Bloch's formula explained
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,
\operatorname{CH}q(X)=\operatorname{H}q(X,Kq(l{O}X))
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
\operatorname{Pic}(X)=H1(X,
. (see also
Picard group.)
The formula for the mixed characteristic is still open.
See also
References
Higher algebraic K-theory: I. In: H. Bass (ed.): Higher K-Theories. Lecture Notes in Mathematics, vol. 341. Springer-Verlag, Berlin 1973.
Notes and References
- For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf
