Rationally isomorphic Azumaya algebras with involution over semilocal Bézout domains

Sofie Beke and Jan Van Geel

Abstract

Let R be a commutative domain in which 2 is invertible, and F its fraction field. Let , σ and ^{}, σ^{}be R algebras with involution. They are said to be rationally isomorphic if _{R}F, σ _{R} id_{F} _{F} ^{}_{R}F, σ^{}_{R} id_{F} . The main result of this paper states that for R a semilocal Bézout domain, rationally isomorphic R algebras with involution are isomorphic.

Let R be a commutative domain in which 2 is invertible, and F its fraction field. Let , σ and ^{}, σ^{}be R algebras with involution. They are said to be rationally isomorphic if RF, σ _{R} id_{F} _{F} ^{}_{R}F, σ^{}_{R} id_{F} . Under which conditions on R can we conclude that , σ and ^{}, σ^{}are already isomorphic as R algebras with involution? If any pair of R algebras with involution that is rationally isomorphic, is already isomorphic over R, we say that R has the RIII property. Consider the linear algebraic group G Aut , σ . The question whether , σ _{R} ^{}, σ^{} translates to the question whether principal homogeneous spaces over G that are trivial over F, are already trivial over R. This problem has been studied for R a discrete valuation ring, and more generally for regular local rings. For these rings it goes back to a conjecture of Grothendieck and Serre, stating that principal G homogeneous spaces over a Noetherian regular integral k scheme X, k a field, that are rationally trivial are locally trivial. In [N], Y. Nisnevich proved the conjecture for schemes of dimension one and regular Henselian local schemes of any dimension. Nisnevich’s work includes, in the case R is a discrete valuation ring, a postive answer to the question considered in this paper. Nisnevich’s proof is based on the relation between étale cohomology and adélic invariants of G. Other ingredients are the fact that the statement holds for complete discrete valuation rings (an unpublished theorem of Tits), and weak approximation (cf. [H]). In [Pa], I. Panin proved a purity theorem on multipliers for Aut , σ , which together with the result for discrete valuations, gives a positive answer to the conjecture for automorphism groups of algebras with involution.

In the rest of the introduction, we will always assume that 2 is invertible in the rings we consider. In [B] the first author considers Azumaya algebras with involution under specialisation and the question whether rationally isomorphic algebras with involution have isomorphic reductions. The question whether for algebras with involution over valuation rings, rational isomorphism implies isomorphism, is natural in this context. In [B, (8.7)] it is shown that Henselian valuation rings have the RIII property. In this paper we will show that semilocal Bézout domains have the RIII property. We will first treat the case of finite Krull dimension and then show that the general case can be reduced to this case. Our result includes in particular that valuation rings have the RIII property. Although this also covers the case of discrete valuation rings, and more generally semilocal principal ideal domains, we will treat this class of rings (the Noetherian case) separately, since the proof then simplifies in different places. Furthermore, the role played by a Cassels–Pfister type result for algebras with involution becomes transparent. Using the relation between ε hermitian spaces and adjoint involutions we obtain, as a corollary to our main result, that “rational similarity implies similarity” for ε hermitian spaces over an Azumaya algebra with involution without zero divisors over a semilocal Bézout domain. In particular this implies that rationally similar symmetric or skew–symmetric bilinear spaces over semilocal Bézout domains, are similar.

The structure of the paper is as follows. In the first section we state the problem formally and recall from [B] some reduction that can be made in order to show the RIII property holds. We also show that proving that the RIII property holds for two involutions on a fixed algebra, is equivalent to a statement on multipliers. We then prove certain characterisations of the multipliers of an algebra with involution over a semilocal Bézout domain of finite Krull dimension. This characterisation is then used to prove the RIII property for such rings. In section 3 the characterisation is first obtained in the Noetherian case (discrete valuation rings and semilocal principal ideal domains), using that valuation rings are elementary divisor domains, and a Cassels–Pfister type result for algebras with involution over semilocal principal ideal domains, which is proven in section 2. In section 4 we treat the general case of semilocal Bézout domains, first considering the case of finite Krull dimension. The characterisation of the multipliers is then obtained using some result for Henselian valuation rings (treated in [B]), and a norm argument based on Paulo Ribenboim’s approximation theorem for valuations.

We fix some notation for the rest of the article. F will denote a field of characteristic different from 2. R will be a commutative semilocal domain in which 2 is invertible, with fraction field F, and S will be either equal to R or to a commutative separable R algebra that is free of dimension 2 as an R module. In the latter case, we call S a free separable quadratic R algebra. If S is a domain, we denote its fraction field by K.

Let T be a commutative ring and M a finitely generated, free T module. We will use the term dimension for the rank of M over T, and denote this by dim_{T} M .

1 Hermitian spaces, algebras with involution and multipliers

In this section we define R algebras with involution and elaborate on the properties of rationally isomorphic R algebras with involution, also by interpreting them as adjoint involutions of hermitian spaces. Therefore, we zoom in on some properties of hermitian spaces first.

Let C be a (not necessarily commutative) ring with unit. We assume 2 C^{}. Let θ be an involution on C, i.e. an anti–automorphism of C of order at most 2. Let ε 1. An ε hermitian module over C, θ is a pair V, h where V is a finitely generated, projective right C module, and h V V C a bi–additive map such that for all x, y V and all α, β C, the following hold: h xα, yβ θ α h x, y β and h y, x εθ h x, y . The form h is called hermitian if ε 1 and skew–hermitian if ε 1. If θ id_{C}, h is called a bilinear form. Let V^{}Hom_{C} V, C . This is a left C module. Define the right C module ^{θ}V^{}by ^{θ}V^{}^{θ}φ φ V^{}with the operations ^{θ}φ ^{θ}ψ ^{θ} φ ψ , ^{θ}φ α ^{θ} θ α φ for all φ, ψ V^{}and all α C. Then h is called non–singular if the adjoint transformation

h V ^{θ}V^{}x ^{θ}φ, where φ y h x, y for all y V,

is an isomorphism of right C modules. We call V, h an ε hermitian space if h is non–singular.

Suppose V is free over C with basis e_{1}, …, e_{n} . Then h defines a matrix C_{h} h e_{i}, e_{j} _{i,j} M_{n} Δ . Define the dual basis ^{#} e_{1}^{#}, …, e_{n}^{#} by the property e_{i}^{#} e_{j} δ_{ij}. Then ^{θ}e_{1}^{#}, …,^{θ}e_{n}^{#} is a C basis for^{θ}V^{}. The matrix of ĥwith respect to the bases , ^{#} is given by εC_{h}. Hence, ĥis an isomorphism if and only C_{h} is invertible. If h is non–singular, we may consider the elements θe_{1}^{#}, …,^{θ}e_{n}^{#} as elements of V.

