at 8:02
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    $\begingroup$ It does, since obvious embedding of $K_0(AT)\to K_0(Mot_{num})$ (the categories are semi-simple!) factors through $K_0(AT)\to K_0(Chow)$. $\endgroup$
    – Mikhail Bondarko
    Commented Aug 22, 2018 at 13:58

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