In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Witt group of a
field, named after
Ernst Witt, is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
whose elements are represented by
symmetric bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
s over the field.
Definition
Fix a field ''k'' of
characteristic not equal to 2. All
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s 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.
[Milnor & Husemoller (1973) p. 14] Each class is represented by the
core form of a
Witt decomposition.
[Lorenz (2008) p. 30]
The Witt group of ''k'' is the abelian group ''W''(''k'') of
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es 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.
[Milnor & Husemoller (1973) p. 65] 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
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
.
[
The elements of finite order in the Witt group have order a power of 2;][Lorenz (2008) p. 37][Milnor & Husemoller (1973) p. 72] the torsion subgroup
In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group ...
is the kernel of the functorial map from ''W''(''k'') to ''W''(''k''py), where ''k''py is the Pythagorean closure of ''k'';[Lam (2005) p. 260] it is generated by the Pfister forms with a non-zero sum of squares.[Lam (2005) p. 395] If ''k'' is not formally real, then the Witt group is torsion, with exponent a power of 2.[ The height of the field ''k'' is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.][Lam (2005) p. 395]
Ring structure
The Witt group of ''k'' can be given a commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
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 one assumes that ''k'' is of characteristic not equal to 2, so that one may identify symmetric bilinear forms and quadratic forms.
The kernel of the rank mod 2 homomorphism is a prime ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
, ''I'', of the Witt ring[Milnor & Husemoller (1973) p. 66] termed the ''fundamental ideal''.[ The ]ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
s from ''W''(''k'') to Z correspond to the field orderings of ''k'', by taking signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
with respective to the ordering.[Lorenz (2008) p. 31] The Witt ring is a Jacobson ring.[Lorenz (2008) p. 35] It is a Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
if and only if there are finitely many square classes; that is, if the squares in ''k'' form a subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
of finite index
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
in the multiplicative group of ''k''.[Lam (2005) p. 32]
If ''k'' is not formally real, the fundamental ideal is the only prime ideal of ''W''[Lorenz (2008) p. 33] and consists precisely of the nilpotent element
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term, along with its sister idempotent, was introduced by Benjamin Peirce i ...
s;[ ''W'' is a ]local ring
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
and has Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
0.[Lam (2005) p. 280]
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 0;[Lorenz (2008) p. 36] ''W'' has Krull dimension 1.[
If ''k'' is a real Pythagorean field then the ]zero-divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
s of ''W'' are the elements for which some signature is 0; otherwise, the zero-divisors are exactly the fundamental ideal.[Lam (2005) p. 282]
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
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
with the orderings ''Xk'' of ''k'' and constitute the minimal prime ideal spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
MinSpec ''W''(''k'') of ''W''(''k''). The bijection is a homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
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.[Lam (2005) p.316]
Examples
* The Witt ring of C, and indeed any algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
or quadratically closed field In mathematics, a quadratically closed field is a field of characteristic not equal to 2 in which every element has a square root.Lam (2005) p. 33Rajwade (1993) p. 230 Examples
* The field of complex numbers is quadratically closed; more ...
, is Z/2Z.[Lam (2005) p. 34]
* The Witt ring of R is Z.[
* The Witt ring of a ]finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
F''q'' with ''q'' odd is Z/4Z if ''q'' ≡ 3 mod 4 and isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
(Z/2Z) 2">'F*''/''F*''2if ''q'' ≡ 1 mod 4.[Lam (2005) p.37]
* The Witt ring of a local field
In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...
with maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
of norm congruent to 1 modulo 4 is isomorphic to the group ring (Z/2Z) 'V''where ''V'' is the Klein 4-group.[Lam (2005) p.152]
* The Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 is (Z/4Z) 2">'C''2where ''C''2 is a cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
of order 2.[Lam (2005) p.152]
* The Witt ring of Q2 is of order 32 and is given by[Lam (2005) p.166]
::.
