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There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-geometric conjectures just to formulate the definition of motives.

There are things that I know about motives on some level, e.g. I know what the Grothendieck ring of varieties is or, roughy, what are the ingredients of the definition of motives.

But, how would you explain the Grothendieck's yoga of motives? What is it referring to?

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There are two excellent answers, hard to select a one... –  Ilya Nikokoshev Oct 26 '09 at 22:40
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up vote 39 down vote accepted

So this is a crazy question, but I will try to give at least a partial answer. This question about the Beilinson regulator is also relevant, and this is also an attempt to reply to the comments of Ilya there. I apologize for simplifying and glossing over some details, see the references for the full story.

First of all, some references: A leisurely but still far from content-free exposition by Kahn on the yoga of motives is available here (in French). For Grothendieck's idea of pure motives, see Scholl: Classical motives, available on his webpage in zipped dvi format. For mixed motives, see this survey article of Levine. There is also lots of stuff in the Motives volumes, edited by Jannsen, Kleiman and Serre, here is the Google Books page. Finally, I would strongly recommend the book by André: Introduction aux motifs - this is has lots of background and "yoga", as well as precise statements about what is known and what one conjectures.

Pure motives

The standard way of explaining what motives are is to say that they form a "universal cohomology theory". To make this a bit more precise, let's start with pure motives. We fix a base field, and consider the category of smooth projective varieties, and various cohomology functors on this category. The precise notion of cohomology functor in this context is given by the axioms for a Weil cohomology theory, see this blog post of mine for more details.

There are (at least) three key points to mention here: one is that a Weil cohomologies are "geometric" theories, as opposed to "absolute". For example, when considering etale cohomology, we are considering the functor given by base changing the variety to the absolute closure of the ground field, and then taking sheaf cohomology with respect to the constant sheaf Z/l for some prime l, in the etale topology. The "absolute" theory here would be the same, but without base changing in the beginning. In the classical literature, and in number theory, the geometric version is the most important, partly because it carries an action of the Galois group of the base field, and hence gives rise to Galois representations. On the other hand, the absolute version is important for example in the work of Rost and Voevodsky on the Bloch-Kato conjecture, and in comparison theorems with motivic cohomology. Similarly, it seems like cohomology theories in general come in geometric/absolute pairs.

The second key point to mention is that the Weil cohomology groups come with "extra structure", such as Galois action or Hodge structure. For example, l-adic cohomology takes values in the category of l-adic vector spaces with Galois action, and Betti cohomology takes values in a suitable category of Hodge structures. A nice reference for some of this is Deligne: Hodge I, in the ICM 1970 volume.

The third key point is that Weil cohomology theories are always "ordinary" in some sense, i.e. in some framework of oriented cohomology theories they would correspond to the additive formal group law (see Lurie: Survey on elliptic cohomology). If we allowed more general (oriented) cohomology theories, the universal cohomology would not be pure motives, but algebraic cobordism.

Now all these cohomology theories are functors on the category of smooth projective varieties, and the idea is that they should all factor through the category of pure motives, and that the category of pure motives should be universal with this property. We know how to construct the category of pure motives, but there is a choice involved, namely choosing an equivalence relation on algebraic cycles, see the article by Scholl above for more details. For many purposes, the most natural choice is rational equivalence, and the resulting notion of pure motives is usually called Chow motives. For a precise statement about the universal property of Chow motives, see André, page 36: roughly (omitting some details), any sensible monoidal contravariant functor on the category of smooth projective varieties, with values in a rigid tensor category, factors uniquely over the category of Chow motives.

Now to the point of realizations raised by Ilya in the question about regulators. Given a category of pure motives with a universal property as above, there must be functors from the category of motives to the category of (pure) Hodge structures, to the category of Q_l vector spaces with Galois action, etc, simply because of the universal property. These functors are called realization functors.

