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In Hanamura's paper Mixed Motives and Algebraic Cycles III

http://intlpress.com/site/pub/files/_fulltext/journals/mrl/1999/0006/0001/MRL-1999-0006-0001-a005.pdf

He proved that if assume Grothendieck's standard conjecture, Murre's conjecture and vanishing conjecture, there is a $t$ structure on Voevodsky's category of mixed motives $\textbf{DM}_{\text{gm}}(k,\mathbb{Q})$, whose heart is an abelian category $\text{MM}(k)$. Each of Betti, etale or de Rham cohomology (realisation) functor \begin{equation} \mathbb{R}\Gamma:\textbf{DM}_{\text{gm}}(k,\mathbb{Q}) \rightarrow D^b(\Lambda-\text{Vec}) \end{equation} (for suitable $\Lambda$) is a $t$-exact functor with respect to this $t$ structure and the natural $t$ structure on $D^b(\Lambda-\text{Vec})$. I have several questions,

1, Is the vanishing conjecture Hanamura assumed in his paper stronger or equivalent to the Beilinson-Soule vanishing conjecture about field $k$?

2, Is $\textbf{DM}_{\text{gm}}(k,\mathbb{Q})$ the bounded derived category of $\text{MM}(k)$?, i.e. $\textbf{DM}_{\text{gm}}(k,\mathbb{Q})=D^b(\text{MM}(k))$? Has this been proved in Hanamura's paper or is it still only a conjecture (even with several conjectures assumed)?

3, Let the restriction of $\mathbb{R}\Gamma$ to $\text{MM}(k)$ be \begin{equation} \Gamma:\text{MM}(k) \rightarrow \Lambda-\text{Vec} \end{equation} Since $\mathbb{R}\Gamma$ is $t$ exact, then $\Gamma$ is an exact functor, therefore the derived functor of $\Gamma$ is well defined. \begin{equation} D\Gamma:D^b(\text{MM}(k)) \rightarrow D^b(\Lambda-\text{Vec}) \end{equation} If 2 is true, if $D\Gamma$ the same as (or equivalent to $\mathbb{R}\Gamma$)? Any references or comments will be sincerely appreciated!

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  • $\begingroup$ For (2), see mathoverflow.net/questions/240474/… $\endgroup$
    – Jon Pridham
    Jul 3, 2017 at 20:23
  • $\begingroup$ Thank you, the example proves the heart $\text{MM}_{k}$ of the $t$ structure of $\text{DM}_{gm}(k,\mathbb{Q})$ is equivalent to Nori's mixed motives $\text{NMM}(k)_{\mathbb{Q}}$, could you elaborate why $\text{DM}_{gm}(k,\mathbb{Q})$ is equivalent to the bounded derived category $D^b(\text{NMM}(k)_{\mathbb{Q}})$? Sorry for these kind of questions, I am a beginner! $\endgroup$
    – Wenzhe
    Jul 3, 2017 at 21:00

2 Answers 2

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1) I believe that the (Murre's) vanishing needed for Hanamura's argument is stronger than the BS conjecture.

2) There are certain standard conditions ensuring that a triangulated category is equivalent to the derived category of the heart of a t-structure on it. Yet I don't known much about them and they were not treated by Hanamura; thus it is not clear which conjectures are needed to answer your question.

3) This should be easy since the category of vector spaces is semi-simple.

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  • $\begingroup$ Thank you very much. Ayoub's in his paper a guide to (etale) motivic sheaves gives an ambiguous remark 5.8, i.e. conjecture A and conjecture B, in his paper "L'algebre de Hopf et le groupe de Galois motiviques d'un corps de caracteristique nulle, I." should imply the t structure conjecture of $\textbf{DA}^{et}_{ct}(S,\Lambda)$ andmore, "by more, we have in mind the property that $\textbf{DA}^{et}_{ct}(S,\Lambda)$ is equivalent to the derived category of the heart of its motivic t-structure ". Do you know whether this "more" has been proved or not? $\endgroup$
    – Wenzhe
    Jun 30, 2017 at 15:15
  • $\begingroup$ It's easy to believe that certain conjectures on the "motivic Hopf algebra" imply the statement you are interested in. Yet I doubt that anybody has written this down carefully. $\endgroup$
    – Mikhail Bondarko
    Jun 30, 2017 at 19:22
  • $\begingroup$ I would recommend you to contact either Leonid Positselski (positselski.narod.ru, positselski at yandex dot ru) or Sasha Beilinson. $\endgroup$
    – Mikhail Bondarko
    Jul 1, 2017 at 6:05
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As I don't have enough rep to comment, I'll have to make my reply to Wenzhe's comment an answer. The example cited in References - Voevodsky motives are the derived category of Nori motives? doesn't just show the heart of the t-structure would be given by Nori's mixed motives, it shows that Voevodsky motives would also have to be the derived category of the heart (the statement $\mathcal{M}_{dg,\mathbb{A}^1}(F,\mathbb{Q})\simeq \mathcal{D}_{dg}(\mathcal{MM}_F)$ in that example).

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