# Grothendieck rings and the Tannakian formalism

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 ?

• I know you are asking about K_0(Var) and Mot, but just as in Will Sawin's answer I'd like to suggest looking at K_0(Mot). In here you can say very concretely what is lost: K_0 sees no difference between split and non-split extensions, and in particular, every mixed motive agrees with the sum of its weight-graded pieces in this K_0. I think you could be very concrete about this in the case of say, hodge structures, K_0 is somehow only seeing the reductive part of the motivic fundamental group. Jul 1, 2021 at 12:32

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.