Let U be a C submodule of V. The orthogonal complement of U, which is equal to x V h x, y 0 for all y U , will be denoted by U^{}. The subspace U is called totally isotropic if U U^{}. An ε hermitian module V, h is called isotropic if it contains a nonzero totally isotropic subspace U, and anisotropic otherwise. Equivalently, V, h is isotropic if there exists an element 0 x V such that h x, x 0. V, h is called hyperbolic if it contains a direct summand U such that U^{}U.

Two ε hermitian modules V, h , V^{}, h^{}over C, θ are called isometric, denoted by V, h V^{}, h^{}or h h^{}, if there is a C linear bijection φ V V^{}such that h x, y h^{}φ x , φ y for all x, y V.

We recall some facts on R algebras with involution from [B].

1.1 Proposition.

(a) Suppose S R. Then S has an R basis 1, z with z^{2} az b, for certain a, b R. Let f x x^{2} ax b R x . Then the discriminant of f x is a unit in R. Furthermore, S is a domain if and only if f x is irreducible in R x .

(b) Suppose R is integrally closed in F. Then S is the integral closure of R in S _{R} F. Furthermore S is a domain if and only if S _{R} F is a field, and the latter is then the fraction field of S. If S is not a domain, then S R R.

Proof. See [B, (2.7)].

1.2 Proposition. There is a unique involution ι on S such that R x S ι x x . If S R then this involution is given by ι S S c dz c d a z .

Proof. See [B, (2.9)].

In the special case where S R R, ι is given by the switch map.

Let be an Azumaya algebra over S and σ an R linear involution on such that, if Z R, then σ restricts to the involution ι on S. If S R, σ is called an involution of the first kind, otherwise, it is called an involution of the second kind. We call the pair , σ an R algebra with involution. If S is not a domain then we call , σ degenerate.

If R is a field, then σ is called isotropic if there exists a nonzero x such that σ x x 0, and anisotropic otherwise. σ is called hyperbolic if there is an idempotent x such that σ x 1 x.

Let R^{}be a commutative semilocal domain that is also an R algebra. We write , σ _{R^{}} R^{}, σ_{R^{}}_{R}R^{}, σ _{R}id_{R^{}}. The center of _{R^{}}is equal to S _{R}R^{}.

1.3 Proposition. Let , σ be an R algebra with involution and let R^{}a commutative semilocal domain that is an R algebra. Then , σ _{R^{}}is an R^{}algebra with involution.

Proof. See [B, (2.10)].

Let , σ and ^{}, σ^{}be R algebras with involution. We say , σ and ^{}, σ^{}are isomorphic over R if there exists an isomorphism φ ^{}of R algebras such that φ σ σ^{}φ. We denote this by , σ _{R} ^{}, σ^{}.

Let be an Azumaya algebra over R. The map sw  ^{op}  ^{op} a, b b, a defines

an involution of the second kind on ^{op}, called the switch involution.

1.4 Proposition. Let be an Azumaya algebra wit center R R. Then there exist Azumaya algebras _{1}, _{2} over R such that _{1} _{2}. Furthermore, if σ is an involution of the second kind on , then _{2} _{1}^{op} and , σ _{R} _{1} _{1}^{op}, sw .

Proof. See [B, (2.11)].

1.5 Corollary. Suppose R is a field and let , σ be a degenerate R algebra with involution. Then σ is hyperbolic.

Proof. By Proposition 1.1 (b), Z R R and Proposition 1.4 yields that there exists an idempotent x such that σ x 1 x.

1.6 Proposition. Let be an Azumaya algebra over S. Then is free as an R module.

Proof. See [B, (2.12)].

In this paper we will work with algebras with involution over a semilocal Bézout domain. These are semilocal domains in which the finitely generated ideals are principal (the defining property of a Bézout domain). Semilocal Bézout domains are exactly those semilocal domains whose localisations at their finitely many maximal ideals are valuation rings, cf. [B, (2.2)].

The following characterisation of semilocal Bézout domains of finite Krull dimension is easily obtained.

1.7 Lemma. R is a semilocal Bézout domain of finite Krull dimension if and only if R is the intersection of finitely many valuation rings of F of finite rank (= Krull dimension).

If R is a semilocal Bezout domain then R algebras with involution have additional properties.

1.8 Proposition. Suppose R is semilocal Bézout domain and assume S is a domain. Then S is a semilocal Bézout domain. Let Δ be an Azumaya algebra over S without zero divisors. Every finitely generated, torsion–free left or right Δ module is free.

Proof. See [B, (2.4), (2.8) and (2.16)].

1.9 Proposition. Suppose R is a semilocal Bézout domain.

(a) Let , θ be an R algebra with involution with center a domain. Let V, h be an ε hermitian space over , θ . There exists a unique involution σ on End_{}V such that σ a θ a for all a Z , and for all x, y V and all f End_{}V we have that

h x, f y h σ f x , y .

We denote this involution by ad_{h}. Then End_{}V , ad_{h} is an R algebra with involution, called the adjoint algebra with involution of h, and denoted by Ad h . If θ is of the first kind (resp. of the second kind), then ad_{h} is of the first kind (resp. of the second kind).

(b) Let , σ be an R algebra with involution with center a domain. Then there exists an R algebra with involution Δ, θ without zero divisors, with Z Δ Z and θ of the same kind as σ, and an ε hermitian space V, h over Δ, θ , such that , σ _{R} Ad h .

Proof. See [B, (4.6)] for (a) and [B, (4.7)] for (b).

1.10 Notation. Let , θ be an R algebra with involution and V, h an ε hermitian module over , θ . We denote by V, h _{F} V_{F}, h_{F} the ε hermitian module over , θ _{F} obtained by extending scalars from R to F.

1.11 Proposition. Let R be a commutative semilocal Bézout domain with fraction field F. Let Δ, θ be an R algebra with involution without zero divisors. The following hold.

(a) Let V, h be an ε hermitian space over Δ, θ . If V, h _{F} is isotropic (metabolic), then V, h is already isotropic (metabolic).

(b) Suppose 2 R^{}. Then hyperbolic ε hermitian spaces of the same dimension over Δ, θ are isometric.

(c) Suppose 2 R^{}. Let V, h , V^{}, h^{}be ε hermitian spaces over Δ, θ . If V, h _{F} _{F} V^{}, h^{}_{F} then V, h V^{}, h^{}.

Proof. See [B, (4.5)]. The proof of (c) uses a general Witt cancellation result of Keller (see [K, (VI.5.7.2)] for more details).

Let , σ and ^{}, σ^{}be two R algebras with involution. We say they have the “rational isomorphism implies isomorphism” property, denoted by RIII, if the following holds:

if , σ _{F} _{F} ^{}, σ^{}_{F} then , σ _{R} ^{}, σ^{}.

We say that R has the RIII property if any pair , σ , ^{}, σ^{}of R algebras with involution has the RIII property. In [B, (8.7)], the author showed that if R is a Henselian valuation ring, then it has the RIII property. In this paper we will extend the results of [B] and show that if R is a semilocal Bézout domain, then it has the RIII property. A crucial ingredient of the proof will be the following result for Henselian valuation rings from [B].

