SIGNATURES OF HERMITIAN FORMS AND THE KNEBUSCH TRACE FORMULA

VINCENT ASTIER AND THOMAS UNGER

Abstract. Signatures of quadratic forms have been generalized to hermitian forms over algebras with involution. In the literature this is done via Morita theory, which causes sign ambiguities in certain cases. In this paper, a hermitian version of the Knebusch Trace Formula is established and used as a main tool to resolve these ambiguities.

1. Introduction

In this paper we study signatures of hermitian forms over central simple algebras with involution of any kind, defined over formally real fields. These generalize the classical signatures of quadratic forms.

Following [3] we do this via extension to real closures and Morita equivalence. This leads to the notion of M-signature of hermitian forms in Section 3.2. We study its properties, make a detailed analysis of the impact of choosing different real closures and different Morita equivalences and show in particular that sign changes can occur. This motivates the search for a more intrinsic notion of signature, where such sign changes do not occur.

In Section 3.3 we define such a signature, the H-signature, which only depends on the choice of a tuple of hermitian forms, mimicking the fact that in quadratic form theory the form ⟨1⟩ always has positive signature. The H-signature generalizes the definition of signature in [3] and is in particular well-defined when the involution becomes hyperbolic after scalar extension to a real closure of the base field, addressing an issue with the definition proposed in [3].

Our main tool is a generalization of the Knebusch Trace Formula to M-signatures of hermitian forms, which we establish in Section 5. In Section 7 we show that the total H-signature of a hermitian form is a continuous map and in Section 8 we prove the Knebusch Trace Formula for H-signatures.

2. Preliminaries

2.1. Algebras with Involution. The general reference for this section is [11, Chapter I].

2000 Mathematics Subject Classification. 16K20, 11E39, 13J30.

Key words and phrases. Central simple algebras, involutions, hermitian forms, signatures, Knebusch frace formula.

Let F be a field of characteristic different from 2. An F-algebra with involution is a pair (A, σ) consisting of a finite-dimensional simple F-algebra A with centre a field K = Z(A) and an F-linear involution σ : A → A. Either σ is of the first kind, in which case K = F and σ|_{K} = id_{K}, or σ is of the second kind, in which case K = F( \sqrt{d}) is a quadratic field extension of F and σ|_{K} is the nontrivial element in the Galois group of K/F.

Consider the F-subspaces

Sym(A, σ) = {a ∈ A | σ(a) = a} and Skew(A, σ) = {a ∈ A | σ(a) = −a}

of A. Then A = Sym(A, σ) ⊕ Skew(A, σ). Assume that σ is of the first kind. Then dim_{F}(A) = m^{2} for some positive integer m. Furthermore, σ is either orthogonal (or, of type +1) if dim_{F} Sym(A, σ) = m(m + 1)/2, or symplectic (or, of type −1) if dim_{F} Sym(A, σ) = m(m − 1)/2. If σ is of the second kind, then dim_{F}(A) = 2m^{2} for some positive integer m and dim_{F} Sym(A, σ) = dim_{F} Skew(A, σ) = m^{2}. Involutions of the second kind are also called unitary.

Let σ and τ be two involutions on A that have the same restriction to K. By the Skolem-Noether theorem they differ by an inner automorphism:

τ = Int(u) ∘ σ

for some u ∈ A^{×}, uniquely determined up to a factor in F^{×}, such that σ(u) = u if σ and τ are both orthogonal, both symplectic or both unitary and σ(u) = −u if one of σ, τ is orthogonal and the other symplectic. Here Int(u)(x) := uxu^{−1} for x ∈ A.

2.2. ε-Hermitian Spaces and Forms. The general references for this section are [10, Chapter I] and [20, Chapter 7], both for rings with involution. Treatments of the central simple and division cases can also be found in [6] and [14], respectively.

Let (A, σ) be an F-algebra with involution. Let ε ∈ {−1, 1}. An ε-hermitian space over (A, σ) is a pair (M, h), where M is a finitely generated right A-module (which is automatically projective since A is semisimple) and h : M × M −→ A is a sesquilinear form such that h(y, x) = εσ(h(x, y)) for all x, y ∈ M. We call (M, h) a hermitian space when ε = 1 and a skew-hermitian space when ε = −1. If (A, σ) is a field equipped with the identity map, we say (skew-) symmetric bilinear space instead of (skew-) hermitian space.

Consider the left A-module M^{∗} = Hom_{A}(M, A) as a right A-module via the involution σ. The form h induces an A-linear map h^{∗} : M → M^{∗}, x ↦ h(x, ·). We call (M, h) nonsingular if h^{∗} is an isomorphism. All spaces occurring in this paper are assumed to be nonsingular. We often simply write h instead of (M, h) and speak of a form instead of a space.

If A = D is a division algebra (so that M ≃ D^{n} for some integer n) such that (D, σ, ε) (F, id_{F}, −1), then h can be diagonalized: there exist invertible elements a_{1}, …, a_{n} ∈ Sym(D, σ) such that, after a change of basis,

h(x, y) = ∑_{i=1}^{n} σ(x_{i})a_{i}y_{i}, ∀x, y ∈ D^{n}.

In this case we use the shorthand notation

h = ⟨a_{1}, …, a_{n}⟩_{σ},

which resembles the usual notation for diagonal quadratic forms. If A is not a division algebra we can certainly consider diagonal hermitian forms defined on free A-modules of finite rank, but some hermitian forms over (A, σ) may not be diagonalizable.

Witt cancellation and Witt decomposition hold for ε-hermitian spaces over (A, σ). Let W_{ε}(A, σ) denote the Witt group of Witt classes of ε-hermitian spaces over (A, σ). When ε = 1 we drop the subscript and simply write W(A, σ). We denote the usual Witt ring of F by W(F). We find it convenient to identify forms over (A, σ) with their classes in W_{ε}(A, σ).

Lemma 2.1. Let (A, σ) be an F-algebra with involution.

(i) If σ is of the first kind, then W_{−1}(A, σ) ≃ W(A, τ) for some involution τ of opposite type to σ.

(ii) If σ is of the second kind, then W_{−1}(A, σ) ≃ W(A, σ).

Proof. (i) Let u ∈ A^{×} be such that σ(u) = −u and define τ := Int(u) ∘ σ. Let h be a skew-hermitian form over (A, σ). Then uh is a hermitian form over (A, τ). The one-to-one correspondence h ↦ uh respects isometries, orthogonal sums and hyperbolicity and so induces the indicated isomorphism.

(ii) Let Z(A) = K = F( \sqrt{d}) for some d ∈ F^{×}. Then σ( \sqrt{d}) = − \sqrt{d}. The one-to-one correspondence h ↦ \sqrt{d}h induces the indicated isomorphism.

2.3. Adjoint Involutions. The general reference for this section is [11, 4.A].

Let (A, σ) be an F-algebra with involution. Let (M, h) be an ε-hermitian space over (A, σ). The algebra End_{A}(M) is again central simple over F since M is finitely generated [11, 1.10]. The involution ad_{h} on End_{A}(M), defined by

h(x, f (y)) = h(ad_{h}(f)(x), y), ∀x, y ∈ M, ∀f ∈ End_{A}(M)

is called the adjoint involution of h. The involutions σ and ad_{h} are of the same kind and σ(α) = ad_{h}(α) for all α ∈ K. In case ad_{h} and σ are of the first kind we also have

type(ad_{h}) = ε type(σ).

Furthermore, every involution on End_{A}(M) is of the form ad_{h} for some ε-hermitian form h over (A, σ) and the correspondence between ad_{h} and h is unique up to a multiplicative factor in F^{×} in the sense that ad_{h} = ad_{λh} for every λ ∈ F^{×}.

Let (A, σ) be an F-algebra with involution. By a theorem of Wedderburn there exists a division algebra D (unique up to isomorphism) with centre Z(A) and a finitedimensional right D-vector space V such that A ≃ End_{D}(V). Thus A ≃ M_{m}(D) for some positive integer m. Furthermore there exists an involution ϑ on D of the same kind as σ and an ε_{0}-hermitian form φ_{0} over (D, ϑ) with ε_{0} ∈ {−1, 1} such that (A, σ) and (End_{D}(V), ad_{φ_{0}}) are isomorphic as algebras with involution. In matrix form ad_{φ_{0}} is described as follows:

ad_{φ_{0}}(X) = Φ_{0}ϑ^{t}(X)Φ_{0}^{−1}, ∀X ∈ M_{m}(D),

where ϑ^{t}(X) := (ϑ(x_{ij}))^{t} for X = (x_{ij}) and Φ_{0} ∈ GL_{m}(D) is the Gram matrix of φ_{0}. Thus ϑ^{t}(Φ_{0}) = ε_{0}Φ_{0}.

2.4. Hermitian Morita Theory. We refer to [2, §1], [5], [6, Chapters 2–3], [10, Chapter I, §9], or [13] for more details.

