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2014-10-19T05:34:24Z

Add a navbox

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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,~~		I am Ardis and was born on 12 April 1973. My hobbies are Antiquing and Boxing.<br><br>Feel free to visit my weblog - sacramento homes for sale ([http://Wiki.Net-Stroy.ru/index.php/%D0%A3%D1%87%D0%B0%D1%81%D1%82%D0%BD%D0%B8%D0%BA:WaylonDeschamps this])
	~~::<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.<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>.~~

	~~The formula for the [[mixed characteristic]~~] ~~is still open.~~

	~~== References ==~~
	~~{{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. ISBN 3-540-06434-6~~

	~~[[Category:Algebraic K-theory]]~~
	~~[[Category:Algebraic geometry]]~~

	~~{{algebra-stub}}~~

132.70.4.118: /* Section 2.2 */

2013-12-23T13:39:39Z

Section 2.2

New page

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,
::<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.<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>.

The formula for the [[mixed characteristic]] is still open.

== References ==
{{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. ISBN 3-540-06434-6

[[Category:Algebraic K-theory]]
[[Category:Algebraic geometry]]

{{algebra-stub}}

Cours d'Analyse - Revision history

en>RekishiEJ: Add a navbox

132.70.4.118: /* Section 2.2 */