1.12 Proposition. Suppose R is a Henselian valuation ring of F. Denote its residue field by κ. Let , σ be an R algebra with involution. Then , σ _{κ} is hyperbolic if and only if , σ _{F} is hyperbolic.

Proof. See [B, (7.7)].

As in [B], we will use this result in the following way. Let be a valuation ring of F. Let F^{s} be a separable closure of F and let ^{s} be an extension of to F^{s}. Let G ρ Gal F^{s} F ρ ^{s} s . Then F^{s} ^{G}, ^{s} F^{s} ^{G} is Henselian by [EP, (3.2.15)]. We denote it by F^{h}, ^{h} , it is called a Henselisation of F, . By [EP, (5.2.5)], F, F^{h}, ^{h} is an immediate extension, i.e. F, and F^{h}, ^{h} have isomorphic value groups and residue fields.

1.13 Corollary. Suppose R is a valuation ring of F. Denote its residue field by κ. Let F^{h}, R^{h} be a Henselisation of F, R . Let , σ be an R algebra with involution. Then , σ _{κ} is hyperbolic if and only if , σ _{F^{h}} is hyperbolic.

Proof. Since R^{h} still has residue field κ, the statement follows immediately from Proposition 1.12.

We recall several reductions of the RIII problem explained in [B].

1.14 Proposition. Suppose R is a semilocal Bézout domain. If pairs of R algebras with involution of the form , σ , , σ^{}, where , σ _{F} _{F} ^{}, σ^{}_{F} via a Z _{F} isomorphism, have the RIII property, then R has the RIII property.

Proof. This is shown in [B, (8.3)] in the case R is a valuation ring, but it goes through for semilocal Bézout domains.

1.15 Proposition. Suppose R is a semilocal Bézout domain. Let , σ , ^{}, σ^{}be a pair of R algebras with involution. Assume that , σ _{F} _{F} ^{}, σ^{}_{F} and Z R R. Then , σ ^{}, σ^{}.

Proof. By Proposition 1.14, in order to show the claim we may assume that ^{}. Since all involutions of the second kind on are isomorphic over R by Proposition 1.4, the statement follows.

1.16 Proposition. Suppose R is a semilocal Bézout domain. Let Δ, θ be an R algebra with involution without zero divisors, and let V, h be an ε hermitian space over Δ, θ . Let , σ Ad h . Let furthermore s ^{}be such that σ s s and let σ^{}Int s σ. Define h^{}V V Δ by h^{}x, y h s^{1} x , y for all x, y, V. Then V, h^{}is an ε hermitian space over Δ, θ such that , σ^{}Ad h^{}, and the following are equivalent:

(i) , σ _{R} , σ^{}through a Z automorphism.

(ii) There exists elements e R^{}and g ^{}such that es σ g g.

(iii) There exists u R^{}such that h^{}_{R} uh.

Furthermore, if u R^{}, then h^{}_{R} uh if and only if there exists an element g ^{}such that us σ g g.

Proof. See [B, (4.9)].

The following proposition is shown in [B, (8.5)] for valuation rings, but the proof goes through for semilocal Bézout domains.

1.17 Proposition. Suppose R is a semilocal Bézout domain and assume S is a domain. Let be an Azumaya algebra over S, and σ, σ^{}involutions of the first or second kind on . Suppose , σ _{F} _{F} , σ^{}_{F}. Then there exists s ^{}such that σ s s and σ^{}Int s σ.

Proposition 1.16 yields the equivalence between similarity of certain hermitian spaces and isomorphism of their adjoint algebras with involution. This result holds in fact without constraints on the hermitian spaces, as is shown below.

1.18 Proposition. Suppose R is a semilocal Bézout domain and let Δ, θ be an R algebra with involution without zero divisors. Let V, h and V^{}, h^{}be two ε hermitian spaces over Δ, θ . Then there exists u R^{}such that h^{}_{R} uh if and only if Ad h and Ad h^{}are isomorphic through a Z Δ isomorphism.

Proof. Suppose first there exists u R^{}and a Δ linear bijection φ V V^{}such that uh x, y h^{}φ x , φ y , for all x, y V. Then one easily checks that End_{Δ} V End_{Δ} V^{}; f φ f φ^{1} defines a Z Δ isomorphism of algebras with involution: Ad h Ad h^{}. Suppose conversely that there exists an isomorphism β Ad h Ad h^{}that is the identity on Z Δ . Then End_{Δ} V End_{Δ} V^{}and hence V and V^{}have the same dimension over Δ. Hence, there is a Δ linear bijection ψ V V^{}. Define an ε hermitian form h̃ V^{}V^{}Δ by h̃ ψ x , ψ y h x, y , for all x, y V. Then h̃ _{R} h and therefore, Ad h̃ _{R} Ad h _{R} Ad h^{} via Z Δ isomorphisms. Since ad_{h^{}}and adh̃ are involutions on End_{Δ} V^{}and adh̃_{F} and ad_{h_{F}^{}} are isomorphic, it follows from Proposition 1.17 that there exists s End_{Δ} V^{}^{}such that adh̃ Int s ad_{h^{}}and ad_{h^{}}s s. Define an ε hermitian form h^{}V^{}V^{}Δ by

h^{}x^{}, y^{}h^{}s^{1} x^{}, y^{},

for all x^{}, y^{}V^{}. One easily checks that ad_{h^{}}adh̃. Furthermore, we have that v { \atop h^{}}^{1} h̃ End_{Δ} V^{}^{}and, by definition of v, h^{}v x^{}, y^{}h̃ x^{}, y^{}for all x^{}, y^{}V^{}. It follows that ad_{h^{}}adh̃ Int v^{1} ad_{h^{}}. Hence, v Z Δ , and we get h̃ θ v h^{}. Since h̃ and h^{}are both ε hermitian, it follows that θ v v and hence v R^{}. It follows that

h^{}_{R} v^{1} h̃ _{R} v^{1}h.

It follows that Ad h^{}and Ad h , and hence Ad h^{}and Ad h^{}are isomorphic through a Z Δ isomorphism. By Proposition 1.16, this means there exists e R^{}, g End_{Δ} V^{}^{}such that

es ad_{h^{}}g g.

For all x^{}, y^{}V^{}, we get that

h^{}x^{}, y^{}h^{}s^{1} x^{}, y^{}eh^{}g^{1} ad_{h^{}}g^{1} x^{}, y^{}eh^{}ad_{h^{}}g^{1} x^{}, ad_{h^{}}g^{1} y^{}.

This means that h^{}_{R} eh^{}. So, putting everything together, we obtain h^{}_{R} e^{1}h^{}_{R} e^{1}v^{1}h, and e^{1}v^{1} R^{}. This yields the statement.

