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,

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

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. For q = 1, one recovers $\operatorname {Pic} (X)=H^{1}(X,{\mathcal {O}}_{X}^{*})$.

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