Given a commutative ring and two -modules and then a map is called quadratic if
for all and .
, for , is a bilinear map.
The only difference between quadratic maps and quadratic forms is the insistence on the codomain instead of a . 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 has no 2-torsion ( implies ) then
defines a symmetric -bilinear map with . In particular if then . This definition is one instance of a polarization (i.e.: substituting a single variable in a formula with and comparing the result with the formula over and separately.) Continuing without -torsion, if is a symmetric -bilinear map (perhaps not a form) then defining determines a quadratic map since
An alternative understanding of is to treat this as the obstruction to being an additive homomorphism. Thus a submodule of for which is a submodule of on which is an additive homomorphism. Of course because of the first condition, is semi-linear on only when is an automorphism of , in particular, if has characteristic 2. When the characteristic of is odd or 0 then if and only if simply because (or up to a multiple depending on conventions). However, in characteristic 2 it is possible for yet . For instance, we can have yet . This is summed up in the following definition:
A subspace of is called totally singular if and totally isotropic if . In odd or 0 characteristic, totally singular subspaces are precisely totally isotropic subspaces.
|Date of creation||2013-03-22 16:27:55|
|Last modified on||2013-03-22 16:27:55|
|Last modified by||Algeboy (12884)|