# Hodge Realisation of Mixed Tate Motives

For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper,

https://www.uni-due.de/~bm0032/publ/TateMotives.pdf

There exists an abelian category of mixed Tate motives $\text{TM}_k$. If $\sigma:k\rightarrow \mathbb{C}$ is an embedding, we have a Hodge realisation functor, $$R_{\sigma}:\text{TM}_k \rightarrow \text{TH}_{\mathbb{Q}}$$ where $\text{TH}_{\mathbb{Q}}$ is the abelian subcategory of $\mathbb{Q}$-mixed Hodge structures generated by Tate objects $\mathbb{Q}$ and is closed under extensions. From Hodge conjecture and conservative conjecture, $R_{\sigma}$ is exact and fully faithful. But since $\text{TM}_k$ is so nice and simple (compared to the category of pure motives or even the conjectured category of mixed motives ), I am wondering whether it has been proved that $R_{\sigma}$ is exact and fully faithful. Any references?

• Proving exactness should be quite easy. On the other hand, I don't think that $R_{\sigma}$ can be fully faithful unless $\sigma$ is an isomorphism. In the latter case the answer to your question depens on the motivic cohomology of complex numbers (with rational coefficients) and probably not much is known about it; cf. mathoverflow.net/questions/269354/…. – Mikhail Bondarko Jul 9 '17 at 15:54