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Now, in the category of effective geometric motives $\mathbf{DM}^{eff}_{gm}= (K^{b} (Cor_{fin}(k))[\{\text{Mayer Vietoris, W}\}^{-1}])^{\#} \hookrightarrow \mathbf{DM}^{eff}(k) =D^{-} (Sh_{Nis})[W^{-1} … In other words why the LHS and RHS above are equal under the embedding of geometric motives into motives? …

this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question. … First of all, why the word "motives" in "Anderson $T$-motives"? Are $T$-motives analogous to motives of arithmetic schemes? What's its number field analogue? Thanks in advance. …

I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem? …

In Motives for modular forms, Scholl constructs motives associated to modular forms . Deligne's also constructed $\ell$-adic representations attached to modular forms of weight $k \geq 2$. … My question is if Scholl construction is compatible with Deligne's construction in the sense that these motives are the underlying geometric objects of Deligne's constructions. …

I have tried reading some texts about motives, mainly motivic cohomology (Bloch's " Lectures on Algebraic Cycles", Voevodsky's "Motivic Cohomology" etc). However some things confused me. … What is $\mathbb{L}$ sitting inside the triangulated category of mixed motives? (I think it's simply $M(1) = \mathbf{Z}(1)= C_* \mathbf{Z}_{tr} (\mathbb{G}_m) [-1]$, but I'm not sure). …

motives. … (V) Voevodsky motives $DM(S, \Lambda)$ for $S$ Noetherian; (etV) Étale Voevodsky motives $DM_{ét} (S, \Lambda)$ for $S$ Noetherian; (L) Levine motives $\mathcal{DM} (k, \Lambda)$ for $k$ a perfect …