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).[1]

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

${\displaystyle \{a_{1},\dots ,a_{n}\}\ }$

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

${\displaystyle p^{n-1}-1.\ }$

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

${\displaystyle s_{d}(V)\ }$

should not be divisible by p², it being known it is divisible by p.