Let (M, h) be an ε-hermitian space over (A, σ). One can show that the algebras with involution (End_{A}(M), ad_{h}) and (A, σ) are Morita equivalent: for every μ ∈ {−1, 1} there is an equivalence between the categories ℋ _{μ}(End_{A}(M), ad_{h}) and ℋ _{εμ}(A, σ) of non-singular μ-hermitian forms over (End_{A}(M), ad_{h}) and non-singular εμ-hermitian forms over (A, σ), respectively (where the morphisms are given by isometry), cf. [10, Chapter I, Theorem 9.3.5]. This equivalence respects isometries, orthogonal sums and hyperbolic forms. It induces a group isomorphism

W_{μ}(End_{A}(M), ad_{h}) ≃ W_{εμ}(A, σ).

The Morita equivalence and the isomorphism are not canonical.

The algebras with involution (A, σ) and (D, ϑ) are also Morita equivalent. An example of such a Morita equivalence is obtained by composing the following three non-canonical equivalences of categories, the last two of which we will call scaling and collapsing. For computational purposes we describe them in matrix form. We follow the approach of [17]:

ℋ _{ε}(A, σ) ⟶ ℋ _{ε}(M_{m}(D), ad_{φ_{0}}) ⟶^{scaling} ℋ _{ε_{0}ε}(M_{m}(D), ϑ^{t}) ⟶^{collapsing} ℋ _{ε_{0}ε}(D, ϑ). (1)

Scaling: Let (M, h) be an ε-hermitian space over (M_{m}(D), ad_{φ_{0}}). Scaling is given by

(M, h) ⟼→ (M, Φ_{0}^{−1}h).

Note that Φ_{0}^{−1} is only determined up to a scalar factor in F^{×} since ad_{φ_{0}} = ad_{λφ_{0}} for any λ ∈ F^{×} and that replacing Φ_{0} by λΦ_{0} results in a different Morita equivalence.

Collapsing: Recall that M_{m}(D) ≃ End_{D}(D^{m}) and that we always have M ≃ (D^{m})^{k} ≃ M_{k, m}(D) for some integer k. Let h : M × M −→ M_{m}(D) be an ε_{0}ε-hermitian form with respect to ϑ^{t}. Then

h(x, y) = ϑ^{t}(x)By, ∀x, y ∈ M_{k, m}(D),

where B ∈ M_{k}(D) satisfies ϑ^{t}(B) = ε_{0}εB, so that B determines an ε_{0}ε-hermitian form b over (D, ϑ). Collapsing is then given by

(M, h) ⟼→ (D^{k}, b).

3. Signatures of Hermitian Forms

3.1. Signatures of Forms: the Real Closed Case. Let R be a real closed field, C = R( \sqrt{−1}) (which is algebraically closed) and H = (−1, −1)_{R} Hamilton’s quaternion division algebra over R. We recall the definitions of signature for the various types of forms (all assumed to be nonsingular) over R, C, R × R and H. We will use them in the definition of the M-signature of a hermitian form over (A, σ) in Section 3.2.

(a) Let b be a symmetric bilinear (or quadratic) form over R. Then b ≃ ⟨α_{1}, …, α_{n}⟩ for some n ∈ and α_{i} ∈ {−1, 1}. We let

signb := ∑_{i=1}^{n} α_{i}.

By Sylvester’s Law of Inertia, signb is well-defined.

(b) Let b be a skew-symmetric form over R. Then b is hyperbolic and we let

signb := 0.

(c) Let h be a hermitian form over (C, −), where \overline{\sqrt{−1}} = − \sqrt{−1}. Then h ≃ ⟨α_{1}, …, α_{n}⟩_{−} for some n ∈ and α_{i} ∈ {−1, 1}. By a theorem of Jacobson [7], h is up to isometry uniquely determined by the symmetric bilinear form b_{h} := 2 × ⟨α_{1}, …, α_{n}⟩ defined over R. We let

signh := \frac{1}{2} signb_{h} = sign⟨α_{1}, …, α_{n}⟩.

(d) Let h be a hermitian form over (R × R, ), where { \atop (x, y)} = (y, x) is the exchange involution. Then h is a torsion form [19, p. 43] and we let

signh := 0.

(e) Let h be a hermitian form over (H, −), where − denotes quaternion conjugation. Then h ≃ ⟨α_{1}, …, α_{n}⟩_{−} for some n ∈ and α_{i} ∈ {−1, 1}. By a theorem of Jacobson [7], h is up to isometry uniquely determined by the symmetric bilinear form b_{h} := 4 × ⟨α_{1}, …, α_{n}⟩ defined over R. We let

signh := \frac{1}{4} signb_{h} = sign⟨α_{1}, …, α_{n}⟩.

(f) Let h be a skew-hermitian form over (H, −), where − denotes quaternion conjugation. Then h is a torsion form [20, Chapter 10, Theorem 3.7] and we let

signh := 0.

Remark 3.1. The signature maps defined in (a), (c) and (e) above give rise to unique group isomorphisms

W(R) ≃ , W(C, −) ≃ , and W(H, −) ≃

such that sign⟨1⟩ = 1, sign⟨1⟩_{−} = 1 and sign⟨1⟩_{−} = 1, respectively. In addition, we have the group isomorphisms

W_{−1}(R, id_{R}) = 0, W_{−1}(R × R, ) ≃ W_{−1}(H, −) ≃ /2 .

See also [8, Chapter I, 10.5] and [19, p. 43].

3.2. The M-Signature of a Hermitian Form. Our approach in this section is inspired by [3, §3.3,§3.4]. We only consider hermitian forms over (A, σ), cf. Lemma 2.1.

Let F be a formally real field and let (A, σ) be an F-algebra with involution. Consider an ordering P ∈ X_{F}, the space of orderings of F. By a real closure of F at P we mean a field embedding ι : F → K, where K is real closed, ι(P) ⊆ K^{2} and K is algebraic over ι(F).

Let h be a hermitian form over (A, σ). Choose a real closure ι : F → F_{P} of F at P, and use it to extend scalars from F to F_{P}:

W(A, σ) −→ W(A ⊗_{F} F_{P}, σ ⊗ id), h ⟼→ h ⊗ F_{P} := (id_{A}⊗ ι)^{∗}(h),

where the tensor product is along ι. The extended algebra with involution (A⊗_{F}F_{P}, σ⊗ id) is Morita equivalent to an F_{P}-algebra with involution (D_{P}, ϑ_{P}), chosen as follows:

(i) If σ is of the first kind, D_{P} is equal to one of F_{P} or H_{P} := (−1, −1)_{F_{P}}. Furthermore, we may choose (D_{P}, ϑ_{P}) = (F_{P}, id_{F_{P}}) in the first case and (D_{P}, ϑ_{P}) = (H_{P}, −) in the second case by Morita theory (scaling).

(ii) If σ is of the second kind, recall that Z(A) = K = F( \sqrt{d}). Now if d <_{P} 0, then D_{P} is equal to F_{P}( \sqrt{−1}), whereas if d >_{P} 0, then D_{P} is equal to F_{P} × F_{P} and A ⊗_{F} F_{P} is a direct product of two simple algebras. Furthermore, we may choose (D_{P}, ϑ_{P}) = (F_{P}( \sqrt{−1}), −) in the first case and (D_{P}, ϑ_{P}) = (F_{P} × F_{P}, ) in the second case, again by Morita theory (scaling).

Note that ϑ_{P} is of the same kind as σ in each case.

The extended involution σ⊗id_{F_{P}} is adjoint to an ε_{P}-hermitian form φ_{P} over (D_{P}, ϑ_{P}) where ε_{P} = −1 if one of σ and ϑ_{P} is orthogonal and the other is symplectic, whereas ε_{P} = 1 if σ and ϑ_{P} are of the same type, i.e. both orthogonal, symplectic or unitary.

Now choose any Morita equivalence

ℳ : ℋ (A ⊗_{F} F_{P}, σ ⊗ id) −→ ℋ _{ε_{P}}(D_{P}, ϑ_{P})  (2)

with (D_{P}, ϑ_{P}) ∈ \{(F_{P}, id_{F_{P}}), (H_{P}, −), (F_{P}( \sqrt{−1}), −), (F_{P} × F_{P}, )\}, which exists by the analysis above. This Morita equivalence induces an isomorphism, which we again denote by ℳ,

ℳ : W(A ⊗_{F} F_{P}, σ ⊗ id) −→^{∼} W_{ε_{P}}(D_{P}, ϑ_{P}).  (3)

Definition 3.2. Let P ∈ X_{F}. Fix a real closure ι : F → F_{P} of F at P and a Morita equivalence ℳ as above. Define the M-signature of h at (ι, ℳ), denoted sign_{ι}^{ℳ} h, as follows:

sign_{ι}^{ℳ} h := signℳ (h ⊗ F_{P}),

where signℳ (h ⊗ F_{P}) can be computed as shown in Section 3.1.

This definition relies on two choices: firstly the choice of the real closure ι : F → F_{P} of F at P and secondly the choice of the Morita equivalence ℳ. Note that there is no canonical choice for ℳ. We now study the dependence of the M-signature on the choice of ι and ℳ.

