# 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 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.^{[1]} Each class is represented by the core form of a Witt decomposition.^{[2]}

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.

^{[3]}Although classes may contain spaces of different dimension, the parity of the dimension is constant across a class and so rk :

*W*(

*k*) →

**Z**/2

**Z**is a homomorphism.

^{[4]}

The elements of finite order in the Witt group have order a power of 2;^{[5]}^{[6]} the torsion subgroup is the kernel of the functorial map from *W*(*k*) to *W*(*k*^{py}), where *k*^{py} is the Pythagorean closure of *k*;^{[7]} it is generated by the Pfister forms [math]\displaystyle{ \langle\!\langle w \rangle\!\rangle = \langle 1, -w \rangle }[/math] with [math]\displaystyle{ w }[/math] a non-zero sum of squares.^{[8]} If *k* is not formally real, then the Witt group is torsion, with exponent a power of 2.^{[9]} The **height** of the field *k* is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.^{[8]}

## Ring structure

The Witt group of *k* can be given a commutative ring structure, by using the tensor product of quadratic 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^{[4]} termed the *fundamental ideal*.^{[10]} The ring homomorphisms from *W*(*k*) to **Z** correspond to the field orderings of *k*, by taking signature with respective to the ordering.^{[10]} The Witt ring is a Jacobson ring.^{[9]} It is a Noetherian ring if and only if there are finitely many square classes; that is, if the squares in *k* form a subgroup of finite index in the multiplicative group of *k*.^{[11]}

If *k* is not formally real, the fundamental ideal is the only prime ideal of *W*^{[12]} and consists precisely of the nilpotent elements;^{[9]} *W* is a local ring and has Krull dimension 0.^{[13]}

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;^{[14]} *W* has Krull dimension 1.^{[13]}

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.^{[5]}^{[15]}

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.

^{[16]}

The *n*-th power of the fundamental ideal is additively generated by the *n*-fold Pfister forms.^{[17]}

## Examples

- The Witt ring of
**C**, and indeed any algebraically closed field or quadratically closed field, is**Z**/2**Z**.^{[18]} - The Witt ring of
**R**is**Z**.^{[18]} - The Witt ring of a finite field
**F**_{q}with*q*odd is**Z**/4**Z**if*q*≡ 3 mod 4 and isomorphic to the group ring (**Z**/2**Z**)[*F**/*F**^{2}] if*q*≡ 1 mod 4.^{[19]} - The Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to the group ring (
**Z**/2**Z**)[*V*] where*V*is the Klein 4-group.^{[20]} - The Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 is (
**Z**/4**Z**)[*C*_{2}] where*C*_{2}is a cyclic group of order 2.^{[20]} - The Witt ring of
**Q**_{2}is of order 32 and is given by^{[21]}

- [math]\displaystyle{ \mathbf{Z}_8[s,t]/\langle 2s,2t,s^2,t^2,st-4 \rangle . }[/math]

## Invariants

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.^{[22]}

### Rank and discriminant

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**/2**Z**. Addition and multiplication are defined by:

- [math]\displaystyle{ (d_1,e_1) + (d_2,e_2) = ((-1)^{e_1e_2}d_1d_2, e_1+e_2) }[/math]
- [math]\displaystyle{ (d_1,e_1) \cdot (d_2,e_2) = (d_1^{e_2}d_2^{e_1}, e_1e_2). }[/math]

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 *I*^{2}.^{[23]} The elements of *Q* may be regarded as classifying graded quadratic extensions of *K*.^{[24]}

### Brauer–Wall group

The triple of discriminant, rank mod 2 and Hasse invariant defines a map from *W*(*K*) to the Brauer–Wall group BW(*K*).^{[25]}

## Witt ring of a local field

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 ⟨*a*_{1},...*a*_{n}⟩ to ⟨*u*_{1},...*u*_{n}⟩ where *u*_{i} is a unit of *K* with image *a*_{i} in *k*. This yields

