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| In [[mathematics]], a '''norm variety''' is a particular type of [[algebraic variety]] ''V'' over a [[field (mathematics)|field]] ''F'', introduced for the purposes of [[algebraic K-theory]] by [[Voevodsky]]. The idea is to relate [[Milnor K-theory]] of ''F'' to geometric objects ''V'', having [[function field of an algebraic variety|function field]]s ''F''(''V'') that 'split' given 'symbols' (elements of Milnor K-groups).<ref name=Suslin2006>{{cite journal|last=Suslin|first=Andrei|coauthors=Seva Joukhovitski|title=Norm varieties|journal=Journal of Pure and Applied Algebra|date=July 2006|volume=2006|issue=1-2|pages=245-276|doi=10.1016/j.jpaa.2005.12.012|url=http://www.sciencedirect.com/science/article/pii/S0022404905003166|accessdate=17 November 2013}}</ref>
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| The formulation is that ''p'' is a given prime number, different from the [[characteristic (algebra)|characteristic]] of ''F'', and a symbol is the class mod ''p'' of an element
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| :<math>\{a_1, \dots, a_n\}\ </math>
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| of the ''n''-th Milnor K-group. A [[field extension]] is said to ''split'' the symbol, if its image in the K-group for that field is 0.
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| The conditions on a norm variety ''V'' are that ''V'' is [[Irreducible variety|irreducible]] and a [[non-singular]] [[complete variety]]. Further it should have [[dimension of an algebraic variety|dimension]] ''d'' equal to
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| :<math>p^{n - 1} - 1.\ </math>
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| The key condition is in terms of the ''d''-th [[Newton polynomial]] ''s''<sub>''d''</sub>, evaluated on the (algebraic) total [[Chern class]] of the [[tangent bundle]] of ''V''. This number
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| :<math>s_d(V)\ </math> | |
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| should not be divisible by ''p''<sup>2</sup>, it being known it is divisible by ''p''.
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| ==Examples==
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| These include (''n'' = 2) cases of the [[Severi–Brauer variety]] and (''p'' = 2) [[Pfister form]]s. There is an existence theorem in the general case (paper of [[Markus Rost]] cited).
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.math.uni-bielefeld.de/~rost/data/nv-ac.pdf Paper by Rost]
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| [[Category:Algebraic varieties]]
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| [[Category:K-theory]]
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Revision as of 23:01, 28 February 2014
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