Bloch's formula
In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for [math]\displaystyle{ K_2 }[/math], 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 [math]\displaystyle{ \mathcal{O}_X }[/math]; that is,
- [math]\displaystyle{ \operatorname{CH}^q(X) = \operatorname{H}^q(X, K_q(\mathcal{O}_X)) }[/math]
where the right-hand side is the sheaf cohomology; [math]\displaystyle{ K_q(\mathcal{O}_X) }[/math] is the sheaf associated to the presheaf [math]\displaystyle{ U \mapsto K_q(U) }[/math], U Zariski open subsets of X. The general case is due to Quillen.[1] For q = 1, one recovers [math]\displaystyle{ \operatorname{Pic}(X) = H^1(X, \mathcal{O}_X^*) }[/math]. (see also Picard group.)
The formula for the mixed characteristic is still open.
References
- ↑ For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf
- 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
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