Invariants
Certain invariants of a quadratic form can be regarded as functions on Witt classes. 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
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
of the field of definition.[Lam (2005) p.119]
Rank and discriminant
A ring is defined 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
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
ring homomorphism from ''W''(''K'') to this obtained by mapping a class to discriminant and rank mod 2. The kernel is ''I''2.[Conner & Perlis (1984) p.12] The elements of ''Q'' may be regarded as classifying graded quadratic extensions of ''K''.[Lam (2005) p.113]
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'').[Lam (2005) p.117]
Witt ring of a local field
Let ''K'' be a complete local field
In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...
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
:
identifying ''W''(''k'') with its image in ''W''(''K'').[Garibaldi, Merkurjev & Serre (2003) p.64]
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 signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
s coming from real embeddings.[Conner & Perlis (1984) p.16]
The symbol ring is defined 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'' is +1. Let 'a'', ''b''be the sequence of Hilbert symbols: it satisfies the conditions on ''f'' just stated.[Conner & Perlis (1984) p.16-17]
Addition and multiplication is defined as follows:
:
:.
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.[Conner & Perlis (1984) p.18]
The symbol ring is a realisation of the Brauer-Wall group.[Lam (2005) p.116]
Witt ring of the rationals
The Hasse–Minkowski theorem implies that there is an injection[Lam (2005) p.174]
:.
One can 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''), one obtains a group homomorphism ∂''p'': W(Q) → W(F''p'') (for ''p'' = 2, ∂2 is defined to be the 2-adic valuation of the discriminant, taken mod 2).
One will then have a split exact sequence
The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits ( ...
[Lam (2005) p.175]
:
which can be written as an isomorphism
:
where the first component is the signature.[Lam (2005) p.178]
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 form a descending filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
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 0. 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 0 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.
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
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorp ...
). There is a natural homomorphism ''GW'' → Z given by dimension: a field is quadratically closed if and only if this is an isomorphism.[Lam (2005) p. 34] The hyperbolic spaces generate an ideal in ''GW'' and the Witt ring ''W'' is the quotient.[Lam (2005) p. 28] The exterior power gives the Grothendieck–Witt ring the additional structure of a λ-ring.[Garibaldi, Merkurjev & Serre (2003) p.63]
Examples
* The Grothendieck–Witt ring of C, and indeed any algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
or quadratically closed field In mathematics, a quadratically closed field is a field of characteristic not equal to 2 in which every element has a square root.Lam (2005) p. 33Rajwade (1993) p. 230 Examples
* The field of complex numbers is quadratically closed; more ...
, is Z.[
* The Grothendieck–Witt ring of R is isomorphic to the group ring Z 2">'C''2 where ''C''2 is a cyclic group of order 2.][
* The Grothendieck–Witt ring of any finite field of odd characteristic is Z ⊕ Z/2Z with trivial multiplication in the second component.][Lam (2005) p.36, Theorem 3.5] 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.[
* The Grothendieck–Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 is Z ''⊕'' Z/4Z ⊕ Z/2Z.][
]
Grothendieck–Witt ring and motivic stable homotopy groups of spheres
Fabien Morel 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").
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. 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.
Generalizations
Witt groups can also be defined in the same way for skew-symmetric forms, and for quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
s, 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''2''k''(''R'') and even-dimensional quadratic ''L''-groups ''L''2''k''(''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
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while An ...
, forming one of the three terms of the surgery exact sequence.
See also
* Reduced height of a field
Notes
References
*
*
*
*
*
*
*
Further reading
* {{cite book , last=Balmer , first=Paul , chapter=Witt groups , editor1-last=Friedlander , editor1-first=Eric M. , editor2-last=Grayson , editor2-first=D. R. , title=Handbook of ''K''-theory , volume=2 , pages=539–579 , publisher=Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, year=2005 , isbn=3-540-23019-X , zbl=1115.19004
External links
Witt rings
in the Springer encyclopedia of mathematics
Quadratic forms