Witt group

Algebra term
Witt groups in the theory of algebraic groups
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.

Fix a field k of characteristic not equal to two. All vector spaces will be assumed to be finite dimensional. We say that 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 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 with a non zero sum of squares. If k is not formally real, then the Witt group is torsion, with exponent a power of 2.

Certain invariants of a quadratic form can be regarded as functions on Witt classes. We have seen that dimension mod 2 is a function on classes: the discriminant is also well defined. The Hasse invariant of a quadratic form is again a well defined function on Witt classes with values in the Brauer group of the field of definition.

We define a ring over K, Q(K), as a set of pairs (d, e) with d in K*/K*2 and e in Z/2Z. Addition and multiplication are defined by:

Then there is a surjective ring homomorphism from W(K) to this obtained by mapping a class to discriminant and rank mod 2. The kernel is I2. The elements of Q may be regarded as classifying graded quadratic extensions of K.

The triple of discriminant, rank mod 2 and Hasse invariant defines a map from W(K) to the Brauer–Wall group BW(K).

Let K be a complete local field with valuation v, uniformiser π and residue field k of characteristic not equal to 2. There is an injection W(k) → W(K) which lifts the diagonal form ⟨a1, an⟩ to ⟨u1, un⟩ where ui is a unit of K with image ai in k. This yields

identifying W(k) with its image in W(K).

Let k be a field of characteristic not equal to 2. The powers of the ideal I of forms of even dimension ("fundamental ideal") in form a descending filtration and one may consider the associated graded ring, that is the direct sum of quotients Let be the quadratic form considered as an element of the Witt ring. Then is an element of I and correspondingly a product of the form

is an element of John Milnor in a 1970 paper proved that the mapping from to that sends to is multilinear and maps Steinberg elements (elements such that for some and such that one has ) to zero. This means that this mapping defines a homomorphism from the Milnor ring of k to the graded Witt ring. Milnor showed also that this homomorphism sends elements divisible by 2 to zero and that it is surjective. In the same paper he made a conjecture that this homomorphism is an isomorphism for all fields k (of characteristic different from 2). This became known as the Milnor conjecture on quadratic forms. The conjecture was proved by Dmitry Orlov, Alexander Vishik and Vladimir Voevodsky in 1996 (published in 2007) for the case , leading to increased understanding of the structure of quadratic forms over arbitrary fields.

The Grothendieck Witt ring GW is a related construction generated by isometry classes of nonsingular quadratic spaces with addition given by orthogonal sum and multiplication given by tensor product. Since two spaces that differ by a hyperbolic plane are not identified in GW, the inverse for the addition needs to be introduced formally through the construction that was discovered by Grothendieck (see Grothendieck group). There is a natural homomorphism GW → Z given by dimension: a field is quadratically closed if and only if this is an isomorphism. The hyperbolic spaces generate an ideal in GW and the Witt ring W is the quotient. The exterior power gives the Grothendieck Witt ring the additional structure of a λ ring.

The Grothendieck Witt ring of C, and indeed any algebraically closed field or quadratically closed field, is Z. The element (1, 0) corresponds to the quadratic form ⟨a⟩ where a is not a square in the finite field. The Grothendieck Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to Z ⊕ (Z/2Z)3. showed that the Grothendieck Witt ring of a perfect field is isomorphic to the motivic stable homotopy group of spheres π0,0(S0,0) (see "A¹ homotopy theory").

Two fields are said to be Witt equivalent if their Witt rings are isomorphic. For global fields there is a local to global principle: two global fields are Witt equivalent if and only if there is a bijection between their places such that the corresponding local fields are Witt equivalent. In particular, two number fields K and L are Witt equivalent if and only if there is a bijection T between the places of K and the places of L and a group isomorphism t between their square class groups, preserving degree 2 Hilbert symbols. In this case the pair (T, t) is called a reciprocity equivalence or a degree 2 Hilbert symbol equivalence. Some variations and extensions of this condition, such as "tame degree l Hilbert symbol equivalence", have also been studied.

Witt groups can also be defined in the same way for skew symmetric forms, and for quadratic forms, and more generally ε quadratic forms, over any * ring R.

The resulting groups (and generalizations thereof) are known as the even dimensional symmetric L groups L2k(R) and even dimensional quadratic L groups L2k(R). The quadratic L groups are 4 periodic, with L0(R) being the Witt group of (1) quadratic forms (symmetric), and L2(R) being the Witt group of (−1) quadratic forms (skew symmetric); symmetric L groups are not 4 periodic for all rings, hence they provide a less exact generalization. L groups are central objects in surgery theory, forming one of the three terms of the surgery exact sequence.