Cohomological definition of the Chow ring Let $X$ be a smooth projective variety over a field $k$. One can define the Chow ring $A^\bullet(X)$ to be the free group generated by irreducible subvarieties, modulo rational equivalence. Multiplication comes from intersection. The problem is, verifying that everything is well-defined is quite a pain. My question is 

Is there a definition of $A^\bullet(X)$ that is purely cohomological?

In other words, is it possible to give a definition of $A^\bullet(X)$ that works for any ringed space? Probably the fact that this definition is equivalent to the usual one will be a theorem of some substance. 
Note: I am not claiming that the usual definition of the Chow ring is "bad" in any way. I just think it would be nice to know if there was a "high-level" approach.

Another result I just ran across. In Hulsbergen's book Conjectures in arithmetic algebraic geometry he mentions the following theorem "of Grothendieck." 
For a general ringed space, let 
$$
  K_0(X)^{(n)} = \{x\in K_0(X)_{\mathbf Q}:\psi^r(x) = r^n x\text{ for all }r\geqslant 1\}
$$
where $\psi^r$ is the $r$-th Adams operation. The theorem states: 

If $X$ is a smooth variety over a field $k$, then $\operatorname{A}^n(X)_{\mathbf Q}\simeq K_0(X)^{(n)}$.

Unfortunately, besides the obvious inference that this result is somewhere in SGA 6, no specific reference is given. 
