A LOCAL-GLOBAL PRINCIPLE FOR ALGEBRAS WITH INVOLUTION AND HERMITIAN FORMS

DAVID W. LEWIS AND THOMAS UNGER

Abstract. Weakly hyperbolic involutions are introduced and a proof is given of the following local-global principle: a central simple algebra with involution of any kind is weakly hyperbolic if and only if its signature is zero for all orderings of the ground field. Also, the order of a weakly hyperbolic algebra with involution is a power of two, this being a direct consequence of a result of Scharlau. As a corollary an analogue of Pfister’s local-global principle is obtained for the Witt group of hermitian forms over an algebra with involution.

1. Introduction

Pfister’s well-known local-global principle states that a nonsingular quadratic form q over a field k (which we assume to be of characteristic different from 2) is a torsion element in the Witt ring W (k) of k if and only if the signature of q is zero for all orderings of k. Furthermore, every torsion element of W (k) has 2-power order.

If W (A, σ) is the Witt group of hermitian forms over some central simple kalgebra with involution of any kind (A, σ), then Scharlau showed that the torsion elements of W (A, σ) have 2-power order.

In this paper, we complement Scharlau’s result by showing that h ∈ W (A, σ) is a torsion element if and only if h has signature zero for all orderings of the ground field, thus obtaining an analogue of Pfister’s local-global principle for hermitian forms. In fact, this will follow from our main theorem which states that if (A, σ) is a central simple k-algebra with involution of any kind, then the signature of σ is zero for all orderings of k if and only if (A, σ) is weakly hyperbolic. As a consequence of Scharlau’s theorem we also get that the order of (A, σ) is a power of 2 when (A, σ) is weakly hyperbolic.

Weakly hyperbolic involutions are a natural generalization of torsion forms and are a special case of weakly isotropic involutions. The latter were studied in [6] as an ingredient of a different local-global principle for algebras with an involution of the first kind.

Date: November 21, 2001.

2000 Mathematics Subject Classification. 16K20, 11E39.

Key words and phrases. Central simple algebras, involutions, hermitian forms, local-global principles.

All involutions on central simple algebras considered in this paper are arbitrary and all forms (quadratic, hermitian, etc.) are assumed to be nonsingular. Standard references are [5] and [10] for the theory of quadratic forms, [4] for central simple algebras with an involution and [8] for real fields.

2. Preliminaries

Let K be a field of characteristic different from two and let A be a central simple K-algebra, equipped with an involution σ. We say that σ is of the first kind if σ|_{K} = 1_{K} and of the second kind otherwise. If σ is of the second kind, let k denote its fixed field. Then K is a quadratic étale extension field of k, i.e. either K = k(\sqrt{α}) for some α ∈ k^{×} or K = k × k. In the latter case, A is no longer simple. Unless stated otherwise, we will always assume that K is a field, for reasons which will become apparent later. To cater for both kinds of involution, we take k as the base field and simply say that (A, σ) is a central simple k-algebra with involution, it being understood that the center is k or K, depending on the situation.

By Wedderburn’s theorem, we can write

A ≅ End_{D}(V),

where D is a central division algebra over k (with center K) and V is some finite dimensional right D-vector space. The degree of A is deg A := \sqrt{dim_{K} A} and the (Schur) index of A is ind A := deg D. We say that A is split if D = K. Any field extension L ⊃ k such that A ⊗_{k} L is split, is called a splitting field of A. In particular, the separable closure k_{s} of k is a splitting field of A.

Let n = deg A. If σ is of the first kind, σ is called orthogonal (resp. symplectic) if the extended involution σ ⊗ 1_{k_{s}} on A ⊗_{k} k_{s} ≅ M_{n}(k_{s}) is adjoint to a symmetric (resp. skew-symmetric) bilinear form. Involutions of the second kind are also called unitary involutions.

It is well-known that σ is the adjoint involution of some non-singular ε-hermitian form (ε = ±1) h : V × V → D, with respect to some involution ϑ on D which is of the same kind as σ. (If σ is of the second kind, then ε = +1 and ϑ(α) = σ(α)∀α ∈ K.) So σ is of the form σ_{h}, where σ_{h} is implicitly defined by

h(x, f(y)) = h(σ_{h}(f)(x), y) for x, y ∈ V and f ∈ End_{D}(V).