Let ι_{1} : F → L_{1} and ι_{2} : F → L_{2} be two real closures of F at P, and let (D_{1}, ϑ_{1}) and ε_{1} play the role of (D_{P}, ϑ_{P}) and ε_{P}, respectively, obtained above when ι is replaced by ι_{1}. Let

ℳ_{1} : ℋ (A ⊗_{F} L_{1}, σ ⊗ id) −→ ℋ _{ε_{1}}(D_{1}, ϑ_{1})

be a fixed Morita equivalence. By the Artin-Schreier theorem [20, Chapter 3,Theorem 2.1] there is a unique isomorphism ρ : L_{1} → L_{2} such that ρ ∘ ι_{1} = ι_{2}. It extends to an isomorphism id ⊗ ρ : (A ⊗_{F} L_{1}, σ ⊗ id) → (A ⊗_{F} L_{2}, σ ⊗ id). The isomorphism ρ also extends canonically to D_{1} ∈ \{L_{1}, (−1, −1)_{L_{1}}, L_{1}( \sqrt{−1}), L_{1} × L_{1}\}. Consider the L_{2}-algebra with involution (D_{2}, ϑ_{2}) := (ρ(D_{1}), ρ ∘ ϑ_{1} ∘ ρ^{−1}). We define ρ(ℳ_{1}) to be the Morita equivalence from (A ⊗_{F} L_{2}, σ ⊗ id) to (D_{2}, ϑ_{2}), described by the following diagram:

\xymatrix{ℋ (A ⊗_{F} L_{1}, σ ⊗ id) \ar[r]^{ℳ_{1}} \ar[d]^{(id⊗ρ)^{∗}} & ℋ _{ε_{1}}(D_{1}, ϑ_{1}) \ar[d]^{ρ^{∗}} \\ ℋ (A ⊗_{F} L_{2}, σ ⊗ id) \ar[r]^{ρ(ℳ_{1})} & ℋ _{ε_{1}}(D_{2}, ϑ_{2})}

Proposition 3.3 (Change of Real Closure). With notation as above we have for every h ∈ W(A, σ),

signℳ_{1}(h ⊗ L_{1}) = signρ(ℳ_{1})(h ⊗ L_{2}),

in other words

sign_{ι_{1}}^{ℳ_{1}} h = sign_{ι_{2}}^{ρ(ℳ_{1})} h.

Proof. The statement is trivially true when ε_{1} = −1, by cases (b), (d) and (f) in Section 3.1, so we may assume that ε_{1} = 1. Consider the diagram

\xymatrix{ & W(A ⊗_{F} L_{1}, σ ⊗ id) \ar[r]^{ℳ_{1}}_{∼} \ar[dd]^{(id⊗ρ)^{∗}} & W(D_{1}, ϑ_{1}) \ar[dd]^{ρ^{∗}} \ar[dr]^{sign} \\ W(A, σ) \ar[ur]^{(id⊗ι_{1})^{∗}} \ar[dr]_{(id⊗ι_{2})^{∗}} &  &  &  \\  & W(A ⊗_{F} L_{2}, σ ⊗ id) \ar[r]^{ρ(ℳ_{1})}_{∼} & W(D_{2}, ϑ_{2}) \ar[ur]_{sign}}

The left triangle commutes by the definition of ρ. The square commutes by the definition of ρ(ℳ_{1}). The right triangle commutes since ρ^{∗}(⟨1⟩_{ϑ_{1}}) = ⟨1⟩_{ϑ_{2}} and by Remark 3.1. The statement follows.

Proposition 3.4 (Change of Morita Equivalence). Let ℳ_{1} and ℳ_{2} be two different Morita equivalences as in (2). Then there exists δ ∈ {−1, 1} such that for every h ∈ W(A, σ),

sign_{ι}^{ℳ_{1}} h = δ sign_{ι}^{ℳ_{2}} h.

Proof. Note that ε_{1} = ε_{2}. The statement is trivially true when ε_{1} = −1, by cases (b),

(d) and (f) in Section 3.1, so we may assume that ε_{1} = 1. The two different Morita equivalences give rise to two different group isomorphisms

m_{i} : W(A ⊗_{F} F_{P}, σ ⊗ id) −→^{∼} (i = 1, 2),

by (3) and Remark 3.1. The map m_{1} ∘ m_{2}^{−1} is an automorphism of and is therefore equal to id_{}or −id_{}.

Propositions 3.3 and 3.4 immediately imply

Corollary 3.5. Let ι_{1} : F → L_{1} and ι_{2} : F → L_{2} be two real closures of F at P and let ℳ_{1} and ℳ_{2} be two different Morita equivalences as in (2). Then there exists δ ∈ {−1, 1} such that for every h ∈ W(A, σ),

sign_{ι_{1}}^{ℳ_{1}} h = δ sign_{ι_{2}}^{ℳ_{2}} h.

The following result easily follows from the properties of Morita equivalence:

Proposition 3.6.

(i) Let h be a hyperbolic form over (A, σ), then

sign_{ι}^{ℳ} h = 0.

(ii) Let h_{1} and h_{2} be hermitian forms over (A, σ), then

sign_{ι}^{ℳ} (h_{1} ⊥ h_{2}) = sign_{ι}^{ℳ} h_{1} + sign_{ι}^{ℳ} h_{2}.

(iii) The M-signature at (ι, ℳ), sign_{ι}^{ℳ} , induces a homomorphism of additive groups

W(A, σ) −→ .

(iv) Let h be a hermitian form over (A, σ) and q a quadratic form over F, then

sign_{ι}^{ℳ} (q ⊗ h) = sign_{P}q · sign_{ι}^{ℳ} h,

where sign_{P}q denotes the usual signature of the quadratic form q at P.

Definition 3.7. Let h be a hermitian form over (A, σ). From Definition 3.2 and Section 3.1 it follows that sign_{ι}^{ℳ} h is automatically zero whenever P belongs to the following subset of X_{F}, which we call set of nil-orderings:

Nil[A, σ] := \begin{cases}\{P ∈ X_{F} | D_{P} = H_{P}\} & if σ is orthogonal \\ \{P ∈ X_{F} | D_{P} = F_{P}\} & if σ is symplectic \\ \{P ∈ X_{F} | D_{P} = F_{P} × F_{P}\} & if σ is unitary \end{cases} ,

where the square brackets indicate that Nil[A, σ] depends only on the Brauer class of A and the type of σ.

3.3. The H-Signature of a Hermitian Form. It follows from Corollary 3.5 that sign_{ι}^{ℳ} is uniquely defined up to a choice of sign. We can arbitrarily choose the sign of the signature of a form at each ordering P. See for instance Remark 3.13 for a way to change sign using Morita equivalence (scaling).

A more intrinsic definition is therefore desirable, in particular when considering the total signature map of a hermitian form X_{F} → since such arbitrary changes of sign would prevent it from being continuous. We are thus led to define a signature that is independent of the choice of ι and ℳ.

Lemma 3.8. Let P ∈ X_{F} \ Nil[A, σ]. Let ι_{1} : F → L_{1} and ι_{2} : F → L_{2} be two real closures of F at P and let ℳ_{1} and ℳ_{2} be two different Morita equivalences as in (2). Let h_{0} ∈ W(A, σ) be such that sign_{ι_{1}}^{ℳ_{1}} h_{0} 0 and let δ_{k} ∈ {−1, 1} be the sign of sign_{ι_{k}}^{ℳ_{k}} h_{0} for k = 1, 2. Then

δ_{1} sign_{ι_{1}}^{ℳ_{1}} h = δ_{2} sign_{ι_{2}}^{ℳ_{2}} h,

for all h ∈ W(A, σ).

Proof. Let δ ∈ {−1, 1} be as in Corollary 3.5. Then, we have for all h ∈ W(A, σ) that sign_{ι_{2}}^{ℳ_{2}} h = δ sign_{ι_{1}}^{ℳ_{1}} h and in particular that sign_{ι_{2}}^{ℳ_{2}} h_{0} = δ sign_{ι_{1}}^{ℳ_{1}} h_{0}. It follows that δ_{1} = δδ_{2}. Thus δ_{1} sign_{ι_{1}}^{ℳ_{1}} h = δδ_{2} sign_{ι_{1}}^{ℳ_{1}} h = δ_{2} sign_{ι_{2}}^{ℳ_{2}} h.

We will show in Theorem 6.4 that there exists a finite tuple H = (h_{1}, …, h_{s}) of diagonal hermitian forms of rank one over (A, σ) such that for every P ∈ X_{F}\Nil[A, σ] there exists h_{0} ∈ H such that sign_{ι}^{ℳ} h_{0} 0.

Definition 3.9. Let h ∈ W(A, σ) and let P ∈ X_{F}. We define the H-signature of h at P as follows: If P ∈ Nil[A, σ], define sign_{P}^{H} h := 0. If P Nil[A, σ], let i ∈ {1, …, s} be the least integer such that sign_{ι}^{ℳ} h_{i} 0 (for any ι and ℳ, cf. Corollary 3.5), let δ_{ι,ℳ} ∈ {−1, 1} be the sign of sign_{ι}^{ℳ} h_{i} and define

sign_{P}^{H} h := δ_{ι,ℳ} sign_{ι}^{ℳ} h.