The next result will be crucial in order to show that semilocal Bézout domains have the RIII property.

1.19 Proposition. Suppose R is a valuation ring of F, with residue field κ. Let , σ be an R algebra with involution. Let e F^{}, s ^{}and g _{F}^{}be such that es σ_{F} g g. Let F^{h}, R^{h} be a Henselisation of F, R . If e F^{2}R^{}then , σ _{F^{h}} and , σ _{κ} are hyperbolic.

Proof. If Z is not a domain then , σ is hyperbolic over F^{h} and κ by Proposition 1.5. See [B, (8.6)] for the case where Z is a domain and combine it with Corollary 1.13.

Let B, τ be an F algebra with involution. An element f B is called a similitude if τ f f F^{}. The set of similitudes for B, τ forms a group, denoted by Sim B, τ . If f Sim B, τ then μ f σ f f F^{}is called a multiplier of B, τ . The multipliers of B, τ form a subgroup of F^{}, denoted by G B, τ .

1.20 Proposition. Let C, θ be an F algebra with involution of any kind. Let V, h̃ be an ε hermitian space over C, θ and let B, τ End_{C} V , adh̃ . Then

G B, τ α F^{}h̃ αh̃ .

Proof. We have that h̃ αh̃if and only if there exist a C linear bijection φ V V such that

h̃ x, y αh̃ φ x , φ y αh̃ τ φ φ x , y , for all x, y V.

The non–singularity of h̃ implies that this equality holds if and only if τ φ φ α^{1}, and hence α^{1} G B, τ . Since G B, τ is a group, it follows that α G B, τ .

1.21 Proposition. Let B, τ be an F algebra with involution of any kind.

(a) If B, τ is hyperbolic then G B, τ F^{}.

(b) Let L F be a finite extension. Then

N_{L F} G B, τ _{L} G B, τ .

Proof. (a) follows using the characterisation of G B, τ in terms of ε hermitian forms from Proposition 1.20. If Z B is a domain, then (b) holds by [KMRT, (12.21)]. If Z B is not a domain, then (b) holds since G B, τ F^{}by (a).

Using the language of multipliers, property (ii) in Proposition 1.16 can be replaced by another (seemingly weaker) property,

1.22 Corollary. In the situation of Proposition 1.16, (i), (ii) and (iii) are also equivalent to

(ii’) There exist elements e G , σ _{F} R^{}and g _{F}^{}such that es σ g g.

Furthermore, if e F^{}is such that there exists an element g such that es σ g g and , σ _{R} , σ^{}, then e G , σ _{F} R^{}.

Proof. For the first statement, it suffices to prove that (ii’) is equivalent to (iii). It follows directly from Proposition 1.16 that (iii) implies (ii’), (since (ii) is stronger). So assume (ii’) holds. By Proposition 1.16, es σ_{F} g g yields that h_{F}^{}_{F} eh_{F}. Write e au with a G , σ _{F} and u R^{}. By Proposition 1.20, we have that h_{F} _{F} ah_{F} and hence, it follows that h_{F}^{}_{F} uh_{F}. By Proposition 1.11 (b), we get that h^{}_{R} uh.

Let e F^{}be such that there exists g such that es σ g g and assume that , σ _{R} , σ^{}. Then g _{F}^{}and Proposition 1.16 yields that h_{F}^{}_{F} eh_{F}. Since , σ _{R} , σ^{} by hypothesis, invoking Proposition 1.16 once more yields that there exists u R^{}such that h_{F}^{}_{F} uh_{F}. It follows that euh_{F} _{F} h_{F}. By Proposition 1.20, the latter means exactly that eu G , σ _{F} .

2 Cassels–Pfister type theorems for involutions and hermitian forms

In this section, we assume R is a semilocal principal ideal domain. We furthermore fix an R algebra with involution without zero divisors Δ, θ , and an ε hermitian space V, h over Δ, θ . Furthermore, we let , σ Ad h and D Δ_{F}. Note that Z Z Δ is a domain.

A (Δ )lattice in V, h _{F} is a finitely generated, right Δ submodule of V_{F} containing a D basis of V_{F}. Lattices in V, h _{F} are torsion–free Δ modules, and hence free Δ modules by Proposition 1.8.

Let be a lattice in V, h _{F}. The dual of is defined as

# v V_{F} h_{F} v, Δ .
is called integral (with respect to h_{F}) if  ^{#} and self–dual or unimodular if  ^{#}.

2.1 Proposition. Let be an integral lattice in V, h _{F} and denote the restriction of h_{F} to by h_{}. The following are equivalent:

(i) , h_{}is an ε hermitian space over Δ, θ .

(ii) ^{#} .

Proof. Let  f_{1}, …, f_{n} be a Δ basis for . Consider the dual basis ^{θ} ^{#}  ^{θ}f_{1}^{#}, …,^{θ}f_{n}^{#} as

elements of V_{F}. Suppose (i) holds. Then the elements of^{θ} ^{#} belong to . Suppose h_{F} v, Δ for some v V_{F}. We can write v _{i 1}^{n} ^{θ}e_{i}^{#}x_{i}, with the x_{i} D. Then h_{F} v, e_{j} _{i 1}^{n} θ_{F} x_{i} δ_{ij} θ_{F} x_{j} Δ and hence v . So, is unimodular.

Suppose ^{#}. Since h_{F} ^{θ}f_{i}^{#}, f_{j} δ_{ij} Δ and is unimodular, we have that ^{θ} ^{#} belongs to . The matrix of ĥ_{}is the matrix of base change from to ^{θ} ^{#}, and hence invertible over Δ. This yields (i).

2.2 Corollary. V is a unimodular lattice in V, h _{F}.

Proof. It is clear that V is an integral lattice in V, h _{F}. Since V, h is a ε-hermitian space, the statement follows from Proposition 2.1.

We refer to [T] for proofs of the following known facts on lattices.

2.3 Proposition.

(a) Let be a lattice in V, h _{F}. Then ^{#} is a lattice in V, h _{F} and ^{##} .

(b) Let _{1}, _{2} be lattices in V, h _{F}. Then

1 _{2} ^{#}  _{1}^{#} _{2}^{#} and _{1} _{2} ^{#}  _{1}^{#} _{2}^{#}.

Note that this implies that the intersection of two lattices in V, h _{F} is again a lattice.

(c) Let _{1}, _{2} be lattices in V, h _{F}. If _{1}  _{2}, then _{2}^{#}  _{1}^{#}.

2.4 Lemma. The unimodular lattices in V, h _{F} are exactly the maximal integral lattices in V, h _{F}.

2.5 Proposition. Let be a unimodular lattice in V, h _{F}. Then there is an isometry u of V, h _{F} such that u V .

Proof. Let h_{}denote the restriction of h_{F} to . The hermitian spaces , h_{}and V, h become isometric over F since they become two representations of the same form. Proposition 1.11 (c) yields that , h_{}and V, h are isometric over R. This means that there is a bijective Δ linear map u V such that h_{}u x , u y h x, y . Extending scalars to F, u defines an isometry of V, h _{F} with u V .