The form h is uniquely determined up to multiplication with a ϑ-invariant factor in K^{×}. (More generally, we can substitute a central simple algebra E for D and a finitely generated right E-module M for V, see [4, 4.A].)

2.1. Hyperbolic Involutions. A right ideal I of A is called isotropic if σ(I)I = 0. The algebra with involution (A, σ), or the involution σ itself, is called isotropic if A contains an isotropic right ideal and anisotropic if σ(x)x = 0 implies x = 0, ∀x ∈ A. We also say that (A, σ) is weakly isotropic if there exist nonzero x_{1}, …, x_{n} ∈ A such that σ(x_{1})x_{1} + ⋯ + σ(x_{n})x_{n} = 0 and strongly anisotropic otherwise.

The notion of a hyperbolic involution was first defined by Bayer-Fluckiger et al. [2] (see also [4, 6.B]): an algebra with involution (A, σ), or the involution σ itself, is called hyperbolic if either the center of A is isomorphic to k × k, or σ is the adjoint involution of some hyperbolic ε-hermitian form. In both cases (A, σ) is hyperbolic if and only if A contains an idempotent e such that σ(e) = 1 − e. This is also equivalent with the existence of an isotropic right ideal I of dimension dim_{k} I = \frac{1}{2} dim_{k} A.

Note that hyperbolic involutions remain hyperbolic under arbitrary scalar extensions and if (A, σ) is hyperbolic and (B, τ) is arbitary, then (A ⊗_{k} B, σ ⊗ τ) is hyperbolic.

2.2. Signatures of Involutions. Suppose that k is a real field. We fix some notation for the rest of this paper. We denote the space of orderings of k by X_{k}. If P ∈ X_{k}, k_{P} will denote the real closure of k with respect to P. Furthermore, we set A_{P} := A ⊗_{k} k_{P}, σ_{P} = σ ⊗ 1_{k_{P}}, etc.

Let P ∈ X_{k}. The signature of an involution was defined by Lewis and Tignol [7] for involutions of the first kind and by Quéguiner [9] for involutions of the second kind as

sig_{P} σ = \begin{cases}\sqrt{sig_{P} T_{σ}} & if σ is of the first kind, \\ \sqrt{\frac{1}{2} sig_{P} T_{σ}} & if σ is of the second kind. \end{cases}

Here T_{σ} denotes the trace quadratic form T_{σ}(x) := Trd_{A}(σ(x)x), ∀x ∈ A, which takes values in k.

If (A, σ) is split with orthogonal involution, (A, σ) ≅ (End_{k}(V), σ_{q}), q being a quadratic form over k, then Lewis and Tignol showed that

sig_{P} σ_{q} = | sig_{P} q|.

Likewise, if (A, σ) is split with unitary involution, (A, σ) ≅ (End_{K}(V), σ_{h}), h being a K/k-hermitian form, then Quéguiner showed that

sig_{P} σ_{h} = | sig_{P} h|.

Clearly, the signature of σ will be zero in the split-symplectic case.

2.3. The n-fold Orthogonal Sum. Let f be a quadratic or K/k-hermitian form, then the notion of n-fold orthogonal sum of f,

⊥^{n} f = n × f = ⟨\underbrace{1, …, 1}_{n}⟩ ⊗ f,

can be extended to the realm of algebras with an involution in a straightforward way:

Definition 2.1. Let (A, σ) be a central simple k-algebra with involution of any kind. The n-fold orthogonal sum ⊞^{n}(A, σ) is defined by

⊞^{n}(A, σ) := (M_{n}(K), ∗) ⊗_{k} (A, σ),

with ∗ the conjugate transpose involution, defined by (a_{ij})^{∗} = (ι(a_{ij}))^{t}, ∀(a_{ij}) ∈ M_{n}(K), where ι is either the nontrivial automorphism of K if σ is of the second kind or the identity otherwise. It is clearly again a central simple k-algebra of degree n · deg A.

