quadratic map (original) (raw)

Given a commutative ring K and two K-modules M and N then a mapq:M→N is called quadratic if

    1. q⁢(α⁢x)=α2⁢q⁢(x) for all x∈M and α∈K.
    1. b⁢(x,y):=q⁢(x+y)-q⁢(x)-q⁢(y), for x,y∈M, is a bilinear map.

The only differencePlanetmathPlanetmath between quadratic maps and quadratic formsMathworldPlanetmath is the insistence on the codomain N instead of a K. So in this way every quadratic form is a special case of a quadratic map. Most of the properties for quadratic forms apply to quadratic maps as well. For instance, if K has no 2-torsion (2⁢x=0 implies x=0) then

2⁢c⁢(x,y)=q⁢(x+y)-q⁢(x)-q⁢(y).

defines a symmetricPlanetmathPlanetmathPlanetmathPlanetmath K-bilinear map c:M×M→N with c⁢(x,x)=q⁢(x). In particular if 1/2∈K thenc⁢(x,y)=12⁢b⁢(x,y). This definition is one instance of a polarization (i.e.: substituting a single variable in a formulaMathworldPlanetmathPlanetmath with x+y and comparing the result with the formula over x and y separately.) Continuing without 2-torsionPlanetmathPlanetmath, if b is a symmetric K-bilinear map (perhaps not a form) then definingqb⁢(x)=b⁢(x,x) determines a quadratic map since

qb⁢(α⁢x)=b⁢(α⁢x,α⁢x)=α2⁢b⁢(x,x)=α2⁢q⁢(x)

and

qb⁢(x+y)-qb⁢(x)-qb⁢(y)=b⁢(x+y,x+y)-b⁢(x,x)-b⁢(y,y)=b⁢(x,y)+b⁢(y,x)=2⁢b⁢(x,y).

Have have no 2-torsion we can recover b form qb. So in odd and 0 characteristic rings we find symmetric bilinear maps and quadratic maps are in 1-1 correspondence.

An alternative understanding of b is to treat this as the obstruction toq being an additivePlanetmathPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Thus a submodule T of M for which b⁢(T,T)=0 is a submodule of M on which q|T is an additive homomorphism. Of course because of the first condition, q is semi-linear on T only when α↦α2 is an automorphismPlanetmathPlanetmath of K, in particular, if K has characteristic 2. When the characteristic of K is odd or 0 then q⁢(T)=0if and only if b⁢(T,T)=0 simply because q⁢(x)=b⁢(x,x) (or up to a 1/2 multipleMathworldPlanetmathPlanetmath depending on conventions). However, in characteristic 2 it is possible for b⁢(T,T)=0 yet q⁢(T)≠0. For instance, we can haveq⁢(x)≠0 yet b⁢(x,x)=q⁢(2⁢x)-q⁢(x)-q⁢(x)=0. This is summed up in the following definition:

A subspacePlanetmathPlanetmathPlanetmath T of M is called totally singular if q⁢(T)=0 and totally isotropic if b⁢(T,T)=0. In odd or 0 characteristic, totally singular subspaces are precisely totally isotropic subspaces.