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.