Remark 2.2. When A is split, the involution ∗ ⊗ σ will be the adjoint of n × f, for an appropriate form f.

Remark 2.3. If (A, σ) is a central simple k-algebra with involution of the first kind, the n-fold orthogonal sum ⊞^{n}(A, σ) was defined in a more intrinsic way in [12], so as to conform with Dejaiffe’s [3] construction of an orthogonal sum of two Morita equivalent algebras with involution of the first kind.

3. The Local-Global Principle for Algebras with Involution

Definition 3.1. The algebra with involution (A, σ) is called weakly hyperbolic if there exists an n ∈ ℕ such that ⊞^{n}(A, σ) is hyperbolic. The order of (A, σ) is the least integer n such that ⊞^{n}(A, σ) is hyperbolic.

The main theorem of this paper reads:

Theorem 3.2. Let (A, σ) be a central simple k-algebra with involution of any kind. Then sig_{P} σ = 0, ∀P ∈ X_{k} if and only if (A, σ) is weakly hyperbolic. Furthermore, the order of (A, σ) is a power of two when (A, σ) is weakly hyperbolic.

Note that when k is not real, there is no notion of signature and (M_{n}(k), t) will be hyperbolic for some positive integer n, so that (A, σ) will be weakly hyperbolic. The main theorem will thus hold trivially. Hence, we assume from now on that k is a real field. In order to prove this theorem, we first need to establish a few lemmas.

Lemma 3.3. The main theorem holds when A is split; A ≅ End_{k}(V).

Proof. When A is split, σ will be the adjoint of a quadratic, a skew-symmetric bilinear, or a K/k-hermitian form f. If f is quadratic, we have proof by Pfister’s local-global principle. If f is skew-symmetric, the statement is trivial. If f is K/k-hermitian, f is completely determined by its trace form q_{f}(x) := f(x, x) ∈ k, ∀x ∈ V (Jacobson’s theorem, see [10, Theorem 10.1.1]), in which case we have proof again by Pfister’s local-global principle.

Lemma 3.4. Let (A, σ) be a central simple k-algebra with involution of any kind and let L be a real closed extension field of k such that A_{L} ≅ M_{n}((−1, −1)_{L}) for some positive integer n. Then the main theorem holds for (A_{L}, σ_{L}).

Proof. If σ_{L} is symplectic, then σ_{L} is the conjugate transpose involution t, where

is quaternion conjugation. In this case, (A_{L}, σ_{L}) is totally anisotropic and σ_{L} is positive definite, so the lemma holds.

If σ_{L} is orthogonal, then σ_{L} is the conjugate transpose involution ˆ t, where ˆ is the orthogonal involution on (−1, −1)_{L}, determined by ˆ1 = 1, î = −i, ĵ = j, k̂ = k. Now sig σ_{L} = 0 and ⊞^{2}(A_{L}, σ_{L}) is hyperbolic, since ⊞^{2}((−1, −1)_{L}, ˆ ) is hyperbolic (the element e := \frac{1}{2}( \begin{matrix}1 & j^{−1} \\ j & 1 \end{matrix} ) satisfies e^{2} = e and ê^{t} = 1 − e.)

If σ_{L} is unitary, then the center of A_{L} is L(\sqrt{−1}), which is algebraically closed, since L is real. Hence A_{L} ≅ M_{2n}(L(\sqrt{−1})) and σ_{L} is the conjugate transpose involution ∗, defined earlier. Now (A_{L}, σ_{L}) is totally anisotropic and σ_{L} is positive definite, so that we are done.

Lemma 3.5. Suppose σ is an anisotropic involution of any kind on a central simple k-algebra A and let L = k(\sqrt{δ}) be a quadratic field extension of k. If the involution σ_{L} = σ ⊗ 1_{L} on A ⊗_{k} L is hyperbolic, then there exists r ∈ A such that r^{2} = δ and σ(r) = −r.

