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After reading the answer to my previous question: What are the different theories that the motivic fundamental group attempts to unify?

I decided to read up on Tannakian formalism.

Given the category of numerical motives, and assuming Conjecture C of the standard conjectures (the one regarding the grading of numerical motives), one can construct a category that will be Tannakian. This will be done by changing the sign of the ``canonical'' morphism $h^iX\otimes h^jX \cong h^jX \otimes h^iX$ for $ij$ odd .

It seems in texts about motives, that the end goal was always to achieve a Tannakian category. But what motivation is there for this? Why would a category that has to do with motives be the category of representations of an affine group scheme? This seems crazy to me. Is this immitative of some easier, more well-understood, theory in which it make sense to relate cohomology with representations?

Also, is it conjectured what this mysterious affine group scheme is, in the case of numerical motives with the adjustment written above?

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    $\begingroup$ Dear Qiaochu, I think the suggestion of your last sentence probably underestimates the strength of Grothendieck's mathematical imagination. Regards, Matthew $\endgroup$ – Emerton Jul 31 '11 at 0:27
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    $\begingroup$ Dear James, The absolute Galois group of the ground field acts on the $\ell$-adic cohomology of any variety. But for fields with big absolute Galois groups (like $\mathbb Q$) not every Galois representation appears in $\ell$-adic cohomology of a vareity, while for fields with small absolute Galois groups (e.g. algebraically closed ones) there is nothing to be gained from this view-point. The Tannakian group of the category of motives is an analogue of the absolute Galois group which is precisely constructed so as to act on the cohomology of varieties, and characterize the ... $\endgroup$ – Emerton Jul 31 '11 at 0:54
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    $\begingroup$ Dear James, Not $Gal(\overline{\mathbb Q_{\ell}})/\mathbb Q_{\ell})$, but $G_k:= Gal(\overline{k}/k)$, if $k$ is the ground field. A standard aspect of $\ell$-adic cohomology of varieties over $k$ is that $G_k$ acts on it. If you're not familiar with this, it would make sense to learn it before pursuing motives further. In any event, there will be a tautological map from $G_k$ to the motivic Tannakian group (to be precise, the motivic Tannakian group computed with respect to $\ell$-adic cohomology as the fibre functor); the existence of this map is equivalent to the statement that $G_k$ ... $\endgroup$ – Emerton Jul 31 '11 at 1:27
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    $\begingroup$ I'm not sure what exactly the original question is asking, but it might be helpful to think of Tannakian categories in terms of their definition, as certain linear tensor categories, rather than as representation categories of pro-affine group schemes. Then it is pretty reasonable to think of a Tannakian category as a natural axiomitization of what it means to do linear algebra. Then the category of motives is just the universal example of a (certain kind of) linearization of the category of varieties. (Continued) $\endgroup$ – JBorger Jul 31 '11 at 3:54
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    $\begingroup$ The fact that once you choose a fiber functor, you can identify your category with a category of representations is (i) secondary and (ii) an essentially category-theoretic result. So, if you want to understand why affine group schemes come up in the study of Tannakian categories, that's a general principle in category theory (Beck's theorem). Or do you want to know why one looks at Tannakian categories in the study of cohomology theories in algebraic geometry, as opposed to some other axiomitization? $\endgroup$ – JBorger Jul 31 '11 at 4:01
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Actually, the category of motives isn't equivalent to the category of representations of an affine group scheme except in characteristic zero, and even there the equivalence depends on the choice of a fibre functor.

The functor to motives is supposed to be a universal cohomology theory. Certainly, one would like the target of a cohomology theory to be at least tannakian.

If you assume the Hodge conjecture, then the affine group scheme attached to the category of abelian motives over $\mathbb{C}$ (that generated by abelian varieties) is more-or-less known --- at least its algebraic quotients are classified.

If you assume the Tate conjecture, then the affine groupoid attached to the category of motives over an algebraic closure of $\mathbb{F}_p$ is more-or-less known.

In the general case nothing is known except that the group is VERY BIG --- for example, over $\mathbb{C}$ it has uncountably many distinct quotients isomorphic to PGL(2).

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  • $\begingroup$ I guess certainly one would like the target category to be Tannakian, but I don't understand why one would expect it. $\endgroup$ – James D. Taylor Jul 30 '11 at 23:37
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    $\begingroup$ Dear anon, I hope you won't mind if I make the last paragraph of your answer slightly less cryptic, by pointing out that every non-CM elliptic curve over $\mathbb C$ gives one of these PGL(2)s. Regards, $\endgroup$ – Emerton Jul 31 '11 at 0:22
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    $\begingroup$ The concept of a motive emerged slowly in Grothendieck's mind in the 1960's. Later he had a student Saavedra develop the theory of Tannakian categories for his thesis. Although there are many naturally occurring examples of Tannakian categories in algebraic geometry, I think for Grothendieck the principal (conjectural) example was motives. To some extent Tannakian categories were developed to give the correct formalism for motives. The idea that the category of motives should be describable by some monster affine group scheme (or groupoid scheme) was probably an early part of G's thinking. $\endgroup$ – anon Jul 31 '11 at 0:39
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    $\begingroup$ Dear anon, I think that your last comment is correct. Grothendieck was also surely motivated by the Galois action on $\ell$-adic cohomology. (Consider Serre's very brief description of motives in his $\ell$-adic representations book as "$\ell$-adic cohomology without the $\ell$".) This leads to the description of the motivic Tannakian group as the "motivic Galois group". Regards, Matthew $\endgroup$ – Emerton Jul 31 '11 at 0:43
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My impression is that the language of Tannakian categories is a nice thing for itself; thus applying it to motives may (and will, see below) yield interesting applications. There are several ways of dealing with Tannakian categories, and I am not sure that dealing with explicit affine group schemes appearing this way is the "main" one (since this group scheme is awfully huge and complicated for motives, and you have to "make the category neutral" to ensure its existence). Also, Tannakian categories give a possibility of dealing with the category of motives "abstractly".

Probably the nicest results on the relation of motives to Tannakian formalism are in the case where the base field is either finite or (at least) algebraic over a finite field; you can find them in Milne's http://www.jmilne.org/math/articles/1994aP.pdf. Firstly, the category of numerical motives is known to be Tannakian in this case; see Proposition 1.1. To say more about it one needs certain "standard" conjectures. In particular, the Tate conjecture allows to describe the category of motives over a finite field almost completely in Corollary 1.16, Proposition 1.17 (these two statements are improved in Propositions 3.7 and 3.8), Proposition 2.6, and Proposition 2.22. Moroever, Theorem 3.13 (cf. also Theorem 3.19) gives a complete description of motives over the algebraic closure $\mathbb{F}$ of a finite field. 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 Tannakian categories (since no "geometric" description of this functor is given).

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