Bloch's formula

In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for ${\displaystyle K_{2}}$, states that the Chow group of a smooth variety X over a field is isomorphic to the cohomology of X with coefficients in the K-theory of the structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$; that is,

${\displaystyle \operatorname {CH} ^{q}(X)=\operatorname {H} ^{q}(X,K_{q}({\mathcal {O}}_{X}))}$

where the right-hand side is the sheaf cohomology; ${\displaystyle K_{q}({\mathcal {O}}_{X})}$ is the sheaf associated to the presheaf ${\displaystyle U\mapsto K_{q}(U)}$, U Zariski open subsets of X. The general case is due to Quillen.[1] For q = 1, one recovers ${\displaystyle \operatorname {Pic} (X)=H^{1}(X,{\mathcal {O}}_{X}^{*})}$. (see also Picard group.)

The formula for the mixed characteristic is still open.

References

1. For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf Archived 2013-12-15 at the Wayback Machine

• 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