- [math]\displaystyle{ W(K) = W(k) \oplus \langle \pi \rangle \cdot W(k) }[/math]

identifying *W*(*k*) with its image in *W*(*K*).^{[26]}

## Witt ring of a number field

Let *K* be a number field. For quadratic forms over *K*, there is a Hasse invariant ±1 for every finite place corresponding to the Hilbert symbols. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.^{[27]}

We define the **symbol ring** over *K*, Sym(*K*), as a set of triples (*d*, *e*, *f* ) with *d* in *K**/*K**^{ 2}, *e* in *Z*/2 and *f* a sequence of elements ±1 indexed by the places of *K*, subject to the condition that all but finitely many terms of *f* are +1, that the value on acomplex places is +1 and that the product of all the terms in *f* in +1. Let [*a*, *b*] be the sequence of Hilbert symbols: it satisfies the conditions on *f* just stated.^{[28]}

We define addition and multiplication as follows:

- [math]\displaystyle{ (d_1,e_1,f_1) + (d_2,e_2,f_2) = ((-1)^{e_1e_2}d_1d_2, e_1+e_2, [d_1,d_2][-d_1d_2,(-1)^{e_1e_2}]f_1f_2) }[/math]
- [math]\displaystyle{ (d_1,e_1,f_1) \cdot (d_2,e_2,f_2) = (d_1^{e_2}d_2^{e_1}, e_1e_2, [d_1,d_2]^{1+e_1e_2}f_1^{e_2}f_2^{e_1}) \ . }[/math]

Then there is a surjective ring homomorphism from *W*(*K*) to Sym(*K*) obtained by mapping a class to discriminant, rank mod 2, and the sequence of Hasse invariants. The kernel is *I*^{3}.^{[29]}

The symbol ring is a realisation of the Brauer-Wall group.^{[30]}

### Witt ring of the rationals

The Hasse–Minkowski theorem implies that there is an injection^{[31]}

- [math]\displaystyle{ W(\mathbf{Q}) \rightarrow W(\mathbf{R}) \oplus \prod_p W(\mathbf{Q}_p) \ . }[/math]

We make this concrete, and compute the image, by using the "second residue homomorphism" W(**Q**_{p}) → W(**F**_{p}). Composed with the map W(**Q**) → W(**Q**_{p}) we obtain a group homomorphism ∂_{p}: W(**Q**) → W(**F**_{p}) (for *p* = 2 we define ∂_{2} to be the 2-adic valuation of the discriminant, taken mod 2).

We then have a split exact sequence^{[32]}

- [math]\displaystyle{ 0 \rightarrow \mathbf{Z} \rightarrow W(\mathbf{Q}) \rightarrow \mathbf{Z}/2 \oplus \bigoplus_{p\ne2} W(\mathbf{F}_p) \rightarrow 0 \ }[/math]

which can be written as an isomorphism

- [math]\displaystyle{ W(\mathbf{Q}) \cong \mathbf{Z} \oplus \mathbf{Z}/2 \oplus \bigoplus_{p\ne2} W(\mathbf{F}_p) \ }[/math]

where the first component is the signature.^{[33]}

## Witt ring and Milnor's K-theory

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 [math]\displaystyle{ W(k) }[/math] form a descending filtration and one may consider the associated graded ring, that is the direct sum of quotients [math]\displaystyle{ I^n/I^{n+1} }[/math]. Let [math]\displaystyle{ \langle a\rangle }[/math] be the quadratic form [math]\displaystyle{ ax^2 }[/math] considered as an element of the Witt ring. Then [math]\displaystyle{ \langle a\rangle - \langle 1\rangle }[/math] is an element of *I* and correspondingly a product of the form

