Английская Википедия:Bloch's formula

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Шаблон:Short description In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for <math>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>\mathcal{O}_X</math>; that is,

<math>\operatorname{CH}^q(X) = \operatorname{H}^q(X, K_q(\mathcal{O}_X))</math>

where the right-hand side is the sheaf cohomology; <math>K_q(\mathcal{O}_X)</math> is the sheaf associated to the presheaf <math>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>\operatorname{Pic}(X) = H^1(X, \mathcal{O}_X^*)</math>. (see also Picard group.)

The formula for the mixed characteristic is still open. Шаблон:Further

References

Шаблон:Reflist

  • Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (ed.): Higher K-Theories. Lecture Notes in Mathematics, vol. 341. Springer-Verlag, Berlin 1973. Шаблон:ISBN


Шаблон:Algebraic-geometry-stub

  1. For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf Шаблон:Webarchive