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The general Tannaka duality in part says that a (quasi-compact, quasi-separated) scheme can be recovered from the tensor category of its quasi-coherent sheaves. Jacob Lurie's paper Tannaka Duality for Geometric Stacks even extends it to certain relative situations involving morphisms of stacks.

This gives rise to the question: if we think of a scheme $X$ as its category of quasi-coherent sheaves $QC_X$, how can we think of the motives associated to $X$ in the same terms? Is there a construction of motives starting with $QC_X$? The reason for asking is that both motives and sheaves are linearizing devices, so the connection between the two might be easier to see.

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    $\begingroup$ What types of motives are you interested in (Chow, André, Nori, Voevodsky, ...)? For Chow, the reasonable thing seems to be to use the isomorphism $\operatorname{CH}^*(X) \otimes \mathbf Q \cong K_0(X) \otimes \mathbf Q$ and work with $K$-theory instead (where $X$ is a smooth projective variety over an algebraically closed field, which is usually where Chow motives takes place). It's possible that a similar picture is available in some of the other cases. $\endgroup$ Nov 15, 2020 at 18:49
  • $\begingroup$ @ R.vanDobbendeBruyn, Voevodsky motives primarily. Thanks for the remark on Chow motives. $\endgroup$
    – Nimas
    Nov 15, 2020 at 18:51
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    $\begingroup$ We may complete R.vanDobbendeBruyn's comment by saying there are theories of non-commutative motives which compare well with Voevodsky's theory of motives: one by Robalo sciencedirect.com/science/article/pii/S0001870814003570 and one by Tabuada sciencedirect.com/science/article/pii/S0001870814002552 However, when we restrict to smooth and proper varieties, this reduces to replacing Chow groups by K-groups indeed. $\endgroup$ Nov 15, 2020 at 19:09
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    $\begingroup$ The fact that $K$-motives determine Chow motives is a non trivial conjecture in D. O. Orlov. Derived categories of coherent sheaves, and motives.Uspekhi Mat. Nauk, 60(6(366)):231–232, 2005 ( see arxiv.org/abs/math/0512620v2 ) which is still open. $\endgroup$ Nov 15, 2020 at 19:09
  • $\begingroup$ @Denis-CharlesCisinski, After reading Tabuada's monograph and some papers on non-commutative motives I realized the implication of your remark and why in fact it contains an answer to my question. $\endgroup$
    – Nimas
    Dec 27, 2020 at 0:26

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