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The main references for this question are

1 : V.Voevodsky's paper Triangulated categories of motives over a field

2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.

Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $\mathit{Hom}$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.

This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the Nisnevich site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}\mathit{Hom}_{\text{geo. motives}}(M(X),R(i)[n]).$$

My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I am interested in the case where $R=M$ with $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with it? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?

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  • $\begingroup$ You can also define the motivic cohomology of $X$ as a Hom in the category $\mathrm{DM}(X)$ of motives over $X$, as defined by Cisinski-Déglise – see Example 11.2.3 in their book Triangulated categories of mixed motives (Springer, 2019). Once in $\mathrm{DM}(X)$, you can use "motivic sheaves on $X$", for example using relative Chow motives (the category of relative Chow motives exists for reasonable base, see the work of Corti-Hanamura and the article "Borel–Moore motivic homology and weight structure on mixed motives" by Fangzhou Jin). $\endgroup$ Commented Jun 12, 2022 at 17:25
  • $\begingroup$ Thank you François, I think your comment can be answer, but it's up to you. $\endgroup$ Commented Jun 12, 2022 at 18:19
  • $\begingroup$ Hi Marsault, I made my comment into an answer. (To be clear I'm not familiar on the technical details on how to map relative Chow motives into DM, but I'm pretty sure this can be done.) $\endgroup$ Commented Jun 12, 2022 at 18:26

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You can also define the motivic cohomology of $X$ as a $\mathrm{Hom}$ in the category $\mathrm{DM}(X)$ of motives over $X$, as defined by Cisinski-Déglise – see Example 11.2.3 in their book Triangulated categories of mixed motives (Springer, 2019).

Once in $\mathrm{DM}(X)$, you can use "motivic sheaves on $X$", for example using relative Chow motives. The category of relative Chow motives exists for reasonable base, see the work of Corti-Hanamura and the article "Borel–Moore motivic homology and weight structure on mixed motives" by Fangzhou Jin.

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