TITLE: Derived Algebraic Geometry and Chow Rings/Chow Motives
QUESTION [29 upvotes]: I recently heard a talk about Chow motives and also read Milne's exposition on motives. If I understand it correctly, the naive definition of the Chow ring would be that it simply consists of all algebraic cycles, but to define a multiplication one needs to impose a certain equivalence relation, either rational or numerical equivalence. 
I wondered myself if an alternative definition using derived algebraic geometry would be possible. Regardless which framework of derived algebraic geometry you use, a feature should be that you can get always the correct intersection/fiber product. Therefore, one might try to define a derived Chow ring by considering 'derived algebraic cycles' (without any equivalence relation). One would probably get a space out of this instead of a set, but this wouldn't necessarily be a bad thing. Also, the associated category of 'derived Chow motives' would then be a simplicial category (or $(\infty,1)$-category).
What I would like to know is the following: Has somebody tried to build such a theory and if not, what are the problems of it or why is it perhaps a bad idea right from the start?

REPLY [17 votes]: I can't give you a complete answer apart from saying that this is definitely a hard problem!
If you have a derived scheme $X$, you can always truncate to get an every-day scheme $t_0 (X)$. On the level of rings, this corresponds to truncating a simplicial ring $A$ to $\pi_0 (A)$. The canonical morphism $A \to \pi_0(A)$ is obviously 0-connected. A result of Waldhausen then tells you that $A$ and $\pi_0(A)$ have isomorphic $K_0$ and $K_1$. 
Now the Chow-Ring is more or less the same as $K_0$ (after tensoring with Q and up to rational equivalence). So I guess what this means is that on the level of cycles it is impossible to tell the difference between a derived scheme and an ordinary scheme, because $CH(X) = CH(t_0 (X)).$  
I think the fancy way of saying this is that the etale topoi of $X$ and $t_0(X)$ are the same, but that is slightly over my head. This result can be found both in Luries thesis and in HAG II by Toen-Vezzosi.
The situation immediately changes though if you work with Bloch´s higher Chow groups! They see the difference starting from $K_2$, since the higher Chow groups are tied to higher $K$-groups via a Riemann-Roch map. So on the level of these cycles you could tell a difference.