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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

{ a 1 , … , a n }   {\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

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

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

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

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

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).

References

  1. Suslin, Andrei; Seva Joukhovitski (July 2006). "Norm varieties". Journal of Pure and Applied Algebra. 2006 (1–2): 245–276. doi:10.1016/j.jpaa.2005.12.012.

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