Some aspects of the algebraic theory of quadratic forms

30 Pages·2010·0.207 MB·English

by  R. Parimala

Some aspects of the algebraic theory of
quadratic forms
R. Parimala
March 14 – March 18, 2009
(Notes for lectures at AWS 2009)
There are many good references for this material including [EKM], [L],
[Pf] and [S].
1 Quadratic forms
Let k be a field with chark (cid:54)= 2.
Definition 1.1. Aquadratic formq: V → k onafinite-dimensionalvector
space V over k is a map satisfying:
1. q(λv) = λ2q(v) for v ∈ V, λ ∈ k.
2. The map b : V ×V → k, defined by
q
1
b (v,w) = [q(v +w)−q(v)−q(w)]
q
2
is bilinear.
We denote a quadratic form by (V,q), or simply as q.
The bilinear form b is symmetric; q determines b and for all v ∈ V,
q q
q(v) = b (v,v).
q
1
For a choice of basis {e ,...,e } of V, b is represented by a symmetric
1 n q
(cid:80)
matrix A(q) = (a ) with a = b (e ,e ). If v = X e ∈ V, X ∈ k,
ij ij q i j 1≤i≤n i i i
then
(cid:88) (cid:88) (cid:88)
q(v) = a X X = a X2 +2 a X X .
ij i j ii i ij i j
1≤i,j≤n 1≤i≤n i<j
Thus q is represented by a homogeneous polynomial of degree 2. Clearly,
every homogeneous polynomial of degree 2 corresponds to a quadratic form
on V with respect to the chosen basis. We define the dimension of q to be
the dimension of the underlying vector space V and denote it by dim(q).
Definition 1.2. Two quadratic forms (V ,q ), (V ,q ) are isometric if there
1 1 2 2
∼
is an isomorphism φ: V → V such that q (φ(v)) = q (v), ∀v ∈ V .
1 2 2 1 1
If A(q ), A(q ) are the matrices representing q and q with respect to
1 2 1 2
bases B and B of V and V respectively, φ yields a matrix T ∈ M (k),
1 2 1 2 n
n = dimV, such that
TA(q )Tt = A(q ).
2 1
In other words, the symmetric matrices A(q ) and A(q ) are congruent. Thus
1 2
isometry classes of quadratic forms yield congruence classes of symmetric
matrices.
Definition 1.3. The form q: V → k is said to be regular if b : V ×V → k
q
is nondegenerate.
Thus q is regular if and only if the map V → V∗ = Hom(V,k), defined by
v (cid:55)→ (w (cid:55)→ b (v,w)), is an isomorphism. This is the case if A(q) is invertible.
q
Henceforth, we shall only be concerned with regular quadratic forms.
Definition 1.4. Let W be a subspace of V and q: V → k be a quadratic
form. The orthogonal complement of W denoted W⊥ is the subspace
W⊥ = {v ∈ V : b (v,w) = 0 ∀ w ∈ W}.
q
Exercise 1.5. Let (V,q) be a regular quadratic form and W a subspace of
V. Then
1. dim(W)+dim(W⊥) = dim(V).
2. (W⊥)⊥ = W.
2
1.1 Orthogonal sums
Let (V ,q ), (V ,q ) be quadratic forms. The form
1 1 2 2
(V ,q ) ⊥ (V ,q ) = (V ⊕V ,q ⊥ q ),
1 1 2 2 1 2 1 2
with q ⊥ q defined by
1 2
(q ⊥ q )(v ,v ) = q (v )+q (v ), v ∈ V , v ∈ V
1 2 1 2 1 1 2 2 1 1 2 2
is called the orthogonal sum of (V ,q ) and (V ,q ).
1 1 2 2
1.2 Diagonalization
Let (V,q) be a quadratic form. There exists a basis {e ,...,e } of V such
1 n
that b (e ,e ) = 0 for i (cid:54)= j. Such a basis is called an orthogonal basis for q
q i j
and, with respect to an orthogonal basis, b is represented by a diagonal
q
matrix.