We can now prove a Cassels–Pfister type theorem for algebras with involution over semilocal principal ideal domains.

2.6 Theorem (Involution CP for semilocal principal ideal domains). Let f _{F} be such that σ_{F} f f . Then there exists an element u _{F}^{}such that σ_{F} u u 1 and uf .

Proof. We follow the proof of the main theorem of [T].

We have that , σ _{F} _{F} End_{D} V_{F} , ad_{h_{F}} . Let f End_{D} V_{F} be such that σ_{F} f f End_{Δ} V . Then we can write f d^{1} f̃for some f̃ End_{Δ} V and d R. For all m, m^{}V, we have that

h_{F} f m , f m^{}h_{F} σ_{F} f f m , m^{}Δ,

since σ_{F} f f m V and V V^{#}.

Note that f V is not necessarily a lattice in V, h _{F}. However, f V dV is a lattice and it is also integral since df End_{Δ} V . Since R is a principal ideal domain, every integral lattice in V, h _{F} is contained in a unimodular lattice. This can be seen as follows. Let _{1} _{2} …be a chain of integral lattices, we have that _{i} _{1}^{#}, for all i. Since _{1}^{#} is a lattice over Δ, Δ is finitely generated over R, and R is a Noetherian ring, it follows that _{1}^{#} is a Noetherian R module. Since all _{i} are R submodules of _{1}^{#}, it follows that the chain _{1} _{2} … must stop. Hence, any integral lattice in V, h _{F} is contained in a maximal integral lattice.

Let be a maximal integral lattice in V, h _{F} containing f V dV. Then is unimodular by Lemma 2.4. Proposition 2.5 implies that there is an isometry u of V, h _{F} such that u V . It follows that f V u V , so u^{1}f V V and hence u^{1}f End_{Δ} V. Since u is an isometry of V, h _{F}, we have that

h_{F} σ_{F} u^{1} u^{1} x , y h_{F} u^{1} x , u^{1} y h_{F} x, y ,

for all x, y V_{F}. It follows that σ_{F} u^{1} u^{1} 1. This proves the statement.

2.7 Corollary (Hermitian CP for semilocal principal ideal domains). If there exists an element x V_{F} such that h_{F} x, x Δ, then there exists an element x^{}V such that h x^{}, x^{} h_{F} x, x .

Proof. Let e_{1}, …, e_{n} be a Δ basis for V. Let δ h_{F} x, x , with x V_{F}. Consider the element f End_{D} V_{F} defined by f e_{1} x and f e_{i} 0, for i 2, …, n. Since h_{F} ad_{h} f f e_{j} , e_{i} h_{F} f e_{j} , f e_{i} 0 for i 1 or j 1, and h_{F} ad_{h} f f e_{1} , e_{1} h_{F} f e_{1} , f e_{1} h_{F} x, x Δ, it follows that h_{F} ad_{h} f f y , y Δ for all y V. Therefore, ad_{h} f f y V^{#} V, and hence ad_{h} f f End_{Δ} V .

By Theorem 2.6 there exists an element u End_{D} V_{F} ^{}with ad_{h} u u 1 such that uf End_{Δ} V . Since ad_{h} u u 1, we have h_{F} u y , u y h_{F} ad_{h} u u y , y h_{F} y, y for all y V_{F} and therefore in particular

h_{F} u x , u x h_{F} x, x δ.

So, we have found an element z u x u f e_{1} V representing δ.

2.8 Remarks.

(a) The conclusion of Theorem 2.6 also holds if R is any valuation ring of F and Δ is a valuation ring of D. The proof of Theorem 2.6 goes through completely provided that every integral lattice is contained in a unimodular (= maximal integral) lattice. If R is not discrete then it is not Noetherian and the proof of the latter fact is fairly technical. It uses the elementary divisor property for finitely generated modules over Δ.

(b) Theorem 2.6 is used in the proof of Proposition 3.5, which will be the crucial result in order to show that semilocal principal ideal domains have the RIII property. The proofs given of the analogous results for algebras with involution over semilocal Bézout domains of finite Krull dimension (see section 4), rely heavily on Proposition 1.19. The proof of this proposition, cf. [B, (8.6)], uses the Cassels–Pfister property as formulated in Corollary 2.7, but for hermitian spaces over Azumaya algebras that are in addition (non– commutative) valuation rings. A direct proof of this CP version is given in [B, (4.16)].

3 Semilocal principal ideal domains

In this section we assume R is a semilocal principal ideal domain. Let , σ be an R algebra with involution. The main result in this section, Proposition 3.5, gives a characterisation of the multipliers of , σ _{F} up to units in R. These multipliers are described by local conditions, i.e. in terms of the discrete valuation rings of F lying over R. As consequence of this characterisation, it follows that R has the RIII property. In particular we obtain a different proof of Nisnevich’ theorem, stating that discrete valuation rings in which 2 is invertible have the RIII property. The results in this section will also be covered by the more general results in the next section, but since the proofs simplify when working with discrete valuation rings, we treat this case separately.

3.1 Lemma. Let B, τ be an F algebra with involution. The following are equivalent.

(i) B, τ is hyperbolic.

(ii) There is an element b B, such that τ b b 0 and dim_{F} bB \frac{1}{2} dim_{F} B.

Proof. See [T, (2.1)].

3.2 Proposition. Let a discrete valuation ring of F. Denote the residue field of by κ. Let , σ be an algebra with involution. Suppose there exist elements e , s ^{}and g _{F}^{} such that es σ g g. If e F^{2} ^{}, then , σ _{κ} is hyperbolic.

Proof. Note that, if Z then , σ _{κ} is degenerate and hence automatically hyperbolic by Proposition 1.5.

Let be a discrete valuation on F with valuation ring . Without loss of generality, we may assume that e 1. For let π be a uniformiser for , then multiplying both sides of σ g g es with an appropriate even power of π and using that σ is the identity on , we obtain πus σ_{F} g^{}g^{}, with u ^{}, g^{}_{F}^{}. By Theorem 2.6, there exists g̃ such that πus σ g̃ g̃. By abuse of notation, we denote g̃ again by g in the rest of the proof. Let n be such that dim_{}dim_{F} _{F} n^{2} if σ is of the first kind, and such that dim_{} dim_{F} _{F} 2n^{2} if σ is of the second kind. Let L Z _{F} be a splitting field of _{F} and let V be a discrete valuation ring of L lying over . Let be a discrete valuation with valuation ring V and let Π be a uniformiser for . Denote the residue field of V by κ̃. We denote the maps κ and the induced map _{κ} by . The commutativity of the diagram

[figure: ; σ; ; _{κ}; σ_{κ}; _{κ}]

implies that \overline{σ g} σ_{κ} \overline{g} and since e 1, it follows that 0 σ_{κ} \overline{g} \overline{g}. So, in order to show that , σ _{κ} is hyperbolic, by Lemma 3.1, it suffices to show that

dim_{κ} \overline{g} _{κ} \frac{1}{2} dim_{κ} _{κ} \frac{n^{2}}{2}.

