Question

Hi there, in recent times I was reading texts about motives, and I want to ask something about Tate motives which is not clear to me (as I came across different definitions in different texts).

Let $V_k$ be the category of (projective) k-varieties. I read that the polynomial ring $\mathbb{Z}[\mathbb{L}] \subset K_0(V_k)$, the latter being the Grothendieck ring of $V_k$, with $\mathbb{L}$ the Lefschetz class $[\mathbb{A}_1]$, "corresponds to" mixed Tate motives generated by the Tate objects $\mathbb{Q}(m)$ in the Grothendieck ring $K_0(M_k)$, $M_k$ being the category of pure $k$-motives. In another article one rather spoke about the subring $\mathbb{Z}[\mathbb{L},\mathbb{L}^{-1}] \subset K_0(M_k)$ ($\mathbb{L}$ now the Lefschetz motive). And in yet another paper I read that mixed Tate motives are defined differently.

My question is: is one of the two first approaches indeed the correct way to see mixed Tate motives ? Or is this a restricted way to define them ?

I am especially interested in the connection between mixed Tate motives and $\mathbb{Z}$-varieties which are polynomial-countable. (When assuming the Tate conjecture, these varieties would have mixed Tate motives, and conversely.)

Thanks !!!

Answer 1

There exists an abelian category of mixed Tate motives M (defined as the heart of some t-structure on the subtriangulated category generated by the $\mathbb{Q}(m)$ in the triangulated category of mixed motives defined by Voevodsky). In this category, every object has a increasing filtration whose quotients are one $\mathbb{Q}(m)$. This implies that at the K-group level, we obtain just the ring $\mathbb{Z}[\mathbb{L}, \mathbb{L}^{-1}]$ which is your second definition. Your first definition is not very coherent : if one take only $\mathbb{Z}[\mathbb{L}]$ then one don't have all the $\mathbb{Q}(m)$. A remark about the restriction of the definition : the existence of the category M (and more important the fact that we know the Ext groups in M) is a much more non-trivial and poweful thing than the description at the K-group level.