Bloch's formula
In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for K 2 {\displaystyle 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 {\displaystyle {\mathcal {O}}_{X}} ; that is, CH q ( X ) = H q ( X , K q ( O X ) ) {\displaystyle \operatorname {CH} ^{q}(X)=\operatorname {H} ^{q}(X,K_{q}({\mathcal {O}}_{X}))} where the right-hand side is the sheaf cohomology; K q ( O X ) {\displaystyle K_{q}({\mathcal {O}}_{X})} is the sheaf associated to the presheaf U ↦ K q ( U ) {\displaystyle U\mapsto 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 ∗ ) {\displaystyle \operatorname {Pic} (X)=H^{1}(X,{\mathcal {O}}_{X}^{*})} .