The commutative diagram

[figure: ; V; κ; κ̃]

induces a commutative diagram

[figure: ; φ; _{}V; _{κ}; ψ; _{κ} _{κ} κ̃.]

We have that _{L} M_{n} L if σ is of the first kind, and by [KMRT, (2.15)], _{L} M_{n} L M_{n} L M_{n} L M_{n} L ^{op} if σ is of the second kind. It follows from [B, (2.20), (2.22)] that V M_{n} V , and hence also _{V}κ̃ M_{n} κ̃ , if σ is of the first kind, and from [B, (2.23)] that _{}V M_{n} V M_{n} V M_{n} V M_{n} V ^{op}, and hence also _{V}κ̃ M_{n} κ̃ M_{n} κ̃ M_{n} κ̃ M_{n} κ̃ ^{op}, if σ is of the second kind. Note that the involution σ_{V} on _{}V then corresponds to the switch involution on M_{n} V M_{n} V ^{op}.

Since

dimκ̃ψ \overline{g} _{κ} κ̃ dimκ̃ \overline{g} 1 _{κ} _{κ} κ̃ dimκ̃ \overline{g} _{κ} _{κ} κ̃ dim_{κ} \overline{g} _{κ}

it suffices to show that

dimκ̃ψ \overline{g} _{κ} κ̃ dimκ̃ \overline{φ g} _{κ} κ̃ \frac{n^{2}}{2}.

Suppose σ is of the first kind. It is well–known that dimκ̃ \overline{φ g} κ̃ rank \overline{φ g} n. Since V is a valuation ring, it is an elementary divisor domain by [Ka, p. 480]. Hence, there are matrices P, Q M_{n} V ^{}such that

φ g P diag d_{1}, …, d_{n} Q,

with d_{1}, …, d_{n} V (see [Ka, p. 465 (1)]). Let C diag d_{1}, …, d_{n} . It follows that \overline{φ g} \overline{P}\overline{C}\overline{Q}. Since \overline{P}, \overline{Q} M_{n} κ̃ ^{}, we have that rank \overline{φ g} rank \overline{C} . The latter is obtained by subtracting the number of \overline{d_{i}} 0 from n. Let us denote this number by ℓ. Then ℓ is equal to the number of d_{i} that are divisible by Π. This number is at most d_{1} … d_{n} det φ g , since det P , det Q V^{}. Taking determinants of the relation eφ s φ σ g φ g σ_{V} φ g φ g yields

e^{n}det φ s det φ g ^{2},

by [KMRT, (2.2)]. Since s ^{}, it follows that det φ s V^{}and hence

det φ g \frac{n}{2}.

So, we get that

dimκ̃ \overline{φ g} _{κ} κ̃ n rank \overline{φ g} n n ℓ n n det φ g n^{2} 2.

Suppose σ is of the second kind. Let g^{}M_{n} V and g^{}M_{n} V ^{op} be such that φ g g^{}, g^{}_{}V M_{n} V M_{n} V ^{op}. It follows that

dimκ̃ \overline{φ g} _{κ}κ̃ dimκ̃ \overline{g^{}}M_{n} κ̃ dimκ̃ \overline{g^{}}M_{n} κ̃ ^{op}.

Invoking the elementary divisor property of V, we find matrices P^{}, Q^{}, P^{}, Q^{}M_{n} V ^{}and elements d_{1}^{}, …, d_{n}^{}, d_{1}^{}, …, d_{n}^{}V such that

g^{}P^{}diag d_{1}^{}, …, d_{n}^{}Q^{}and g^{}P^{}diag d_{1}^{}, …, d_{n}^{}Q^{}.

We have that dimκ̃ \overline{g^{}}M_{n} κ̃ rank \overline{g^{}}n n n ℓ^{}, where ℓ^{}is the number of indices in i 1, …, n such that \overline{d_{i}^{}}0, and dimκ̃ \overline{g^{}}M_{n} κ̃ rank \overline{g^{}}n n n ℓ^{}, where ℓ^{}is the number of indices in i 1, …, n such that \overline{d_{i}^{}}0. We have that ℓ^{}is equal to the number of d_{i}^{}divisible by Π and ℓ^{}is equal to the number of d_{i}^{}divisible by Π. As in the reasoning in the first kind case, we get that

dimκ̃ \overline{g^{}}M_{n} κ̃ n n det g^{}and dimκ̃ \overline{g^{}}M_{n} κ̃ ^{op} n n det g^{}.

So, it follows that

dimκ̃ \overline{φ g} _{κ}κ̃ n n det g^{}n n det g^{}2n^{2} n det g^{}g^{}.  ( )

Let s^{}, s^{}M_{n} V be such that φ s s^{}, s^{}. Since s ^{}, it follows that s^{}, s^{}M_{n} V ^{}. Using the fact that σ_{V} acts as the switch involution on M_{n} V M_{n} V ^{op}, we get es^{}, es^{} g^{}, g^{}g^{}, g^{}g^{}g^{}, g^{}g^{}g^{}g^{}, g^{}g^{}in M_{n} V M_{n} V ^{op}. It follows that es^{}g^{}g^{} in M_{n} V , and hence, taking determinants, e^{n}det s^{}det g^{}g^{}. Applying and using that det s^{}V^{}, it follows that det g^{}g^{}n (since e e 1). Plugging this in into ( ) yields

dimκ̃ \overline{φ g} _{κ}κ̃ 2n^{2} n det g^{}g^{}n^{2} n^{2} 2.

3.3 Corollary. Let be a discrete valuation ring of F. Denote its residue field by κ. Let , σ be an algebra with involution. Let e G , σ _{F} O^{}. If e F^{2} ^{}, then , σ _{κ} is hyperbolic.

Proof. By assumption, there exist u ^{}and g _{F}^{}such that eu σ_{F} g g. Proposition 3.2 yields the statement.

The converse of the corollary, if , σ _{κ} is hyperbolic then every element of F^{}is a multiplier times a unit in , also holds. This will be shown in proposition 3.5.

3.4 Proposition. Let B, τ an F algebra with involution. Let E F be an algebraic field extension such that τ_{E} is hyperbolic. Then there is a finite separable subextension L F over which τ becomes hyperbolic.

Proof. Let e_{1}, …, e_{n} be an F basis for B. Then it is a E basis for B_{E}. Since τ_{E} is hyperbolic, there is an idempotent x B_{E} such that τ_{E} x 1 x. Write x _{i 1}^{n} e_{i}x_{i}, with the x_{i} in E. Then τ already becomes hyperbolic over F x_{1}, …, x_{n} . Since all the x_{i} are algebraic over F, this is a finite extension of F. Since char F 2, we get that τ already becomes hyperbolic over the separable closure of F in F x_{1}, …, x_{n} (see [KMRT, (9.16)]).

