Witt Group - Ring Structure

Ring Structure

The Witt group of k can be given a commutative ring structure, by using the tensor product of two bilinear forms to define the ring product. This is sometimes called the Witt ring W(k), though the term "Witt ring" is often also used for a completely different ring of Witt vectors.

To discuss the structure of this ring we assume that k is of characteristic not equal to 2, so that we may identify symmetric bilinear forms and quadratic forms.

The kernel of the rank mod 2 homomorphism is a prime ideal, I, of the Witt ring termed the fundamental ideal. The ring homomorphisms from W(k) to Z correspond to the field orderings of k, by taking signature with respective to the ordering. The Witt ring is a Jacobson ring. It is a Noetherian ring if and only if the squares in k form a subgroup of finite index in the multiplicative group.

If k is not formally real, the fundamental ideal is the only prime ideal of W and consists precisely of the nilpotent elements; W is a local ring and has Krull dimension 0.

If k is real, then the nilpotent elements are precisely those of finite additive order, and these in turn are the forms all of whose signatures are zero; W has Krull dimension 1.

If k is a real Pythagorean field then the zero-divisors of W are the elements for which some signature is zero; otherwise, the zero-divisors are exactly the fundamental ideal.

If k is an ordered field with positive cone P then Sylvester's law of inertia holds for quadratic forms over k and the signature defines a ring homomorphism from W(k) to Z, with kernel a prime ideal K_P. These prime ideals are in bijection with the orderings X_k of k and constitute the minimal prime ideal spectrum MinSpec W(k) of W(k). The bijection is a homeomorphism between MinSpec W(k) with the Zariski topology and the set of orderings X_k with the Harrison topology.

The n-th power of the fundamental ideal is additively generated by the n-fold Pfister forms.