This is in fact [2, Lemma 3.2] which holds for involutions of any kind, not just for those of the first kind (as it is stated in [2]). Another result from [2] which holds for arbitrary involutions, is their Theorem 3.3:

Theorem 3.6. Let (A, σ) be a central simple k-algebra with involution of any kind and let L = k(\sqrt{δ}) be a quadratic field extension of k. If there exists r ∈ A such that r^{2} = δ and σ(r) = −r, then (A ⊗_{k} L, σ_{L}) is hyperbolic. The converse holds except in the case where A is split, σ is orthogonal and its associated quadratic form has odd Witt index.

Proof. If σ is an involution of the first kind, we refer to [2, Theorem 3.3] for a proof, which can be made to work for involutions of the second kind quite easily, as we will proceed to do now. (The crucial ingredients, Witt cancellation and Witt decomposition, also work in this situation.) So, suppose that σ is unitary and note that the exceptional case does not occur now. Recall that K = k(\sqrt{α}) for some α ∈ k^{×}.

Given r, let t = δ^{−1}r ⊗ \sqrt{δ} ∈ A ⊗_{k} L. Note that t^{2} = 1 and σ_{L}(t) = −t. Therefore, the element e = \frac{1}{2}(1 + t) satisfies e^{2} = e and σ_{L}(e) = 1 − e, so that (A ⊗_{k} L, σ_{L}) is hyperbolic.

Conversely, let V be an irreducible left A-module and D = End_{A}(V), so that A = End_{D}(V). Choose an involution of the second kind on D and let h be a hermitian form on V with respect to which σ is the adjoint involution.

Consider a Witt decomposition

(V, h) ≅ (V_{1}, h_{1}) ⊥ (V_{0}, h_{0}),

where (V_{1}, h_{1}) is hyperbolic and (V_{0}, h_{0}) is anisotropic. Let σ_{i} denote the adjoint involution with respect to h_{i} on End_{D}(V_{i}) for i = 0, 1. By Witt cancellation, (V_{0}, h_{0}) becomes hyperbolic over L, hence Lemma 3.5 yields an element r_{0} ∈ End_{D}(V_{0}) such that r_{0}^{2} = δ and σ_{0}(r_{0}) = −r_{0}.

By [2, Theorem 2.2], End_{D}(V_{1}) contains a σ_{1}-invariant subalgebra M such that (M, σ_{1}|_{M}) ≅ (M_{2}(K), Θ), where Θ is defined by

Θ ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}\overline{d} & \overline{b} \\ \overline{c} & \overline{a} \end{matrix} ) , ∀a, b, c, d ∈ K.

Let r_{1} ∈ M ⊂ End_{D}(V_{1}) denote the image of ( \begin{matrix}0 & δ\sqrt{α} \\ \frac{1}{\sqrt{α}} & 0 \end{matrix} ), then σ_{1}(r_{1}) = −r_{1}

and r_{1}^{2} = δ.

Finally, regarding V = V_{1} ⊕ V_{0}, define r ∈ A = End_{D}(V) by

r(x, y) = (r_{1}(x), r_{0}(y)).

It is straightforward to check that σ(r) = −r and r^{2} = δ.

Proof of Theorem 3.2. Suppose first that (B, τ) := ⊞^{n}(A, σ) is hyperbolic for some n. Since hyperbolic involutions remain hyperbolic under arbitrary scalar extensions, (B_{P} , τ_{P}) is hyperbolic for every ordering P of k. Since (B_{P} , τ_{P}) is either split or a matrix algebra over (−1, −1)_{k_{P}} and since signatures do not change under scalar extension, we obtain sig_{P} τ = 0, ∀P ∈ X_{k} from Lemmas 3.3 and 3.4. This implies sig_{P} σ = 0, ∀P ∈ X_{k}.

Conversely, suppose for the sake of contradiction that (A, σ) has zero signature at each ordering of the field k, but that (A, σ) is not weakly hyperbolic. By Zorn’s Lemma, there exists a maximal algebraic extension field L of k for which (A_{L}, σ_{L}) is not weakly hyperbolic, see Remark 3.8.

