Norm variety

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, ..., a_n}

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

p^{n − 1} − 1.

The key condition is in terms of the the d-th Newton polynomial s_d, evaluated on the (algebraic) total Chern class of the tangent bundle of V. This number

s_d(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).