Norm variety
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In mathematics, a norm variety is a particular type of algebraic variety V over a 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 fields F(V) that 'split' given 'symbols' (elements of Milnor K-groups).
The formulation is that p is a given prime number, different from the characteristic of F, and a symbol is the class mod p of an element
- {a1, ..., an}
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.
The conditions on a norm variety V are that V is irreducible and a non-singular complete variety. Further it should have dimension d equal to
- pn − 1 − 1.
The key condition is in terms of the the d-th Newton polynomial sd, evaluated on the (algebraic) total Chern class of the tangent bundle of V. This number
- sd(V)
should not be divisible by p2, it being known it is divisible by p.
[edit] Examples
These include (n = 2) cases of the Severi-Brauer variety and (p = 2) Pfister forms. There is an existence theorem in the general case (paper of Markus Rost cited).
