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| 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 cohomology of ''X'' with coefficients in K-theory of the structure sheaf <math>\mathcal{O}_X</math>; that is,
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| ::<math>\operatorname{CH}^q(X) = \operatorname{H}^q(X, K_q(\mathcal{O}_X))</math>
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| 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.<ref>For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf</ref> For ''q'' = 1, one recovers <math>\operatorname{Pic}(X) = H^1(X, \mathcal{O}_X^*)</math>.
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| The formula for the [[mixed characteristic]] is still open.
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| == References ==
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| {{reflist}}
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| *[[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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| [[Category:Algebraic K-theory]]
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| [[Category:Algebraic geometry]]
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| {{algebra-stub}}
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Latest revision as of 06:34, 19 October 2014
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