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. 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.