If {e ,...,e } is an orthogonal basis of q and q(e ) = d , we write q =
1 n i i
(cid:104)d ,...,d (cid:105). In this case, V = ke ⊕···⊕ke is an orthogonal sum and q|ke
1 n 1 n i
is represented by (cid:104)d (cid:105). Thus every quadratic form is diagonalizable.
i
1.3 Hyperbolic forms
Definition 1.6. A quadratic form (V,q) is said to be isotropic if there is a
nonzero v ∈ V such that q(v) = 0. It is anisotropic if q is not isotropic. A
quadratic form (V,q) is said to be universal if it represents every nonzero
element of k.
Example 1.7. The quadratic form X2 − Y2 is isotropic over k. Suppose
(V,q) is a regular form which is isotropic. Let v ∈ V be such that q(v) = 0,
v (cid:54)= 0. Since q is regular, there exists w ∈ V such that b (v,w) (cid:54)= 0. After
q
scaling we may assume b (v,w) = 1. If q(w) (cid:54)= 0, we may replace w by
q
w +λv, λ = −1q(w), and assume that q(w) = 0. Thus W = kv ⊕kw is a
2
2-dimensional subspace of V and q|W is represented by (0 1) with respect to
1 0
{v,w}.
3
Definition 1.8. A binary quadratic form isometric to (k2,(0 1)) is called a
1 0
hyperbolic plane. A quadratic form (V,q) is hyperbolic if it is isometric
to an orthogonal sum of hyperbolic planes. A subspace W of V such that q
restricts to zero on W and dimW = 1 dimV is called a Lagrangian.
2
Every regular quadratic form which admits a Lagrangian can easily be
seen to be hyperbolic.
Exercise 1.9. Let (V,q) be a regular quadratic form and (W,q|W) a regular
form on the subspace W. Then (V,q) →∼ (W,q|W) ⊥ (W⊥,q|W⊥).
Let (V,q) be a quadratic form. Then
V = {v ∈ V : b (v,w) = 0 ∀ w ∈ V}
0 q
is called the radical of V. If V is any complementary subspace of V in V,
1 0
then q|V is regular and (V,q) = (V ,0) ⊥ (V ,q|V ). Note that V is regular if
1 0 1 1
and only if the radical of V is zero. If (V,q) is any quadratic form, we define
the rank of q to be the dimension of V/V⊥. Of course if (V,q) is regular,
then rank(q) = dim(q).
Theorem 1.10 (Witt’s Cancellation Theorem). Let (V ,q ), (V ,q ), (V,q)
1 1 2 2
be quadratic forms over k. Suppose
∼
(V ,q ) ⊥ (V,q) = (V ,q ) ⊥ (V,q).
1 1 2 2
∼
Then (V ,q ) = (V ,q ).
1 1 2 2
The key ingredient of Witt’s cancellation theorem is the following.
Proposition 1.11. Let (V,q) be a quadratic form and v,w ∈ V with q(v) =
∼
q(w) (cid:54)= 0. Then there is an isometry τ: (V,q) → (V,q) such that τ(v) = w.
Proof. Let q(v) = q(w) = d (cid:54)= 0. Then
q(v +w)+q(v −w) = 2q(v)+2q(w) = 4d (cid:54)= 0.
Thus q(v + w) (cid:54)= 0 or q(v − w) (cid:54)= 0. For any vector u ∈ V with q(u) (cid:54)= 0,
define τ : V → V by
u
2b (z,u)u
q
τ (z) = z − .
u
q(u)
Then τ is an isometry called the reflection with respect to u.
u
Suppose q(v − w) (cid:54)= 0. Then τ : V → V is an isometry of V which
v−w
sends v to w. Suppose q(v +w) (cid:54)= 0. Then τ ◦τ sends v to w.
w v+w
4
Remark 1.12. The orthogonal group of (V,q) denoted by O(q) is the set of
isometries of V onto itself. This group is generated by reflections. This is
seen by an inductive argument on dim(q), using the above proposition.
