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
, 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
; that is,
![{\displaystyle \operatorname {CH} ^{q}(X)=\operatorname {H} ^{q}(X,K_{q}({\mathcal {O}}_{X}))}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where the right-hand side is the sheaf cohomology;
is the sheaf associated to the presheaf
, U Zariski open subsets of X. The general case is due to Quillen.[1] For q = 1, one recovers
. (see also Picard group.)
The formula for the mixed characteristic is still open.
See also
References
- ^ For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf Archived 2013-12-15 at the Wayback Machine
- Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (ed.): Higher K-Theories. Lecture Notes in Mathematics, vol. 341. Springer-Verlag, Berlin 1973. ISBN 3-540-06434-6