Steinberg symbol explained

In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of fields. It is named after mathematician Robert Steinberg.

For a field F we define a Steinberg symbol (or simply a symbol) to be a function

( ⋅ , ⋅ ) : F ^{*} x F ^{*} → G , where G is an abelian group, written multiplicatively, such that( ⋅ , ⋅ ) is bimultiplicative;

• if

a + b = 1 then (a,b) = 1 .

The symbols on F derive from a "universal" symbol, which may be regarded as taking values in

F ^{*} ⊗ F ^{*} / \langle a ⊗ 1- a \rangle . By a theorem of Matsumoto, this group is K _{2} F and is part of the Milnor K-theory for a field.

Properties

If (⋅,⋅) is a symbol then (assuming all terms are defined)

(a, - a) = 1 ;(b, a) = (a, b) ^{-1} ;(a, a) = (a, -1 ) is an element of order 1 or 2;(a, b) = (a + b, - b/a) .

Examples

• The trivial symbol which is identically 1.
• The Hilbert symbol on F with values in defined by^{[1] [2]}

(a,b) = \begin{cases} 1 ,& if z ^{2=ax} ^{2+by} ^{2hasanon-zerosolution(x,y,z)\in} F ^{3;\\-1,&ifnot.\end{cases}}

• The Contou-Carrère symbol is a symbol for the ring of Laurent power series over an Artinian ring.

Continuous symbols

If F is a topological field then a symbol c is weakly continuous if for each y in F^∗ the set of x in F^∗ such that c(x,y) = 1 is closed in F^∗. This makes no reference to a topology on the codomain G. If G is a topological group, then one may speak of a continuous symbol, and when G is Hausdorff then a continuous symbol is weakly continuous.^[3]

The only weakly continuous symbols on R are the trivial symbol and the Hilbert symbol: the only weakly continuous symbol on C is the trivial symbol.^[4] The characterisation of weakly continuous symbols on a non-Archimedean local field F was obtained by Moore. The group K₂(F) is the direct sum of a cyclic group of order m and a divisible group K₂(F)^m. A symbol on F lifts to a homomorphism on K₂(F) and is weakly continuous precisely when it annihilates the divisible component K₂(F)^m. It follows that every weakly continuous symbol factors through the norm residue symbol.^[5]

See also

• Steinberg group (K-theory)

References

• Book: P.E. . Conner . R. . Perlis . A Survey of Trace Forms of Algebraic Number Fields . Series in Pure Mathematics . 2 . World Scientific . 1984 . 9971-966-05-0 . 0551.10017 .
• Book: Lam, Tsit-Yuen . Tsit Yuen Lam

. Introduction to Quadratic Forms over Fields . 67 . . Tsit Yuen Lam . American Mathematical Society . 2005 . 0-8218-1095-2 . 132–142 . 1068.11023 .

• Book: Milnor . John Willard . John Milnor . Introduction to algebraic K-theory . . Princeton, NJ . 0349811 . 1971 . 0237.18005 . Annals of Mathematics Studies . 72 .
• Robert . Steinberg . Robert Steinberg . Générateurs, relations et revêtements de groupes algébriques . Colloq. Théorie des Groupes Algébriques . Bruxelles . 1962 . Gauthier-Villars . 113–127 . 0153677 . 0272.20036 . French .

External links

• Steinberg symbol at the Encyclopaedia of Mathematics

Notes and References

1. Book: Serre, Jean-Pierre . Jean-Pierre Serre

   . Jean-Pierre Serre . A Course in Arithmetic . . Berlin, New York . . 7 . 978-3-540-90040-5 . 1996 .

2. Milnor (1971) p.94
3. Milnor (1971) p.165
4. Milnor (1971) p.166
5. Milnor (1971) p.175