Theorem 1.13 (Witt’sdecomposition). Let (V,q) be a quadratic form. Then
there is a decomposition
(V,q) = (V ,0) ⊥ (V ,q ) ⊥ (V ,q )
0 1 1 2 2
where V is the radical of q, q = q|V is anisotropic and q = q|V is hy-
0 1 1 2 2
perbolic. If (V,q) = (V ,0) ⊥ (W ,f ) ⊥ (W ,f ) with f anisotropic and f
0 1 1 2 2 1 2
hyperbolic, then
∼ ∼
(V ,q ) = (W ,f ), (V ,q ) = (W ,f ).
1 1 1 1 2 2 2 2
Remark 1.14. A hyperbolic form (W,f) is determined by dim(W); for if
dim(W) = 2n, (W,f) ∼= nH, where H = (k2,(0 1)) is the hyperbolic plane.
1 0
From now on, we shall assume (V,q) is a regular quadratic form. We
denote by q the quadratic form (V ,q ) in Witt’s decomposition which is
an 1 1
determined by q up to isometry. We call 1 dim(V ) the Witt index of q. Thus
2 2
∼
any regular quadratic form q admits a decomposition q = q ⊥ (nH), with
an
q anisotropic and H denoting the hyperbolic plane. We also sometime
an
denote by Hn the sum of n hyperbolic planes.
2 Witt group of forms
2.1 Witt groups
We set
W(k) = {isomorphism classes of regular quadratic forms over k}/ ∼
where the Witt equivalence ∼ is given by:
there exist r, s ∈ Z such that
(V ,q ) ∼ (V ,q ) ⇐⇒ .
1 1 2 2 (V ,q ) ⊥ Hr ∼= (V ,q ) ⊥ Hs
1 1 2 2
5
W(k) is a group under orthogonal sum:
[(V ,q )] ⊥ [(V ,q )] = [(V ,q ) ⊥ (V ,q )].
1 1 2 2 1 1 2 2
The zero element in W(k) is represented by the class of hyperbolic forms.
For a regular quadratic form (V,q), (V,q) ⊥ (V,−q) has Lagrangian
W = {(v,v) : v ∈ V}
so that (V,q) ⊥ (V,−q) ∼= Hn, n = dim(V). Thus, [(V,−q)] = −[(V,q)] in
W(k).
It follows from Witt’s decomposition theorem that every element in W(k)
is represented by a unique anisotropic quadratic form up to isometry. Thus
W(k) may be thought of as a group made out of isometry classes of aniso-
tropic quadratic forms over k.
The abelian group W(k) admits a ring structure induced by tensor prod-
∼
uct on the associated bilinear forms. For example, if q = (cid:104)a ,...,a (cid:105) and q
1 1 n 2
∼
is a quadratic form, then q ⊗q = a q ⊥ a q ⊥ ··· ⊥ a q .
1 2 1 2 2 2 n 2
Definition 2.1. Let I(k) denote the ideal of classes of even-dimensional
quadratic forms in W(k). The ideal I(k) is called the fundamental ideal.
In(k) stands for the nth power of the ideal I(k).
Definition 2.2. Let P (k) denote the set of isomorphism classes of forms of
n
the type
(cid:104)(cid:104)a ,...,a (cid:105)(cid:105) := (cid:104)1,a (cid:105)⊗···⊗(cid:104)1,a (cid:105).
1 n 1 n
Elements in P (k) are called n-fold Pfister forms.
n
The ideal I(k) is generated by the forms (cid:104)1,a(cid:105), a ∈ k∗. Moreover, the
ideal In(k) is generated additively by n-fold Pfister forms. For instance, for
n = 2, the generators of I2(k) are of the form
∼
(cid:104)a,b(cid:105)⊗(cid:104)c,d(cid:105) = (cid:104)1,ac,ad,cd(cid:105)−(cid:104)1,cd,−bc,−bd(cid:105) = (cid:104)(cid:104)ac,ad(cid:105)(cid:105)−(cid:104)(cid:104)cd,−bc(cid:105)(cid:105)
Example 2.3. If k = C, every 2-dimensional quadratic form over k is
isotropic.
W(k) ∼= Z/2Z
[(V,q)] (cid:55)→ dim(V) (mod 2)
is an isomorphism.
