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I will begin by giving a rough sketch of my understanding of motives.

In many expositions about motives (for example, www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined to be a category such that every Weil cohomology (viewed as a functor) factors through it. This does not define the category uniquely, nor does it imply that it exists.

There are two concrete candidates that we can construct. The category of Chow motives, which is well-defined, is trivially a category of motives. However, it has some bad properties. For example, it is not Tannakian. The second candidate is the category of numerical motives. It too is well-defined, however it is only conjectured that it is category of motives (i.e., that every Weil cohomology factors through this category). This conjecture is closely related to (or perhaps even equivalent to?) Grothendieck's standard conjectures. That would be desirable, because the category of numerical motives is very well-behaved.

Furthermore, the original motivation for motives is that Grothendieck has proven that if the category of numerical motives is indeed a category of motives, then the Weil conjectures are correct.

So far, even though I a murky on many of the details, I follow the storyline.

Question

Where does "motivic cohomology" (in the sense of, for example, www.claymath.org/library/monographs/cmim02.pdf) fit into this story?

I know that motivic cohomology has something to do with Milnor K-theory, but that is more or less where my understanding of the context of motivic cohomology ends. If motives are already an abstract object that generalizes cohomology, what does motivic cohomology signify? What is the motivation for defining it? What is the context in which it arose?

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Classically, Grothendieck's motives are only the pure motives, meaning abelian-ish things which capture the (Weil-cohomology-style) $H^i$ of smooth, projective varieties. To see the relationship with motivic cohomology, one should extend the notion of motive so that non-pure (i.e. "mixed") motives are allowed, these mixed motives being abelian-ish things which capture the $H^i$ of arbitrary varieties. The main novelty with mixed motives is that the (conjectural) abelian category of them is not semi-simple -- in fact every mixed motive should be a (generally non-trivial) iterated extension of pure motives, these extensions essentially coming from compactification and resolution of singularities, as in the story of mixed Hodge structures.

Then once one thinks of mixed motives, a natural direction of study (or speculation, as the case may be...) is that of determining all possible extensions (or iterated extensions) between two motives. And that's what motivic cohomology is, essentially: the study of these Ext groups. More formally, every variety $X$ should determine an object $C(X)$ in the bounded derived category of mixed motives, collecting together all the various mixed motives $H^i(X)$, and the $(i,j)^{th}$ motivic cohomology of $X$ is (up to twisting conventions) the abelian group of maps from the unit object to $C(X)$ \ $[i](j)$ (the $j^th$ Tate twist of the $i^th$ shift of $C(X)$) in the derived category of mixed motives.

Now, there are a few points to make here. The first is that, though the above motivation and definition of motivic cohomology rely on an as-yet-conjectural abelian category of mixed motives, one can, independently of any conjectures, define a triangulated category which, as far as anyone can tell, behaves as if it were the bounded derived category of this conjectural abelian category. The most popular such definition, because of its simplicity and relative workability, is Voevodsky's. So the basic theory and many basic results on motivic cohomology are unconditional.

Another thing to say is that, as always, matters with motives are illuminated by considering realization functors. Let me single out the $\ell$-adic etale realization, since its extension from pure to mixed motives is straightforward (unlike for Hodge structures): any mixed motive, just as any pure motive, yields a finite-dimensional $\ell$-adic vector space with a continuous action of the absolute Galois group of our base field. It then "follows" (in our conjectural framework... or actually follows, without quotation marks, in Voevodsky's framework) that the $(i,j)^{th}$ motivic cohomology of X maps to the abelian group of maps from the unit object to $C^{et}(X)$ \ $[i](j)$ in the bounded derived category of $\ell$-adic Galois representations. But this abelian group of maps is just the classical (continuous) $\ell$-adic etale cohomology $H^i(X(j))$ of the variety $X$, making this latter group the natural target of an $\ell$-adic etale "realization" map from motivic cohomology.

So here comes the third point: note that this is the etale cohomology of $X$ itself, not of the base change from $X$ to its algebraic closure. So this etale cohomology group mixes up arithmetic information and geometric information, and the same is true of motivic cohomology in general. (Think especially of the case $X=pt$: the motivic cohomology of a point admits a generally nontrivial realization map to the $\ell$-adic Galois cohomology of the base field.) For example, it is expected (e.g. by Grothendieck -- see http://www.math.jussieu.fr/~leila/grothendieckcircle/motives.pdf for this and more) that for an abelian variety $A$ over an ``arithmetic'' base field $k$, the most interesting part of the motivic cohomology $H^(2,1)(A)$ (again my twists may be off...), by which I mean the direct summand which classifies extensions of $H^1(A)$ by $H^1(G_m)$, should identify with the rationalization of the abelian group of $k$-rational points of the dual abelian variety of $A$, the map being given by associating to such $k$-rational point the mixed motive given as $H^1$ of the total space of the corresponding $G_m$-torsor on $A$. And in this case, the above "realization" map to $\ell$-adic etale cohomology is closely related to the classical Kummer-style map used in the proof of the Mordell-Weil theorem.

So in a nutshell: motivic cohomology is very related to motives, since morally it classifies extensions of motives. But it is of a different nature, since it is an abelian group rather than an object of a more exotic abelian category; and it's also quite different from a human standpoint in that we know how to define it unconditionally. Finally, motivic cohomology realizes to Galois cohomology of a variety itself, rather than to the base change of such a variety to the algebraic closure.

Hope this was helpful in some way.

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It does indeed help! –  Makhalan Duff Jul 22 '12 at 1:26
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I loved the «from a human standpoint» :-) –  Mariano Suárez-Alvarez Jul 22 '12 at 1:57
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