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In the Kontsevich's 'change of variables for motivic integral' rule does one consider any possible choices for the corresponding canonical divisors, or is it necessary to fix certain representatives i …

Let $M$ be a $R$-linear Chow motif over a field $k$ that is perfect but not necessarily algebraically closed. Can one prove that $M$ is not a direct summand of itself (that is, $M\not\cong M\bigoplus …

Yes, it is.:) Gysin triangles (along with orientable cohomology theories) were studied in detail in several papers of Deglise. For example, have a look at section 4 of http://perso.ens-lyon.fr/frederi …

Well, everything follows from the fact that the total direct image $Rf_*\mathbb{Q}$ splits as the direct sum of its (shifted, co)homology sheaves (so, it is "formal"). Now, this statement can be prove …

On the other hand, to obtain an abelian category motives you should "enforce" certain long exact sequences (of symbols of the form $H^i(V)$), and I doubt that any general formalism allows this. …

Now, the latter statement is a consequence of the existence of a (Chow) weight structure on $DMT(k;R)$ whose heart consists on Chow-Tate motives; these are direct sums of $R(i)[2i]=\mathbb L^{\otimes i …

The problem is that is really difficult to define mixed motives (even over an algebraically closed field). … Note that the original idea of Grothendieck is that motives should be "defined in terms of varieties". And this is more or less the case for triangulated motivic categories. …

For any $c>0$ does there exist an object $C$ of Voevodsky's $DM^{eff}_-$ (over the field of complex numbers) such that for any $i\le c$ and any smooth variety $X$ the $i$-th singular cohomology of $X$ …

As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. …

$) exists if $f$ is smooth (thus, it should be finite and separable if you consider motives over fields). …

I am interested in certain properties Chow motives that seem to be (more or less) "classical" (so, I mostly need references; the base field may be assumed to be algebraically closed). … So, consider the full subcategories $C_{\le 1}$ and $C_{\ge 2}$ of effective Chow motives with rational coefficients consisting of objects whose ($\mathbb{Q}_l$-adic for l being a fixed prime distinct …

Any characteristic zero field is an inductive limit of fields that can be embedded into complex numbers (i.e., those of characteristic 0 and of cardinality at most continuum). Hence the assumption t …

Since motives of affine lines are "trivial", the latter morphism yields the isomorphism in question. … If you want to study "ordinary motives", look at Hodge structures, yes. …

(1) As pointed out by Marc Hoyois, there is a natural homomorphism $K_0(Var)\to K_0(DM^{gm})$, where $DM^{gm}$ is the category of geometric Voevodsky motives with coefficients in any (commutative unital … ) ring $R$; you should only assume (at our current level of knowledge on the resolution of singularities) that the base field characteristic $p$ is invertible in $R$ (to put Borel-Moore motives into $DM …

Also, Tannakian categories give a possibility of dealing with the category of motives "abstractly". … Lastly, Theorem 4.22 gives a very funny functor from the so-called CM-motives over the algebraic closure of $\mathbb{Q}$ into motives over $\mathbb{F}$; this result crucially depends on the language of …