Steinberg symbol

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 $(\cdot ,\cdot ):F^{*}\times F^{*}\rightarrow G$, where $G$ is an abelian group, written multiplicatively, such that for all $a,b,c\in F^{*}$,

• $(ab,c)=(a,c)(b,c)$ and $(a,bc)=(a,b)(a,c)$, and
• if $a+b=1$ then $(a,b)=1$.

The first condition is sometimes referred to as $(\cdot ,\cdot )$ being bimultiplicative, while the second condition is known as the Steinberg property.

The symbols on $F$ derive from a "universal" symbol, which may be regarded as taking values in $F^{*}\otimes F^{*}/\langle a\otimes 1-a\rangle$. By a theorem of Hideya Matsumoto, this group is $K_{2}F$ and is part of the Milnor K-theory for a field.

Properties

If $(\cdot ,\cdot )$ is a symbol on $F$, then for all $a,b\in F^{*}$ we have

• $(a,-a)=1$;
• $(b,a)=(a,b)^{-1}$;
• $(a,a)=(a,-1)$ 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 {±1} defined by

$(a,b)={\begin{cases}1,&{\mbox{ if }}z^{2}=ax^{2}+by^{2}{\mbox{ has a non-zero solution }}(x,y,z)\in F^{3};\\-1,&{\mbox{ if not.}}\end{cases}}$

• 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.

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. 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.

See also

• Steinberg group (K-theory)

References

• Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. Vol. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017.
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. pp. 132–142. ISBN 0-8218-1095-2. Zbl 1068.11023.
• Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. Vol. 72. Princeton, NJ: Princeton University Press. MR 0349811. Zbl 0237.18005.
• Steinberg, Robert (1962). "Générateurs, relations et revêtements de groupes algébriques". Colloq. Théorie des Groupes Algébriques (in French). Bruxelles: Gauthier-Villars: 113–127. MR 0153677. Zbl 0272.20036.

External links

• Steinberg symbol at the Encyclopaedia of Mathematics