Is there a higher Grothendieck ring of varieties?

Question

Fix a field $k$. The Grothendieck ring $K_0(\mathrm{Var}_k)$ of varieties over $k$ is defined as the quotient of the free abelian group on isomorphism classes of algebraic varieties by the scissor relation. The notation suggests that there might be higher $K$-groups $K_i(\mathrm{Var}_k)$ as well, but naive attempt at defining such an object fails as $K_0(\mathrm{Var})$ is not defined as $K_0$ of an exact additive category. Is there a reasonable definition of these groups nonetheless?

Answer 1

Torsten Ekedahl proposed a definition of higher Grothendieck groups of varieties. Unfortunately it seems that he never wrote anything down on this topic before passing away. Torsten had quite a large number of unfinished mathematical manuscripts and projects. I don't know what happened to them, although surely someone at Stockholm University has taken care of them.

I saw him give a talk about higher Grothendieck groups of varieties at Gerard van der Geer's birthday conference on Schiermonnikoog in 2010. The abstract is available online: http://www-irm.mathematik.hu-berlin.de/~ortega/schierm/

“Higher Grothendieck groups of varieties”

We shall (slightly) modify the setup of Waldhausen’s definition of higher K-theory in order to introduce higher Grothendieck groups of varieties such that the 0’th group is the ordinary Grothendieck group of varieties. We then establish some basic properties of these groups that generalise basic properties for the ordinary Grothendieck groups.

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Edit: I found my (sketchy) notes from the 2010 talk!!! Any mistakes in what follows are my own; I know close to nothing about K-theory today and I knew literally nothing in 2010. Anyone who is more knowledgeable than me is very welcome to edit the following:

He begins by recalling the definition of $\newcommand{\Var}{\mathbf{Var}}K_0(\Var_S)$ for a base scheme $S$. There is nothing new here. Then he declares his intention to define higher K-groups.

He makes a list of properties that such higher K-groups should have (I might not have written down all of them):

• There should exist products $K_i(\Var_S) \times K_j(\Var_S) \to K_{i+j}(\Var_S)$ extending the usual product when $i=j=0$.
• For $f \colon S \to T$ there should be $f^\ast \colon K_i(\Var_T) \to K_i(\Var_S)$ resp. $f_\ast \colon K_i(\Var_S) \to K_i(\Var_T)$. When $i=0$ these should be given by fibered product, resp. by composing the structure morphism with $f$.
• Functoriality and projection formula: $(fg)_\ast = f_\ast g_\ast$, $(fg)^\ast = f^\ast g^\ast$, $x\cdot f_\ast y = f_\ast (f^\ast x \cdot y)$.

So far we could just set the higher K-groups to be zero. We want a non-triviality condition.

• Consider the functor $\newcommand{\Finset}{\mathbf{Finset}}\Finset \to \Var_k$, $$ A \mapsto \coprod_A \mathrm{Spec}(k).$$ (DP: for this bullet point the notes change from $\Var_S$ to $\Var_k$. Perhaps at this point it becomes necessary to work over a field.) This should induce $K_i(\Finset) \to K_i(\Var_k)$. Recall that the K-groups of $\Finset$ are the stable homotopy groups of spheres.

• When $k= \mathbf F_q$ there is a functor $\Var_k \to \Finset$, $$ X \mapsto X(k).$$ The composition $K_i(\Finset) \to K_i(\Var_k) \to K_i(\Finset)$ should be the identity.

• When $k = \mathbf C$ there should exist a map $K_i(\Var_\mathbf{C}) \to K_i(\mathbf Z)$ such that the composition $K_i(\Finset) \to K_i(\Var_\mathbf{C}) \to K_i(\mathbf Z)$ is the "standard one" (DP: at this point in my notes I wrote ?!!)

He goes on to discuss generally how to define algebraic K-theory. You want a suitable category, such that the homotopy groups of its nerve are the K-theory groups. He mentions Quillen's Q construction but says that he will follow Waldhausen's approach. Waldhausen's idea is to associate a "simplicial category" $\newcommand{\C}{\mathscr C}s\C$ to a category $\C$. He notes that there is a subtlety here, in that the simplicial identities $d_i d_{j} = d_{j-1}d_i$ etc. need to be strict.

For $\Var_k$ he defines $(s \C)_n$ to be the category with objects $$ \varnothing \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X_n $$ where all injections are closed immersions of $k$-varieties, morphisms are isomorphisms of such diagrams. All the $d_i$ are "what you expect" except for $d_0$, which maps to $$ \varnothing \hookrightarrow X_2 \setminus X_1 \hookrightarrow X_3 \setminus X_1 \hookrightarrow \cdots \hookrightarrow X_n \setminus X_1. $$For each $n$, $N((s\C)_n)$ is a simplicial set. $N(s\C)$ is a bisimplicial set, so more or less a simplicial set. We define $$ K_i(\Var_k) = \pi_{i+1} N(s\Var_k).$$

He goes on to discuss Waldhausen's additivity theorem. Consider the category of pairs $X \hookrightarrow Y$ of closed immersions. There are three functors to $\Var_k$ mapping to $X, Y$ and $Y \setminus X$ respectively. These give three functors $K_i(\Var \hookrightarrow \Var) \to K_i(\Var)$ and the additivity theorem says that two of these sum to the third.

Claim: He can prove this theorem for his definition of K-groups.

He notes that all his constructions mirror those of Waldhausen for topological spaces. The biggest difference is that Waldhausen's uses the existence of a quotient $Y/X$ for $X \hookrightarrow Y$. In particular one needs to give a different proof of the additivity theorem but this is possible. My notes end here.

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Of course the first question one asks is whether this definition agrees with the one due to Inna Zakharevich, that Clark Barwick linked to in a comment.