- [math]\displaystyle{ \langle\langle a_1,\ldots ,a_n\rangle\rangle = (\langle a_1\rangle - \langle 1\rangle)\cdots (\langle a_n\rangle - \langle 1\rangle) }[/math]

is an element of [math]\displaystyle{ I^n. }[/math] John Milnor in a 1970 paper ^{[34]} proved that the mapping from [math]\displaystyle{ (k^*)^n }[/math] to [math]\displaystyle{ I^n/I^{n+1} }[/math] that sends [math]\displaystyle{ (a_1,\ldots ,a_n) }[/math] to [math]\displaystyle{ \langle\langle a_1,\ldots ,a_n\rangle\rangle }[/math] is multilinear and maps Steinberg elements (elements such that for some [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] such that [math]\displaystyle{ i\ne j }[/math] one has [math]\displaystyle{ a_i+a_j=1 }[/math]) 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^{[35]} in 1996 (published in 2007) for the case [math]\displaystyle{ \textrm{char}(k)=0 }[/math], leading to increased understanding of the structure of quadratic forms over arbitrary fields.

## Grothendieck-Witt ring

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.^{[18]} The hyperbolic spaces generate an ideal in *GW* and the Witt ring *W* is the quotient.^{[36]} The exterior power gives the Grothendieck-Witt ring the additional structure of a λ-ring.^{[37]}

### Examples

- The Grothendieck-Witt ring of
**C**, and indeed any algebraically closed field or quadratically closed field, is**Z**.^{[18]} - The Grothendieck-Witt ring of
**R**is isomorphic to the group ring**Z**[*C*_{2}], where*C*_{2}is a cyclic group of order 2.^{[18]} - The Grothendieck-Witt ring of any finite field of odd characteristic is
**Z**⊕**Z**/2**Z**with trivial multiplication in the second component.^{[38]}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**/2**Z**)^{3}.^{[20]} - The Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 it is
**Z'**/4*⊕*Z**Z**⊕**Z**/2**Z**.^{[20]}

## Grothendieck-Witt ring and motivic stable homotopy groups of spheres

Fabien Morel^{[39]}^{[40]} showed that the Grothendieck-Witt ring of a perfect field is isomorphic to the motivic stable homotopy group of spheres π_{0,0}(S^{0,0}) (see "A¹ homotopy theory").

## Witt equivalence

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.^{[41]} 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**.^{[42]} Some variations and extensions of this condition, such as "tame degree *l* Hilbert symbol equivalence", have also been studied.^{[43]}

## Generalizations

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 *L*^{2k}(*R*) and even-dimensional quadratic *L*-groups *L*_{2k}(*R*). The quadratic *L*-groups are 4-periodic, with *L*_{0}(*R*) being the Witt group of (1)-quadratic forms (symmetric), and *L*_{2}(*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.

