Witt group

In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.

Definition

Fix a field k of characteristic not equal to 2. All vector spaces will be assumed to be finite-dimensional. Two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector. Each class is represented by the core form of a Witt decomposition.

The Witt group of k is the abelian group W(k) of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms. It is additively generated by the classes of one-dimensional forms. Although classes may contain spaces of different dimension, the parity of the dimension is constant across a class and so rk: W(k) → Z/2Z is a homomorphism.

The elements of finite order in the Witt group have order a power of 2; the torsion subgroup is the kernel of the functorial map from W(k) to W(kpy), where kpy is the Pythagorean closure of k; it is generated by the Pfister forms

⟨

⟨

w

⟩

⟩

=

⟨

1

,

−

w

⟩

{\displaystyle \langle \!\langle w\rangle \!\rangle =\langle 1,-w\rangle }

with

w

{\displaystyle w}

a non-zero sum of squares. If k is not formally real, then the Witt group is torsion, with exponent a power of 2.