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? 



Is there a higher Grothendieck ring of varieties?
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.


