Cours d'Analyse

In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for $K_2$, states that the Chow group of a smooth variety X over a field is isomorphic to cohomology of X with coefficients in K-theory of the structure sheaf ${\mathcal{𝒪}}_X$; that is,

${\mathrm{CH}}^q(X)={\mathrm{H}}^q(X,K_q({\mathcal{𝒪}}_X))$

where the right-hand side is the sheaf cohomology; $K_q({\mathcal{𝒪}}_X)$ is the sheaf associated to the presheaf $U\mapsto K_q(U)$, U Zariski open subsets of X. The general case is due to Quillen. For q = 1, one recovers $\mathrm{Pic}(X)=H^1(X,{\mathcal{𝒪}}_X^{*})$.

The formula for the mixed characteristic is still open.

References

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• Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (ed.): Higher K-Theories. Lecture Notes in Mathematics, vol. 341. Springer-Verlag, Berlin 1973. ISBN 3-540-06434-6