6
Example 2.4. If k = R, every quadratic form q is represented by
(cid:104)1,...,1,−1,...,−1(cid:105)
with respect to an orthogonal basis. The number r of +1’s and the number
s of −1’s in the diagonalization above are uniquely determined by the iso-
morphism class of q. The signature of q is defined as r − s. The signature
yields a homomorphism sgn: W(R) → Z which is an isomorphism.
2.2 Quadratic forms over p-adic fields
Let k be a finite extension of the field Q of p-adic numbers. We call k a non-
p
dyadic p-adic field if p (cid:54)= 2. The field k has a discrete valuation v extending
the p-adic valuation on Q . Let π be a uniformizing parameter for v and κ
p
the residue field for v. The field κ is a finite field of characteristic p (cid:54)= 2. Let
u be a unit in k∗ such that u ∈ κ is not a square. Then
k∗/k∗2 = {1,u,π,uπ}.
Since κ is finite, every 3-dimensional quadratic form over κ is isotropic. By
Hensel’s lemma, every 3-dimensional form (cid:104)u ,u ,u (cid:105) over k, with u units
1 2 3 i
in k is isotropic. Since every form q in k has a diagonal representation
(cid:104)u ,...,u (cid:105) ⊥ π(cid:104)v ,...,v (cid:105),
1 r 1 s
if r or s exceeds 3, q is isotropic. In particular every 5-dimensional quadratic
form over k is isotropic. Further, up to isometry, there is a unique quadratic
form in dimension 4 which is anisotropic, namely,
(cid:104)1,−u,−π,uπ(cid:105).
This is the norm form of the unique quaternion division algebra H(u,π)
over k (cf. section 2.3).
2.3 Central simple algebras and the Brauer group
Recall that a finite-dimensional algebra A over a field k is a central simple
algebra over k if A is simple (has no two-sided ideals) and the center of A
is k. Recall also that for a field k,
Br(k) = {Isomorphism classes of central simple algebras over k}/ ∼
7
∼
where the Brauer equivalence ∼ is given by: A ∼ B if and only if M (A) =
n
M (B) for some integers m,n. The pair (Br(k),⊗) is a group. The inverse
m
of [A] is [Aop] where Aop is the opposite algebra of A: the multiplication
structure, ∗, on Aop is given by a ∗ b = ba. We have a k-algebra isomor-
phism φ: A⊗Aop −∼→ End (A) induced by φ(a⊗b)(c) = acb. The identity
k
element in Br(k) is given by [k]. By Wedderburn’s theorem on central sim-
ple algebras, the elements of Br(k) parametrize the isomorphism classes of
finite-dimensional central division algebras over k.
For elements a,b ∈ k∗, we define the quaternion algebra H(a,b) to be
the 4-dimensional central simple algebra over k generated by {i,j} with the
relations i2 = a, j2 = b, ij = −ji. This is a generalization of the standard
Hamiltonian quaternion algebra H(−1,−1). The algebra H(a,b) admits a
canonical involution¯: H(a,b) → H(a,b) given by
α+iβ +jγ +ijδ = α−iβ −jγ −ijδ
This involution gives an isomorphism H(a,b) ∼= H(a,b)op; in particular,
H(a,b) has order 2 in Br(k), where Br(k) denotes the 2-torsion subgroup of
2 2
theBrauergroupofk. ThenormformforthisalgebraisgivenbyN(x) = xx,
which is a quadratic form on H(a,b) represented with respect to the orthog-
onal basis {1,i,j,ij} by (cid:104)1,−a,−b,ab(cid:105) = (cid:104)(cid:104)−a,−b(cid:105)(cid:105).
2.4 Classical invariants for quadratic forms
Let (V,q) be a regular quadratic form. We define dim(q) = dim(V) and
dim (q) = dim(V) modulo 2. We have a ring homomorphism dim : W(k) →
2 2
Z/2Z. We note that I(k) is the kernel of dim . This gives an isomorphism
2
dim : W(k)/I(k) −∼→ Z/2Z.
2
Let disc(q) = (−1)n(n−1)/2[det(A(q))] ∈ k∗/k∗2. Since A(q) is deter-
mined up to congruence, det(A(q)) is determined modulo squares. We have
disc(H) = 1, and disc(q) induces a group homomorphism
disc: I(k) → k∗/k∗2
which is clearly onto. It is easy to verify that ker(disc) = I2(k). Thus the
discriminant homomorphism induces an isomorphism I(k)/I2(k) → k∗/k∗2.