## See also

- Reduced height of a field

## Notes

- ↑ Milnor & Husemoller (1973) p. 14
- ↑ Lorenz (2008) p. 30
- ↑ Milnor & Husemoller (1973) p. 65
- ↑
^{4.0}^{4.1}Milnor & Husemoller (1973) p. 66 - ↑
^{5.0}^{5.1}Lorenz (2008) p. 37 - ↑ Milnor & Husemoller (1973) p. 72
- ↑ Lam (2005) p. 260
- ↑
^{8.0}^{8.1}Lam (2005) p. 395 - ↑
^{9.0}^{9.1}^{9.2}Lorenz (2008) p. 35 - ↑
^{10.0}^{10.1}Lorenz (2008) p. 31 - ↑ Lam (2005) p. 32
- ↑ Lorenz (2008) p. 33
- ↑
^{13.0}^{13.1}Lam (2005) p. 280 - ↑ Lorenz (2008) p. 36
- ↑ Lam (2005) p. 282
- ↑ Lam (2005) pp. 277–280
- ↑ Lam (2005) p.316
- ↑
^{18.0}^{18.1}^{18.2}^{18.3}^{18.4}Lam (2005) p. 34 - ↑ Lam (2005) p.37
- ↑
^{20.0}^{20.1}^{20.2}^{20.3}Lam (2005) p.152 - ↑ Lam (2005) p.166
- ↑ Lam (2005) p.119
- ↑ Conner & Perlis (1984) p.12
- ↑ Lam (2005) p.113
- ↑ Lam (2005) p.117
- ↑ Garibaldi, Merkurjev & Serre (2003) p.64
- ↑ Conner & Perlis (1984) p.16
- ↑ Conner & Perlis (1984) p.16-17
- ↑ Conner & Perlis (1984) p.18
- ↑ Lam (2005) p.116
- ↑ Lam (2005) p.174
- ↑ Lam (2005) p.175
- ↑ Lam (2005) p.178
- ↑ Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms",
*Inventiones Mathematicae***9**(4): 318–344, doi:10.1007/BF01425486, ISSN 0020-9910 - ↑ Orlov, Dmitry; Vishik, Alexander; Voevodsky, Vladimir (2007), "An exact sequence for
*K*with applications to quadratic forms",_{*}^{M}/2*Annals of Mathematics***165**(1): 1–13, doi:10.4007/annals.2007.165.1 - ↑ Lam (2005) p. 28
- ↑ Garibaldi, Merkurjev & Serre (2003) p.63
- ↑ Lam (2005) p.36, Theorem 3.5
- ↑ , On the motivic stable π
_{0}of the sphere spectrum, In: Axiomatic, Enriched and Motivic Homotopy Theory, pp. 219–260, J.P.C. Greenlees (ed.), 2004 Kluwer Academic Publishers. - ↑ Fabien Morel,
**A**^{1}-Algebraic topology over a field. Lecture Notes in Mathematics 2052, Springer Verlag, 2012. - ↑ Perlis, R.; Szymiczek, K.; Conner, P.E.; Litherland, R. (1994). "Matching Witts with global fields". in Jacob, William B..
*Recent advances in real algebraic geometry and quadratic forms*. Contemp. Math..**155**. Providence, RI: American Mathematical Society. pp. 365–387. ISBN 0-8218-5154-3. - ↑ Szymiczek, Kazimierz (1997). "Hilbert-symbol equivalence of number fields".
*Tatra Mt. Math. Publ.***11**: 7–16. - ↑ Czogała, A. (1999). "Higher degree tame Hilbert-symbol equivalence of number fields".
*Abh. Math. Sem. Univ. Hamburg***69**: 175–185. doi:10.1007/bf02940871.

## References

- Conner, Pierre E.; Perlis, Robert (1984).
*A Survey of Trace Forms of Algebraic Number Fields*. Series in Pure Mathematics.**2**. World Scientific. ISBN 9971-966-05-0. - Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003).
*Cohomological invariants in Galois cohomology*. University Lecture Series.**28**. Providence, RI: American Mathematical Society. ISBN 0-8218-3287-5. - Lam, Tsit-Yuen (2005).
*Introduction to Quadratic Forms over Fields*. Graduate Studies in Mathematics.**67**. American Mathematical Society. ISBN 0-8218-1095-2. - Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics,**211**(Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4 - Lorenz, Falko (2008).
*Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics*. Springer. ISBN 978-0-387-72487-4. - Milnor, John; Husemoller, Dale (1973).
*Symmetric Bilinear Forms*. Ergebnisse der Mathematik und ihrer Grenzgebiete.**73**. Springer-Verlag. ISBN 3-540-06009-X. - Witt, Ernst (1936), "Theorie der quadratischen Formen in beliebigen Korpern",
*Journal für die reine und angewandte Mathematik***176**(3): 31–44

## Further reading

- Balmer, Paul (2005). "Witt groups". in Friedlander, Eric M.; Grayson, D. R..
*Handbook of*K*-theory*.**2**. Springer-Verlag. pp. 539–579. ISBN 3-540-23019-X.

## External links

- Witt rings in the Springer encyclopedia of mathematics

Original source: https://en.wikipedia.org/wiki/ Witt group.
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