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 O X ; that is,

where the right-hand side is the sheaf cohomology; K q ( O X ) is the sheaf associated to the presheaf U ↦ K q ( U ) , U Zariski open subsets of X. The general case is due to Quillen. For q = 1, one recovers Pic ⁡ ( X ) = H 1 ( X , O X ∗ ) . (see also Picard group.)

The formula for the mixed characteristic is still open.