Observe that L must be real, because if not, then (M_{n}(L), t) will be hyperbolic for some positive integer n, so that ⊞^{n}(A_{L}, σ_{L}) is hyperbolic. Also, L cannot be a splitting field for A, because the local-global principle works in the split case (Lemma 3.3).

Next we will show that L is euclidean by showing that ±1 are the only two square classes in L. Suppose for the sake of contradiction that there exists α ∈ L such that ±α is not a square in L. Then L(\sqrt{α}) and L(\sqrt{−α}) are each quadratic extension fields of L. By maximality of L we have that (A, σ) must become weakly hyperbolic on extension to each of these two fields. Thus there exists an integer n for which the n-fold orthogonal sum of (A, σ) becomes hyperbolic over both L(\sqrt{α}) and L(\sqrt{−α}). Let us write (B, τ) := ⊞^{n}(A_{L}, σ_{L}).

Applying Theorem 3.6 to the two quadratic extensions above, yields two elements r, s ∈ B such that r^{2} = α, τ(r) = −r, s^{2} = −α and τ(s) = −s. Now we will show that the sum of two copies of (B, τ) is hyperbolic (which is a contradiction and hence L must be euclidean). To show this, note that the sum of two

copies of (B, τ) is (M_{2}(B), t ⊗ τ) and let e = \frac{1}{2} ( \begin{matrix}1 & rs^{−1} \\ sr^{−1} & 1 \end{matrix} ). Then it is easy to check that e^{2} = e and t ⊗ τ(e) = 1 − e. Therefore, the involution t ⊗ τ is hyperbolic. Thus we have proved that L is euclidean.

In fact, L is real closed, since L cannot have any proper odd degree extensions by [4, Corollary 6.16]. But then Q := (−1, −1)_{L} is the only division algebra over L. Hence, A_{L} is an endomorphism algebra over Q (since L is not a splitting field of A). Since the local-global principle holds for such algebras (Lemma 3.4), we have a contradiction and conclude that (A, σ) has to be weakly hyperbolic.

The second statement is a direct consequence of Scharlau’s result [11] that the torsion in the Witt group of central simple algebras with involution is 2primary.

Remark 3.7. Whenever there exists an invertible element y ∈ B with τ(y) = −y^{−1} one can get that the sum of two copies of (B, τ) is hyperbolic by defining

e = \frac{1}{2} ( \begin{matrix}1 & y^{−1} \\ y & 1 \end{matrix} ). When B is split and τ is the adjoint of some quadratic form

ϕ, the condition τ(y) = −y^{−1} reduces to the fact that ϕ is isometric to −ϕ, which of course implies that the sum of two copies of ϕ is hyperbolic.

Remark 3.8. The reasoning behind the use of Zorn’s Lemma is analogous to what happens in the proof of the “(iv) ⟹ (v)” direction of [10, Theorem 2.7.1]: Let \overline{k} be an algebraic closure of k and let M be the set of all intermediate fields M such that A_{M} := A ⊗_{k} M is not weakly hyperbolic. The set M is ordered by inclusion. For a totally ordered chain of subfields M_{i} in M, we let A_{i} := A ⊗_{k} M_{i} and M := ⋃ M_{i}.

Assume for the sake of contradiction that (A_{M}, σ_{M}) is weakly hyperbolic. This means that there exists a square matrix X with entries in A_{M} such that X^{2} = X and σ_{M} ⊗ ∗(X) = I − X. These are matrix equations with finitely many coefficients from A_{M}. So all coefficients must lie in A_{i} for some i. Thus M_{i} would not belong to M, a contradiction. Therefore M ∈ M and we have found an upper bound. By Zorn’s Lemma, M has a maximal element.

