Bloch's formula

{{short description|Result in algebraic K-theory relating Chow groups to cohomology}}

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 $\backslash mathcal\{O\}\_X$; that is,

::$\backslash operatorname\{CH\}^q(X)\; =\; \backslash operatorname\{H\}^q(X,\; K\_q(\backslash mathcal\{O\}\_X))$

where the right-hand side is the sheaf cohomology; $K\_q(\backslash mathcal\{O\}\_X)$ is the sheaf associated to the presheaf $U\; \backslash mapsto\; K\_q(U)$, U Zariski open subsets of X. The general case is due to Quillen.For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf {{Webarchive|url=https://web.archive.org/web/20131215101615/http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf |date=2013-12-15 }} For q = 1, one recovers $\backslash operatorname\{Pic\}(X)\; =\; H^1(X,\; \backslash mathcal\{O\}\_X^*)$. (see also Picard group.)

The formula for the mixed characteristic is still open.