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Bloch's formula

From Wikipedia, the free encyclopedia

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,

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

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References

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  1. ^ 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