8
The next invariant for quadratic forms is the Clifford invariant. To each
quadraticform(V,q)wewishtoconstructacentralsimplealgebracontaining
V whose multiplication on elements of V satisfies v ·v = q(v). The smallest
such algebra (defined by a universal property) will be the Clifford algebra.
Definition 2.5. The Clifford algebra C(q) of the quadratic form (V,q) is
T(V)/I , whereI isthetwo-sidedidealinthetensoralgebraT(V)generated
q q
by {v ⊗v −q(v),v ∈ V}.
The algebra C(q) has a Z/2Z gradation C(q) = C (q)⊕C (q) induced by
0 1
the gradation T(V) = T (V)⊕T (V), where
0 1
(cid:77) (cid:77)
T (V) = V⊗i and T (V) = V⊗i.
0 1
i≥0,ieven i≥1,iodd
If dim(q) is even, then C(q) is a central simple algebra over k. If dim(q)
is odd, C (q) is a central simple algebra over k. The Clifford algebra C(q)
0
comes equipped with an involution τ defined by τ(v) = −v, v ∈ V. Thus, if
dim(q) is even, C(q) determines a 2-torsion element in Br(k).
Definition 2.6. The Clifford invariant c(q) of (V,q) in Br(k) is defined as
(cid:40)
[C(q)], if dim(q) is even
c(q) =
[C (q)], if dim(q) is odd
0
TheCliffordinvariantinducesahomomorphismc: I2(k) → Br(k), Br(k)
2 2
again denoting the 2-torsion in the Brauer group of k. The very first case of
theMilnorconjecture(seesection3)states: cissurjectiveandker(c) = I3(k).
Theorem 2.7 (Merkurjev [M1]). The map c induces an isomorphism
I2(k)/I3(k) ∼= Br(k)
2
Example 2.8. Let q ∼= ⊗n (cid:104)(cid:104)−a ,−b (cid:105)(cid:105) ∈ I2(k). Then
i=1 i i
∼
c(q) = ⊗ H
1≤i≤n i
where H = H(a ,b ).
i i i
9
(cid:78)
Exercise 2.9. Given H , a tensor product of n quaternion algebras
1≤i≤n i
over k, show that there is a quadratic form q over k of dimension 2n+2 such
∼ (cid:78)
that c(q) = H .
1≤i≤n i
Thus the image of I2(q) in Br(k) is spanned by quaternion algebras. It
2
was a longstanding question whether Br(k) is spanned by quaternion alge-
2
bras. Merkurjev’s theorem answers this question in the affirmative; further,
it gives precise relations between quaternion algebras in Br(k).
2
3 Galois cohomology and the Milnor conjec-
ture
¯ ¯
Let Γ = Gal(k|k), k denoting the separable closure of k, be the absolute
k
Galois group of k. The group
Γ = lim Gal(L|k)
k ←−
L⊂k¯,L|k finiteGalois
is a profinite group. A discrete Γ -module M is a continuous Γ -module for
k k
the discrete topology on M and the profinite topology on Γ . For a discrete
k
Γ -module M, we define Hn(k,M) as the direct limit of the cohomology of
k
the finite quotients
Hn(k,M) = lim Hn(Gal(L|k),MΓL).
−→
L⊂k¯,L|k finiteGalois
Suppose char(k) (cid:54)= 2 and M = µ . The module µ has trivial Γ action.
2 2 k
We denote this module by Z/2Z. We have
H0(k,Z/2Z) = Z/2Z
H1(k,Z/2Z) ∼= k∗/k∗2
H2(k,Z/2Z) ∼= Br(k)
2
These can be seen from the Kummer exact sequence of Γ -modules:
k
0 −→ µ −→ k¯∗ −·→2 k¯∗ −→ 0
2
and noting that H1(Γ ,k¯∗) = 0 (Hilbert’s Theorem 90) and H2(Γ ,k¯∗) =
k k
Br(k).
10