By assumption R is a semilocal principal ideal domain. The localisation of R at a maximal ideal is a valuation ring that is moreover a principal ideal domain, and therefore a discrete valuation ring.

3.5 Proposition. Let , σ be an R algebra with involution. Let e F^{}. The following conditions are equivalent:

(i) e G , σ _{F} R^{}.

(ii) For each discrete valuation ring of F lying over R such that e F^{2} ^{}, we have that , σ becomes hyperbolic over the residue field of .

Proof. That (i) implies (ii) follows Corollary 3.3.

Assume that (ii) holds. Let π_{1}, …, π_{r} be generators for the (finitely many) maximal ideals of R, and let _{1}, …, _{r} be the localisations of R at π_{1} , …, π_{r} respectively. These are discrete valuation rings and R _{1} … _{r}. Since R is a unique factorisation domain, there exist u R^{} and α_{1}, …, α_{r} such that e uπ_{1}^{α_{1}} π_{r}^{α_{r}}. Since F^{2} G , σ _{F} and G , σ _{F} is a group, in order to prove (i) holds, it suffices to show that for each i 1, …, r such that e F^{2} _{i}^{}, i.e. α_{i} is odd, there exists a generator π̃_{i} for the prime ideal π_{i} with π̃_{i} G , σ _{F} . Let _{1}, …, _{r} be arbitrary such that e F^{2} ^{}. Let π π_{1}, …, π_{r} be the prime element corresponding to and denote the residue field of by κ. Let F^{h}, ^{h} be a Henselisation of F, . Recall that this is an immediate extension and therefore κ^{h} κ. Therefore, for any subfield F L F^{h}, the residue field of ^{h} _{L} is also isomorphic to κ. Since , σ _{κ}is hyperbolic by hypothesis, it follows from Corollary 1.13 that , σ _{F^{h}} is hyperbolic as well. Proposition 3.4 yields that there is a finite separable subextension F L F^{h} over which σ becomes hyperbolic. Let V ^{h} _{L}. Let R^{}be the integral closure of R in L. Since R is a Dedekind domain and L F is a finite separable extension, R^{}is also a Dedekind domain by [FT, (II.5)]. Furthermore, R^{}is the intersection of the valuation rings of L lying over _{1}, …, _{r}. There are only finitely many such valuation rings and therefore R^{}is a semilocal Dedekind domain, and hence a principal ideal domain by [L2, (I.15)]. Let Π be the principal prime ideal of R^{}corresponding to the valuation ring V. Taking norms of ideals, we have that N Π N Π π ^{f_{V}}, with f_{V} κ_{V} κ the relative residue degree, cf. [L2, (I.22)]. Since, as we saw above, κ_{V} κ, it follows that f_{V} 1, and hence, π̃ N Π is a generator for the prime ideal π . Since , σ _{L} is hyperbolic, G , σ _{L} L^{}by Proposition 1.21 (a). Invoking part (b) of the same Proposition then implies that π̃ G , σ _{F} . This proves the statement.

3.6 Remark. If the center of is not a domain then the properties (i) and (ii) in Proposition 3.5 both hold for trivial reasons. For if , σ is degenerate then , σ _{F} is also degenerate and hence hyperbolic. Proposition 1.21 (a) yields that G , σ _{F} F^{}. Furthermore, , σ remains degenerate over the valuation rings lying over R and hence, , σ is automatically hyperbolic over the residue fields of these valuation rings.

3.7 Remark. In the next section it is shown that Proposition 3.5 holds more generally for semilocal Bezout domains of finite Krull dimension. Although the proof presented here for semilocal principal ideal domains has the same structure as the one that will be given in the next section, it is somewhat simpler. First of all for discrete valuations we need not invoke the Henselisation to prove that (i) implies (ii). For the converse we do use the Henselisation (Corollary 1.13) but the norm argument in the above proof is simpler than the norm argument we will present in the next section, since we can use the properties of norms of ideals in prinicipal ideal domains. This makes the norm argument much more direct.

Using Proposition 3.5together with the reductions made in section 1, we can now show semilocal principal ideal domains have the RIII property.

3.8 Theorem. Suppose R is a semilocal principal ideal domain. Then R has the RIII property.

Proof. Let , σ , ^{}, σ^{}be a pair of R algebras with involution such that , σ _{F} _{F} , σ^{}_{F}^{}. By Proposition 1.14, we may assume that ^{}and that , σ _{F} _{F} , σ^{}_{F}^{}are isomorphic through a Z _{F} isomorphism. Furthermore, by Proposition 1.15, we may assume that Z is a domain. By Proposition 1.16, there exist nonzero elements e F^{}, s ^{}, g _{F}^{}such that es σ_{F} g g. Combining Propositions 3.2, 3.5 and Corollary 1.22 yields that , σ _{R} , σ^{}.

4 Semilocal Bézout domains

In this section we assume R is a semilocal Bézout domain. Let , σ be an R algebra with involution. We will show that the characterisation of the elements in G , σ _{F} R^{}, as obtained Proposition 3.5 for semilocal principal ideal domains, holds more generally for semilocal Bézout domains of finite Krull dimension. As a consequence we obtain our main result saying that semilocal Bézout domains of finite Krull dimension have the RIII property. Using this we can then prove that R also has the RIII property if it does not necessarily have finite Krull dimension.

In order to obtain a local description of the elements in G , σ _{F} R^{}if R has finite Krull dimension, the hyperbolicity result in the Henselian case (see Proposition 1.19) will play an important role. The norm argument on ideals from the previous section will be replaced by a norm argument on elements, based on the following strong approximation theorem of Paulo Ribenboim.

4.1 Theorem. Let E be a field and _{1}, …, _{m} pairwise incomparable valuations on E with respective valuation rings _{1}, …, _{m}. For i 1, …, m, let Γ_{i} be the value group of _{i}. Let V_{ij} be the smallest overring of _{i} and _{j} in E and let Δ_{ij} be the convex subgroup of Γ_{i} such that Γ_{i} Δ_{ij} is the value group of V_{ij}. Then Γ_{i} Δ_{ij} Γ_{j} Δ_{ji}. Let θ_{ij} be the quotient map Γ_{i} Γ_{i} Δ_{ij}. Let γ_{1}, …, γ_{m} Γ_{1} … Γ_{m} be such that θ_{ij} γ_{i} θ_{ji} γ_{j} under the identification Γ_{i} Δ_{ij} Γ_{j} Δ_{ji}. Then there exists an element x E such that _{i} x γ_{i} for i 1, …, m.

Proof. See [R, Théorème 5’].

If _{1}, …, _{m} are pairwise independent valuation rings, then Δ_{ij} Γ_{i} and one gets the well– known classical approximation theorem.

