# Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the litterature quite hard to follow -"from experts, for experts".

Voevodsky in "Triangulated categories of motives over a field" [4.3.8] shows that there is no reasonable t-structure on his category $DM^{eff}_{gm}(k)$ (for $k=\mathbb Q$ or most fields, characterized it seems by a certain cohomological dimension condition which flies above my head) but I do not quite understand the meaning of his proof, in particular how he came up with it, though I more or less follow individual steps. Can anyone help me?

Related to this are Bruno Kahn's remarks in his review article in the "Handbook of K-theory" that the etale version of $DM_{gm,et}^{eff}(k)$, with Homsets isomorphic to $DM^{eff}_{gm}(k)$'s after tensoring by $\mathbb Q$ (using some motivic scissoring magic I presume), should have a t-structure whose heart should be (equivalent to) Nori's category. Kahn says this is related to the Hodge conjecture (not "Hodge-type standard"), can anyone flesh this relation out here?

So I wonder what happens when we pass from $gm$ to $gm,et$. Does anybody know? Is it related to (Serre-type) supersingularity-based counterexamples to the existence of a Weil cohomology theory with $\mathbb Q$ or $\mathbb Q_p$ coefficients?

Also, what has been done to relate the cohomological t-structure on the bounded derived category of Nori mixed motives to $DM_{gm,et}^{eff}(k)$? ($DM_{gm}^{eff}(k)$ and $DM_{gm,et}^{eff}(k)$ have a canonical functor to $D^b(NMM(k))$, which should be an equivalence after tensoring with $\mathbb Q$ -Beilinson's "mixed motives" conjecture.)

Are these questions related to CM lifting results of Chai-Conrad-Oort? Do their constructions explain why there is no t-structure for Nisnevich triangulated motives but there is for etale triangulated motives? Should Voevodsky's example be interpreted in that context?

As you see I am starting to divagate, so I would greatly appreciate some enlightenment.

I would also appreciate any comment on the feeling (probably quite ill-informed) that the t-structure on $DM_{gm,et}^{eff}(k)$ should not be too hard to construct but rather a matter of technical mastery of the algebra/arithmetic involved. And along this line, would the existence of that t-structure yield insight on the Tate and Hodge conjectures, or other conjectures on algebraic cycles? Those seem harder but to confirm (and following Kahn's remark mentioned above): does the Hodge conjecture imply the existence of the motivic t-structure on $DM_{gm,et}^{eff}(k)$?

Finally, I hesitate asking more but... Has anything been tried regarding "bootstrapping" the thorough understanding we have of t-structures on Tate motives to construct t-structures on larger triangulated categories of motives? I think I remember something from Déglise, I have to check... And have those constructions on Tate motives been related to the Nori-Kontsevich tannakian philosophy -e.g. to justify/formalize a hypothetical bootstrapping?

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You asked many hard questions. Concerning one of them, the Hodge conjecture "only" gives an embedding of the category of pure homological motives into Hodge structures. So I don't see how this will help with the $t$-structure, but I could be wrong. –  Donu Arapura Dec 14 '11 at 14:09

I will try to give some answers.

Voevodsky proved that there could be no 'reasonable' motivic $t$-structure for motives with INTEGRAL coefficients (over a non-algebraically closed field); note that there seems to be no nice $t$-structure already for the (very small!) triangulated subcategory of Artin's motives (generated by motives of varieties of dimension $0$). Most people believe that for the 'rational hull' of this category i.e. for motives with coefficients in $\mathbb{Q}$ the motivic $t$-structure does exist, both for the category of effective geometric motives and for the whole category of geometric motives. Now, for motives with rational coefficients there is no difference between etale and Nisnevich versions of Voevodsky's motives. There is a difference only for motives with integral coefficients; in this setting there is a 'reasonable' $t$-structure for etale Artin motives. Serre's example seems to have nothing to do with this question.

The existence of the motivic t-structure is certainly a very hard conjecture (already for motives with rational coefficients). You need a long list of conjectures on algebraic cycles in order to deduce it. You can find such a list here: http://mrlonline.org/mrl/1999-006-001/1999-006-001-005.pdf Note that the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0, but it says nothing about the Murre's conjectures.

Some remarks. 1. Hanamura considers his own category of motives; yet I proved that it is (anti)-equivalent to Voevodsky's motives. 2. There is also the category of Nori's motives. As far as I remember, this is an abelian category. Yet it is certainly a very hard question whether the derived category for it is equivalent to Voevodsky's motives. 3. Another interesting reference is http://arxiv.org/PS_cache/arxiv/pdf/1006/1006.1116v2.pdf

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Other 'easy' subcategories of $DM$ are $1$-motives (here you can look at arxiv.org/abs/1009.1900 and at arxiv.org/PS_cache/arxiv/pdf/1102/1102.0579v2.pdf) and Artin-Tate motives (see math.uiuc.edu/K-theory/0918 and arxiv.org/PS_cache/arxiv/pdf/1006/1006.4343v5.pdf). –  Mikhail Bondarko Dec 15 '11 at 10:00
Mikhail, your answer was quite instructive. If I may add one more thing, a difficulty with Nori's category is that the details had not been written down in an accessible form before. But now there are various accounts in front.math.ucdavis.edu/0801.0261, front.math.ucdavis.edu/1105.0865 and front.math.ucdavis.edu/1111.5146. –  Donu Arapura Dec 15 '11 at 13:49
I will try to learn a little more about all of this, to avoid saying silly things as much as possible. Regarding the equivalence of the derived category of Nori's abelian category, it follows from (and so is equivalent to -I think) Beilinson's original conjectures on mixed motives (his "higher regulators" paper I think). Again, in his handbook survey Kahn says that Nori's mixed motives should be the heart of a t-structure defined integrally on etale DM of Voevodsky -i.e. even before tensoring with $\mathbb Q$. And thanks for pointing Belinson's article. Fortunately I have 0 character left. –  plm Dec 16 '11 at 1:09
I do not think that 'mixed motivic' conjectures say much about Nori's motives. This matter seems to be related with periods; cf. ihes.fr/~maxim/TEXTS/Periods.pdf Note that Beilinson's conjectures only predict that singular cohomology is a fibre functor; I believe that this information is not sufficient to recover Nori's motives. About Kahn's text. I think that this is his own conjecture. Probably, it does not contradict the known facts (in particular, for the 'easy' subcategories of motives mentioned above); yet it cannot follow from 'motivic' conjectures since they are 'rational'. –  Mikhail Bondarko Dec 16 '11 at 4:55