Lemma 3.8 shows that this definition is independent of the choice of ι and ℳ (but it does depend on the choice of H).

A choice of Morita equivalence which is convenient for computations of signatures is given by (1) with (A, σ) replaced by (A ⊗_{F} F_{P}, σ ⊗ id). We denote this Morita equivalence by 𝒩 and now describe the induced isomorphisms of Witt groups:

\xymatrix{W(A ⊗_{F} F_{P}, σ ⊗ id) \ar[r]^{ξ_{P}^{∗}} & W(M_{m}(D_{P}), ad_{φ_{P}}) \ar[r]^{scaling} & W(M_{m}(D_{P}), ϑ_{P}^{t}) \ar[r]^{collapsing} & W(D_{P}, ϑ_{P}) \\ h ⊗ F_{P} \ar@{|->}[r] & ξ_{P}^{∗}(h ⊗ F_{P}) \ar@{|->}[r] & Φ_{P}^{−1}ξ_{P}^{∗}(h ⊗ F_{P}) \ar@{|->}[r] & 𝒩 (h ⊗ F_{P}),}

(4)

where h is a hermitian form over (A, σ), P ∈ X_{F} \ Nil[A, σ] (so that ε_{P} = 1), ξ_{P}^{∗} is the group isomorphism induced by some fixed isomorphism

ξ_{P} : (A ⊗_{F} F_{P}, σ ⊗ id) −→^{∼} (M_{m}(D_{P}), ad_{φ_{P}}),

and Φ_{P} is the Gram matrix of the form φ_{P}. Observe that signφ_{P} can be computed as in Section 3.1.

Lemma 3.10. Let P ∈ X_{F} \ Nil[A, σ], let ι : F → F_{P} be a real closure of F and let 𝒩 and φ_{P} be as above. Then sign_{ι}^{𝒩} ⟨1⟩_{σ} = signφ_{P}.

Proof. We extend scalars from F to F_{P} via ι, ⟨1⟩_{σ} ⟼→ ⟨1⟩_{σ} ⊗F_{P} = ⟨1⊗1⟩_{σ⊗id} and push ⟨1 ⊗ 1⟩_{σ⊗id} through the sequence (4),

⟨1 ⊗ 1⟩_{σ⊗id} ⟼→ ξ_{P}^{∗}(⟨1 ⊗ 1⟩_{σ⊗id} ) = ⟨ξ_{P}(1 ⊗ 1)⟩_{ad_{φ_{P}}} = ⟨I_{m}⟩_{ad_{φ_{P}}} ⟼→ Φ_{P}^{−1}⟨I_{m}⟩_{ad_{φ_{P}}} = ⟨Φ_{P}^{−1}⟩_{ϑ_{P}}^{t}.

(Note that ξ_{P}(1 ⊗ 1) = I_{m}, the m × m-identity matrix in M_{m}(D_{P}) since ξ_{P} is an algebra homomorphism.) By collapsing, the matrix Φ_{P}^{−1} now corresponds to a quadratic form over F_{P}, a hermitian form over (F_{P}( \sqrt{−1}), −) or a hermitian form over (H_{P}, −). In either case Φ_{P}^{−1} is congruent to Φ_{P}. Thus sign_{ι}^{𝒩} ⟨1⟩_{σ} = signφ_{P}.

Remark 3.11. It follows from Lemma 3.10 that the signature defined in [3, §3.3, §3.4] is actually sign_{P}^{H} with H = (⟨1⟩_{σ}). It is now clear that this signature cannot be computed when sign_{ι}^{𝒩} ⟨1⟩_{σ} = 0, i.e. when σ ⊗ id_{F_{P}} ≃ ad_{φ_{P}} is hyperbolic. In contrast, if we take H = (h_{1}, …, h_{s}), as described before Definition 3.9 we are able to compute the signature in all cases. Note that we may choose h_{1} = ⟨1⟩_{σ}, so that Definition 3.9 generalizes the definition of signature in [3, §3.3, §3.4].

Example 3.12. Let (A, σ) = (M_{4}( ), ad_{φ}), where φ = ⟨1, −1, 1, −1⟩. Then signφ = 0. Consider the rank one hermitian forms

h = ⟨( \begin{matrix}1 \\ & 1 \\ & & −1 \\ & & & 1 \end{matrix})⟩_{σ} , h_{1} = ⟨( \begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix})⟩_{σ} , and h_{2} = ⟨( \begin{matrix}1 \\ & −1 \\ & & 1 \\ & & & −1 \end{matrix} )⟩_{σ}

over (A, σ). Then sign^{𝒩} h = −2, sign^{𝒩} h_{1} = 0 and sign^{𝒩} h_{2} = 4, where we suppressed the index ι since is real closed. Let H_{1} = (h_{1}) and H = (h_{1}, h_{2}), then sign^{H_{1}} h is not defined, whereas sign^{H} h = −2. Observe that taking H = (h_{1}, −h_{2}) instead would result in sign^{H} h = 2.

Remark 3.13. Let a ∈ A^{×} be such that σ(a) = εa with ε ∈ {−1, 1}. The Morita equivalence scaling by a,

ℋ (A, σ) −→ ℋ _{ε}(A, Int(a) ∘ σ), h ⟼→ ah

induces an isomorphism

ζ_{a} : W(A, σ) −→ W_{ε}(A, Int(a) ∘ σ), h ⟼→ ah.

It is clear that

sign_{ι}^{ℳ} h = sign_{ι}^{ℳ ∘(ζ_{a}^{−1}⊗id)} ah.

Consider the special case where a ∈ F^{×}. Thus ε = 1 and Int(a) ∘ σ = σ. Assume that a <_{P} 0. Then

sign_{ι}^{ℳ} ζ_{a}(h) = signℳ (ah ⊗ F_{P}) = signℳ (−h ⊗ F_{P}) = − sign_{ι}^{ℳ} h,

where the last equality follows from Proposition 3.6(iii). The same computation shows that sign_{P}^{H} ζ_{a}(h) = − sign_{P}^{H} h for any choice of H. Thus, scaling by a changes the sign of the signature, which is contrary to what is claimed in [3, p. 662].

Remark 3.14. For any choice of H, P and ι as in Definition 3.9, there exists a Morita equivalence ℳ′ such that sign_{P}^{H} h = sign_{ι}^{ℳ′} h for any h ∈ W(A, σ) (i.e. such that sign_{ι}^{ℳ′} h_{i} > 0 with h_{i} as in Definition 3.9). Indeed, for ℳ as in Definition 3.9, it suffices to take ℳ′ = δ_{ι,ℳ} ℳ.

It remains to be shown that a tuple H as described just before Definition 3.9 does exist. In order to reach this conclusion we first need to develop more theory in Sections 4 and 5.

4. Signatures of Involutions

Let (A, σ) be an F-algebra with involution. Consider the involution trace form

T_{σ} : A × A −→ K, (x, y) ⟼→ Trd_{A}(σ(x)y),

where Trd_{A} denotes the reduced trace of A. If σ is of the first kind, T_{σ} is a symmetric bilinear form over K = F. If σ is of the second kind, T_{σ} is a hermitian form over (K, σ|_{K}). Let P ∈ X_{F}. The signature of the involution σ at P is defined by

sign_{P}σ := \sqrt{sign_{P}T_{σ}}

and is a nonnegative integer, since sign_{P}T_{σ} is always a square; cf. Lewis and Tignol [15] for involutions of the first kind and Quéguiner [18] for involutions of the second kind. We call the involution σ positive at P if sign_{P}σ 0.

Example 4.1.

(i) Let (A, σ) = (M_{n}(F), t), where t denotes transposition. Then T_{σ} ≃ n^{2}×⟨1⟩. Hence sign_{P}σ = n for all P ∈ X_{F}.

(ii) Let (A, σ) = ((a, b)_{F}, −), where − denotes quaternion conjugation. Then T_{σ} ≃ ⟨2⟩ ⊗ ⟨1, −a, −b, ab⟩. Hence sign_{P}σ = 2 for all P ∈ X_{F} such that a <_{P} 0, b <_{P} 0 and sign_{P} = 0 for all other P ∈ X_{F}.

(iii) Let (A, σ) = (F( \sqrt{d}), −), where − denotes conjugation. Then T_{σ} ≃ ⟨1⟩_{σ}. We have sign_{P}⟨1⟩_{σ} = \frac{1}{2} sign_{P}⟨1, −d⟩, cf. [20, Chapter 10, Examples 1.6(iii)]. Hence sign_{P}σ = 0 for all P ∈ X_{F} such that d >_{P} 0 and sign_{P}σ = 1 for all P ∈ X_{F} such that d <_{P} 0.

Remark 4.2. Let (A, σ) and (B, τ) be two F-algebras with involution.

(i) Consider the tensor product (A⊗_{F}B, σ⊗τ). Then T_{σ⊗τ} ≃ T_{σ} ⊗T_{τ} and so sign_{P}(σ⊗ τ) = (sign_{P}σ)(sign_{P}τ) for all P ∈ X_{F}.

(ii) If (A, σ) ≃ (B, τ), then T_{σ} ≃ T_{τ} so that sign_{P}σ = sign_{P}τ for all P ∈ X_{F}.

Remark 4.3. Pfister’s local-global principle holds for algebras with involution (A, σ) and also for hermitian forms h over such algebras, [16].

Remark 4.4. The map signσ is continuous from X_{F} (equipped with the Harrison topology, see [12, Chapter VIII 6] for a definition) to (equipped with the discrete topology). Indeed: define the map √ on by setting \sqrt{k} = −1 if k is not a square in . Since is equipped with the discrete topology, this map is continuous. Since T_{σ} is a symmetric bilinear form or a hermitian form over (K, σ|_{K}), the map signT_{σ} is continuous from X_{F} to (by [12, VIII, Proposition 6.6] and [20, Chapter 10, Example 1.6(iii)]). Thus, by composition, signσ = \sqrt{signT_{σ}} is continuous from X_{F} to .

Lemma 4.5. Let P ∈ X_{F}. If P ∈ Nil[A, σ], then

sign_{P}σ = signφ_{P} = 0.

Otherwise,

sign_{P}σ = λ_{P}|signφ_{P}|,

where λ_{P} = 1 if (D_{P}, ϑ_{P}) = (F_{P}, id_{F_{P}}) or (D_{P}, ϑ_{P}) = (F_{P}( \sqrt{−1}), −) and λ_{P} = 2 if (D_{P}, ϑ_{P}) = (H_{P}, −).

Proof. This is a reformulation of [15, Theorem 1] and part of its proof for involutions of the first kind and [18, Proposition 3] for involutions of the second kind.

Lemma 4.6. Let (M, h) be a hermitian space over (A, σ), let P ∈ X_{F}, let ι : F → F_{P} be a real closure of F at P and let ℳ be a Morita equivalence as in (2). If P ∈ Nil[A, σ], then

sign_{P}ad_{h} = sign_{ι}^{ℳ} h = 0.

Otherwise,

sign_{P}ad_{h} = λ_{P}|sign_{ι}^{ℳ} h|,

with λ_{P} as defined in Lemma 4.5. In particular,

sign_{ι}^{ℳ} h = 0 ⇔ sign_{P}ad_{h} = 0.

Proof. Assume first that P ∈ Nil[A, σ]. Then sign_{ι}^{ℳ} h = 0. Consider the adjoint involution ad_{h} on End_{A}(M). Since h is hermitian, σ and ad_{h} are of the same type. Furthermore, A and End_{A}(M) are Brauer equivalent by [11, 1.10]. Thus Nil[A, σ] = Nil[End_{A}(M), ad_{h}]. By Lemma 4.5 we conclude that sign_{P}ad_{h} = 0.

Next, assume that P ∈ X_{F} \ Nil[A, σ]. Without loss of generality we may replace F by F_{P}. Consider a Morita equivalence

ℳ′: ℋ (A, σ) −→ ℋ (D, ϑ)

with (D, ϑ) = (F, id), (D, ϑ) = (H, −) or (D, ϑ) = (F( \sqrt{−1}), −). Let (N, b) be the hermitian space over (D, ϑ) corresponding to (M, h) under ℳ′. Then sign^{ℳ′} h = signb. By [2, Remark 1.4.2] we have (End_{A}(M), ad_{h}) ≃ (End_{D}(N), ad_{b}) so that signad_{h} = signad_{b}. By [15, Theorem 1] and [18, Proposition 3] we have signad_{b} = λ |signb| with λ = 1 if (D, ϑ) = (F, id) or (D, ϑ) = (F( \sqrt{−1}), −) and λ = 2 if (D, ϑ) = (H, −). We conclude that signad_{h} = λ |sign^{ℳ′} h| = λ |sign^{ℳ} h|, where the last equality follows from Corollary 3.5.

5. The Knebusch Trace Formula for M-Signatures

We start with two preliminary sections in order not to overload the proof of Theorem 5.1 below.

5.1. Hermitian Forms over a Product of Rings with Involution. Let

(A, σ) = (A_{1}, σ_{1}) × ⋯ × (A_{t}, σ_{t}),

where A, A_{1}, …, A_{t} are rings and σ, σ_{1}, …, σ_{t} are involutions. We write an element a ∈ A indiscriminately as (a_{1}, …, a_{t}) or a_{1}+⋯ +a_{t}with a_{i} ∈ A_{i} for i = 1, …, t. Writing 1_{A} = (e_{1}, …, e_{t}), the elements e_{1}, …, e_{t} are central idempotents, and the coordinates of a ∈ A are given by

A −→ A_{1} × ⋯ × A_{t}, a ⟼→ (ae_{1}, …, ae_{t}).

Note that e_{i}e_{j} = 0 whenever i j. We assume that σ(1) = 1 and thus σ(e_{i}) = e_{i} for i = 1, …, t.

Let M be an A-module and let h : M × M → A be a hermitian form over (A, σ). Following [9, Proof of Lemma 1.9] we can write

M = ∐_{i=1}^{t} Me_{i}, m = (me_{1}, …, me_{t}),

where ∐ _{i=1}^{t} Me_{i} is the A-module with set of elements ∏ _{i=1}^{t} Me_{i}, whose sum is defined coordinate by coordinate and whose product is defined by (m_{1}e_{1}, …, m_{t}e_{t})a = (m_{1}e_{1}a_{1}, …, m_{t}e_{t}a_{t}) for m_{1}, …, m_{t} ∈ M and a ∈ A.

Define h_{i} = h|_{Me_{i}}. Then h_{i}(xe_{i}, ye_{i}) = σ(e_{i})h(x, y)e_{i} = h(x, y)e_{i}^{2} = h(x, y)e_{i} and h_{i} : Me_{i} × Me_{i} → A_{i} is a hermitian form over (A_{i}, σ_{i}). We also have

\begin{aligned}h(xe_{1} + ⋯ + xe_{t}, ye_{1} + ⋯ + ye_{t}) & = ∑_{i,j=1}^{t} h(xe_{i}, ye_{j})e_{i}e_{j} \\ & = ∑_{i=1}^{t} h(xe_{i}, ye_{i})e_{i} \end{aligned}

which proves that h = h_{1} ⊥ …⊥ h_{t}.

5.2. Algebraic Extensions and Real Closures. We essentially follow [20, Chapter 3, Lemma 2.6, Lemma 2.7, Theorem 4.4].

Let P ∈ X_{F} and let F_{P} denote a real closure of F at P. Let L be a finite extension of F. Writing L = F[X]/(R) for some R ∈ F[X] and R = R_{1} ⋯ R_{t} as a product of pairwise distinct irreducibles in F_{P}[X] with degR_{1} = ⋯= degR_{r} = 1 and degR_{r+1} = ⋯= degR_{t} = 2, we obtain canonical F_{P}-isomorphisms L ⊗_{F} F_{P} ≃ F_{P}[X]/(R_{1} ⋯ R_{t}) and

L ⊗_{F} F_{P} −→^{ω} F_{1} × ⋯ × F_{t},  (5)

where F_{i} = F_{P}[X]/(R_{i}) is a real closed field for 1 ≤ i ≤ r and is algebraically closed for r + 1 ≤ i ≤ t. We write 1 = (e_{1}, …, e_{t}) in F_{1} × ⋯ × F_{t} and define ω_{i}(x) = ω(x)e_{i} for x ∈ L ⊗_{F} F_{P}, the projection of ω(x) on its i-th coordinate.

Let ι_{0} : F_{P} → L ⊗_{F} F_{P} be the canonical inclusion. Then ω_{i} ∘ ι_{0} : F_{P} → F_{i} is an isomorphism of fields and of F_{P}-modules for 1 ≤ i ≤ r. In particular, for 1 ≤ i ≤ r, F_{i} is naturally an F_{P}-module of dimension one, and it is easily seen that Tr_{F_{i}/F_{P}} = (ω_{i} ∘ ι_{0})^{−1}. It follows that Tr_{F_{i}/F_{P}} is an isomorphism of fields.

Let ι_{1} : L → L ⊗_{F} F_{P} be the canonical inclusion. Then ω_{i}∘ ι_{1} : L → F_{i} (i = 1, …, r) denote the r different embeddings of ordered fields corresponding to the orderings Q_{i} on L that extend P. In other words, if {Q_{1}, …, Q_{r}} are the different extensions of P to L, then for every 1 ≤ i ≤ r, the map

\xymatrix{L \ar[r]^{ι_{1}} & L ⊗_{F} F_{P} \ar[r]^{ω_{i}} & F_{i}}

is a real closure of L at Q_{i}. Since Tr_{F_{i}/F_{P}} is an isomorphism of fields, it follows that the map

\xymatrix{L \ar[r]^{ι_{1}} & L ⊗_{F} F_{P} \ar[r]^{ω_{i}} & F_{i} \ar[r]^{^{Tr^{Fi/FP}}} & F_{P}}

is also a real closure of L at Q_{i}.

5.3. The Knebusch Trace Formula. Let (A, σ) be an F-algebra with involution. Let L/F be a finite extension. The trace Tr_{L/F} : L → F induces an A-linear homomorphism

Tr_{A⊗_{F}L} = id_{A}⊗ Tr_{L/F} : A ⊗_{F} L −→ A

which induces a group homomorphism (transfer map)

Tr_{A⊗_{F}L}^{∗} : W(A ⊗_{F} L, σ ⊗ id) −→ W(A, σ), (M, h) ⟼→ (M, Tr_{A⊗_{F}L} ∘ h),

cf. [1, p. 362].

The following theorem is an extension of a result due to Knebusch [8, Proposition 5.2], [20, Chapter 3, Theorem 4.5] to F-algebras with involution. The proof follows the general lines of Knebusch’s original proof.

Theorem 5.1. Let P ∈ X_{F}. Let L/F be a finite extension of ordered fields and let h be a hermitian form over (A ⊗_{F} L, σ ⊗ id). Fix a real closure ι : F → F_{P} and a Morita equivalence ℳ as in (2). Then, with notation as in Section 5.2,

sign_{ι}^{ℳ} (Tr_{A⊗_{F}L}^{∗}h) = ∑_{i=1}^{r} sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} h.

Proof. By definition of signature we have

sign_{ι}^{ℳ} (Tr_{A⊗_{F}L}^{∗}h) = signℳ [(Tr_{A⊗_{F}L}^{∗}h) ⊗_{F} F_{P}].  (6)

Consider the commutative diagram

\xymatrix{L \ar[r]^{^{Tr^{L/F}}} \ar[d]^{⊗_{F}F_{P}} & F \ar[d]^{⊗_{F}F_{P}} \\ L ⊗_{F} F_{P} \ar[r]^{_{Tr_{L⊗FP/FP}}} & F_{P}}

It induces a commutative diagram

\xymatrix{A ⊗_{F} L \ar[r]^{^{Tr^{A⊗L}}} \ar[d]^{⊗_{F}F_{P}} & A \ar[d]^{⊗_{F}F_{P}} \\ A ⊗_{F} L ⊗_{F} F_{P} \ar[r]^{id_{A}⊗Tr_{L⊗FP/FP}} & A ⊗_{F} F_{P}}

which in turn induces a commutative diagram of Witt groups

\xymatrix{W(A ⊗_{F} L, σ ⊗ id) \ar[r]^{^{Tr{∗ \atop A⊗L}}} \ar[d] & W(A, σ) \ar[d] \\ W(A ⊗_{F} L ⊗_{F} F_{P}, σ ⊗ id ⊗ id) \ar[r]^{(id_{A}⊗Tr_{L⊗FP/FP})^{∗}} & W(A ⊗_{F} F_{P}, σ ⊗ id)}

where the vertical arrows are the canonical restriction maps. Thus

signℳ [(Tr_{A⊗_{F}L}^{∗}h) ⊗_{F} F_{P}] = signℳ [(id_{A}⊗ Tr_{L⊗F_{P}/F_{P}})^{∗}(h ⊗_{F} F_{P})].  (7)

With reference to Section 5.2 consider the diagram

\xymatrix{A ⊗_{F} L ⊗_{F} F_{P} \ar[r]^{id_{A}⊗ω} \ar[d]^{id_{A}⊗_{F}Tr_{L⊗F_{P}/F_{P}}} & A ⊗_{F} (F_{1} × ⋯ × F_{t}) \ar[r]^{∼} \ar[d]^{id_{A}⊗Tr_{F_{1}×··· ×Ft/F_{P}}} & (A ⊗_{F} F_{1}) × ⋯ × (A ⊗_{F} F_{t}) \ar[d]^{∑ _{i=1}^{t} id_{A}⊗Tr_{F_{i}/F_{P}}} \\ A ⊗_{F} F_{P} \ar[r]^{id} & A ⊗_{F} F_{P} \ar[r]^{id} & A ⊗_{F} F_{P}}

where commutativity of the first square follows from the isomorphism (5) of F_{P}- algebras, whereas commutativity of the second square follows from [4, p. 137]. We push h ⊗ F_{P} through the induced commutative diagram of Witt groups:

[figure: h ⊗ F_{P}; (id_{A}⊗ω)^{∗}; h′; h′_{1} ⊥ …⊥ h′_{t}; (id_{A}⊗ Tr_{L⊗F_{P}/F_{P}})^{∗}(h ⊗ F_{P}); id; ∑ _{i=1}^{t}(id_{A}⊗ Tr_{F_{i}/F_{P}})^{∗}(h′_{i})]

where the image of h′equals the orthogonal sum h′_{1} ⊥ …⊥ h′_{t}by Section 5.1. Thus

signℳ [(id_{A}⊗ Tr_{L⊗F_{P}/F_{P}})^{∗}(h ⊗_{F} F_{P})] = ∑_{i=1}^{t} signℳ [(id_{A}⊗ Tr_{F_{i}/F_{P}})^{∗}(h′_{i})].  (8)

We have to consider the following two cases:

Case 1: Assume that 1 ≤ i ≤ r. Observe that h′_{i} = (id_{A}⊗ (ω_{i} ∘ ι_{1}))^{∗}(h). Then

signℳ [(id_{A}⊗ Tr_{F_{i}/F_{P}})^{∗}(h′_{i})] = signℳ [(id_{A}⊗ (Tr_{F_{i}/F_{P}}))^{∗} ∘ (id_{A}⊗ (ω_{i} ∘ ι_{1}))^{∗}(h)].

The form (id_{A}⊗ (ω_{i} ∘ ι_{1}))^{∗}(h) is defined over A ⊗_{F} F_{i}, and the commutative diagram

\xymatrix{ & F_{P} \ar[dd]^{ω_{i}∘ι_{0}} \\ F_{i} \ar[ur]^{_{Tr_{F_{i}/F_{P}}}} \ar[dr]_{id} \\  & F_{i}}

together with Proposition 3.3 gives

signℳ [(id_{A}⊗ (Tr_{F_{i}/F_{P}}))^{∗} ∘ (id_{A}⊗ (ω_{i} ∘ ι_{1}))^{∗}(h)] = sign(ω_{i} ∘ ι_{0})(ℳ)[(id_{A}⊗ (ω_{i} ∘ ι_{1}))^{∗}(h)] (9)

Case 2: Assume that r + 1 ≤ i ≤ t. Since F_{i} is algebraically closed, it follows from Morita theory that the Witt group W(A ⊗_{F} F_{i}, σ ⊗ id) is torsion and so h′_{i} is a torsion form. Therefore (id_{A}⊗ Tr_{F_{i}/F_{P}})^{∗}(h′_{i}) is also a torsion form and thus has signature zero.

We conclude that equations (6)-(9) yield the Knebusch Trace Formula.

6. Existence of Forms with Nonzero Signature

Theorem 6.1. Let (A, σ) be an F-algebra with involution and let P ∈ X_{F} \ Nil[A, σ]. Fix a real closure ι : F → F_{P} and a Morita equivalence ℳ as in (2). There exists a hermitian form h over (A, σ) such that sign_{ι}^{ℳ} h 0.

Proof. Let P ∈ X_{F} \ Nil[A, σ]. Then A ⊗_{F} F_{P} ≃ M_{m}(D_{P}) for some m ∈ , where D_{P} = F_{P}, H_{P} or F_{P}( \sqrt{−1}) if σ is orthogonal, symplectic or unitary, respectively. In each case there exists a positive involution τ on M_{m}(D_{P}) (namely, transposition, conjugate transposition and quaternion conjugate transposition, respectively, cf. Example 4.1). By Lemma 4.6 the hermitian form ⟨1⟩_{τ} over (M_{m}(D_{P}), τ) has nonzero signature since τ is the adjoint involution of the form ⟨1⟩_{τ}. After scaling we obtain a rank one hermitian form h_{0} over (A ⊗_{F} F_{P}, σ ⊗ id) such that signℳ (h_{0}) 0 by Proposition 3.4. The form h_{0} is already defined over a finite extension L of ι(F), contained in F_{P}. Thus we can consider h_{0} as a form over (A ⊗_{F} L, σ ⊗ id) and we have signℳ (h_{0} ⊗ F_{P}) 0. In other words, if Q_{1} is the ordering L ∩ F_{P}^{×2} on L, then for any real closure κ_{1} : L → L_{1} of L at Q_{1} and any Morita equivalence ℳ_{1} as in (2), but starting from ℋ (A ⊗_{F} L_{1}, σ ⊗ id), we have

sign_{κ_{1}}^{ℳ_{1}} h_{0} 0

by Corollary 3.5.

Let X = {Q ∈ X_{L} | P ⊂ Q}. By [20, Chapter 3, Lemma 2.7], X is finite, say X = {Q_{1}, Q_{2}, …, Q_{r}}. Thus there exist a_{2}, …, a_{r} ∈ L^{×} such that

{Q_{1}} = H(a_{2}, …, a_{r}) ∩ X.

Consider the Pfister form q := ⟨⟨a_{2}, …, a_{r}⟩⟩ = ⟨1, a_{2}⟩⊗⋯ ⊗⟨1, a_{r}⟩. Then sign_{Q_{1}} q = 2^{r−1} and sign_{Q_{ℓ}}q = 0 for ℓ 1. It follows that sign_{κ_{1}}^{ℳ_{1}} (q ⊗ h_{0}) = sign_{Q_{1}} q · sign_{κ_{1}}^{ℳ_{1}} h_{0} 0 and sign_{κ_{ℓ}}^{ℳ_{ℓ}} (q ⊗ h_{0}) = 0 for ℓ 1, where κ_{ℓ} : L → L_{ℓ} is any real closure of L at Q_{ℓ} and ℳ_{ℓ} is any Morita equivalence as in (2), but starting from ℋ (A ⊗_{F} L_{ℓ}, σ ⊗ id).

Now Tr_{A⊗_{F}L}^{∗}(q ⊗ h_{0}) is a hermitian form over (A, σ). By the trace formula, Theorem 5.1, we have

sign_{ι}^{ℳ} Tr_{A⊗_{F}L}^{∗}(q ⊗ h_{0}) = ∑_{i=1}^{r} sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} (q ⊗ h_{0}) = sign_{ω_{1}∘ι_{1}}^{(ω_{1}∘ι_{0})(ℳ)} (q ⊗ h_{0}) 0.

Taking h := Tr_{A⊗_{F}L}^{∗}(q ⊗ h_{0}) proves the theorem.

Corollary 6.2. Let (A_{1}, σ_{1}) and (A_{2}, σ_{2}) be F-algebras with involution of the same type. Assume that A_{1} and A_{2} are Brauer equivalent. Let P ∈ X_{F}, let ι : F → F_{P} be a real closure of F at P and let ℳ_{ℓ} be any Morita equivalence as in (2), but starting from ℋ (A_{ℓ} ⊗_{F} F_{P}, σ_{ℓ} ⊗ id) for ℓ = 1, 2. Then the following statements are equivalent:

(i) sign_{ι}^{ℳ_{1}} h = 0 for all hermitian forms h over (A_{1}, σ_{1}); (ii) sign_{ι}^{ℳ_{2}} h = 0 for all hermitian forms h over (A_{2}, σ_{2}); (iii) sign_{P}ϑ = 0 for all involutions ϑ on A_{1} of the same type as σ_{1}; (iv) sign_{P}ϑ = 0 for all involutions ϑ on A_{2} of the same type as σ_{2}.

Proof. By Theorem 6.1, the first two statements are equivalent to P ∈ Nil[A_{1}, σ_{1}] = Nil[A_{2}, σ_{2}]. Thus (i) ⇔ (ii).

(i) ⇒ (iii) Let ϑ be as in (iii). Then ϑ = ad_{⟨1⟩_{ϑ}}and ϑ = Int(a)∘σ_{1} for some invertible a ∈ Sym(A_{1}, σ_{1}). Thus, with notation as in Remark 3.13 and using Lemma 4.6 and Proposition 3.4 we have for any Morita equivalence ℳ,

sign_{P}ϑ = λ_{P}| sign_{ι}^{ℳ} ⟨1⟩_{ϑ}|
= λ_{P}| sign_{ι}^{ℳ ∘(ζ_{a}⊗id)} ζ_{a}^{−1}(⟨1⟩_{ϑ})|
= λ_{P}| sign_{ι}^{ℳ_{1}} ζ_{a}^{−1}(⟨1⟩_{ϑ})|
= λ_{P}| sign_{ι}^{ℳ_{1}} ⟨a^{−1}⟩_{σ_{1}} |

which is zero by the assumption.

(ii) ⇒ (iv): This is the same proof as (i) ⇒ (iii) after replacing (A_{1}, σ_{1}) by (A_{2}, σ_{2}). For the remainder of the proof we may assume without loss of generality that A_{2} is

a division algebra and that A_{1} ≃ M_{m}(A_{2}) for some m ∈ .

(iii) ⇒ (iv): Let ϑ be any involution on A_{2}. By the assumption, sign_{P}(t ⊗ ϑ) = 0, where t denotes the transpose involution. Since t is a positive involution, it follows that sign_{P}ϑ = 0, cf. Remark 4.2.

(iv) ⇒ (ii): The assumption implies that sign_{ι}^{ℳ_{2}} h = 0 for every hermitian form h of dimension 1 over (A_{2}, σ_{2}), which implies (ii) since all hermitian forms over (A_{2}, σ_{2}) can be diagonalized.

Remark 6.3. Note that statement (iii) in Corollary 6.2 is equivalent to: sign_{ι}^{ℳ_{1}} h = 0 for all diagonal hermitian forms h of rank one over (A_{1}, σ_{1}).

Theorem 6.4. Let (A, σ) be an F-algebra with involution. There exists a finite set H = {h_{1}, …, h_{s}} of diagonal hermitian forms of rank one over (A, σ) such that for every P ∈ X_{F} \ Nil[A, σ], real closure ι : F → F_{P} and Morita equivalence ℳ as in (2) there exists h ∈ H such that sign_{ι}^{ℳ} h 0.

Proof. For every P ∈ X_{F}, choose a real closure ι_{P} : F → F_{P} and a Morita equivalence ℳ_{P} as in (2). By Corollary 3.5 we may assume without loss of generality and for the sake of simplicity that the map ι_{P} is an inclusion. The algebra A ⊗_{F} F_{P} is isomorphic to a matrix algebra over D_{P}, where D_{P} ∈ \{F_{P}, H_{P}, F_{P}( \sqrt{−1}), F_{P}× F_{P}\}. There is a finite extension L_{P} of F, L_{P} ⊂ F_{P} such that A ⊗_{F} L_{P} is isomorphic to a matrix algebra over E_{P}, where E_{P} ∈ \{L_{P}, (−1, −1)_{L_{P}}, L_{P}( \sqrt{−1}), L_{P} × L_{P}\}, and P extends to L_{P}. Let

U_{P} := {Q ∈ X_{F} | Q extends to L_{P}}.

Since P ∈ U_{P} we can write

X_{F} = ⋃_{P∈X_{F}}U_{P}.

We know from [20, Chapter 3, Lemma 2.7, Theorem 2.8] that sign_{Q}(Tr_{L_{P}/F}^{∗}⟨1⟩) is the number of extensions of Q to L_{P}. Thus,

U_{P} = (sign(Tr_{L_{P}/F}^{∗}⟨1⟩))^{−1}({1, …, k}),

where k is the dimension of the quadratic form Tr_{L_{P}/F}^{∗}⟨1⟩, and so U_{P} is clopen in X_{P} (and in particular compact). Therefore, and since X_{F} is compact, there exists a finite number of orderings P_{1}, …, P_{ℓ} in X_{F} such that

X_{F} = ⋃_{i=1}^{ℓ} U_{P_{i}}.

Now let P ∈ {P_{1}, …, P_{ℓ}} and let L_{P} be as before. By Theorem 6.1 we have that for every Q ∈ U_{P} \ Nil[A, σ] there exists a hermitian form h_{Q} over (A, σ) such that sign_{ι_{Q}}^{ℳ_{Q}} h_{Q} 0. By Corollary 6.2 and Remark 6.3 we may assume that h_{Q} is diagonal of rank one. Consider the total signature map

μ_{Q} : X_{F} −→ , P ⟼→ sign_{ι_{P}}^{ℳ_{P}} h_{Q}.

Then

U_{P} \ Nil[A, σ] = ⋃_{Q∈U_{P}\Nil[A,σ]}μ_{Q}^{−1}( \ {0}).  (10)

Consider the continuous map

λ_{P} : X_{L_{P}} −→ X_{F}, R ⟼→ R ∩ F.

We have Q ∈ U_{P} \ Nil[A, σ] if and only if some extension Q′ of Q to L_{P} is in X_{L_{P}} \ Nil[A⊗_{F}L_{P}, σ⊗id] (this follows from the fact that the ordered fields (F, Q) and (L_{P}, Q′) have a common real closure) if and only if Q ∈ λ(X_{L_{P}} \ Nil[A ⊗_{F} L_{P}, σ ⊗ id]).

We observe that X_{L_{P}} \ Nil[A ⊗_{F} L_{P}, σ ⊗ id] is clopen and compact since Nil[A ⊗_{F} L_{P}, σ ⊗ id] is either or the whole of X_{L_{P}}, which follows from the fact that A ⊗_{F} L_{P} is a matrix algebra over one of L_{P}, (−1, −1)_{L_{P}}, L_{P}( \sqrt{−1}), L_{P} × L_{P}.

Hence,

U_{P} \ Nil[A, σ] = λ(X_{L_{P}} \ Nil[A ⊗_{F} L_{P}, σ ⊗ id])

is compact and thus closed. Thus U_{P} ∩ Nil[A, σ] is open in U_{P}. Using (10) we can write

U_{P} = (U_{P} ∩ Nil[A, σ])∪ ⋃_{Q∈U_{P}\Nil[A,σ]}μ_{Q}^{−1}( \ {0}).

Now μ_{Q}^{−1}( \ {0}) = (signad_{h_{Q}})^{−1}( \ {0}) by Lemma 4.6, which is open since signad_{h_{Q}} is continuous by Remark 4.4. Thus, since U_{P} is compact, there exist Q_{1}, …, Q_{t} ∈ U_{P} \ Nil[A, σ] such that

U_{P} = (U_{P} ∩ Nil[A, σ])∪ ⋃_{i=1}^{t} μ_{Q_{i}}^{−1}( \ {0}).

In other words, for every Q ∈ U_{P}\Nil[A, σ] one of sign_{ι_{Q}}^{ℳ_{Q}} h_{Q_{i}} (i = 1, …, t) is nonzero. Now let H_{P} = {h_{Q_{1}}, …, h_{Q_{t}}}. Letting H = ⋃ _{i=1}^{ℓ} H_{P_{i}} finishes the proof.

Corollary 6.5. Let (A, σ) be an F-algebra with involution. The set Nil[A, σ] is clopen in X_{F}.

Proof. By Theorem 6.4 we have Nil[A, σ] = ⋂ _{i=1}^{s}{P ∈ X_{F} | sign_{ι_{P}}^{ℳ_{P}} h_{i} = 0}. The result follows from Lemma 4.6 and Remark 4.4.

At this stage we have established all results that are needed for the definition of the H-signature in Definion 3.9. In the final two sections we show that the total Hsignature of a hermitian form is continuous and we reformulate the Knebusch Trace Formula in terms of the H-signature.

7. Continuity of the Total H-Signature Map of a Hermitian Form

Let h be a hermitian form over (A, σ). With reference to Definition 3.9 we denote by sign^{H} h the total H-signature map of h:

X_{F} −→ , P ⟼→ sign_{P}^{H} h.

Lemma 7.1. Let H = (h_{1}, …, h_{s}) be as in Definition 3.9. There is a finite partition of X_{F} into clopens

X_{F} = Nil[A, σ] ˙∪ ˙⋃_{i=1}^{ℓ} Z_{i},

such that for every i ∈ {1, …, ℓ} one of the total H-signature maps sign^{H} h_{1}, …, sign^{H} h_{s} is constant non-zero on Z_{i}.

Proof. For r = 1, …, s, let

Y_{r} := {P ∈ X_{F} | sign_{P}^{H} h_{i} = 0, i = 1, …, r}.

By Lemma 4.6 we have

Y_{r} = ⋂_{i=1}^{r} {P ∈ X_{F} | sign_{P}ad_{h_{i}} = 0}.

Thus each Y_{r} is clopen.

We have Y_{0} := X_{F} ⊇ Y_{1} ⊇ ⋯⊇ Y_{s−1} ⊇ Y_{s} = Nil[A, σ] and therefore,

X_{F} = (Y_{0} \ Y_{1}) ˙∪(Y_{1} \ Y_{2}) ˙∪ ⋯ ˙∪(Y_{s−1} \ Y_{s}) ˙∪ Nil[A, σ].

Let r ∈ {0, …, s − 1} and consider Y_{r} \ Y_{r+1}. By the definition of Y_{1}, …, Y_{s} the map sign^{H} h_{r+1} is never 0 on Y_{r}\Y_{r+1}. Furthermore, since the rank of h_{r+1} is finite, sign^{H} h_{r+1} only takes a finite number of values k_{1}, …, k_{m}.

Now observe that there exists a λ ∈ {1, 2} such that

sign^{H} h_{r+1} = \frac{1}{λ} signad_{h_{r+1}}

on Y_{r} \ Y_{r+1}. This follows from Lemma 4.6 and Definition 3.9 for P ∈ Y_{r} \ Y_{r+1}. Therefore,

(sign^{H} h_{r+1})^{−1}(k_{i}) ∩ (Y_{r} \ Y_{r+1}) = (signad_{h_{r+1}})^{−1}(λk_{i}) ∩ (Y_{r} \ Y_{r+1}),

which is clopen by Remark 4.4. It follows that Y_{r} \ Y_{r+1} is covered by finitely many disjoint clopen sets on which the map sign^{H} h_{r+1} has constant non-zero value. The result follows since the sets Y_{r} \ Y_{r+1} for r = 0, …, s − 1 form a partition of X_{F} \ Nil[A, σ].

Theorem 7.2. Let h be a hermitian form over (A, σ). The total H-signature of h,

sign^{H} h : X_{F} −→ , P ⟼→ sign_{P}^{H} h

is continuous.

Proof. We use the notation and the conclusion of Lemma 7.1. Since Nil[A, σ] and the sets Z_{i} are clopen, it suffices to show that (sign^{H} h)|_{Z_{i}} is continuous for every i = 1, …, ℓ.

Let i ∈ {1, …, ℓ}, k_{i} ∈ \ {0} and j ∈ {1, …, s} be such that sign^{H} h_{j} = k_{i} on Z_{i}. Let k ∈ . Then

((sign^{H} h)|_{Z_{i}})^{−1}(k)= {P ∈ Z_{i} | sign_{P}^{H} h = k}
= {P ∈ Z_{i} | k_{i} sign_{P}^{H} h = k_{i}k}
= {P ∈ Z_{i} | k_{i} sign_{P}^{H} h = k sign_{P}^{H} h_{j}}
= {P ∈ Z_{i} | sign_{P}^{H}(k_{i} × h ⊥ k × h_{j}) = 0}.

It follows from Lemma 4.6 that

((sign^{H} h)|_{Z_{i}})^{−1}(k) = {P ∈ Z_{i} | sign_{P}ad_{k_{i}×h⊥k×h_{j}} = 0},

which is clopen by Remark 4.4.

8. The Knebusch Trace Formula for H-Signatures

Theorem 8.1. Let H = (h_{1}, …, h_{s}) be as in Definition 3.9. Let P ∈ X_{F}. Let L/F be a finite extension of ordered fields and let h be a hermitian form over (A ⊗_{F} L, σ ⊗ id). Then, with H ⊗ L := (h_{1} ⊗ L, …, h_{s} ⊗ L), we have

sign_{P}^{H}(Tr_{A⊗_{F}L}^{∗}h) = ∑_{P⊆Q∈X_{L}}sign_{Q}^{H⊗L} h.

Proof. By Theorem 5.1 (and using its notation), we know that

sign_{ι}^{ℳ} (Tr_{A⊗_{F}L}^{∗}h) = ∑_{i=1}^{r} sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} h,

for any ι and ℳ. Fix a real closure ι : F → F_{P} and choose a Morita equivalence ℳ such that sign_{ι}^{ℳ} = sign_{P}^{H} (cf. Remark 3.14). We only have to check that sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} = sign_{Q_{i}}^{H⊗L} for i = 1, …, r.

By definition of ℳ, there is a k ∈ {1, …, s} such that sign_{ι}^{ℳ} h_{j} = 0 for 1 ≤ j ≤ k − 1 and sign_{ι}^{ℳ} h_{k} > 0. To check that sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} = sign_{Q_{i}}^{H⊗L} for i = 1, …, r, it suffices to check that sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} (h_{j} ⊗ L) = 0 for j = 1, …, k − 1 and sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} (h_{k} ⊗ L) > 0. This follows from the fact that

sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} (h_{ℓ} ⊗ L) = sign_{ι}^{ℳ} h_{ℓ} for every 1 ≤ ℓ ≤ s,

which we verify in the remainder of the proof.

By definition,

sign_{ω_{i}∘ι_{1}}^{(ω_{i}∘ι_{0})(ℳ)} (h_{ℓ} ⊗ L) = sign(ω_{i} ∘ ι_{0})(ℳ)[(id_{A}⊗ (ω_{i} ∘ ι_{1}))^{∗}(h_{ℓ} ⊗ L)].

Consider the commutative diagram

\xymatrix{ & F_{P} \ar[r]^{ι_{0}} & L ⊗_{F} F_{P} \ar[r]^{ω_{i}} & F_{i} \ar[dd]^{id} \\ F \ar[ur]^{ι} \ar[dr] \\  & L \ar[r]_{ι_{1}} & L ⊗_{F} F_{P} \ar[r]_{ω_{i}} & F_{i}}

Thus, by Proposition 3.3,

sign(ω_{i} ∘ ι_{0})(ℳ)[(id_{A}⊗ (ω_{i} ∘ ι_{1}))^{∗}(h_{ℓ}⊗ L)] = sign(ω_{i} ∘ ι_{0})(ℳ)[(id_{A}⊗ (ω_{i} ∘ ι_{0} ∘ ι))^{∗}(h_{ℓ})].

Finally the commutative diagram

\xymatrix{ & F_{i} \ar[dd]^{(ω_{i}∘ι_{0})^{−1}} \\ F \ar[ur]^{ω_{i}∘ι_{0}∘ι} \ar[dr]_{ι} \\  & F_{P}}

together with Proposition 3.3 yields

sign(ω_{i} ∘ ι_{0})(ℳ)[(id_{A}⊗ (ω_{i} ∘ ι_{0} ∘ ι))^{∗}(h_{ℓ})] = signℳ [(id_{A}⊗ ι)^{∗}(h_{ℓ})] = sign_{ι}^{ℳ} h_{ℓ},

which concludes the proof.

Acknowledgements

The authors wish to thank the referee, whose comments helped improve the original version of this paper.

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School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

E-mail address: vincent.astier@ucd.ie, thomas.unger@ucd.ie