Mixed motives

It seems like all the cohomology functors one typically considers can be defined not only for smooth projective varieties, but also for more general varieties. The right notion of cohomology here seems to be axiomatized by some version of the Bloch-Ogus axioms. One could again hope for a category which has a similar universal property as above, but now with respect to all varieties. This category would be the category of mixed motives, and in the standard conjectural framework, one hopes that it should be an abelian category. It is not clear whether this category exists or not, but see Levine's survey above for a discussion of some attempts to construct it, by Nori and others. If we had such a category, a suitable universal property would imply that there are realization functors again, now to "mixed" categories, for example mixed Hodge structures. The realization functors would induce maps on Ext groups, and a suitable such map would be the Beilinson regulator, from some Ext groups in the category of mixed motives (i.e. motivic cohomology groups) to the some Ext groups which can be identified with Deligne-Beilinson cohomology.

We do not have the abelian category of mixed motives, but we have an excellent candidate for its derived category: this is Voevodsky's triangulated categories of motives. They are also presented very well in the survey of Levine. A really nice recent development is the work of Déglise and Cisinski, in which they construct these triangulated categories over very general base schemes (I think Voevodsky's original work was mainly focused on fields, at least he only proved nice properties over fields).

To end by reconnecting to the Beilinson conjectures, there is extremely recent work of Jakob Scholbach (submitted PhD thesis, maybe on the arXiv soon) which seems to indicate that the Beilinson conjectures should really be formulated in the setting of the Déglise-Cisinski category of motives over Z, rather than the classical setting of motives over Q.

The yoga of motives involves far more than what I have mentioned so far, for example things related to periods and special values of L-functions, the standard conjectures, and the idea of motivic (and maybe even "cosmic") Galois groups, but all this could maybe be the topic for another question, some other day :-)

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That's amazing! I'll read the whole thing tomorrow (it's late night here, time to go off) but you're already obviously highly knowledgeable and motivated to write. –  Ilya Nikokoshev Oct 23 '09 at 22:54
    
It's such a lot of things to learn that I'll probably be posting lots of questions on the topic of motives this month. –  Ilya Nikokoshev Oct 26 '09 at 22:36
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In one sentence: the theory of cohomology theories on algebraic varieties and the idea that there is a universal such thing.

Of course, this is not a very satisfying answer, unless we specify what a cohomology theory is. Examples are l-adic cohomologies, singular cohomology, de Rham cohomology, Deligne cohomology, rigid (or Monsky-Washnitzer) cohomology. The idea is that any computation which seems to hold in all these nice cohomology theories should be motivic (which means that it should be obtained from the analogous computation in the (conjectural) category of motives by the suitable realization functor): example of such computations are those which involve only intersection theory (using cup products of cycle classes). Conjecturally, the theory of motives is essentially determined by intersection theory of schemes, while higher Chow groups (i.e. motivic cohomology) should be to motives what Deligne cohomology is to mixed Hodge structures.

1) Pure motives --- Historically, such a cohomology theory was thought as one which behaves like singular cohomology (with rational coefficients) or de Rham cohomology on smooth and projective varieties over complex numbers, so that we would have cycle classes, the Künneth formula, Gysin maps as well as the projective bundle formula (hence Poincaré duality), and, as a consequence, a Lefschetz fixed point formula (i.e. everything needed to do intersection theory there). If the cohomology theory takes its values in the category of (graded) vector spaces over some field of characteristic zero, this leaded to the notion of Weil cohomology (they were named after Weil because of his insights that the existence of such a cohomology for smooth and projective varieties over a field of characteristic p>0 would imply the Weil conjectures, i.e. the good behaviour of the zeta functions associated to the Frobenius action). However, there is not any universal Weil cohomology: over finite fields, the existence of l-adic cohomologies would imply that this universal theory would be with coefficients in the category of Q-vector spaces, and it is known that that there is no Q-linear Weil cohomology for varieties over a field k which contains a non trivial extension of the field with p elements (this follows from a computation of Serre which shows that supersingular elliptic curves over such a k cannot be realized with Q-linear coefficients). (One of) the observations of Grothendieck was that, in practice, Weil cohomology theories takes their value in more complex categories, namely tannakian categories (e.g. Galois representations, mixed Hodge structures), which made him conjecture the existence of a universal cohomology theory with values in a tannakian category. His candidate for this universal tannakian category is the category of pure motives up to numerical equivalence (which is completely determined by intersection theory in classical Chow groups of smooth and projective varieties over a field).

