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

• $(ab,c)=(a,c)(b,c)$ 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 $F^{\ast }\otimes F^{\ast }/⟨a\otimes 1-a⟩$. 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 $a,b\in F^{\ast }$ 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

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

• 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

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External links

• Steinberg symbol at the Encyclopaedia of Mathematics