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.
