Bloch's formula

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 the cohomology of X with coefficients in the K-theory of the structure sheaf ${\mathcal {O}}_{X}$ ; that is,

\operatorname {CH} ^{q}(X)=\operatorname {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.^[1] For q = 1, one recovers $\operatorname {Pic} (X)=H^{1}(X,{\mathcal {O}}_{X}^{*})$ . (see also Picard group.)

The formula for the mixed characteristic is still open.

References

↑ For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf

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

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[1] For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf