VALUATIONS ON ALGEBRAS WITH INVOLUTION

J.-P. TIGNOL AND A.R. WADSWORTH

Valuations are a major tool for the study of the structure of division algebras. The purpose of this work is to introduce a notion that plays a similar role for central simple algebras with involution, and to prove analogues for this notion to fundamental results on valuations on division algebras.

Since the definition of a (Schilling) valuation implies an absence of zero divisors, the only central simple algebras that can have valuations are division algebras. Given a division algebra D finitedimensional over its center F, it is natural to view valuations on a D as extensions of valuations on F, since valuations on fields are abundant and their theory is well-developed. But not every valuation v on F extends to D. In the extension question, Henselian valuations play a special role. Schilling proved in [Sch, pp. 53–54] that if v on F is Henselian, then v has an extension to a valuation on D, and this extension is unique. Much later it was proved by Ershov [Er_{1}] and Wadsworth [W] that for any valuation v on F, v extends to D if and only if it satisfies a Henselian-like condition with respect to the field extensions of F within D; they also proved that when v extends to D the extension is unique. Another fundamental criterion was proved by Morandi [M]: v on F extends to a valuation on D if and only if D remains a division algebra after scalar extension to the Henselization F_{h} of F for v. We will prove analogues for central simple algebras with involutions to these theorems of Schilling, Ershov-Wadsworth, and Morandi.

An involution on a central simple algebra A is a ring-anti-automorphism σ such that σ^{2} = id_{A}. As Weil suggested in [We], the theory of central simple algebras with involution is a natural sibling to the theory of central simple algebras, since the associated automorphism groups are the basic types of classical groups. In each setting there is a notion of anisotropic object, corresponding to when the associated automorphism group is anisotropic as an algebraic group. The anisotropic central simple algebras are the division algebras. An involution σ on a central simple algebra A is anisotropic just when the equation σ(x)x = 0 holds only for x = 0. In earlier work [TW] we have developed the theory of gauges, which are a kind of value functions for central simple algebras. (The definition of a gauge is recalled at the end of this introduction.) For a central simple algebra A with involution σ, we define a σ-special gauge to be a gauge φ on A satisfying the condition^{1} that φ(σ(x)x) = 2φ(x) for all x ∈ A. A σ-special gauge for an algebra with involution is our analogue to a valuation on a division algebra. If A has a σ-special gauge, then σ is easily seen to be anisotropic. If v is a Henselian valuation on the σ-invariant part F of Z(A) and σ is anisotropic, we show in Th. 2.2 that there is a unique σ-invariant gauge φ on A extending v, and φ is a σ-special gauge. When v on F is not Henselian, we show in Th. 6.1 that there is a σ-special gauge φ on A extending v if and only if the anisotropic involution σ remains anisotropic after scalar extension to the Henselization of F with respect to v; furthermore, there is only one such φ. Our results require tame ramification and exclude orthogonal involutions if the residue characteristic is 2; see the statements of Theorems 2.2 and 6.1 for the precise conditions required.

A gauge φ on a central simple algebra A induces a filtration on A which yields an associated graded ring gr(A), analogous to what one has with a valuation on a field or a division ring. The graded structure is intrinsic to the definition of a gauge, and is used heavily throughout this paper. The degree 0 part of gr(A), denoted A_{0}, is the residue ring of the “valuation ring” of A determined by the gauge φ; A_{0} is always a semisimple Z(A)_{0}-algebra, but not simple in general. If σ is an involution on A and φ is invariant under σ, then σ induces involutions σ̃ on gr(A) and σ_{0} on A_{0}. We show in Prop. 1.1 and Remark 2.5(1) that a σ-invariant gauge φ is σ-special if and only if σ̃ is anisotropic, if and only if σ_{0} is anisotropic. We also prove an analogue of a theorem of Springer: when the base field is Henselian, an involution σ is isotropic if and only if its residue involution σ_{0} is isotropic (Cor. 2.3). This criterion is applied to show that under specified valuation-theoretic conditions, an anisotropic involution remains anisotropic after certain scalar extensions (Cor. 3.6).

The first author is partially supported by the F.R.S.–FNRS (Belgium). The second author would like to thank the first author and UCL for their hospitality during several visits while this paper was developing.

1Notice the similarity with the definition of C^{∗}-algebras, cf. [Dix, Déf. 1.3.1].

An outline of this paper is as follows: In Section 1, we discuss in general terms the compatibility of a value function with an involution, relating that notion to a compatibility condition between norms and hermitian forms defined in [RTW]. In Section 2, we restrict to the case of Henselian valuations and give the proofs of Th. 2.2 and Cor. 2.3. Some applications to scalar extensions (in particular Cor. 3.6) are given in Section 3. Sections 4 and 5 prepare the ground for the extension of our results to the non-Henselian case in Section 6. The main problem is to analyze how the condition for the existence of a splitting base of a value function (which is a critical part of the definition of a gauge) behaves under restriction of scalars; this is done in Section 5. In Section 4, we investigate this condition for the composition of value functions. This is used in Section 6 in the proof of Th. 6.1 by induction on the rank of valuations.

For the convenience of the reader, we now review the basic notions of value functions, norms, and gauges introduced in [RTW] and [TW]. Throughout the paper, we fix a divisible totally ordered abelian group Γ, which will contain the values of all the valuations and the degrees of all the gradings we consider. Thus, a valued field (F, v) is a pair consisting of a field F and a valuation v: F → Γ ∪ {∞}. The group v(F^{×}) of values of F is denoted by Γ_{F}, and the residue field by \overline{F}. We use analogous notation for valuations on division rings.

Let (F, v) be a valued field. A v-value function on an F -vector space V is a map α: V → Γ ∪ {∞} such that

(i) α(x) = ∞ if and only if x = 0; (ii) α(x + y) ≥ min(α(x), α(y)) for x, y ∈ V; (iii) α(xc) = α(x) + v(c) for all x ∈ V and c ∈ F.

The v-value function α is called a norm if V is finite-dimensional and contains a base (e_{i})_{i=1}^{n} such that

α(∑_{i=1}^{n} e_{i}c_{i}) = min_{1≤i≤n}(α(e_{i}c_{i})) for c_{1}, …, c_{n} ∈ F .

Such a base is called a splitting base of V for α. A v-value function φ on an F -algebra A is surmultiplicative if φ(1) = 0 and φ(xy) ≥ φ(x) + φ(y) for x, y ∈ A.

The valuation v defines a filtration on F: for γ ∈ Γ we set

F^{≥γ} = {x ∈ F | v(x) ≥ γ}, F^{>γ} = {x ∈ F | v(x) > γ}, and F_{γ} = F^{≥γ}/F^{>γ}.

The associated graded ring is

gr(F) = ⨁_{γ∈Γ} F_{γ}.

It is called a graded field because every nonzero homogeneous element in gr(F) is invertible. Likewise, every v-value function α on an F -vector space V defines a filtration, and the associated graded structure gr_{α}(V) is a graded module over gr(F), which we call a graded vector space. It is a free module, whose rank is called its dimension. The value function is a norm if and only if dim_{gr(F)}(gr_{α}(V)) = dim_{F} (V) < ∞, see [RTW, Cor. 2.3]. Every nonzero element x ∈ V has an image x̃ in gr_{α}(V) defined by

x̃ = x + V^{>α(x)} ∈ V_{α(x)}.

We also set ˜0 = 0 ∈ gr_{α}(V). If φ is a surmultiplicative v-value function on an F -algebra A, then gr_{φ}(A) is an algebra over gr(F), in which multiplication is defined by

ãb̃ = ab + V^{>φ(a)+φ(b)} = \begin{cases}ãb & if φ(ab) = φ(a) + φ(b), \\ 0 & if φ(ab) > φ(a) + φ(b), \end{cases} for a, b ∈ A.

Now, suppose A is a finite-dimensional simple F -algebra. We denote by [A:F] its dimension and by Z(A) its center. A surmultiplicative v-value function φ on A is called a v-gauge if it satisfies the following conditions:

(i) φ is a v-norm, i.e., [A:F] = [gr_{φ}(A): gr(F)]; (ii) gr_{φ}(A) is a graded semisimple gr(F)-algebra, i.e., it does not contain any nonzero nilpotent

homogeneous two-sided ideal.

The v-gauge φ is said to be tame if Z(gr_{φ}(A)) = gr_{φ}(Z(A)) and Z(gr_{φ}(A)) is separable over gr(F). If the residue characteristic is 0, then every v-gauge is tame, see [TW, Cor. 3.6].

1. Special gauges

Let (F, v) be a valued field and let A be an F -algebra. An F -linear involution on A is an F -linear map σ: A → A such that

(i) σ(x + y) = σ(x) + σ(y) for x, y ∈ A; (ii) σ(xy) = σ(y)σ(x) for x, y ∈ A; (iii) σ^{2}(x) = x for x ∈ A.

(The F -linearity implies that σ|_{F} = id_{F}.) A surmultiplicative v-value function φ: A → Γ ∪ {∞} is said to be invariant under σ if

φ(σ(x)) = φ(x) for all x ∈ A. (1.1)

The involution then preserves the filtration on A defined by φ. Therefore, it induces an involution σ̃ on gr_{φ}(A) such that

σ̃(x̃) = σ̃(x) for all x ∈ A.

As in [KMRT, §6.A], we say that the involution σ is anisotropic if there is no nonzero element x ∈ A such that σ(x)x = 0. Likewise, σ̃ is said to be anisotropic if there is no nonzero homogeneous element ξ ∈ gr_{φ}(A) such that σ̃(ξ)ξ = 0. Clearly, if σ̃ is anisotropic, then σ is anisotropic.

Proposition 1.1. Let φ be a surmultiplicative v-value function and σ an F -linear involution on A. The following conditions are equivalent:

(a) φ(σ(x)x) = 2φ(x) for all x ∈ A;

(b) φ is invariant under σ, and σ̃ is anisotropic.

They imply that if x, y ∈ A satisfy σ(x)y = 0 or xσ(y) = 0, then

φ(x + y) = min(φ(x), φ(y)).  (1.2)

Moreover, when these equivalent conditions hold, σ is anisotropic and the gr(F)-algebra gr_{φ}(A) contains no nonzero homogeneous nil left or right ideal.

Proof. (a) ⇒ (b): If σ(x)x = 0, then condition (a) implies that φ(x) = ∞, so x = 0. Thus, σ is anisotropic. By surmultiplicativity, we have

φ(σ(x)x) ≥ φ(σ(x)) + φ(x) for all x ∈ A.

Therefore, (a) implies φ(x) ≥ φ(σ(x)) for all x ∈ A. Substituting σ(x) for x in this inequality, we obtain φ(σ(x)) ≥ φ(x) for all x ∈ A. Therefore, φ is invariant under σ, and condition (a) can be reformulated as φ(σ(x)x) = φ(σ(x)) + φ(x) for all x ∈ A. Thus, it implies

σ̃(x)x̃ = (σ(x)x)^{∼} for all x ∈ A,

whence σ̃ is anisotropic, as σ is anisotropic.

(b) ⇒ (a): For all x ∈ A we have

σ̃(x̃)x̃ = \begin{cases}(σ(x)x)^{∼} & if φ(σ(x)x) = φ(σ(x)) + φ(x), \\ 0 & if φ(σ(x)x) > φ(σ(x)) + φ(x). \end{cases}

Condition (b) implies that the first case always occurs. Hence, for all x, φ(σ(x)x) = φ(σ(x)) + φ(x) = 2φ(x).

For the rest of the proof, assume (a) and (b) hold. Then clearly σ is anisotropic. Also, for x, y ∈ A we have by surmultiplicativity

φ(σ(x) · (x + y)) ≥ φ(σ(x)) + φ(x + y) = φ(x) + φ(x + y).  (1.3)

If σ(x)y = 0, then

φ(σ(x) · (x + y)) = φ(σ(x)x) = 2φ(x).  (1.4)

By combining (1.3) and (1.4), we obtain φ(x) ≥ φ(x + y). Similarly, interchanging x and y we get φ(y) ≥ φ(x + y), hence

min(φ(x), φ(y)) ≥ φ(x + y).

The reverse inequality holds by definition of a value function, hence (1.2) is proved when σ(x)y = 0. If xσ(y) = 0, we substitute σ(x) for x and σ(y) for y in the arguments above, obtaining

φ(σ(x) + σ(y)) = min(φ(σ(x)), φ(σ(y))).

Equation (1.2) follows since φ ∘ σ = φ.

To complete the proof, suppose I ⊂ gr_{φ}(A) is a homogeneous nil left (resp. right) ideal and ξ ∈ I is a nonzero homogeneous element. Let η = σ̃(ξ)ξ (resp. η = ξσ̃(ξ)). Then η ∈ I is σ̃-symmetric, homogeneous, and nonzero since σ̃ is anisotropic. Since I is nil, we may find k ≥ 1 such that η^{k} ≠ 0 and η^{k+1} = 0. For ζ = η^{k} we have

σ̃(ζ)ζ = ζ^{2} = η^{2k} = 0,

so ζ = 0, a contradiction. □

Definition 1.2. A surmultiplicative v-value function φ on a central simple algebra A with involution σ is called σ-special if it satisfies the conditions (a) and (b) of Prop. 1.1.

For use in §§3 and 6, we record how involution invariance of value functions behaves with respect to tensor products. Recall from [TW, Prop. 1.23, (1.25)] that if V is a finite-dimensional F -vector space with a v-norm α and W is an F -vector space with v-value function β, then there is a v-value function α ⊗ β on V ⊗_{F} W uniquely determined by the condition that the map (x ⊗ y)^{∼} ↦ x̃ ⊗ ỹ (for x ∈ V and y ∈ W) defines an isomorphism of graded vector spaces

Ω: gr_{α⊗β}(V ⊗_{F} W) ⟶^{∼} gr_{α}(V) ⊗_{gr(F)} gr_{β}(W).  (1.5)

In particular,

(α ⊗ β)(x ⊗ y) = α(x) + β(y) for all x ∈ V and y ∈ W .  (1.6)

The value function α ⊗ β can be defined as follows: take any splitting base (e_{i})_{i=1}^{n} for α on V; then,

(α ⊗ β)( ∑_{i=1}^{n} e_{i} ⊗ y_{i} ) = min_{1≤i≤n}(α(e_{i}) + β(y_{i})) for any y_{1}, …, y_{n} ∈ W .

Furthermore, analogous to [TW, Cor. 1.26], if (W, β) is a valued field extending (F, v), then α ⊗ β is a β-norm on V ⊗_{F} W and (1.5) is a gr_{β}(W)-vector space isomorphism.

Proposition 1.3. Let σ and τ be F -linear involutions on F -algebras A and B respectively, and let φ (resp. ψ) be a surmultiplicative v-value function on A (resp. B) invariant under σ (resp. τ). Suppose A is finite-dimensional and φ is a v-norm. Then, φ⊗ψ is a surmultiplicative v-value function on A⊗_{F} B invariant under the involution σ ⊗ τ, and the canonical isomorphism Ω of (1.5) is an isomorphism of graded gr(F)-algebras with involution,

(gr_{φ⊗ψ}(A ⊗_{F} B), { \atop σ ⊗ τ}) ⟶^{∼} (gr_{φ}(A) ⊗_{gr(F)} gr_{ψ}(B), σ̃ ⊗ τ̃).

Proof. Let (e_{i})_{i=1}^{n} be a splitting base of A for φ. For x, y ∈ A ⊗_{F} B we may write

x = ∑_{i=1}^{n} e_{i} ⊗ x_{i}  and  y = ∑_{j=1}^{n} e_{j} ⊗ y_{j}  for some x_{1}, …, y_{n} ∈ B.

Then,

(φ ⊗ ψ)(xy) = (φ ⊗ ψ)(∑_{i,j} e_{i}e_{j} ⊗ x_{i}x_{j}) ≥ min_{1≤i,j≤n}((φ ⊗ ψ)(e_{i}e_{j} ⊗ x_{i}y_{j})) = min_{i,j}(φ(e_{i}e_{j}) + ψ(x_{i}y_{j})).

Since φ and ψ are surmultiplicative, we have φ(e_{i}e_{j}) ≥ φ(e_{i})+φ(e_{j}) and ψ(x_{i}y_{j}) ≥ ψ(x_{i})+ψ(y_{j}), hence

(φ ⊗ ψ)(xy) ≥ min_{i,j}(φ(e_{i}) + φ(e_{j}) + ψ(x_{i}) + ψ(y_{j})) = min_{i}(φ(e_{i}) + ψ(x_{i})) + min_{j}(φ(e_{j}) + ψ(y_{j})).

The right side is (φ ⊗ ψ)(x) + (φ ⊗ ψ)(y), so

(φ ⊗ ψ)(xy) ≥ (φ ⊗ ψ)(x) + (φ ⊗ ψ)(y).

Since moreover (φ ⊗ ψ)(1 ⊗ 1) = φ(1) + ψ(1) = 0, surmultiplicativity of φ ⊗ ψ is proved.

To show that the gr_{F} -vector space isomorphism Ω is a ring isomorphism, we check this for Ω^{−1}. The canonical F -algebra homomorphisms ι_{A}: A → A ⊗_{F} B, a ↦ a ⊗ 1 and ι_{B}: B → A ⊗_{F} B, b ↦ 1 ⊗ b, are value-preserving. Hence, they induce gr(F)-algebra homomorphisms ι̃_{A}: gr_{α}(A) → gr_{α⊗β}(A ⊗_{F} B), ã ↦ { \atop a ⊗ 1}, and ι̃_{B}: gr_{β}(B) → gr_{α⊗β}(A ⊗_{F} B), b̃ ↦ { \atop 1 ⊗ b}. For any a ∈ A and b ∈ B, we have from (1.6),

{ \atop a ⊗ 1} · { \atop 1 ⊗ b} = [(a ⊗ 1) · (1 ⊗ b)]^{∼} = { \atop a ⊗ b} = [(1 ⊗ b) · (a ⊗ 1)]^{∼} = { \atop 1 ⊗ b} · { \atop a ⊗ 1}.

Thus, im(ι̃_{B}) centralizes im(ι̃_{A}) in gr_{α⊗β}(A⊗_{F} B). So, there is an induced gr(F)-algebra homomorphism gr_{α}(A) ⊗_{gr(F)} gr_{β}(B) → gr_{α⊗β}(A ⊗_{F} B) given by ã ⊗ b̃ ↦ { \atop a ⊗ 1} · { \atop 1 ⊗ b} = { \atop a ⊗ b}. The description of Ω preceding (1.5) shows that this algebra homomorphism is Ω^{−1}.

To prove φ ⊗ ψ is invariant under σ ⊗ τ, we first show that (σ(e_{i}))_{i=1}^{n} also is a splitting base of A for φ. Take any c_{1}, …, c_{n} ∈ F. Then, as φ is invariant under σ and the c_{i} are central in A and fixed under σ,

φ( ∑_{i=1}^{n} σ(e_{i})c_{i}) = φ(σ( ∑_{i=1}^{n} σ(e_{i})c_{i})) = φ( ∑_{i=1}^{n} e_{i}c_{i}) = min_{1≤i≤n} (φ(e_{i}) + v(c_{i})) = min_{1≤i≤n} (φ(σ(e_{i})) + v(c_{i})).

Thus, (σ(e_{i}))_{i=1}^{n} is a splitting base for φ. With the notation above, we then have

(φ ⊗ ψ)((σ ⊗ τ)(x)) = (φ ⊗ ψ)(∑_{i=1}^{n} σ(e_{i}) ⊗ τ(x_{i})) = min_{1≤i≤n}(φ(σ(e_{i})) + ψ(τ(x_{i}))).

Since φ is invariant under σ and ψ under τ, we have

min_{1≤i≤n}(φ(σ(e_{i})) + ψ(τ(x_{i}))) = min_{1≤i≤n}(φ(e_{i}) + ψ(x_{i})) = (φ ⊗ ψ)(x).

Therefore, φ ⊗ ψ is invariant under σ ⊗ τ. To complete the proof, observe that for a ∈ A and b ∈ B we have

{ \atop σ ⊗ τ}({ \atop a ⊗ b}) = (σ(a) ⊗ τ(b))^{∼} = (σ̃ ⊗ τ̃)(ã ⊗ b̃),

hence the involution σ̃ ⊗ τ̃ corresponds to { \atop σ ⊗ τ} under the canonical isomorphism (1.5). □

The following special case will be particularly useful:

Corollary 1.4. Let A be a finite-dimensional F -algebra with an F -linear involution σ and let (K, v_{K}) be any valued field extension of (F, v). If φ is a surmultiplicative v-norm on A which is invariant under σ, then φ ⊗ v_{K} is a surmultiplicative v_{K}-norm on A ⊗_{F} K invariant under the involution σ ⊗ id_{K}, and (σ ⊗ id_{K})^{∼} ≅ σ̃ ⊗ id_{gr(K)} under the canonical isomorphism (1.5).

Proof. It suffices to note that ĩd_{K} = id_{gr(K)} and that the canonical isomorphism is an isomorphism of gr(K)-algebras, see [TW, Cor. 1.26]. □

Now, assume A is simple and finite-dimensional, and let n = deg A, so [A:Z(A)] = n^{2}. Recall from [KMRT] that involutions on A are classified into two kinds and three types: an involution σ is of the first kind if σ|_{Z(A)} = id_{Z(A)}; otherwise it is of the second kind. Involutions of the second kind are also said to be of unitary type (or simply unitary). To define the type of an involution σ of the first kind we consider the subspaces of symmetric and of symmetrized elements in A, defined by

Sym(A, σ) = {x ∈ A | σ(x) = x}  and  Symd(A, σ) = {x + σ(x) | x ∈ A}.

The involution σ is of symplectic type (or simply symplectic) if either char(F) ≠ 2 and dim_{Z(A)} Sym(A, σ) = \frac{1}{2}n(n − 1) or char(F) = 2 and 1 ∈ Symd(A, σ). Involutions of the first kind that are not symplectic are said to be of orthogonal type (or simply orthogonal). If σ is orthogonal, then dim_{Z(A)} Sym(A, σ) = \frac{1}{2}n(n + 1). The same terminology is used for involutions on graded simple algebras.

Proposition 1.5. Let σ be an F -linear involution on a finite-dimensional simple F -algebra A and let g be a tame v-gauge on A that is invariant under σ. Suppose F is the subfield of Z(A) fixed under σ.

If σ is unitary, two cases may arise:

– if the valuation v extends uniquely from F to Z(A), then gr_{g}(A) is a graded simple gr(F)-algebra and σ̃ is a unitary involution;

– if the valuation v has two different extensions to Z(A), then gr_{g}(A) is a direct product of two graded central simple gr(F)-algebras, which are exchanged under σ̃.

If σ is symplectic, then σ̃ is a symplectic involution on the graded central simple gr(F)-algebra gr_{g}(A).

If σ is orthogonal and char(\overline{F}) ≠ 2, then σ̃ is an orthogonal involution on the graded central simple gr(F)-algebra gr_{g}(A).

Proof. Suppose first that σ is unitary, so Z(A)/F is a quadratic extension. By [TW, Cor. 2.5], the number of simple components of gr(A) equals the number of extensions of v to Z(A). Therefore, to complete the description of σ̃ it suffices to show that σ̃ does not identically fix Z(gr(A)) = gr(Z(A)). Since the Galois group G(Z(A)/F) acts transitively on the set of extensions of v to Z(A), see [EP, Th. 3.2.15, p. 64], if there are two such extensions, then σ|_{Z(A)} must permute them; then σ̃ permutes the corresponding components of gr(Z(A)). So, we may assume that v has a unique extension to Z(A). Then, gr (Z(A)) is a graded field separable over gr(F), and [gr (Z(A)): gr(F)] = [Z(A):F] = 2 since g|_{Z(A)} is a norm, by [RTW, Prop. 2.5]. If char(\overline{F}) ≠ 2 we can find z ∈ Z(A) nonzero such that σ(z) = −z, hence σ̃(z̃) = −z̃ ≠ z̃. If char(\overline{F}) = 2 the separability of gr(Z(A)) over gr(F) implies by [HW_{1}, Th. 3.11, Def. 3.4] that Γ_{Z(A)} = Γ_{F} and Z(A)_{0} is separable over F_{0}; furthermore, [Z(A)_{0}:F_{0}] = [Z(A):F] = 2 since g|_{Z(A)} is a norm, by [RTW, Prop. 2.5]. So Z(A) is unramified Galois over F, hence the non-trivial automorphism σ|_{Z(A)} induces a non-trivial automorphism of the residue algebra Z(A)_{0}, by [E, Th. 19.6, p. 124], showing that σ̃|_{gr(Z(A))} is nontrivial.

Suppose next that σ is of the first kind, so Z(A) = F. For x ∈ A we have x̃+σ̃(x̃) = (x+σ(x))^{∼} or 0. On the other hand, σ(x) = x implies σ̃(x̃) = x̃. Therefore, the following inclusions are clear:

gr(Sym(A, σ)) ⊆ Sym(gr(A), σ̃),  Symd(gr(A), σ̃) ⊆ gr(Symd(A, σ)).  (1.7)

If char(\overline{F}) ≠ 2 (hence char(F) ≠ 2) we have

Sym(A, σ) = Symd(A, σ)  and  Sym(gr(A), σ̃) = Symd(gr(A), σ̃),

so the inclusions in (1.7) above yield gr(Sym(A, σ)) = Sym(gr(A), σ̃). Since the type of an involution can be determined from the dimension of the space of symmetric elements, it follows that σ̃ has the same type as σ.

To complete the proof, suppose char(\overline{F}) = 2 and σ is symplectic, and let n = deg A. Since σ̃ is of the first kind we have

dim_{gr(F)} Symd(gr(A), σ̃) = \frac{1}{2}n(n − 1).

On the other hand, since σ is symplectic we have

dim_{F} Symd(A, σ) = \frac{1}{2}n(n − 1)

(independently of whether char(F) = 2). Since g is a norm we have

dim_{F} Symd(A, σ) = dim_{gr(F)} gr(Symd(A, σ)),

hence Symd(gr(A), σ̃) = gr(Symd(A, σ)). Since 1 ∈ Symd(A, σ), it follows that ˜1 ∈ Symd(gr(A), σ̃), hence σ̃ is symplectic. □

Remark. If σ is orthogonal and char(\overline{F}) = 2, the involution σ̃ may be symplectic, as the following example shows: let (F, v) be a valued field with char(F) = 0 and char(\overline{F}) = 2, and let A = M_{2}(F). Define an orthogonal involution σ on A by

σ ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}d & b \\ c & a \end{matrix})

and a v-gauge g by

g ( \begin{matrix}a & b \\ c & d \end{matrix} ) = min(v(a), v(b), v(c), v(d)).

This gauge is clearly invariant under σ. We have gr_{g}(A) = M_{2}(gr(F)) with the entrywise grading, and

( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) + σ̃ ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ Symd(gr_{g}(A), σ̃).

Therefore, σ̃ is symplectic.

Henceforth, we systematically avoid orthogonal involutions in characteristic 2.

In [RTW, §3], a notion of compatibility is defined between norms and hermitian forms. In the rest of this section, we relate that notion of compatibility with the invariance of value functions under involutions.

Let D be a finite-dimensional division F -algebra with an F -linear involution τ. Suppose v extends to a valuation w on D invariant under τ and let V be a finite-dimensional right D-vector space. Consider a nondegenerate hermitian form h: V × V → D with respect to τ, and a w-norm α on V. The dual norm α^{♯} is defined by

α^{♯}(x) = min{ w(h(x, y)) − α(y) | y ∈ V, y ≠ 0}  for x ∈ V ,  (1.8)

see [RTW, §3]. The norm α is said to be compatible with h if and only if α^{♯} = α (see [RTW, Prop. 3.5]). This is the condition needed in order for h to induce a nondegenerate graded hermitian form on gr_{α}(V). On the simple algebra End_{D}(V) there is the involution ad_{h} adjoint to h, defined by

h(ad_{h}(f)(x), y) = h(x, f(y)) for all x, y ∈ V.

There is also the well-defined surmultiplicative v-value function End(α) on End_{D}(V) defined by

End(α)(f) = min{α(f(x)) − α(x) | x ∈ V, x ≠ 0}.

Recall that End(α) is a v-gauge if and only if w on D is defectless over v, see [TW, Prop. 1.19].

Proposition 1.6. The value functions End(α) and End(α^{♯}) are related by

End(α) ∘ ad_{h} = End(α^{♯}).  (1.9)

Moreover, the following conditions are equivalent:

(a) End(α) is invariant under ad_{h};

(b) End(α^{♯}) = End(α);

(c) α − α^{♯} is constant on V;

(d) there is a constant γ ∈ Γ such that α − γ is compatible with h.

Proof. Let (e_{i})_{i=1}^{n} be a splitting base of V for α. The h-dual base (e_{i}^{♯})_{i=1}^{n} for V is a splitting base for α^{♯}, by [RTW, Lemma 3.4]. Fix some f ∈ End_{D} V, and let

f(e_{j}^{♯}) = ∑_{i=1}^{n} e_{i}^{♯}d_{ij} for some d_{ij} ∈ D.

Then, computation yields

ad_{h}(f)(e_{j}) = ∑_{i=1}^{n} e_{i}τ(d_{ji}).

We may compute End(α^{♯})(f) using the splitting base (e_{i}^{♯})_{i=1}^{n}, and End(α)(ad_{h}(f)) using the splitting base (e_{i})_{i=1}^{n}, obtaining

End(α^{♯})(f) = min_{1≤i,j≤n}(α^{♯}(e_{i}^{♯}) + w(d_{ij}) − α^{♯}(e_{j}^{♯})),
End(α)(ad_{h}(f)) = min_{1≤i,j≤n}(α(e_{i}) + w(τ(d_{ji})) − α(e_{j})).

Equation (1.9) follows since α^{♯}(e_{i}^{♯}) = −α(e_{i}), see [RTW, Lemma 3.4].

The equivalence of (a) and (b) readily follows from (1.9), and the equivalence of (b) and (c) from [TW, Prop. 1.22].

(c) ⇔ (d): By the definition of the dual norm in (1.8), for any constant γ in the divisible group Γ, (α − γ)^{♯} = α^{♯} + γ. Therefore, α − γ is compatible with h if and only if (α − γ)^{♯} = α − γ, which holds if and only if α − α^{♯} = 2γ. □

Suppose the equivalent conditions of Prop. 1.6 hold, and write simply g_{α} for End(α). Recall from [TW, Prop. 1.19] that the graded algebra gr_{g_{α}}(End_{D} V) may be identified with End_{gr(D)}(gr_{α}(V)) so that for f ∈ End_{D} V the element f̃ ∈ gr_{g_{α}}(End_{D} V) is viewed as the map f̃: gr_{α}(V) → gr_{α}(V) defined by

f̃(x̃) = \begin{cases}f̃(x) & if α(f(x)) = α(x) + g_{α}(f), \\ 0 & if α(f(x)) > α(x) + g_{α}(f). \end{cases}

On the other hand, after adding a constant if necessary, we may assume α is compatible with h; hence we may define a graded hermitian form

h̃: gr_{α}(V) × gr_{α}(V) → gr_{w}(D)

(with respect to the involution τ̃) as follows: for x, y ∈ V,

h̃(x̃, ỹ) = h(x, y) + D^{>α(x)+α(y)} = \begin{cases}{ \atop h(x, y)} & if w(h(x, y)) = α(x) + α(y), \\ 0 & if w(h(x, y)) > α(x) + α(y). \end{cases}

This hermitian form is well-defined and nondegenerate (see [RTW, Remark 3.2]), so we may consider the adjoint involution adh̃ on End_{gr_{w}(D)}(gr_{α}(V)) = gr_{g_{α}}(End_{D} V).

Proposition 1.7. Assuming α is compatible with h, the involution ãd_{h} on gr_{g_{α}}(End_{D} V) is the adjoint involution of h̃ under the identification above; i.e.,

ãd_{h} = adh̃ .

In particular, the value function g_{α} = End(α) is σ-special (see Def. 1.2) if and only if h̃ is anisotropic.

Proof. To verify the equality of graded involutions, it suffices to show, for all x, y ∈ V and f ∈ End_{D} V,

h̃(adh̃(f̃)(x̃), ỹ) = h̃(ãd_{h}(f̃)(x̃), ỹ).

From the definition of adh̃, it is equivalent to prove

h̃(x̃, f̃(ỹ)) = h̃(ãd_{h}(f̃)(x̃), ỹ).  (1.10)

Since α is compatible with h, Prop. 1.6 shows g_{α} is invariant under ad_{h}; hence, g_{α}(ad_{h}(f)) = g_{α}(f). Therefore, each side of (1.10) lies in D_{ϵ}, where ϵ = α(x) + α(y) + g_{α}(f). Suppose w(h(x, f(y))) = ϵ. Then, necessarily α(f(y)) = g_{α}(f) + α(y), and the left side of (1.10) equals h(x, f(y))^{∼}. But since h(x, f(y)) = h(ad_{h}(f)(x), y), we then also have g_{α}( ad_{h}(f)(x)) = g_{α}( ad_{h}(f)) + α(x), and the right side of (1.10) becomes h(ad_{h}(f)(x), y)^{∼}. So, (1.10) then holds. But, if w(h(x, f(y))) > ϵ, then each side of (1.10) is 0. Thus, the equality (1.10) holds in all cases, so that ãd_{h} = adh̃.

Since g_{α} is invariant under ad_{h}, Prop. 1.1(b) holds if and only if ãd_{h} is anisotropic. But, the involution ãd_{h} = adh̃ is anisotropic if and only if its associated graded hermitian form h̃ is anisotropic. This is proved analogously to the ungraded case [KMRT, §6.A], using the fact that h̃ is anisotropic if and only if h̃(x̃, x̃) ≠ ˜0 for all nonzero x ∈ V, as remarked in [RTW, p. 101]. □

2. Henselian valuations

Throughout this section, (F, v) is a Henselian valued field and A is a finite-dimensional simple F - algebra with an involution σ. We let K = Z(A) and assume F is the subfield of K fixed by σ. (Thus, A is central over F if σ|_{Z(A)} = id_{Z(A)}). We assume A is tame over F, which means that A is split by the maximal tamely ramified extension of K, and that K is tame over F. Moreover, if char(\overline{F}) = 2 we assume σ is not an orthogonal involution.

Proposition 2.1. With the hypotheses above, every v-gauge on A is tame. Furthermore, there exist v-gauges on A that are invariant under σ.

Proof. We may represent A = End_{D} V for some finite-dimensional right vector space V over a central division K-algebra D. Since v is Henselian, it extends uniquely to a valuation w on D (see for instance [Sch, p. 53, Th. 9] or [W, Th.]). Since A is tame over F, by [TW, Prop. 1.19] D must also be tame over F; hence by [TW, Prop. 1.12 and 1.13] w is a tame v-gauge. Therefore, every v-gauge on A is tame, by [TW, Th. 3.1].

If A is split and σ is symplectic, then σ = ad_{b} for some alternating bilinear form b on V, see [KMRT, §4.A]. Choose a symplectic base B = (e_{i}, f_{i})_{i=1}^{n} of V for b and define a v-norm α on V by

α(∑_{i=1}^{n} e_{i}λ_{i} + f_{i}μ_{i}) = min_{1≤i≤n}(v(λ_{i}), v(μ_{i})) for λ_{1}, …, μ_{n} ∈ F ,

i.e., B is a splitting base for α on V, and each α(e_{i}) = α(f_{j}) = 0. The v-norm α on V induces the v-gauge End(α) on End_{D}(V). For g ∈ End_{D}(V), if g has matrix (c_{ij}) relative to B, then End(α)(g) = min_{1≤i,j≤n} v(c_{ij}). The matrix for σ(g) has the same set of entries up to sign as (c_{ij}), though the entries are relocated. Hence, End(α) is invariant under σ. We exclude this case of A split and σ symplectic for the rest of the proof. We may then choose an F -linear involution θ on D of the same type as σ and an even hermitian form h on V with respect to θ such that σ = ad_{h}, see [KMRT, (4.2)]. By [RTW, Cor. 3.6], there exists a w-norm α on V that is compatible with h. By [TW, Prop. 1.19], End(α) is a v-gauge; by Prop. 1.6, this gauge is invariant under σ. □

Theorem 2.2. With the hypotheses of this section, if σ is anisotropic, then for the Henselian valuation v on F there is a unique σ-special value function φ on A for v. This φ is a tame v-gauge and its value set Γ_{A} lies in the divisible hull of Γ_{F}. It is the unique v-gauge on A invariant under σ.

Proof. We use the same notation as in the proof of Prop. 2.1, representing A = End_{D}(V) as in that proof. Since σ is anisotropic, it is not a symplectic involution on a split algebra. Therefore, it is the adjoint involution of some even Hermitian form h on V with respect to an involution θ on D of the same type as σ, see [KMRT, Th. (4.2)]. The form h is anisotropic since σ is anisotropic. By [RTW, Th. 4.6 and Prop. 4.2], the map α: V → \frac{1}{2}Γ_{D} ∪ \{∞\} defined by

α(x) = \frac{1}{2}w(h(x, x))  (2.1)

is a w-norm on V that is compatible with h, and the residue form h̃ is anisotropic. Prop. 1.6 then shows that φ = End(α) is a surmultiplicative v-value function on A that is invariant under σ, and Prop. 1.7 shows that φ Is σ-special. Since A is tame over F, the valuation w is a v-gauge on D by [TW, Prop. 1.13], hence φ is a tame v-gauge by [TW, Prop. 1.19]. Its value set obviously lies in the divisible hull of Γ_{D}, which is also the divisible hull of Γ_{F}.

To prove uniqueness, suppose φ_{1} and φ_{2} are each σ-special value functions on A for v. To show that φ_{1} = φ_{2}, we argue by induction on the matrix size ms(A), which is defined as the dimension of V in the representation A = End_{D}(V).

Suppose first that A is a division algebra. For any subfield L ⊆ A fixed elementwise under σ we have

φ_{i}(x^{2}) = φ_{i}(σ(x)x) = 2φ_{i}(x) for all x ∈ L and i = 1, 2,

hence for nonzero ξ = x^{φ_{i}} ∈ gr_{φ_{i}}(L) we have ξ^{2} = x̃^{2}^{φ_{i}} ≠ 0. Therefore, gr_{φ_{i}}(L) is semisimple. By [TW, Prop. 1.8], it follows that φ_{1} and φ_{2} coincide with the unique valuation on L extending v. (The extension of v to L is unique because (F, v) is Henselian.) For any x ∈ A, the product σ(x)x lies in a subfield of A fixed under σ, so φ_{1}(σ(x)x) = φ_{2}(σ(x)x). Therefore,

φ_{1}(x) = \frac{1}{2}φ_{1}(σ(x)x) = \frac{1}{2}φ_{2}(σ(x)x) = φ_{2}(x).

The claim is thus proved if ms(A) = 1.

Suppose next that ms(A) > 1. We may then find in A a symmetric idempotent e ≠ 0, 1. (Representing A = End_{D}(V) as above, we have dim_{D} V > 1 and we may take for e the orthogonal projection onto any nonzero proper subspace of V.) Let f = 1 − e. The involution σ restricts to eAe and fAf, and ms(eAe), ms(fAf) < ms(A). By the induction hypothesis, the restrictions of φ_{1} and φ_{2} coincide on eAe and fAf. For any x ∈ A, we have σ(xe)xe ∈ eAe and σ(xf)xf ∈ fAf, hence

φ_{1}(σ(xe)xe) = φ_{2}(σ(xe)xe) and φ_{1}(σ(xf)xf) = φ_{2}(σ(xf)xf).

Since φ_{1} and φ_{2} are σ-special value functions, Prop. 1.1 shows that

φ_{1}(xe) = φ_{2}(xe) and φ_{1}(xf) = φ_{2}(xf).  (2.2)

On the other hand, we have xeσ(xf) = 0 and xe + xf = x, hence Prop. 1.1 also yields

φ_{1}(x) = min(φ_{1}(xe), φ_{1}(xf)) and φ_{2}(x) = min(φ_{2}(xe), φ_{2}(xf)).

By (2.2), it follows that φ_{1}(x) = φ_{2}(x).

Now, suppose g is a gauge on A that is invariant under σ. By [TW, Th. 3.1] we may find a w-norm β on V such that g = End(β). Up to the addition of a constant, we may assume β is compatible with h in view of Prop. 1.6. But the norm α of (2.1) is the only w-norm on V that is compatible with h by [RTW, Prop. 4.2], so β = α and g = End(α). □

If g is a v-gauge on A that is invariant under σ, we denote by σ_{0} the 0-component of σ̃. Thus, σ_{0} is an involution on the \overline{F} -algebra A_{0} = A^{≥0}/A^{>0}, which may be viewed as the residue algebra of A. The algebra A_{0} is semisimple, but not necessarily simple, see [TW, §2]. Note that if A_{0} = B_{1} ×…×B_{k} with the B_{i} simple, then an involution τ on A_{0} is anisotropic if and only if τ(B_{i}) = B_{i} and τ|_{B_{i}} is anisotropic for 1 ≤ i ≤ k.

Corollary 2.3. With the hypotheses of this section, if g is a v-gauge on A that is invariant under σ, the following conditions are equivalent:

(a) σ is anisotropic;

(b) σ̃ is anisotropic;

(c) σ_{0} is anisotropic.

Proof. The implication (a) ⇒ (b) readily follows from Th. 2.2, and the implications (b) ⇒ (a) and (b) ⇒ (c) are clear. To prove (c) ⇒ (b), suppose ξ ∈ gr_{g}(A) is a nonzero homogeneous element such that σ̃(ξ)ξ = 0. Every element η ∈ (ξ gr_{g}(A)) ∩ A_{0} satisfies σ̃(η)η = σ_{0}(η)η = 0. Therefore, σ_{0} is isotropic if (ξ gr_{g}(A)) ∩ A_{0} ≠ {0}. The corollary thus follows from the following general result:

Lemma 2.4. Let A be a graded simple algebra finite-dimensional over a graded field K, and let I ⊆ A be a homogeneous right ideal. Then, there is a homogeneous idempotent e ∈ A of degree 0 such that I = eA.

Proof. By [HW_{2}, Prop. 1.3], we may identify A = End_{D}(V) for some graded division K-algebra D and some finite-dimensional graded D-vector space V. Let W = ∑ im(f), with the sum taken over all homogeneous f ∈ I. Then, W is a graded D-subspace of V and, just as in the ungraded case, I = Hom_{D}(V, W). Take any graded D-subspace Y of V, such that Y is complementary to W, and let e: V → W be the projection of V onto W along Y. Then, the idempotent e is a degree-preserving graded homomorphism, so e ∈ A_{0}. Clearly, I = eA. □

Remarks 2.5. (1) In Cor. 2.3, the hypothesis that (F, v) is Henselian is used only to prove that (a) implies (b) and (c); the implications (c) ⟺ (b) ⇒ (a) hold without this hypothesis (nor any tameness assumption).

(2) Corollary 2.3 may be regarded as a version of Springer’s theorem for involutions. In a slightly different form, it has already been proved by Larmour [L, Th. 4.5]: to see this, observe that the residue involutions defined by Larmour are the direct summands of our residue involution σ_{0} for a suitable gauge.

If the involution σ is isotropic, we may still define up to isomorphism an anisotropic kernel (A, σ)_{an} in such a way that if A = End_{D} V and σ = ad_{h}, then (A, σ)_{an} ≅ (End_{D} V_{0}, ad_{h_{0}}) where (V_{0}, h_{0}) is an anisotropic kernel of (V, h), see [DLT], and [D] for involutions of the second kind. The same construction holds for graded simple algebras with involution.

Theorem 2.6. Let σ_{1}, σ_{2} be F -linear involutions on A such that σ_{1}|_{Z(A)} = σ_{2}|_{Z(A)}. For the Henselian valuation v on F, let g_{1}, g_{2} be v-gauges on A invariant under σ_{1} and σ_{2} respectively. If char(\overline{F}) = 2, assume neither σ_{1} nor σ_{2} is orthogonal. The following conditions are equivalent:

(a) the algebras with involution (A, σ_{1}) and (A, σ_{2}) are isomorphic;

(b) the graded algebras with anisotropic involution (gr_{g_{1}}(A), σ̃_{1})_{an} and (gr_{g_{2}}(A), σ̃_{2})_{an} are isomorphic.

Proof. It follows from Prop. 1.5 that σ_{i} and σ̃_{i} are of the same type. Therefore, (a) and (b) each imply that σ_{1} and σ_{2} are of the same type. If A is split and σ_{1}, σ_{2} are symplectic, then gr_{g_{1}}(A) and gr_{g_{2}}(A) are split and σ̃_{1}, σ̃_{2} are symplectic, hence hyperbolic. In this case, (a) and (b) both hold trivially. For the rest of the proof, we exclude this case and fix a representation A = End_{D} V where V is a right vector space over a central division K-algebra D. We also fix an involution θ on D of the same type as σ_{1} and σ_{2}, and non-degenerate even hermitian forms h_{1}, h_{2} on V with respect to θ such that

σ_{1} = ad_{h_{1}}  and  σ_{2} = ad_{h_{2}} .

As observed in the proof of Prop. 2.1, the valuation v extends uniquely to a valuation w on D. By [TW, Th. 3.1] and Prop. 1.6 we may also find norms α_{1} and α_{2} on V that are compatible with h_{1} and h_{2} respectively, such that

g_{1} = End(α_{1}) and g_{2} = End(α_{2}),

hence

gr_{g_{1}}(A) = End_{gr(D)}(gr_{α_{1}}(V)) and gr_{g_{2}}(A) = End_{gr(D)}(gr_{α_{2}}(V)).

It then follows from Prop. 1.7 that

σ̃_{1} = adh̃_{1}  and  σ̃_{2} = adh̃_{2};

hence, denoting by (V_{1}, k_{1}) and (V_{2}, k_{2}) the anisotropic kernels of (gr_{α_{1}}(V), ˜(h_{1})) and (gr_{α_{2}}(V), ˜(h_{2})) respectively,

(gr_{g_{1}}(A), σ̃_{1})_{an} ≅ (End_{gr(D)}(V_{1}), ad_{k_{1}}) and (gr_{g_{2}}(A), σ̃_{2})_{an} ≅ (End_{gr(D)}(V_{2}), ad_{k_{2}}).

If (a) holds, then h_{1} and h_{2} are similar. Scaling h_{2} by a factor in F^{×}, we may assume h_{1} ≅ h_{2}. By [RTW, Th. 3.11], the anisotropic kernels of h̃_{1} and h̃_{2} are isometric, hence (b) holds.

Conversely, if (b) holds, then the anisotropic kernels of h̃_{1} and h̃_{2} are similar. Scaling h_{2} by a factor in F^{×}, we may assume that they are isometric. By [RTW, Th. 4.6], it follows that h_{1} and h_{2} are isometric, hence (a) holds. □

Corollary 2.7. With the hypotheses of this section, up to Witt-equivalence the graded algebra with involution (gr_{g}(A), σ̃) depends only on the Witt-equivalence class of (A, σ), and not on the choice of the invariant v-gauge g.

3. Scalar extensions of involutions

As an application of the results of §2, we consider a basic case of the problem of determining when an anisotropic involution remains anisotropic over a scalar extension.

Let σ be an F -linear involution on a finite-dimensional simple algebra A over a field F. Assume v is a valuation on F and A carries a v-gauge g invariant under σ. For any extension (L, v_{L}) of (F, v), we may consider the v_{L}-gauge g ⊗ v_{L} on the L-algebra A_{L} = A ⊗_{F} L. By Prop. 1.3, this v_{L}-gauge is invariant under the involution σ ⊗ id_{L} on A_{L}. If the “residue” involution (σ ⊗ id_{L})_{0} is anisotropic, then σ ⊗ id_{L} is anisotropic by Cor. 2.3 and Remark 2.5(1), and the converse holds if v_{L} is Henselian and A_{L} is tame over L, unless σ is orthogonal and char(\overline{F}) = 2. We consider below a case where this residue can be explicitly calculated.

We first recall some facts which will be used repeatedly below. Let α be a surmultiplicative v-norm on a finite-dimensional algebra A over a field F with valuation v. If e is an idempotent of A with α(e) = 0 and N is any F -subspace of A, then e^{2} = ẽ in gr(A) and by [TW, Lemma 1.7],

gr(eN) = ẽgr(N) and gr(Ne) = gr(N)ẽ in gr(A).  (3.1)

If e ≠ 1, let f = 1 − e. Then, α(f) ≥ min (α(1), α(e)) = 0, but since f^{2} = f, α(f) ≤ 0. So, α(f) = 0, hence f̃ = ˜1 − ẽ in gr(A), and hence

gr(A) = ẽgr(A) ⊕ f̃gr(A) = gr(eA) ⊕ gr(fA).

Therefore, by [RTW, Remark 2.6], the direct sum decomposition A = eA ⊕ fA is a splitting decomposition, i.e., α(a) = min (α(ea), α(fa)) for any a ∈ A. Likewise A = Ae ⊕ Af is a splitting decomposition.

Recall also that an element s ∈ A^{×} is said to be α-stable if α(s^{−1}) = −α(s). For such an s we have by [TW, Lemma 1.3 and (1.5)],

α(as) = α(sa) = α(a) + α(s), hence ã s̃ = ãs and s̃ ã = s̃a for every a ∈ A.  (3.2)

We now make some general observations on the tensor product of valuations. Let L/F be a finite separable field extension. Recall that the separability idempotent of L is the idempotent e ∈ L ⊗_{F} L determined uniquely by the conditions that

e · (x ⊗ 1) = e · (1 ⊗ x)  for all x ∈ L  (3.3)

and the multiplication map L ⊗_{F} L → L carries e to 1, see for instance [KMRT, Prop. (18.10)]. The separability of L/F implies that the bilinear trace form

T: L × L → F, T (x, y) = Tr_{L/F} (xy)

is nondegenerate.

Proposition 3.1. Suppose v: F → Γ ∪ {∞} is a valuation that extends uniquely to a valuation v_{L} on L, and that the valued field extension (L, v_{L}) of (F, v) is tame. Then v_{L} is a v-norm on L which is compatible with the bilinear trace form T, and (v_{L} ⊗ v_{L})(e) = 0.

Proof. Since v_{L} is the unique valuation extending v to L and since the extension is defectless, it follows that v_{L} is a v-norm (indeed, a v-gauge) on L, see [TW, Cor. 1.9].

We claim that v(Tr_{L/F} (x)) ≥ v_{L}(x) for all x ∈ L^{×}. To see this, consider a Galois closure M of L over F and an extension v_{M} of v to M. For every F -linear embedding ι: L ↪ M the composition v_{M} ∘ι is a valuation on L extending v, hence v_{M} ∘ ι = v_{L}. Since Tr_{L/F} (x) = ∑ ι(x), where the sum extends over all embeddings ι: L ↪ M, we have

v(Tr_{L/F} (x)) = v_{M}(∑_{ι} ι(x)) ≥ min_{ι}(v_{M} ∘ ι(x)) = v_{L}(x),

proving the claim. It follows that for all x, y ∈ L^{×},

v(T (x, y)) ≥ v_{L}(x) + v_{L}(y).  (3.4)

To show that v_{L} is compatible with T, it remains to show that for any x ∈ L^{×} there exists y ∈ L^{×} for which equality holds in (3.4). For this, it suffices to show that there exists ℓ ∈ L^{×} such that v(Tr_{L/F} (ℓ)) = v_{L}(ℓ), since equality then holds in (3.4) with y = ℓx^{−1}. For every ℓ ∈ L^{×} with v_{L}(ℓ) = 0 we have

\overline{Tr_{L/F} (ℓ)} = |Γ_{L}:Γ_{F} | · Tr_{\overline{L}/\overline{F}} (\overline{ℓ})  (3.5)

by [Er_{2}, p. 65, Cor. 1]. (Ershov assumes his valuation is Henselian; but the result carrries over to the situation here: Let F_{h} be the Henselization of F with respect to v. Since the unique extension of v to L is defectless, for any compositum of L with F_{h} we have [L · F_{h}:F_{h}] ≥ [\overline{L}:\overline{F}]|Γ_{L}:Γ_{F} | = [L:F]. Hence, L ⊗_{F} F_{h} is a field, and (3.5) holds for L/F because it holds for (L ⊗_{F} F_{h})/F_{h}.)

Since L/F is tame, the residue extension \overline{L}/\overline{F} is separable and char(\overline{F}) does not divide |Γ_{L}:Γ_{F} |. Therefore, we may find ℓ ∈ L such that v_{L}(ℓ) = 0 and Tr_{\overline{L}/\overline{F}} (\overline{ℓ}) ≠ 0. Then (3.5) shows that \overline{Tr_{L/F} (ℓ)} ≠ 0, hence

v(Tr_{L/F} (ℓ)) = 0 = v_{L}(ℓ).

Therefore, v_{L} is compatible with T; it thus coincides with its dual norm v_{L}^{♯}.

To complete the proof, we compute (v_{L} ⊗ v_{L})(e). Let (ℓ_{i})_{i=1}^{n} be a splitting F -base of L for v_{L}, and let (ℓ_{i}^{♯})_{i=1}^{n} be the dual base for the form T. By [KMRT, Prop. (18.12)] we have,

e = ∑_{i=1}^{n} ℓ_{i} ⊗ ℓ_{i}^{♯},

hence,

(v_{L} ⊗ v_{L})(e) = min_{1≤i≤n}(v_{L}(ℓ_{i}) + v_{L}(ℓ_{i}^{♯})).

Now, for all i = 1, …, n we have v_{L}(ℓ_{i}^{♯}) = v_{L}^{♯}(ℓ_{i}^{♯}) = −v_{L}(ℓ_{i}) by [RTW, Lemma 3.4]. Therefore, (v_{L} ⊗ v_{L})(e) = 0. □

Continuing with the same notation and hypotheses as in Prop. 3.1, we now assume further that the extension L/F is Galois. Let G denote its Galois group. Since v_{L} is the unique extension of v to L, v_{L} ∘ ι = v_{L} for any ι ∈ G, and hence ι induces a graded gr(F)-automorphism ι̃ of gr(L). For ι ∈ G, let

e_{ι} = (id ⊗ι)(e) ∈ L ⊗_{F} L,

and let ẽ_{ι} be the image of e_{ι} in gr(L ⊗_{F} L), which is canonically identified with gr(L) ⊗_{gr(F)} gr(L) by Prop. 1.3.

Lemma 3.2. The elements (e_{ι})_{ι∈G} form a family of orthogonal idempotents such that ∑_{ι∈G} e_{ι} = 1. They are the primitive idempotents of L ⊗_{F} L. They satisfy (v_{L} ⊗ v_{L})(e_{ι}) = 0 and

e_{ι} · (x ⊗ 1) = e_{ι} · (1 ⊗ ι(x))  for x ∈ L.  (3.6)

Likewise, for any y ∈ gr(L),

ẽ_{ι} · (y ⊗ ˜1) = ẽ_{ι} · (˜1 ⊗ ι̃(y)) in gr(L) ⊗_{gr(F)} gr(L).  (3.7)

Moreover, (ι ⊗ ι)(e) = e for ι ∈ G.

Proof. Equation (3.3) shows that e · (L ⊗_{F} L) = e · (L ⊗ 1) ≅ L. Since L is a field, e must be a primitive idempotent. Equation (3.6) readily follows by applying id ⊗ι to each side of (3.3). For equation (3.7), it suffices to verify the equality when y is homogeneous and nonzero. But then y = x̃ for some nonzero x ∈ L. Both x ⊗ 1 and 1 ⊗ ι(x) are v_{L} ⊗ v_{L}-stable in L ⊗_{F} L, as defined preceding (3.2) above. Hence, using equations (3.2) and (3.6),

ẽ_{ι} · (x̃ ⊗ ˜1) = ẽ_{ι} · ({ \atop x ⊗ 1}) = [e_{ι} · (x ⊗ 1)] = [e_{ι} · (1 ⊗ ι(x))] = ẽ_{ι} · ({ \atop 1 ⊗ ι(x)}) = ẽ_{ι} · (˜1 ⊗ ι̃(x̃)).

Since e is a primitive idempotent, it is clear that each e_{ι} is also a primitive idempotent. For ι, κ ∈ G and x ∈ L, as L ⊗_{F} L is commutative we have

e_{ι}e_{κ} · (1 ⊗ [κ(x) − ι(x)]) = e_{ι}e_{κ} · (1 ⊗ κ(x)) − e_{κ}e_{ι} · (1 ⊗ ι(x)) = (e_{ι}e_{κ} − e_{κ}e_{ι}) · (x ⊗ 1) = 0.

For ι ≠ κ, if we choose x ∈ L with ι(x) ≠ κ(x), then 1⊗[κ(x)−ι(x)] is a unit of L⊗_{F} L; hence, e_{ι}e_{κ} = 0.

As observed in the proof of Prop. 3.1, we have e = ∑_{i=1}^{n} ℓ_{i} ⊗ ℓ_{i}^{♯} if (ℓ_{i})_{i=1}^{n} is an F -base of L and (ℓ_{i}^{♯})_{i=1}^{n} is the dual base for the bilinear form T. It follows that e_{ι} = ∑_{i=1}^{n} ℓ_{i} ⊗ ι(ℓ_{i}^{♯}) for ι ∈ G, hence

∑_{ι∈G} e_{ι} = ∑_{i=1}^{n} ℓ_{i} ⊗ Tr_{L/F} (ℓ_{i}^{♯}).  (3.8)

Since (ℓ_{i}^{♯})_{i=1}^{n} is the dual base of (ℓ_{i})_{i=1}^{n}, we have

x = ∑_{i=1}^{n} ℓ_{i} Tr_{L/F} (ℓ_{i}^{♯}x) for x ∈ L.

In particular, ∑_{i=1}^{n} ℓ_{i} Tr_{L/F} (ℓ_{i}^{♯}) = 1, and equation (3.8) yields ∑_{ι∈G} e_{ι} = 1. So, the e_{ι} are all the primitive idempotents of L ⊗_{F} L.

Since v_{L} is the unique valuation extending v to L, we have v_{L} ∘ ι = v_{L} for all ι ∈ G, hence

(v_{L} ⊗ v_{L})(e_{ι}) = (v_{L} ⊗ v_{L})(e) = 0 for all ι ∈ G.

Finally, it is clear that (ι ⊗ ι)(e) satisfies the same equation (3.3) as e and is carried to 1 by the multiplication map L ⊗_{F} L → L. Since these properties determine e uniquely, we have (ι ⊗ ι)(e) = e for all ι ∈ G. □

It is well-known (cf. [P, pp. 256–257, Lemma b]) that the primitive idempotents of L⊗_{F} L are indexed by G and satisfy (3.6). The further properties of the e_{ι} given in Lemma 3.2 will be useful in what follows.

Now, assume further that L ⊆ D for some finite-dimensional division F -algebra D, and that v extends to a valuation v_{D} on D such that D/F is defectless; i.e., v_{D} is a v-norm on D. The restriction of v_{D} to L is then the unique valuation v_{L} extending v. We will use the idempotents (e_{ι})_{ι∈G} to analyze extensions of involutions from D to D ⊗_{F} L. Let C be the centralizer C_{D}(L). Viewing D as a right C-vector space, we have the canonical isomorphism

η: D ⊗_{F} L ⟶^{∼} End_{C}(D),  (3.9)

which carries d ⊗ ℓ to the map x ↦ dxℓ for d, x ∈ D and ℓ ∈ L. For ι ∈ G, consider the following C-subspace of D:

D_{ι} = {d ∈ D | ℓd = d ι(ℓ) for all ℓ ∈ L}.

Since ι on L is induced by an inner automorphism of D by Skolem-Noether, D_{ι} ≠ {0}. Since in addition, D_{id} = C and D_{κ} · D_{ι} ⊆ D_{ικ} for all ι, κ ∈ G, we must have dim_{C}(D_{ι}) = 1 for each ι.

Lemma 3.3. We have D⊗_{F} L = ⨁_{ι∈G} e_{ι} (D⊗1) and D = ⨁_{ι∈G} D_{ι}, and these direct sums are splitting decompositions of D ⊗_{F} L and D with respect to v_{D} ⊗ v_{L} and v_{D}, respectively. More precisely, we have

(v_{D} ⊗ v_{L})( ∑_{ι∈G} e_{ι} · (x_{ι} ⊗ 1)) = min_{ι∈G}(v_{D}(x_{ι})) for x_{ι} ∈ D,

and

v_{D}( ∑_{ι∈G} y_{ι}) = min_{ι∈G}(v_{D}(y_{ι})) for y_{ι} ∈ D_{ι}.

Furthermore, for all ι, κ ∈ G,

e_{ι}(D ⊗_{F} L)e_{κ} = e_{ι}(D_{κ^{−1}ι} ⊗ 1).

Proof. Let A = D ⊗_{F} L and α = v_{D} ⊗ v_{L}. Since (e_{ι})_{ι∈G} is a family of orthogonal idempotents with ∑_{ι∈G} e_{ι} = 1 and α(e_{ι}) = 0 for each ι, the collection (ẽ_{ι})_{ι∈G} is a family of orthogonal idempotents in gr(A) with ∑_{ι∈G} ẽ_{ι} = ˜1. Hence, using (3.1),

A = ⨁_{ι∈G} e_{ι}A and gr(A) = ⨁_{ι∈G} ẽ_{ι} gr(A) = ⨁_{ι∈G} gr(e_{ι}A).  (3.10)

Likewise, for any ι ∈ G,

e_{ι}A = ⨁_{κ∈G} e_{ι}Ae_{κ} and gr(e_{ι}A) = ⨁_{κ∈G} gr(e_{ι}A)ẽ_{κ} = ⨁_{κ∈G} gr(e_{ι}Ae_{κ}).  (3.11)

In view of (3.6), we have e_{ι}·(1⊗L) = e_{ι}·(L⊗1), hence e_{ι}·(D⊗_{F} L) = e_{ι}·(D⊗1). For any nonzero x ∈ D, α(x ⊗ 1) = v_{D}(x) and α((x ⊗ 1)^{−1}) = α(x^{−1} ⊗ 1) = −v_{D}(x). So, x ⊗ 1 is α-stable, and (3.2) applies. Since (3.10) shows that the direct sum A = ⨁_{ι∈G} e_{ι}A is a splitting decomposition of A for α, it follows using (3.2) that for any x_{ι} ∈ D,

α( ∑_{ι∈G} e_{ι} · (x_{ι} ⊗ 1)) = min_{ι∈G} (α(e_{ι} · (x_{ι} ⊗ 1))) = min_{ι∈G} (α(e_{ι}) + α(x_{ι} ⊗ 1)) = min_{ι∈G} (v_{D}(x_{ι})).

To prove the rest, we use the canonical isomorphism η of (3.9). For each ι ∈ G, let π_{ι} = η(e_{ι}), which is a projection in End_{C}(D). By (3.6) and the commutativity of L ⊗_{F} L, for any ℓ ∈ L and d ∈ D,

ℓ · π_{ι}(d) = η((ℓ ⊗ 1)e_{ι})(d) = η((1 ⊗ ι(ℓ))e_{ι})(d) = π_{ι}(d) · ι(ℓ).

Hence, π_{ι}(D) ⊆ D_{ι}. Since im(π_{ι}) is a nonzero C-subspace of the 1-dimensional C-vector space D_{ι}, it follows that im(π_{ι}) = D_{ι}. Because (π_{ι})_{ι∈G} is a family of orthogonal idempotents of End_{C}(D) such that ∑_{ι∈G} π_{ι} = id_{D}, we have D = ⨁_{ι∈G} D_{ι}; furthermore, each π_{ι} is the projection of D onto D_{ι} parallel to ⨁_{κ≠ι} D_{κ}. Thus, for any ι, κ ∈ G, π_{ι} End_{C}(D)π_{κ} consists of those endomorphisms sending D_{κ} to D_{ι} and D_{λ} to {0} for λ ≠ κ. For any λ ∈ G, since D_{κ^{−1}ι}D_{λ} ⊆ D_{λκ^{−1}ι}, we have

[π_{ι} ∘ η(D_{κ^{−1}ι} ⊗ 1)](D_{λ}) = π_{ι}(D_{κ^{−1}ι}D_{λ}) ⊆ π_{ι}(D_{λκ^{−1}ι}) ⊆ \{ \begin{matrix}D_{ι}, & if λ = κ; \\ \{0\}, & if λ ≠ κ. \end{matrix}

Hence, π_{ι} ∘ η(D_{κ^{−1}ι} ⊗ 1) ⊆ π_{ι} End_{C}(D)π_{κ}. By applying η^{−1}, this yields

e_{ι}(D_{κ^{−1}ι} ⊗ 1) ⊆ e_{ι}Ae_{κ} for all ι, κ ∈ G.  (3.12)

Now, fix ι ∈ G. We have seen that e_{ι}A = e_{ι}(D ⊗ 1). The F -epimorphism ρ_{ι}: D → e_{ι}A given by d ↦ e_{ι}(d ⊗ 1) is clearly injective; ρ_{ι} is also norm-preserving, as α(e_{ι}) = 0 and d ⊗ 1 is stable in A for each nonzero d ∈ D. Since D = ⨁_{κ∈G} D_{κ^{−1}ι}, we have

⨁_{κ∈G} e_{ι}Ae_{κ} = e_{ι}A = ρ_{ι}(D) = ⨁_{κ∈G} ρ_{ι}(D_{κ^{−1}ι}) = ⨁_{κ∈G} e_{ι}(D_{κ^{−1}ι} ⊗ 1).  (3.13)

This shows that the inclusions in (3.12) must all be equalities. It follows from (3.11) above that the direct sum decomposition ⨁_{κ∈G} e_{ι}Ae_{κ} is a splitting decomposition of e_{ι}A. Therefore, by applying the norm-preserving map ρ_{ι}^{−1} to the terms in (3.13), it follows that ⨁_{κ∈G} D_{κ} is a splitting decomposition of D. □

While D ⊗_{F} L is simple, the degree 0 part (D ⊗_{F} L)_{0} of gr(D ⊗_{F} L) is in general only semisimple. The value sets Γ_{D_{ι}} of the D_{ι} encode how (D ⊗_{F} L)_{0} decomposes: Since each D_{ι} is a 1-dimensional C-subspace of D and v_{D}|_{D_{ι}} is a v_{D}|_{C}-norm on D_{ι}, each Γ_{D_{ι}} is a coset of Γ_{C} in Γ_{D}. Therefore, there is a well-defined map

ψ: G → Γ_{D}/Γ_{C} given by ψ(ι) = Γ_{D_{ι}}.

Because D_{ι} · D_{κ} ⊆ D_{κι} and Γ_{D} is abelian, ψ is a group homomorphism, which is surjective since D = ⨁_{ι∈G} D_{ι} is a splitting decomposition of D by Lemma 3.3. So, | ker(ψ)| = |G|/|Γ_{D}:Γ_{C}| = [D:C]/|Γ_{D}:Γ_{C}|, which shows that

ψ is injective if and only if D is totally ramified over C.  (3.14)

Lemma 3.4. Let A = D ⊗_{F} L. Then, each ẽ_{ι} is a primitive idempotent of A_{0}, and

A_{0} = ⨁_{ι∈G} ẽ_{ι}(A_{0}) = ⨁_{ι∈G} ẽ_{ι}(D_{0} ⊗ 1). For any ι, κ ∈ G,

ẽ_{ι}A_{0}ẽ_{κ} = \{ \begin{matrix}ẽ_{ι}((D_{κ^{−1}ι})_{0} ⊗ 1) ≠ 0, & if ψ(κ) = ψ(ι); \\ 0, & if ψ(κ) ≠ ψ(ι). \end{matrix}

Proof. We saw in (3.10) that gr(A) = ⨁_{ι∈G} ẽ_{ι} gr(A). Moreover, as gr(A) = gr(D) ⊗_{gr(F)} gr(L) and ẽ_{ι}(1 ⊗ gr(L)) = ẽ_{ι}( gr(L) ⊗ 1) by (3.7), we have ẽ_{ι} gr(A) = ẽ_{ι}( gr(D) ⊗ 1). So, for the degree 0 components we have A_{0} = ⨁_{ι∈G} ẽ_{ι}(A_{0}) = ⨁_{ι∈G} ẽ_{ι}(D_{0} ⊗ 1). Similarly, for ι, κ ∈ G, by (3.1) and Lemma 3.3

ẽ_{ι} gr(A)ẽ_{κ} = gr(e_{ι}Ae_{κ}) = gr (e_{ι}(D_{κ^{−1}ι} ⊗ 1)) = ẽ_{ι} gr(D_{κ^{−1}ι} ⊗ 1) = ẽ_{ι}( gr(D_{κ^{−1}ι}) ⊗ 1).

Hence, for the degree 0 components,

ẽ_{ι}(A_{0})ẽ_{κ} = ẽ_{ι}((D_{κ^{−1}ι})_{0} ⊗ 1).

If ψ(κ) ≠ ψ(ι), then ψ(κ^{−1}ι) is a nonzero element of Γ_{D}/Γ_{C}, so (D_{κ^{−1}ι})_{0} = {0}. If ψ(κ) = ψ(ι), then (D_{κ^{−1}ι})_{0} ≠ 0, and since nonzero elements of D_{κ^{−1}ι} ⊗ 1 are stable, (3.2) yields ẽ_{ι}((D_{κ^{−1}ι})_{0} ⊗ 1) ≠ {0}. If κ = ι, then (D_{κ^{−1}ι})_{0} ⊗ 1 = C_{0} ⊗ 1, so ẽ_{ι}(A_{0})ẽ_{ι} = ẽ_{ι}(C_{0} ⊗ 1). Since C ⊗ 1 centralizes e_{ι} ∈ L ⊗_{F} L, C_{0} ⊗ 1 centralizes ẽ_{ι}. Hence, ẽ_{ι}(A_{0})ẽ_{ι} ≅ C_{0} ⊗ 1 ≅ C_{0}. Since C_{0} is a division ring, ẽ_{ι} is a primitive idempotent of A_{0}. □

Now, assume σ is an F -linear involution on D which stabilizes L, and therefore restricts to an automorphism σ_{L} of L, and let ι ∈ G be such that ι^{2} = id. Then σ ⊗ ι is an involution on D ⊗_{F} L. Since the valuation v_{D} extending v to D is unique by [W, Th.], it is invariant under σ. Likewise, v_{L} is invariant under ι, hence v_{D} ⊗ v_{L} is invariant under σ ⊗ ι by Prop. 1.3.

Proposition 3.5. The involution σ ⊗ ι on D ⊗_{F} L is isotropic unless σ_{L} = ι and ι lies in the center Z(G) of G. If σ_{L} = ι ∈ Z(G) and D is totally ramified over C_{D}(L), then σ ⊗ ι is anisotropic.

Proof. For κ ∈ G we have

(σ_{L} ⊗ ι)(e_{κ}) = (σ_{L} ⊗ ικ)(e) = (id_{L} ⊗ικσ_{L}) ∘ (σ_{L} ⊗ σ_{L})(e).

Since (σ_{L} ⊗ σ_{L})(e) = e by Lemma 3.2, it follows that

(σ_{L} ⊗ ι)(e_{κ}) = e_{ικσ_{L}}.

If ι ≠ σ_{L} or if ι = σ_{L} and ι ∉ Z(G), we may find κ ∈ G such that ικσ_{L} ≠ κ, hence

(σ ⊗ ι)(e_{κ}) · e_{κ} = e_{ικσ_{L}} · e_{κ} = 0.

Therefore, σ ⊗ ι is isotropic.

Now assume σ_{L} = ι and ι ∈ Z(G). So, (σ ⊗ ι)(e_{κ}) = e_{κ} for all κ ∈ G; hence, in (D ⊗_{F} L)_{0}, (σ ⊗ ι)_{0}(ẽ_{κ}) = ẽ_{κ}. Assume further that D is totally ramified over C = C_{D}(L). Then ψ is injective by (3.14), so by Lemma 3.4, ẽ_{ι}(D⊗_{F} L)_{0} ẽ_{κ} = 0 whenever κ ≠ ι. Hence, (D ⊗_{F} L)_{0} = ⨁_{ι∈G} ẽ_{ι}(D ⊗_{F} L)_{0} ẽ_{ι}. Since (σ ⊗ ι)_{0} maps each direct summand to itself and each summand is a division ring, (σ ⊗ ι)_{0} is anisotropic. It follows from Cor. 2.3 (see also Remark 2.5(1)) that σ ⊗ ι is anisotropic. □

Corollary 3.6. Let D be a central division algebra over a field F. Assume v is a valuation on F which extends to a valuation on D so that D is tame over F. Let σ be an involution of the first kind on D and let L ⊆ D be a subfield Galois over F, consisting of σ-symmetric elements. If D is totally ramified over C_{D}(L), then the involution σ ⊗ id_{L} on D ⊗_{F} L is anisotropic.

Proof. This is immediate from Prop. 3.5. □

Remarks 3.7. (a) The assumption in Cor. 3.6 that D is totally ramified over C_{D}(L) holds whenever D is totally ramified over F. In this case we do not have to assume that L is Galois over F. For, since v extends to D, it follows from a theorem of Morandi [M] that D remains a division ring after scalar extension to a Henselization F_{h} of F for v. Therefore, we may assume that F is Henselian. The extension L/F is then Galois, since it is tame and totally ramified.

(b) Another case in which D is totally ramified over C_{D}(L) occurs whenever the subfield L of D is unramified over F and \overline{L} ⊆ Z(\overline{D}).

(c) Another way to obtain the information about (D ⊗_{F} L)_{0} needed in the proof of Prop. 3.5 is to prove that if the F -central division ring D has a valuation tame over F and L is any subfield of D containing F, and C = C_{D}(L), then the canonical isomorphism D ⊗_{F} L ≅ End_{C}(D) is norm-preserving; so this induces a graded isomorphism gr(D ⊗_{F} L) ≅ gr ( End_{C}(D)) ≅ End_{gr(C)} ( gr(D)).

4. Composition of value functions

Let v: F → Γ ∪ {∞} be a valuation on a field F, and let Δ ⊂ Γ be a convex subgroup, i.e., if 0 ≤ γ ≤ δ with γ ∈ Γ and δ ∈ Δ, then γ ∈ Δ. Let Λ = Γ/Δ, and let ε: Γ → Λ be the canonical map. The ordering on Γ induces a total ordering on Λ such that for γ_{1}, γ_{2} ∈ Γ, if γ_{1} ≤ γ_{2}, then ε(γ_{1}) ≤ ε(γ_{2}). Consequently,

if ε(γ_{2}) < ε(γ_{1}), then γ_{2} < γ_{1}.  (4.1)

Because Γ is assumed to be divisible, Δ and Λ are also divisible. By composing v with ε, we obtain a coarser valuation on F,

w = ε ∘ v: F → Λ ∪ {∞}.

Let \overline{F}^{v} (resp. \overline{F}^{w}) denote the residue field of F for the valuation v (resp. w). The valuation v induces a valuation

u: \overline{F}^{w} → Δ ∪ \{∞\},

with residue field

\overline{\overline{F}^{w}}^{u} = \overline{F}^{v},

see [EP, pp. 44–45].

Now, let V be an F -vector space and let α: V → Γ ∪ {∞} be a v-value function. Composition with ε yields a w-value function

β = ε ∘ α: V → Λ ∪ {∞}.

Each λ ∈ Λ = Γ/Δ is a coset of Δ, and may therefore be viewed as a subset of Γ. For x ∈ V, we have by definition

β(x) = λ ∈ Λ if and only if α(x) ∈ λ ⊂ Γ.

For λ ∈ Λ, let

V^{β≥λ} = {x ∈ V | β(x) ≥ λ}, V^{β>λ} = {x ∈ V | β(x) > λ}, V_{λ}^{β} = V^{β≥λ}/V^{β>λ}.

The group V_{λ}^{β} is an \overline{F}^{w}-vector space.

Lemma 4.1. If x, y ∈ V^{β≥λ} satisfy x ≡ y ≢ 0 mod V^{β>λ}, then α(x) = α(y).

Proof. We have β(x − y) > λ = β(y). Since β = ε ∘ α, (4.1) shows that α(x − y) > α(y). Hence, α(x) = min (α(x − y), α(y)) = α(y). □

In view of this lemma, we may define

α_{λ}: V_{λ}^{β} → λ ∪ \{∞\} by x + V^{β>λ} ↦ \begin{cases}α(x) & if β(x) = λ, \\ ∞ & if β(x) > λ. \end{cases}

Clearly, α_{λ} is a u-value function on V_{λ}^{β}. For γ ∈ λ we have

(V_{λ}^{β})_{γ}^{α_{λ}} = V_{γ}^{α}.

Therefore,

gr_{α}(V) = ⨁_{λ∈Λ} gr_{α_{λ}}(V_{λ}^{β}) where gr_{α_{λ}}(V_{λ}^{β}) = ⨁_{γ∈λ} V_{γ}^{α}  while  gr_{β}(V) = ⨁_{λ∈Λ} V_{λ}^{β}.

Now, let

Γ_{F} = v(F^{×}) ⊆ Γ,  Δ_{F} = Δ ∩ Γ_{F} ⊆ Δ,  Λ_{F} = w(F^{×}) = Γ_{F} /Δ_{F} ⊆ Λ.

These groups are the value groups of, respectively, v, u, and w. Similarly, let

Γ_{V} = α(V \ {0}) ⊆ Γ and Λ_{V} = β(V \ {0}) ⊆ Λ.

For each λ ∈ Λ_{V}, let also

λ_{V} = α_{λ}(V_{λ}^{β} \ {0}) ⊆ λ.

Clearly, λ_{V} = λ ∩ Γ_{V}. Note Γ_{F} (resp. Δ_{F}, resp. Λ_{F}) is a subgroup of Γ (resp. Δ, resp. Λ), while Γ_{V} (resp. Λ_{V}, resp. λ_{V} for λ ∈ Λ_{V}) is a union of cosets of Γ_{F} (resp. Λ_{F}, resp. Δ_{F}). We denote by |Γ_{V} :Γ_{F} | the cardinality of the set of cosets of Γ_{F} in Γ_{V}, and define likewise |Λ_{V} :Λ_{F} | and |λ_{V} :Δ_{F} | for λ ∈ Λ_{V}.

Lemma 4.2. If dim_{F} V is finite, then |Γ_{V} :Γ_{F} |, |Λ_{V} :Λ_{F} |, and |λ_{V} :Δ_{F} | for λ ∈ Λ_{V} are finite. If λ_{1}, …, λ_{r} ∈ Λ_{V} are representatives of the various cosets of Λ_{V} modulo Λ_{F}, then

|Γ_{V} :Γ_{F} | = ∑_{i=1}^{r}|(λ_{i})_{V} :Δ_{F} |.

Proof. By [RTW, Prop. 2.2] we have

|Γ_{V} :Γ_{F} | ≤ dim_{F} V, |Λ_{V} :Λ_{F} | ≤ dim_{F} V,

and also

|λ_{V} :Δ_{F} | ≤ dim\overline{F}^{w} V_{λ} ≤ dim_{F} V for λ ∈ Λ_{V} .

For i = 1, …, r, let γ_{i1}, …, γ_{is_{i}} ∈ (λ_{i})_{V} ⊆ Γ be representatives of the various cosets of (λ_{i})_{V} modulo Δ_{F}. Thus,

(λ_{i})_{V} = ∐_{j=1}^{s_{i}}(γ_{ij} + Δ_{F}),

where ∐ denotes the disjoint union. For γ ∈ Γ_{V}, we have ε(γ) ∈ Λ_{V}, hence

ε(γ) = λ_{i} + w(a) for some i ∈ {1, …, r} and some a ∈ F^{×}.

It follows that γ − v(a) ∈ (λ_{i})_{V}; hence,

γ − v(a) = γ_{ij} + v(b) for some j ∈ {1, …, s_{i}} and some b ∈ F^{×}.

This shows that γ ≡ γ_{ij} mod Γ_{F}, hence

Γ_{V} = ⋃_{i=1}^{r} ⋃_{j=1}^{s_{i}} (γ_{ij} + Γ_{F}).  (4.2)

To complete the proof, it suffices to show the union is disjoint. If γ_{ij} ≡ γ_{kℓ} mod Γ_{F} for some i, j, k, ℓ, then ε(γ_{ij}) ≡ ε(γ_{kℓ}) mod Λ_{F}, hence i = k since ε(γ_{ij}) = λ_{i} and ε(γ_{kℓ}) = λ_{k}. Moreover, from ε(γ_{ij}) = ε(γ_{kℓ}) it follows that γ_{ij} − γ_{kℓ} ∈ Δ, hence γ_{ij} ≡ γ_{kℓ} mod Γ_{F} implies γ_{ij} ≡ γ_{kℓ} mod Δ_{F}, hence also j = ℓ. □

Proposition 4.3. Suppose dim_{F} V is finite, and let λ_{1}, …, λ_{r} ∈ Λ_{V} be representatives of the various cosets of Λ_{V} modulo Λ_{F}. The following conditions are equivalent:

(i) α is a norm; (ii) β is a norm and α_{λ} is a norm for all λ ∈ Λ_{V}; (iii) β is a norm and α_{λ_{i}} is a norm for i = 1, …, r.

Proof. Use the same notation as in the lemma. For simplicity, denote α_{i} = α_{λ_{i}} and V_{i} = V_{λ_{i}}^{β} for i = 1, …, r, and V_{ij} = V_{γ_{ij}}^{α} for i = 1, …, r and j = 1, …, s_{i}, and use the notation [V :F] for dim_{F} V. From (4.2) it follows that

[gr_{α}(V): gr_{v}(F)] = ∑_{i=1}^{r} ∑_{j=1}^{s_{i}} [V_{ij}:\overline{F}^{v}].  (4.3)

Likewise,

[gr_{β}(V): gr_{w}(F)] = ∑_{i=1}^{r} [V_{i}:\overline{F}^{w}]  (4.4)

and

[gr_{α_{i}}(V_{i}): gr_{u}(\overline{F}^{w})] = ∑_{j=1}^{s_{i}} [V_{ij}:\overline{F}^{v}]  for i = 1, …, r.  (4.5)

If α_{k} is not a norm for some k ∈ {1, …, r}, then

[V_{k}:\overline{F}^{w}] > [gr_{α_{k}}(V_{k}): gr_{u}(\overline{F}^{w})].

On the other hand, we have

[V_{i}:\overline{F}^{w}] ≥ [gr_{α_{i}}(V_{i}): gr_{u}(\overline{F}^{w})] for all i;

hence, by (4.4) and (4.5),

[gr_{β}(V): gr_{w}(F)] > ∑_{i=1}^{r} [gr_{α_{i}}(V_{i}): gr_{u}(\overline{F}^{w})] = ∑_{i=1}^{r} ∑_{j=1}^{s_{i}} [V_{ij}:\overline{F}^{v}].

In view of (4.3), it follows that [gr_{β}(V): gr_{w}(F)] > [gr_{α}(V): gr_{v}(F)]. Since [V :F] ≥ [gr_{β}(V): gr_{w}(F)], we have [V :F] > [gr_{α}(V): gr_{v}(F)], hence α is not a norm.

If each α_{i} is a norm, then [V_{i}:\overline{F}^{w}] = [gr_{α_{i}}(V_{i}): gr_{u}(F)] for i = 1, …, r, hence (4.4), (4.5), and (4.3) yield

[gr_{β}(V): gr_{w}(F)] = ∑_{i=1}^{r} ∑_{j=1}^{s_{i}} [V_{ij}:\overline{F}^{v}] = [gr_{α}(V): gr_{v}(F)].

It follows that α is a norm if and only if β is a norm. We have thus proved (i) ⟺ (iii). Since any λ ∈ Λ can be chosen as a representative of its coset, the arguments above also show (i) ⇒ (ii). Since (ii) ⇒ (iii) is clear, the proof is complete. □

To set Prop. 4.3 in perspective, we relate the graded vector spaces gr_{α}(V) and gr_{β}(V) by means of a value-function-like map

α_{∗}: gr_{β}(V) → Γ ∪ {∞}

defined as follows: for ξ ∈ gr_{β}(V), ξ ≠ 0, let ℓ(ξ) be the homogeneous component of ξ of least degree, and let λ = deg(ℓ(ξ)), so ℓ(ξ) ∈ V_{λ}^{β}; then let

α_{∗}(ξ) = α_{λ}(ℓ(ξ)) ∈ λ ⊆ Γ.

Let also α_{∗}(0) = ∞. For x ∈ V we thus have

α_{∗}(x^{β}) = α(x),  (4.6)

where x^{β} denotes the image of x in gr_{β}(V).

A similar construction applies to the valuation v, and yields a map

v_{∗}: gr_{w}(F) → Γ ∪ {∞},

which satisfies the same properties as a valuation, and such that the image v_{∗}(ρ) of any nonzero ρ ∈ gr_{w}(F) depends only on its homogeneous component of least degree. The map α_{∗} deserves the name of a graded v_{∗}-value function since it satisfies the following properties:

(i) α_{∗}(ξ) = ∞ if and only if ξ = 0; if ξ ≠ 0, then α_{∗}(ξ) = α_{∗}(ℓ(ξ)) and ε ∘ α_{∗}(ξ) = deg ℓ(ξ); (ii) α_{∗}(ξ + η) ≥ min(α_{∗}(ξ), α_{∗}(η)) for ξ, η ∈ gr_{β}(V); (iii) α_{∗}(ξρ) = α_{∗}(ξ) + v_{∗}(ρ) for ξ ∈ gr_{β}(V) and ρ ∈ gr_{w}(F).

We may thus consider the associated graded structure gr_{α_{∗}}(gr_{β}(V)). If x ∈ V satisfies β(x) = λ and α(x) = γ, we may identify

(x + V^{β>λ}) + gr_{β}(V)^{α_{∗}>γ} = x + V^{α>γ};

thus

gr_{α_{∗}}(gr_{β}(V)) = gr_{α}(V).  (4.7)

We define α_{∗} to be a graded v_{∗}-norm if [gr_{α_{∗}}( gr_{β}(V)): gr_{v_{∗}}(gr_{w}(F))] = [gr_{β}(V): gr_{w}(F)]. It is easy to check that this holds if and only if each α_{λ} is a u-norm. By an argument analogous to the one in [RTW, Prop. 2.5] for ungraded norms, one can check that if α_{∗} is a graded norm, then for any graded subspace W of gr_{β}(V), α_{∗}|_{W} is a graded norm on W. Consequently, by dimension count, the functor gr_{α_{∗}}( ) preserves strict inclusions of graded subspaces of gr_{β}(V). Prop. 4.3 may be rephrased as follows: α is a v-norm if and only if β is a w-norm and α_{∗} is a graded v_{∗}-norm. Indeed, if (e_{i})_{i=1}^{n} is a splitting base of V for α, then it is also a splitting base for β, and (e_{i}^{β})_{i=1}^{n} is a splitting base of gr_{β}(V) for α_{∗}.

We now apply this construction to a finite-dimensional F -algebra A. If α: A → Γ ∪ {∞} is a surmultiplicative v-value function, then the coarser w-value function β = ε∘α is clearly surmultiplicative, and the map α_{∗} is also surmultiplicative, by an easy calculation using (4.6). The notions of gauge and tame gauge for graded norms are defined analogously to the ungraded cases.

Proposition 4.4. The map α is a v-gauge (resp. a tame v-gauge) if and only if β is a w-gauge (resp. a tame w-gauge) and α_{∗} is a graded v_{∗}-gauge (resp. a tame graded v_{∗}-gauge).

Proof. Prop. 4.3 already shows that α is a v-norm if and only if β is a w-norm and α_{∗} is a graded v_{∗}-norm. We noted above that α is surmultiplicative if and only if β and α_{∗} are surmultiplicative.

Suppose α is a v-gauge. Since gr_{α_{∗}}(gr_{β}(A)) = gr_{α}(A) and gr_{α}(A) is semisimple, it follows that gr_{β}(A) is semisimple. For, if I is a nontrivial nilpotent homogeneous left ideal of gr_{β}(A), then gr_{α_{∗}}(I) is a nontrivial nilpotent homogeneous left ideal of gr_{α_{∗}}(gr_{β}(A)). Thus, β is a w-gauge. Also, gr_{α_{∗}}(gr_{β}(A)) is semisimple by hypothesis, hence α_{∗} is a graded v_{∗}-gauge. Conversely, if β is a w-gauge and α_{∗} is a graded v_{∗}-gauge, then α is a v-gauge since gr_{α}(A) = gr_{α_{∗}}(gr_{β}(A)).

Assume now that α is a v-gauge. For the centers we have the obvious inclusions

gr_{α}(Z(A)) = gr_{α_{∗}}(gr_{β}(Z(A))) ⊆ gr_{α_{∗}}(Z(gr_{β}(A))) ⊆ Z(gr_{α_{∗}}(gr_{β}(A))) = Z(gr_{α}(A)). (4.8)

Thus, Z(gr_{α}(A)) = gr_{α}(Z(A)) if and only if we have equalities throughout (4.8); since gr_{α_{∗}}( ) preserves strict inclusions, this holds if and only if

gr_{β}(Z(A)) = Z(gr_{β}(A)) and gr_{α_{∗}}(Z(gr_{β}(A))) = Z(gr_{α_{∗}}(gr_{β}(A))).

Assume we have these equalities. Let Z = Z(A), which is a direct product of fields, as A is semisimple. The separability condition on the graded center required for tameness holds for α if and only if it holds for α_{∗}, since they have the same graded rings. Suppose now that gr_{β}(Z) is not separable over gr_{w}(F). Because gr_{β}(A) is semisimple, its center gr_{β}(Z) is a direct product C_{1} × … × C_{k} of graded fields, and some C_{j} must not be separable over gr_{w}(F). By [HW_{1}, Prop. 3.7, Prop. 3.5] there is a graded field T with gr_{w}(F) ⊆ T ⫋ C_{j} and C_{j} purely inseparable over T. So,

gr_{v}(F) = gr_{α_{∗}}( gr_{w}(V)) ⊆ gr_{α_{∗}}(T) ⫋ gr_{α_{∗}}(C_{j}),

and gr_{α_{∗}}(C_{j}) is purely inseparable over gr_{α_{∗}}(T). Now, gr_{α}(Z) = gr_{α_{∗}}( gr_{β}(Z)) = ∏_{i=1}^{k} gr_{α_{∗}}(C_{i}). Since gr_{α_{∗}}(C_{j}) is purely inseparable over gr_{α_{∗}}(T), it cannot be separable over gr_{v}(F), so gr_{α}(Z) is not separable over gr_{v}(F). Thus, gr_{α}(Z) is separable over gr_{v}(F) if and only gr_{α_{∗}}( gr_{β}(Z)) is separable over gr_{v_{∗}}( gr_{w}(F)) and gr_{β}(Z) is separable over gr_{w}(F). Therefore, α is a tame v-gauge if and only if β is a tame w-gauge and α_{∗} is a tame graded v_{∗}-gauge. □

5. Descent of norms

Throughout this section, we fix the following notation: V is a finite-dimensional vector space over a field F, and v: F → Γ ∪ {∞} is a valuation. Let (F_{h}, v_{h}) be a Henselization of (F, v). If α: V ⊗_{F} F_{h} → Γ ∪ {∞} is a v_{h}-norm, then clearly α|_{V} : V → Γ ∪ {∞} is a v-value function, but not necessarily a v-norm unless Γ has rank one, see Prop. 5.4 and Ex. 5.6. In this section, we give an inductive criterion for α|_{V} to be a v-norm when Γ is the divisible hull of Γ_{F} and the rank rk(Γ) is finite, see Prop. 5.5.

We first discuss the descent problem in a general context: let (K, v_{K}) be an arbitrary valued field extension of (F, v), and let α: V ⊗_{F} K → Γ ∪ {∞} be a v_{K}-norm. We identify V with its canonical image in V ⊗_{F} K. For any x ∈ V and c ∈ K we have

α(x ⊗ c) = α((x ⊗ 1) · c) = α|_{V} (x) + v_{K}(c).

Therefore, for any γ ∈ Γ_{V,α|_{V}} and δ ∈ Γ_{K} the usual F -bilinear map V × K → V ⊗_{F} K sends V^{≥γ} × K^{≥δ} into (V ⊗_{F} K)^{≥γ+δ}. Likewise, V^{>γ} × K^{≥δ} and V^{≥γ} × K^{>δ} map into (V ⊗_{F} K)^{>γ+δ}. Consequently, there is a well-defined induced map V_{γ} × K_{δ} → (V ⊗_{F} K)_{γ+δ} given by (x̃, c̃) ↦ { \atop x ⊗ c}. The direct sum of these maps over all such γ, δ yields a map gr_{α|_{V}} (V) × gr_{v_{K}}(K) → gr_{α}(V ⊗_{F} K), which is clearly gr_{v}(F)-bilinear, hence there is a canonical map

χ: gr_{α|_{V}} (V) ⊗_{gr_{v}(F)} gr_{v_{K}}(K) → gr_{α}(V ⊗_{F} K)

which maps x̃ ⊗ c̃ to { \atop x ⊗ c} for x ∈ V and c ∈ K.

On the other hand, recall from Sec. 1 (see (1.5)) that if α|_{V} is a v-norm on V, then there is a canonical isomorphism of gr_{v_{K}}(K)-vector spaces

ρ: gr_{α|_{V} ⊗v_{K}}(V ⊗_{F} K) ⟶^{∼} gr_{α|_{V}} (V) ⊗_{gr_{v}(F)} gr_{v_{K}}(K)

which maps { \atop x ⊗ c} to x̃ ⊗ c̃ for x ∈ V and c ∈ K.

Lemma 5.1. The following conditions are equivalent:

(a) α|_{V} is a v-norm on V and α = α|_{V} ⊗ v_{K}.

(b) V contains a K-splitting base of the norm α on V ⊗_{F} K.

(c) α|_{V} is a v-norm and the canonical map χ is injective.

When these conditions hold, the map χ is a graded isomorphism, which is the inverse of ρ, and Γ_{V ⊗_{F} K,α} = Γ_{V,α|_{V}} + Γ_{K,v_{K}}.

Proof. (b) ⇒ (a) If B = (e_{i})_{i=1}^{n} ⊆ V is a splitting base for α on V ⊗_{F} K, then B is clearly also a splitting base for α|_{V} on V. So, α|_{V} is a v-norm. Furthermore, by the definition of α|_{V} ⊗ v_{K}, we have for any k_{1}, …, k_{n} ∈ K,

\begin{aligned}(α|_{V} ⊗ v_{K})( ∑_{i=1}^{n} e_{i} ⊗ k_{i}) & = min_{1≤i≤n} (α|_{V} (e_{i}) + v_{K}(k_{i})) = min_{1≤i≤n} (α(e_{i} ⊗ 1) + v_{K}(k_{i})) \\ & = α( ∑_{i=1}^{n}(e_{i} ⊗ 1) · k_{i}) = α(∑_{i=1}^{n} e_{i} ⊗ k_{i}), \end{aligned}

showing that α|_{V} ⊗ v_{K} = α.

(a) ⇒ (c) When (a) holds, α|_{V} is a norm, and χ is clearly the inverse of ρ, so χ is injective.

(c) ⇒ (b) Suppose (c) holds. Let (e_{i})_{i=1}^{n} be an F -splitting base for α|_{V} on V. Then, by [RTW, Cor. 2.3(ii)] ẽ_{1}, …, ẽ_{n} are gr_{v}(F)-linearly independent in gr_{α|_{V}} (V). Hence, ẽ_{1}⊗˜1, …, ẽ_{n}⊗˜1 are gr_{v_{K}}(K)-

linearly independent in gr_{α|_{V}} (V) ⊗_{gr_{v}(F)} gr_{v_{K}}(K). By the injectivity of χ the χ(ẽ_{i} ⊗ ˜1) = { \atop e_{i} ⊗ 1} are gr_{v_{K}}(K)-linearly independent in gr_{α}(V ⊗_{F} K). But, since α and α|_{V} are norms,

[gr_{α}(V ⊗_{F} K): gr_{v_{K}}(K)] = [V ⊗_{F} K:K] = [V :F] = n.

So, ({ \atop e_{i} ⊗ 1})_{i=1}^{n} is a homogeneous gr_{v_{K}}(K)-vector space base of gr_{α}(V ⊗_{F} K), hence (e_{i} ⊗ 1)_{i=1}^{n} is a K-splitting base for α on V ⊗_{F} K by [RTW, Cor. 2.3(ii)].

When the conditions (a) – (c) hold, we have

Γ_{V ⊗_{F} K,α} = Γ_{V ⊗_{F} K,α|_{V} ⊗v_{K}} = Γ_{V,α|_{V}} + Γ_{K,v_{K}}

and the map χ is the inverse of ρ, so χ is an isomorphism. □

Note that under the hypotheses of Lemma 5.1 if α|_{V} is a norm then α ≥ α|_{V} ⊗ v_{K}. For, if (e_{i})_{i=1}^{n} is a v-splitting base for α|_{V} on V, then for any k_{1}, …, k_{n} ∈ K,

α( ∑_{i=1}^{n} e_{i} ⊗ k_{i}) ≥ min_{1≤i≤n} (α(e_{i} ⊗ k_{i})) = min_{1≤i≤n} (α(e_{i}) + v_{K}(k_{i})) = (α|_{V} ⊗ v_{K})( ∑_{i=1}^{n} e_{i} ⊗ k_{i}).

We next show that the inequality α ≥ α|_{V} ⊗ v_{K} is actually an equality when K is immediate over F, but not in general.

Corollary 5.2. Let (K, v_{K}) be an immediate valued field extension of (F, v) and let α: V ⊗_{F} K → Γ ∪ {∞} be a v_{K}-norm. If α|_{V} is a norm, then α = α|_{V} ⊗ v_{K}. So, the canonical map gr_{α|_{V}} (V) → gr_{α}(V ⊗_{F} K) is an isomorphism gr_{α|_{V}} (V) ≅ gr_{α}(V ⊗_{F} K), and Γ_{V} = Γ_{V ⊗_{F} K}.

Proof. Since v_{K} is immediate over v, we have gr_{v_{K}}(K) = gr_{v}(F), so the canonical map χ of Lemma 5.1(c) is just the injection gr_{α|_{V}} (V) ↪ gr_{α}(V ⊗_{F} K) arising from the canonical inclusion V ↪ V ⊗_{F} K. Thus, the corollary follows from Lemma 5.1, using Γ_{K} = Γ_{F} for the last assertion. □

Example 5.3. Let (K, v_{K}) be an extension of (F, v) with \overline{F} ⫋ \overline{K}. Let ξ ∈ K be such that v_{K}(ξ) = 0 and \overline{ξ} ∉ \overline{F}, and let V be a 2-dimensional F -vector space with base (e_{1}, e_{2}). Let f = e_{1} ⊗1+e_{2} ⊗ξ ∈ V ⊗_{F} K, and consider the v_{K}-norm α on V ⊗_{F} K with splitting base (e_{1} ⊗ 1, f) such that

α(e_{1} ⊗ 1) = 0  and  α(f) > 0.

Then, as e_{2} = (f − e_{1})ξ^{−1}, we have for c_{1}, c_{2} ∈ F

α|_{V} (e_{1}c_{1} + e_{2}c_{2}) = α((e_{1}(c_{1} − ξ^{−1}c_{2}) + fc_{2}) = min (α(e_{1}) + v_{K}(c_{1} − ξ^{−1}c_{2}), α(f) + v_{K}(c_{2}))
= min(v(c_{1}), v(c_{2})).

Hence, (e_{1}, e_{2}) is a v-splitting base of V for α|_{V}, showing that α|_{V} is a v-norm on V. However, α|_{V} ⊗ v_{K} < α since (α|_{V} ⊗ v_{K})(f) = min (v_{K}(1), v_{K}(ξ)) = 0 < α(f). Thus, the first condition in Lemma 5.1(a) holds, but not the second. The Lemma shows that V does not contain any splitting base for the v_{K}-norm α on V ⊗_{F} K. Also, the second condition in part (c) of the Lemma fails, since the canonical map χ satisfies

χ(ẽ_{1} ⊗ 1 + ẽ_{2} ⊗ ξ̃) = { \atop e_{1} ⊗ 1} + { \atop e_{2} ⊗ ξ} = 0.

We now turn to the descent problem posed at the beginning of this section, for (K, v_{K}) = (F_{h}, v_{h}) a Henselization of (F, v). The rank one case is easy:

Proposition 5.4. Let α: V ⊗_{F} F_{h} → Γ∪{∞} be a v_{h}-norm, let γ ∈ Γ, and suppose im(α) ⊆ γ + (Γ_{F} ⊗_{ℤ} ℚ). If rk(Γ_{F}) = 1, then α|_{V} is a v-norm and α = α|_{V} ⊗ v_{h}.

Proof. Let (e_{i})_{i=1}^{n} be an arbitrary F -base of V and let x ∈ V ⊗_{F} F_{h},

x = ∑_{i=1}^{n} e_{i} ⊗ k_{i} for some k_{i} ∈ F_{h}.

Since rk(Γ_{F}) = 1, the field F is dense in F_{h} for the topology of the valuation v_{h}: see [Er_{3}, §1.6] or use the fact that F is dense in its completion F̂ and that F_{h} embeds in F̂ by [E, Th. 17.18]. Furthermore, Γ_{F} is dense in its divisible hull Γ_{F} ⊗_{ℤ} ℚ. For each i, 1 ≤ i ≤ n, since α(x) − α(e_{i} ⊗ 1) ∈ Γ_{F} ⊗_{ℤ} ℚ, we may therefore find an element f_{i} ∈ F such that

v_{h}(k_{i} − f_{i}) > α(x) − α(e_{i} ⊗ 1).

Let y = ∑_{i=1}^{n} e_{i} ⊗ f_{i} = ∑_{i=1}^{n} e_{i}f_{i} ⊗ 1 ∈ V. Then,

α(x − y) = α(∑_{i=1}^{n} e_{i} ⊗ (k_{i} − f_{i})) ≥ min_{1≤i≤n}(α(e_{i} ⊗ (k_{i} − f_{i}))) = min_{1≤i≤n} (α(e_{i}) + v_{h}(k_{i} − f_{i})) > α(x).

Hence,

x̃ = ỹ ∈ gr_{α|_{V}} (V).

This proves that the monomorphism gr_{α|_{V}} (V) ↪ gr_{α}(V ⊗_{F} F_{h}) is an isomorphism. Hence, as α is a norm,

[gr_{α|_{V}} (V) : gr_{v}(F)] = [gr_{α}(V ⊗_{F} F_{h}) : gr_{v_{h}}(F_{h})] = [(V ⊗_{F} F_{h}):F_{h}] = [V :F],

which shows that α|_{V} is a v-norm. □

Now, suppose Γ = Γ_{F} ⊗_{ℤ} ℚ, with rk(Γ) > 1, and suppose Γ contains a convex subgroup Δ of rank 1. As in §4, we consider the canonical map ε: Γ → Γ/Δ = Λ and the coarser valuation

w = ε ∘ v: F → Λ ∪ {∞}.

Let (F_{h,v}, v_{h}) be a Henselization of (F, v) and (F_{h,w}, w_{h}) a Henselization of (F, w). Let also

y = ε ∘ v_{h}: F_{h,v} → Λ ∪ {∞}.

By [EP, Cor. 4.1.4, p. 90], the valuation y is Henselian, hence we may assume (F_{h,w}, w_{h}) ⊆ (F_{h,v}, y). Let α: V ⊗_{F} F_{h,v} → Γ ∪ {∞} be a v_{h}-norm, and let

β = ε ∘ α: V ⊗_{F} F_{h,v} → Λ ∪ {∞}.

By Prop. 4.3, the map β is a y-norm.

Proposition 5.5. If β|_{V} is a w-norm and β = β|_{V} ⊗ y, then α|_{V} is a v-norm on V and α = α|_{V} ⊗ v_{h}.

Proof. As observed in §4, the valuation v induces a valuation u on the residue field \overline{F}^{w},

u: \overline{F}^{w} → Δ ∪ \{∞\}.

Note that the value group of u is Δ_{F} = Γ_{F} ∩ Δ and, as Δ is divisible and torsion-free,

Δ_{F} ⊗_{ℤ} ℚ = (Γ_{F} ⊗_{ℤ} ℚ) ∩ (Δ ⊗_{ℤ} ℚ) = Γ ∩ Δ = Δ.

Let Λ_{V} = β|_{V} (V \ {0}) ⊆ Λ. Clearly, β|_{V} = ε ∘ (α|_{V}). In order to show α|_{V} is a norm, it therefore suffices, by Prop. 4.3, to show that each map

(α|_{V})_{λ}: V_{λ}^{β|_{V}} → λ ∪ {∞},  for λ ∈ Λ_{V} ,  (5.1)

is a u-norm. To simplify notation, we write V_{λ} for V_{λ}^{β|_{V}}. Note that the canonical inclusion V ↪ V ⊗_{F} F_{h,v} is compatible with the respective value functions β|_{V} and β so yields an injection V_{λ} ↪ (V ⊗_{F} F_{h,v})_{λ}^{β}; let V′_{λ} denote the image of V_{λ}. Then, clearly α_{λ}|_{V′_{λ}} ≅ (α|_{V})_{λ}.

Let u_{h}: \overline{F_{h,v}}^{y} → Δ ∪ \{∞\} be the valuation induced by v_{h}. As observed by Morandi [M, p. 239], (\overline{F_{h,v}}^{y}, u_{h}) is a Henselization of (\overline{F}^{w}, u). Since α is a v_{h}-norm, Prop. 4.3 shows that

α_{λ}: (V ⊗_{F} F_{h,v})_{λ}^{β} → λ ∪ {∞}

is a u_{h}-norm for every λ ∈ Λ_{V}. Since (F_{h,v}, y) is an inertial extension of (F_{h,w}, w_{h}) by [M, p. 239], we have

gr_{y}(F_{h,v}) ≅ gr_{w_{h}}(F_{h,w}) ⊗_{gr_{w_{h}}(F_{h,w})_{0}} gr_{y}(F_{h,v})_{0} = gr_{w}(F) ⊗_{gr_{w}(F)_{0}} gr_{y}(F_{h,v})_{0}.

Because β = β|_{V} ⊗ y, this yields graded isomorphisms

gr_{β}(V ⊗_{F} F_{h,v}) ≅ gr_{β|_{V}} (V) ⊗_{gr_{w}(F)} gr_{y}(F_{h,v}) ≅ gr_{β|_{V}} (V) ⊗_{gr_{w}(F)_{0}} gr_{y}(F_{h,v})_{0}
≅ gr_{β|_{V}} (V) ⊗\overline{F}^{w} \overline{F_{h,v}}^{y}.

For any λ ∈ Λ_{F}, when we restrict these graded isomorphisms to the λ-component we obtain the \overline{F_{h,v}}^{y}- vector space isomorphism

ψ: (V ⊗_{F} F_{h,v})_{λ}^{β} ⟶^{∼} V_{λ} ⊗\overline{F}^{w} \overline{F_{h,v}}^{y}.

Let α̂ = α_{λ} ∘ ψ^{−1} : V_{λ} ⊗\overline{F}^{w} \overline{F_{h,v}}^{y} → λ ∪ \{∞\}, which is the u_{h}-value function on im(ψ) corresponding to α_{λ} on the domain of ψ. Since α_{λ} is a u_{h}-norm, so is α̂. Because (\overline{F_{h,v}}^{y}, u_{h}) is a Henselization of (\overline{F}^{w}, u) and λ is a coset of Δ = Δ_{F} ⊗_{ℤ} ℚ, which has rank 1, with Δ_{F} the value group of u, Prop. 5.4 applies to α̂, and shows that α̂|_{V_{λ}} is a u-norm. Note that ψ maps the V′_{λ} defined above after (5.1) to the copy of V_{λ} in im(ψ). So, α_{λ}|_{V′_{λ}} ≅ α̂|_{V_{λ}}. But, we saw above that (α|_{V})_{λ} ≅ α_{λ}|_{V′_{λ}}. Since α̂|_{V_{λ}} is a u-norm, these isomorphisms show that (α|_{V})_{λ} is also a u-norm. Thus, by Prop. 4.3 α|_{V} is a v-norm; then α = α|_{V} ⊗ v_{h} by Cor. 5.2. □

The following is an example of a norm on a Henselization that does not descend to a norm.

Example 5.6. Let k be any field with char(k) ≠ 2, and let F = k(x, y) with x and y algebraically independent over k. Let v be the valuation on F obtained by restriction from the canonical Henselian valuation on k((x))((y)), so Γ_{F} = ℤ × ℤ and \overline{F} = k. Let (F_{h}, v_{h}) be a Henselization of (F, v). Let A = (\frac{1+x, y}{F} ), a quaternion division algebra over F, and let A_{h} = A⊗_{F} F_{h}. The algebra A_{h} is split since

1 + x ∈ F_{h}^{×2}. Therefore, we may find v_{h}-gauges on A_{h} that are unramified, in the sense that Γ_{A_{h}} = Γ_{F}. Fix such a v_{h}-gauge α. We claim that α|_{A} is not a v-norm on A.

Suppose the contrary. Then gr_{α|_{A}}(A) = gr_{α}(A_{h}) by Lemma 5.2, so α|_{A} is a v-gauge. Consider the convex subgroup Δ = ℤ × {0} ⊆ Γ_{F} and the canonical epimorphism

ε: Γ_{F} → Γ_{F} /Δ = ℤ.

Let w = ε ∘ v: F → ℤ ∪ {∞}, which is the y-adic valuation on F, and β = ε ∘ α: A_{h} → ℤ ∪ {∞}. Proposition 4.4 shows that β|_{A} is a tame w-gauge on A. However, the y-adic valuation w extends to A, so by [TW, Cor. 3.4] β|_{A} is the (unique) valuation on A that extends w. In particular, if j ∈ A satisfies j^{2} = y we must have β|_{A}(j) = \frac{1}{2}. This is a contradiction since β|_{A} is unramified.

6. Non-Henselian valuations

Let (F, v) be a valued field and let A be a finite-dimensional simple F -algebra with an involution σ. Let K = Z(A), and assume F is the subfield of K fixed under σ. Fix a Henselization (F_{h}, v_{h}) of (F, v).

Theorem 6.1. Suppose A is split by the maximal tamely ramified extension of F_{h}. Moreover, if char(\overline{F}) = 2 suppose that σ is not an orthogonal involution.Then, the following conditions are equivalent:

(a) σ ⊗ id_{F_{h}} is an anisotropic involution on A ⊗_{F} F_{h};

(b) there exists a σ-special v-gauge φ on A i.e., φ(σ(x)x) = 2φ(x) for all x ∈ A.

When they hold, φ is the unique v-gauge on A that is invariant under σ, it is tame, and its value group lies in the divisible hull of Γ_{F}.

Proof. Let A_{h} = A ⊗_{F} F_{h} and σ_{h} = σ ⊗ id_{F_{h}}. If φ is a σ-special v-gauge on A, then by Prop. 1.1 φ is invariant under σ and σ̃ is anisotropic on gr_{φ}(A). By Cor. 1.4, φ ⊗ v_{h} is invariant under σ_{h}. Since gr_{φ⊗v_{h}}(A_{h}) ≅ gr_{φ}(A) ⊗_{gr_{v}(F)} gr_{v_{h}}(F_{h}) ≅ gr_{φ}(A) and σ̃_{h} ≅ σ̃, it follows that σ̃_{h} is anisotropic, hence σ_{h} must also be anisotropic, proving (b) ⇒ (a).

Now, suppose (a) holds. Let φ_{1} and φ_{2} be v-gauges on A that are each invariant under σ. Then, by Cor. 1.4 each φ_{i}⊗v_{h} is a surmultiplicative v_{h}-norm on A_{h} which is invariant under σ_{h}. Moreover, φ_{i}⊗v_{h} is a gauge on A_{h} since gr_{φ_{i}⊗v_{h}}(A_{h}) ≅ gr_{φ_{i}}(A) ⊗_{gr_{v}(F)} gr_{v_{h}}(F_{h}) ≅ gr_{φ_{i}}(A) and φ_{i} is a gauge on A. Since σ_{h} is assumed anisotropic, the uniqueness part of Th. 2.2 (applied to σ_{h} on A_{h}) yields φ_{1} ⊗v_{h} = φ_{2} ⊗v_{h}, hence φ_{1} = φ_{2}. Th. 2.2 also shows that φ_{1} ⊗v_{h} is tame and satisfies φ_{1}(σ_{h}(x)x) = 2φ_{1}(x) for all x ∈ A_{h}, hence φ_{1} is tame and satisfies condition (b). Furthermore, Γ_{A,φ_{1}} = Γ_{A_{h},φ_{1}⊗v_{h}} which lies in the divisible hull of Γ_{F_{h}} = Γ_{F} by Th. 2.2.

Thus, it only remains to prove the existence of a v-gauge on A invariant under σ, assuming σ_{h} is anisotropic. Note first that K ⊗_{F} F_{h} is a field. For, otherwise, as K is Galois over F with [K:F] = 2, K⊗_{F} F_{h} would be a direct sum of two fields, and the nontrivial F_{h}-automorphism σ|_{K⊗_{F} F_{h}} must permute the two primitive idempotents of K ⊗_{F} F_{h}, call them e_{1} and e_{2}. Then, σ_{h}(e_{1})e_{1} = e_{2}e_{1} = 0; but, this cannot happen as σ_{h} is anisotropic. Since K ⊗_{F} F_{h} is a field and K = Z(A), A_{h} ≅ A ⊗_{K} (K ⊗_{F} F_{h}) is a central simple K ⊗_{F} F_{h}-algebra.

Because A_{h} is simple, σ_{h} is anisotropic, and v_{h} is Henselian, Th. 2.2 yields a σ_{h}-invariant v_{h}-gauge φ_{h} on A_{h} whose value set lies in the divisible hull of Γ_{F_{h}} = Γ_{F}. The restriction φ = φ_{h}|_{A} is clearly a σ-invariant v-value function whose value set lies in the divisible hull of Γ_{F}. Henceforth, we may thus assume Γ = Γ_{F} ⊗_{ℤ} ℚ. If we show that φ is a v-norm, then Cor. 5.2 yields φ_{h} = φ ⊗ v_{h}, so gr_{φ}(A) = gr_{φ⊗v_{h}}(A_{h}) = gr_{φ_{h}}(A_{h}), hence φ is a v-gauge, and the proof will be complete.

Suppose first that rk(Γ_{F}) < ∞. We then argue by induction on rk(Γ_{F}). If rk(Γ_{F}) = 1, then Prop. 5.4 shows that φ is a v-norm. So, we may assume rk(Γ_{F}) > 1. Let Δ ⊆ Γ be a convex subgroup of rank 1 and let

ε: Γ → Λ = Γ/Δ

be the canonical epimorphism. Let w = ε ∘ v and y = ε ∘ v_{h}, to agree with the notation of §5. So, w has value group Λ_{F} = (Γ_{F} + Δ)/Δ, and Λ = Λ_{F} ⊗_{ℤ} ℚ, which has rank rk(Γ_{F}) − 1. Let (F_{h,w}, w_{h}) ⊆ (F_{h}, y) be a Henselization of (F, w). Since σ_{h} is anisotropic, its restriction σ ⊗id_{F_{h,w}} is an anisotropic involution on the subring A⊗_{F} F_{h,w} of A_{h}. Since A_{h} ≅ (A⊗_{F} F_{h,w})⊗_{F_{h,w}} F_{h} and A_{h} is simple, A⊗_{F} F_{h,w} must also be simple. Therefore, Th. 2.2 applies, yielding a w_{h}-gauge ψ_{h} on A ⊗_{F} F_{h,w} invariant under σ ⊗ id_{F_{h,w}}. By induction, ψ_{h}|_{A} is a w-gauge on A invariant under σ. The same argument as for φ_{1} above shows that the gauge ψ|_{A} is tame. Therefore, by [TW, Cor. 1.26] ψ_{h}|_{A} ⊗ y is a y-gauge on A_{h}, which is σ_{h}- invariant by Cor. 1.4. But, ε ∘ φ_{h} is also a y-gauge on A_{h}, by Prop. 4.4 since φ is a gauge, and ε ∘ φ_{h} is invariant under σ_{h} because φ_{h} is. By the uniqueness given in Th. 2.2, it follows that ε ∘ φ_{h} = ψ_{h}|_{A} ⊗ y. Restricting to A, we also have ε ∘ φ = ψ_{h}|_{A}, which is a w-gauge so a w-norm on A. Furthermore, (ε ∘ φ) ⊗ y = ψ_{h}|_{A} ⊗ y = ε ∘ φ_{h}. Prop. 5.5 with α = φ_{h} then shows that φ is a v-norm. The theorem is thus proved if rk(Γ_{F}) < ∞.

For the rest of the proof, assume that Γ_{F} has infinite rank. Let (a_{i})_{i=1}^{n} be an F -base of A. Write a_{i}a_{k} = ∑_{l} c_{ikl}a_{l} for some c_{ikl} ∈ F and σ(a_{i}) = ∑_{k} d_{ik}a_{k} for some d_{ik} ∈ F. Let F_{0} be the prime subfield of F, and let F_{1} = F_{0}({c_{ikl}, d_{ik} | 1 ≤ i, k, l ≤ n}) ⊆ F. Let A_{1} be the F_{1}-span of the a_{i}, which is an F_{1}-algebra. We have A_{1} ⊗_{F_{1}} F = A and σ restricts to an involution σ_{1} on A_{1}. Now, let (e_{i})_{i=1}^{n} be a splitting base of A_{h} for the v-norm φ_{h}. We need to enlarge F_{1} to capture the e_{i} in the Henselization: let L be any field with F_{1} ⊆ L ⊆ F and L finitely generated over F_{1}, and let v_{L} = v|_{L}. Since F_{h} is Henselian, there is a unique Henselization (L_{h}, v_{L,h}) of (L, v_{L}) inside (F_{h}, v_{h}) by [EP, Th. 5.2.2(2), p. 121]. Because F is the direct limit of such fields L, the direct limit over such L of the (L_{h}, v_{L,h}) is a Henselian valued field (M, v_{M}) with F ⊆ M ⊆ F_{h} and v_{h}|_{M} = v_{M}. Therefore, (M, v_{M}) = (F_{h}, v_{h}) by the uniqueness of the Henselization. Since A_{h} = A_{1} ⊗_{F_{1}} M, there is a field F_{2} finitely generated over F_{1} (hence also over F_{0}) such that e_{1}, …, e_{n} ∈ A_{1} ⊗_{F_{1}} (F_{2})_{h}. Let

A_{2} = A_{1} ⊗_{F_{1}} F_{2} ⊆ A, A_{2,h} = A_{1} ⊗_{F_{1}} (F_{2})_{h} = A_{2} ⊗_{F_{2}} (F_{2})_{h} ⊆ A_{h}.

Note that A_{2} is a simple F_{2}-algebra since A = A_{F_{2}} ⊗_{F_{2}} F and A is simple. Let σ_{2} = σ|_{A_{2}}, which is an involution on A_{2}, and let σ_{2,h} = σ_{h}|_{A_{2,h}} = σ_{2} ⊗ id_{(F_{2})_{h}}, which is an anisotropic involution on A_{2,h}. Let φ_{2} = φ_{h}|_{A_{2}} and φ_{2,h} = φ_{h}|_{A_{2,h}}. Since A_{h} = A_{2,h} ⊗_{(F_{2})_{h}} F_{h} and e_{1}, …, e_{n} ∈ A_{2,h}, Lemma 5.1 says that φ_{2,h} is a v_{F_{2},h}-norm on A_{2,h} and φ_{h} = φ_{2,h} ⊗ v_{h}. Now, F_{2} is finitely generated over the prime field F_{0}, so rk(Γ_{F_{2},v_{F_{2}}} ) ≤ trdeg(F_{2}/F_{0}) < ∞ by [B, §10.3, Cor. 2]. Since φ_{2,h} is a v_{F_{2},h}-norm, the finite rank case shows that φ_{2} is a v_{F_{2}}-norm on A_{2}; then, φ_{2,h} = φ_{2} ⊗ v_{F_{2},h} by Cor. 5.2. Hence,

φ_{2} ⊗ v_{h} = (φ_{2} ⊗ v_{F_{2},h}) ⊗ v_{h} = φ_{2,h} ⊗ v_{h} = φ_{h}.

Therefore, as φ_{2} is a norm, φ_{h}|_{A} = (φ_{2} ⊗ v_{h})|_{A_{2}⊗_{F_{2}}F} = φ_{2} ⊗ v, which is a norm since it is a scalar extension of the norm φ_{2}. □

Corollary 6.2. With the hypotheses on A, σ, and v as in Th. 6.1, let φ be a v-gauge on A which is invariant under σ. Then,

(a) If the residue involution σ_{0} is anisotropic, then φ is the unique σ-special v-gauge on A.

(b) If σ_{0} is isotropic, then there is no σ-special v-gauge on A.

Proof. Let (F_{h}, v_{h}) be a Henselization of (F, v), and let A_{h} = A⊗_{F} F_{h} and σ_{h} = σ⊗id_{F_{h}}. Let φ_{h} = φ⊗v_{h}, a surmultiplicative value function on A_{h} which is invariant under the involution σ_{h}, by Cor. 1.4. The graded isomorphisms gr_{φ_{h}}(A_{h}) ≅ gr_{φ}(A) ⊗_{gr_{v}(F)} gr_{v_{h}}(F_{h}) ≅ gr_{φ}(A) show that φ_{h} is a gauge on A_{h}, and σ̃_{h} ≅ σ̃ and (σ_{h})_{0} ≅ σ_{0}. (a) If σ_{0} is anisotropic, then so is (σ_{h})_{0}, and so also is σ_{h} by Cor. 2.3. Th. 6.1 then shows that φ is a σ-special v-gauge and is the unique such v-gauge on A, proving (a). For (b), we prove the contrapositive: If there were a σ-special v-gauge ψ for A then the uniqueness in Th. 6.1 shows that φ = ψ. Hence, σ_{h} is anisotropic by Th. 6.1, so (σ_{h})_{0} is anisotropic by Cor. 2.3, which implies σ_{0} is anisotropic as well. □

Example 6.3. Even when there is no σ-special v gauge on A, there may still be tame v-gauges on A invariant under σ, but they need not be unique. For example, let A the quaternion division algebra (−1, −1)_{ℚ} over the field of rational numbers, and let v be the 3-adic valuation on ℚ. Let (1, i, j, k) be the quaternion base of A with i^{2} = j^{2} = −1 and k = ij = −ji. As shown in [TW, Ex. 1.16], a v-gauge φ can be defined on A by

φ(a_{0} + a_{1}i + a_{2}j + a_{3}k) = min(v(a_{0}), v(a_{1}), v(a_{2}), v(a_{3})).

Clearly, the residue algebra of A for φ is A_{0} = (−1, −1)_{𝔽_{3}} ≅ M_{2}(𝔽_{3}). The v-gauge φ is obviously invariant under the conjugation involution σ on A. (This is the involution with σ(i) = −i and σ(j) = −j, which is the unique symplectic involution on A.) Since A is a division algebra, σ must be anisotropic. The residue involution σ_{0} is the conjugation involution on A_{0}, which is isotropic, since σ_{0}(t)t = Nrd_{A_{0}}(t) for any t in the split quaternion algebra A_{0}. So, by Cor. 6.2(b) there is no σ-special v-gauge on A. For any unit u ∈ A^{×} the map φ_{u} defined by

φ_{u}(x) = φ(uxu^{−1}) for x ∈ A

is a v-gauge on A, and Prop. 1.17 of [TW] shows that φ_{u} = φ if and only if ũ is invertible in gr_{φ}(A), which is not a graded division ring. But, for every u ∈ A^{×}, since φ is invariant under σ and σ(u)u is central,

φ_{u}(σ(x)) = φ(σ[(σ(u^{−1})xσ(u)]) = φ(σ(u^{−1})xσ(u)) = φ_{u}([σ(u)u]^{−1}x[σ(u)u]) = φ_{u}(x),

showing that φ_{u} is invariant under σ.

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Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

   E-mail address: jean-pierre.tignol@uclouvain.be

Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA

   E-mail address: arwadsworth@ucsd.edu