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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.

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The Bloch formula (proved by Quillen, special cases by Bloch) holds for any regular scheme $X$ of finite type over a field: the group of algebraic cycles of codimension $i$ modulo rational equivalence is isomorphic to the cohomology group $H^i(X,\mathcal{K}_i)$ where $\mathcal{K_i}$ is the sheaf associated to the presheaf that sends each open subset $U$ to $K_i(U)$, where $K_i$ is the $K$-theory functor in the sense of Quillen. One can use this to define the intersection product.

Note: $K_i$ is the functor on schemes that sends $X$ to the abelian group $K_i(X)$ where $K_i$ is defined as $K_i$ of the exact category $P(X)$ of vector bundles on $X$. If $f:X\to Y$ is a morphism of schemes the inverse image functor $f^*:P(Y)\to P(X)$ is an exact functor inducing a morphism of $K$-groups $K_i(Y)\to K_i(X)$. In particular, if $U\subseteq X$ is an open subscheme then this provides the restriction, whence $U\mapsto K_i(U)$ is really a presheaf, so there is an associated sheafification.

Of course, defining $K_i(\mathcal{A})$ for an exact category $A$ requires some machinery, but the definition in Quillen's paper assumes only a basic knowledge of category theory and a bit of homological algebra. If the homotopy theory of categories is a bit fast there try Weibel's K-Book (available on his website), Ch. 4, Section 3.

The Quillen's paper "Higher Algebraic K-Theory I", Theorem 5.19 for the original proof. An excellent introduction is Gillet's paper "K-Theory and Intersection Theory" which is in the great "Handbook of K-Theory".

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  • $\begingroup$ I'll display my ignorance by admitting I don't know what the $K$-theory functor in the sense of Quillen is, but I guess the key point for this question is whether the sheaf $\mathcal{K}_i$ really makes sense for arbitrary ringed spaces, as the OP asked. But that isn't to carp: I like this answer a lot, +1. I'm just wondering if the OP's question is a bit too broad. $\endgroup$ – user64687 Oct 29 '13 at 16:59
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    $\begingroup$ @AsalBeagDubh: I'm pretty sure you can define $K_i(X)$ for $X$ n arbitrary ringed space. After all, the notion of a coherent $\mathcal{O}_X$-module makes sense in that generality (see ncatlab.org/nlab/show/coherent+sheaf). Since $K_i(X)$ is defined in terms of the category $\mathsf{Coh}(X)$, there shouldn't be any problem. $\endgroup$ – Daniel Miller Oct 29 '13 at 17:09
  • $\begingroup$ Jason: this is a fantastic answer! Gillet's paper is very nice. $\endgroup$ – Daniel Miller Oct 29 '13 at 17:13
  • $\begingroup$ I added a few more words, including one more refernece to the K-book (url: math.rutgers.edu/~weibel/Kbook.html). $\endgroup$ – user2055 Oct 29 '13 at 17:16
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    $\begingroup$ @DanielMiller: ok, I retract my worries! I think I was a bit perturbed by the idea of whether the Chow groups of some terrible ringed space could have any reasonable properties, but I suppose that's sort of an orthogonal question. $\endgroup$ – user64687 Oct 29 '13 at 17:29
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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.

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