2) Mixed motives --- But this is only a small part of the story (or, if you wish, of the yoga). Cohomology theories, like l-adic cohomology or de Rham cohomology, are not defined only for smooth and projective varieties, and they don't come alone: they come with a whole bunch a derived categories of coefficients (in our examples, l-adic sheaves and D-modules), which have very strong functoriality properties, reflecting dualities and gluing data (expressing decompositions into a closed a subscheme and its open complement) as well as nice descent properties (mainly étale and proper descent). The idea is that any computation or construction which involves only these functorialities (known as the "6 Grothendieck operations") and which holds in all the known examples should be motivic as well (in particular, intersection theory should appear naturally from there; non trivial structures on cohomology groups, like weight filtrations for instance, should also be explained by these functorialities). I mean that there should exists a theory of motivic sheaves which should be the universal system of coefficients M over schemes (not necessarily over a field). Given another system of coefficients A (like l-adic sheaves) we should get tensor exact functors M(X) --> A(X) (for any scheme X) which are compatible with Grothendieck's 6 operations (i.e. pullbacks, direct images with or without compact support, etc). These realization functors are also conjectured to be faithful. All the regulator maps are expected to come from such realization functors. At last, the category of pure motives mentioned above should be a full tensor subcategory of the abelian category of mixed motives over the ground field.

The existence of such motivic sheaves has been conjectured in some way or another by Grothendieck, Deligne, and Beilinson. However, as they noticed themselves, we can weaken these requirements by replacing the categories of coeffcients by their derived categories D(A), and only require that we have triangulated categories of mixed motives over schemes (without asking that they are derived categories of an abelian category). The good news are then that, if we allow these categories of coefficients to be abstract triangulated categories, then such a universal functorial theory of mixed motives over arbitrary schemes is not completely out of reach: the work of Voevodsky, Suslin, Levine, Morel, Ayoub and al. on homotopy theory of schemes makes it already quite close to us: this theory allows to produce triangulated categories DM(X) such that triangulated tensor functors DM(X) --> D(A)(X) actually exist (and are compatible with Grothendieck's 6 operations), while the Hom's in DM compute exactly higher Chow groups (but we don't know if they are conservative, as expected). Hence a significant part of the Yoga is becoming actual mathematics nowdays, via the homotopy theory of schemes.

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I think I'll be posting now more questions to be able to understand such a deep topic. –  Ilya Nikokoshev Oct 25 '09 at 21:43
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I too would recommend to look into André's book very much, and several articles by Deligne, esp. "Hodge I", "Valeurs de fonctions de L et Périods Integrales", "A quoi servent les motifs?". I found Nekovar's slides and Barbieri-Viale's "Pamphlet" usefull too.

Edit: Goncalo Tabuada held a talk on "the construction of the categories of noncommutative motives (pure and mixed) in the spirit of Drinfeld Kontsevich's noncommutative algebraic geometry program. In the process, I will present the first conceptual characterization of Quillen's higher K-theory since Quillen's foundational work in the 70's" (link). Edit: New preprints (1, 2)

Edit: Nori's unpublished notes on motives.

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Most of Tabuada's work is published, and all of it is available on the arXiv. The first paper, about a universal triangulated category computing K-theory of dg categories is: Higher K-theory via universal invariants, Duke Math. J. 145 (2008), 121-206 The analogous corepresentability theorem for non-connective K-theory is proved in: Non-connective K-theory via universal invariants, Compositio Math. 147 (2011), 1281-1320. The paper in which the category of motives à la Kontsevich is described explicitely is Symmetric monoidal structure on Non-commutative motives, arXiv:1001.0228 –  Denis-Charles Cisinski Jul 31 '11 at 19:44
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