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It seems to be a well known fact that algebraic K-theory can be used to understand intersection theory, at least for varieties (or stacks!) over a field. A first glimpse of this result seems to be Bloch's formula (Quillen, 1972): $$ \operatorname H^p(X, \mathcal{K}_p) \cong \operatorname{CH}^p (X) $$ where $\mathcal{K}_p$ is the Zariski-sheafification of $K$-theory. This even induces an isomorphism of the respective graded algebras!

But as beautiful as this formula is, I've got a question:

can we actually extract information about the chow groups and its intersection products using this formula?

Of course, $K$-theory is notoriously complicated, but sheafifying it might make things more amenable (I think $\mathcal{K}_p = \underline{\mathbf{Z}}$ and $\mathcal{K}_1 = \mathcal{O}_X^\times$).

One example on how this might help, it seems that (Guillet, 1984) used the LHS to define an intersection theory for algebraic stacks, but it also seems to me (and I might be very wrong) that (Vistoli, 1989) approach bypassing this is what has become more prominent nowadays, and I don't know if Guillet's approach allows us to deduce anything that Vistoli's cannot.

On a similar note, Quillen says on his paper on higher algebraic K theory that the left hand side of Bloch's formula is "manifestly contraviariant in X"; which may be one advantage, although I don't quite see how to prove this (maybe it follows from earlier remarks - i didn't read the entire paper yet....). It seems that Quillen has convinced himself by this point that the LHS will be important for future work.

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  • $\begingroup$ Just a remark on the "manifest" contravariance in $X$: it follows from the functoriality of cohomology because $\mathcal K_p$ is by definition a presheaf on all schemes. If one regards it as a presheaf of chain complexes in degree $0$ and Zariski-sheafify in the homotopical sense (i.e., take derived sections), one obtains a functor $R\Gamma(-,\mathcal K_p): Sch^{op} \to D(\mathbb Z)_{\leq 0}$, and one can then take cohomology groups. $\endgroup$
    – Marc Hoyois
    Jul 19, 2021 at 10:27

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Not an answer, but as you wrote "Of course, K-theory is notoriously complicated, but sheafifying it might make things more amenable", let me point out that Bloch's formula is also true for the Milnor $K$-theory sheaf instead of the Quillen $K$-theory sheaf. This does not help at all to compute any Chow group, but it makes it possible (and in easy examples possible in a concrete fashion) to write down an algebraic cycles as a representative in Cech cohomology of the Milnor $K$-theory sheaf. So this makes things even more amenable. Nonetheless, I am not sure this has any profound use. It is however definitely easier to work with such cycle representatives than trying to make a Cech representative of a Quillen K-theory sheaf explicit.

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