Grothendieck rings and the Tannakian formalism

Question

I understand that the Tannakian formalism (which I only "know" extremely superficially) is very important for the theory of motives. I guess "the" conjectural category of motives should be Tannakian, or a full subcategory of a Tannakian category (I hope what I write makes sense).

Now let $\mathcal{V}_k$ be the category of $k$-varieties with $k$ a field, and consider the Grothendieck ring $K_0(\mathcal{V}_k)$. It is the quotient of the free abelian group generated by the isomorphism classes $[X]$ of $k$-varieties by the relations $[A] = [A \setminus C] + [C]$, where $C$ is a closed $k$-subvariety in the $k$-variety $A$; the fiber product over $k$ then induces a ring structure.

As you may know, some aspects of motives over $k$ can be seen in $K_0(\mathcal{V}_k)$, but many others can not.

My question is: how much does $K_0(\mathcal{V}_k)$ "capture" from the (full subcategory of the) Tannakian category which the category of $k$-motives should be ?

Answer 1

The conjectural category of motives, being an abelian category, has its own notion of $K$-theory, $K_0 ( \operatorname{Mot}_k)$.

This would be the quotient of the free abelian group generated by isomorphism classes of motives by the relations $[B] = [A]+[C]$ when $0 \to A \to B \to C \to 0$ is an exact sequence. (A similar construction works for the non-conjectural triangulated category of motives.)

The ring $K_0 ( \operatorname{Mot}_k)$ contains a lot of the information of the category of motives $\operatorname{Mot}_k$, but not all.

There is a natural map $K_0 (\mathcal V_k) \to K_0 ( \operatorname{Mot}_k)$ that sends the class of a smooth projective variety to its motive.

The fact that this map is not an isomorphism could be interpreted to say that $K_0 (\mathcal V_k)$ loses some of the information in $K_0 ( \operatorname{Mot}_k)$, or alternately that it contains some information about varieties that is missing in $K_0 ( \operatorname{Mot}_k)$. This second perspective is justified by the observation that, despite the seemingly similar definition, it is often more difficult to work with in practice.

I don't know of any reason to believe that there is overlap between the information lost when passing from $\operatorname{Mot}_k$ to $K_0(\operatorname{Mot}_k)$ and the information lost when passing from $K_0(\mathcal V_k)$ to $K_0(\operatorname{Mot}_k)$, but there might be some overlap.

In summary, the Grothendieck group of varieties is removed by two steps from the category of motives, and one of the steps causes a clear loss of information while the other step is more ambiguous and arguably involves a gain of information.