4. The Local-Global Principle for Hermitian Forms

Let k be a real closed field and let K be either k(\sqrt{−1}) or (−1, −1)_{k}. Let W (K, ) denote the Witt group of hermitian forms over K with respect to the canonical involution on K (i.e. the nontrivial automorphism of k(\sqrt{−1}) or quaternion conjugation). Given h ∈ W (K, ), let q_{h} denote the trace form of h, defined by q_{h}(x) := h(x, x) for all x ∈ K. In both cases q_{h} takes values in the ground field k. Let ⟨α_{1}, …, α_{n}⟩ be a diagonalization of h. Then α_{i} ∈ k^{×} (1 ≤ i ≤ n) in both cases and

q_{h} = \begin{cases}2 × ⟨α_{1}, …, α_{n}⟩ & if K = k(\sqrt{−1}), \\ 4 × ⟨α_{1}, …, α_{n}⟩ & if K = (−1, −1)_{k}. \end{cases}

As is well-known (see e.g. [10, 10.1.6]), the signature of ⟨α_{1}, …, α_{n}⟩ is an invariant of the hermitian form h in both cases, aptly called the signature of h. To summarize,

sig h = \begin{cases}\frac{1}{2} sig q_{h} & if K = k(\sqrt{−1}), \\ \frac{1}{4} sig q_{h} & if K = (−1, −1)_{k}. \end{cases}

Now let k be a real field and let (A, σ) be a central simple k-algebra (with involution of any kind). Let W (A, σ) denote the Witt group of hermitian forms over A with respect to σ. The previous observations allow us to define the signature of h with respect to the ordering P, sig_{P} h, by going over to the real closure k_{P} of k with respect to P and using Morita theory. We will not elaborate this here, but refer instead to [1] for a recent exposition. It turns out that sig_{P} h can be defined in a meaningful way in the following situations:

• σ unitary;

• σ orthogonal and A_{P} split;

• σ symplectic and A_{P} ≅ matrices over (−1, −1)_{k_{P}}. In the other cases, sig_{P} h is defined to be equal to zero. (This includes the situation where K_{P} = k_{P} × k_{P}, which, together with the fact that algebras with involution with center k × k are by definition hyperbolic, prompted us to exclude this possibility from our considerations.)

The main point for us is the observation that

sig_{P} h = 0  ⟺  sig_{P} σ_{h} = 0,

where σ_{h} is the adjoint involution of h. This can be deduced quite easily from the work of Lewis–Tignol and Quéguiner mentioned earlier.

Given h ∈ W (A, σ), it is now a simple matter of going over to the adjoint involution σ_{h} on End_{A}(V), in order to obtain the following consequence of our main theorem:

Theorem 4.1. Let h ∈ W (A, σ). Then h is a torsion element if and only if sig_{P} h = 0 for all P ∈ X_{k}. Every torsion element of W (A, σ) has 2-power order.

The first statement of the theorem can be translated into a familiar looking exact sequence:

0 ⟶ W_{t}(A, σ) ⟶ W (A, σ) ⟶^{(sig_{P})} ∏_{P ∈X_{k}} W (A_{P} , σ_{P}),

where W_{t}(A, σ) denotes the torsion subgroup of W (A, σ) and W (A_{P} , σ_{P}) is either 0, ℤ or ℤ/2ℤ, depending on what happens at the ordering P.

We note again that the second statement of the theorem is nothing new. It is due to Scharlau [11], a fact which we already acknowledged in the proof of our main theorem. To be more precise, Scharlau proved that if A is a finitedimensional semisimple k-algebra with involution σ such that σ|_{k} = 1_{k}, then W_{t}(A, σ) is 2-primary [11, Theorem 5.1(i)]. If we denote the Grothendieck group of hermitian forms over (A, σ) by Ŵ(A, σ) in this situation, then the following exact sequence can be deduced from [11, Theorem 5.2]:

0 ⟶ Ŵ_{t}(A, σ) ⟶ Ŵ(A, σ)⟶^{r^{∗}} ∏_{P ∈X_{k}} Ŵ(A_{P} , σ_{P}),

where r^{∗} is the canonical extension homomorphism.

Acknowledgements

Part of this research is funded by the TMR research network (ERB FMRX CT-97-0107) on “K-theory and algebraic groups”.

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Department of Mathematics, University College Dublin, Belfield, Dublin 4, IRELAND

E-mail address: david.lewis@ucd.ie

Department of Mathematics, University College Dublin, Belfield, Dublin 4, IRELAND

E-mail address: thomas.unger@ucd.ie