4.2 Corollary. Let E be a field and _{1}, …, _{m} pairwise independent valuations with respective value groups Γ_{1}, …, Γ_{m}. Let γ_{1}, …, γ_{m} Γ_{1} … Γ_{m}. Then there exists an element x E such that _{i} x γ_{i} for i 1, …, m.

Proof. See [EP, (2.4.1)], where a stronger form is given.

4.3 Proposition. Different valuation rings of rank 1 of the same field are independent.

Proof. See [EP, (2.3.2)].

4.4 Lemma. Let E F be a finite field extension. Let ^{}be a valuation ring of E lying over . Let β be a prime ideal of ^{}different from the maximal ideal of ^{}. Then β is a prime ideal of that is not maximal.

Proof. Since β is a prime ideal of ^{}, it is clear that β is a prime ideal of . Let T be the integral closure of in E. Then T is the intersection of the finitely many valuation rings of E lying over . Let ^{}be the maximal ideal of ^{}. Then ^{}^{}T is a maximal ideal of T and ^{}T_{^{}}by [EP, (3.2.6), (3.2.7)]. By [L1, (IX.1.11)], β T is a maximal ideal if and only if β is a maximal ideal. Suppose β T is a maximal ideal. Then it is necessarily equal to ^{}, since ^{}can only contain one maximal ideal of T by [EP, (3.2.7)]. Then β contains 1 ^{}T_{^{}}, which is equal to ^{}, and hence, β ^{}, a contradiction. So, it follows that β is not maximal.

4.5 Lemma. Let be a valuation ring of F and let , σ be an algebra with involution. Let F^{h}, ^{h} be a Henselisation of F, . If _{F^{h}}, σ_{F^{h}} is hyperbolic then there exists a finite subextension F M F^{h} with the following properties:

(a) σ_{M} is hyperbolic;

(b) let be a valuation on F with valuation ring and let _{1} ^{h} _{M}, _{2}, …, _{m} be the different valuations on M extending . Then for i 2, …, m, there exist n_{i} such that for all x M

N_{M F} x _{1} x _{i 2}^{m} n_{i} _{i} x .

Proof. Since _{F^{h}}, σ_{F^{h}} is hyperbolic, Proposition 3.4 implies the existence of a finite subextension F L F^{h} such that _{L}, σ_{L} is hyperbolic.

Let N F be the Galois closure of L in F^{s}. Let ^{s} be a valuation on F^{s} extending . Let ^{h} ^{s} _{F^{h}} and ^{s} _{N}. Since N F is a Galois extension, all valuation rings of N lying over are conjugate to _{}by [EP, (3.2.15)]. Let H τ Gal N F τ _{}_{}and let M be the fixed field of H. Then M N F^{h} (for an explicit argument see the proof of [EP, (5.2.5)]). So, we have that L M and therefore, σ_{M} is hyperbolic. Let ρ_{1} id_{N}, ρ_{2}, …, ρ_{t} be a set of representatives for the right cosets of Gal N F H. By Galois theory, the restrictions of the ρ_{i} to M are exactly the different F embeddings of M in F^{s}. We have that ρ_{i} _{M} is a valuation on M with valuation ring ρ_{i}^{1} _{}M. It is possible

that ρ_{i}^{1}  _{}M ρ_{j}^{1}  _{}M for i j, but by the proof of [EP, (3.3.1)], it follows that
M ρ_{i}^{1}  _{}M if i 1. This means that _{M}  ρ_{i} _{M} if i 1. We have that
1  ^{h} _{M}  _{M} and _{2}, …, _{m}  ρ_{2} _{M}, …, ρ_{t} _{M} , and hence, _{i}  _{1} if i 1.

Let x M. Then N_{M F} x xρ_{2} x ρ_{t} x by definition.

N_{M F} x  N_{M F} x  xρ_{2} x ρ_{t} x  x  ρ_{2} x … ρ_{t} x .

By the reasoning above, it follows that there exist n_{1}, …, n_{m} such that

N_{M F} x _{1} x _{i 2}^{m} n_{i} _{i} x .

Let V_{ij} be as in Theorem 4.1 for some pair i, j , i j. Then _{ij} E Γ_{i} Δ_{ij}; x _{i} x mod Δ_{ij} defines a valuation on _{ij} on E with valuation ring V_{ij}. Let γ_{i} Γ_{i} as in Theorem 4.1 and let b E be such that _{i} b γ_{i}. Then θ_{ij} γ_{i} _{ij} b .

4.6 Lemma. Let _{1}, …, _{r} be different valuation rings of F and let R _{1} … _{r}. Suppose x F is such that x F^{2} _{i}^{}for i 1, …, r. Then x F^{2}R^{}.

Proof. Let _{1}, …, _{r} be valuations on F with respective valuation rings _{1}, …, _{r}. Denote their respective value groups by Γ_{1}, …, Γ_{r}. The hypothesis implies that there exists a tuple γ_{1}, …, γ_{r} Γ_{1} … Γ_{r} such that _{1} x 2γ_{1}, …, _{r} x 2γ_{r}. In the notations of Theorem 4.1, we have that θ_{ij} 2γ_{i} _{ij} x _{ji} x θ_{ji} 2γ_{j} . Since the θ_{ij} Γ_{i} Γ_{i} Δ_{ij} are group homomorphisms and since Γ_{i} Δ_{ij} is an ordered abelian group and therefore torsion–free, it follows that θ_{ij} γ_{i} θ_{ji} γ_{j} . Therefore, we can apply Theorem 4.1 to find an element a F such that _{1} a γ_{1}, …, _{r} a γ_{r}. Then _{i} a^{2}x^{1} 0 for i 1, …, r, which means that a^{2}x^{1} R^{}. This proves the claim.

4.7 Proposition. Suppose R has Krull dimension 1. Let , σ be an R algebra with involution. Let e F^{}. If for each maximal ideal of R such that e F^{2}R_{}^{}, we have that , σ becomes hyperbolic over R , then e G , σ _{F} R^{}.

Proof. Let _{1}, …, _{r} be the maximal ideals of R. For i 1, …, r, we denote R_{_{i}} by _{i}. Note that the residue field of _{i} is isomorphic to R _{i}. Let _{1}, …, _{r} be valuations on F with respective valuation rings _{1}, …, _{r}. Let V_{e} i 1, …, r e F^{2} _{i}^{}.

For each j V_{e}, we have that σ becomes hyperbolic over the residue field of _{j} by assumption, and hence, σ becomes hyperbolic over a Henselisation F_{j}^{h}, _{j}^{h} of F, _{j} by Corollary 1.13. Then there exists a finite subextension F M_{j} F_{j}^{h} with the properties as in Proposition 4.5. Let _{j1}, …, _{jn_{j}} be the set of valuations on M_{j} lying over some _{i}, starting with the ones lying over _{j}, that is, let ℓ_{j} 1, …, n_{j} be such that _{j1}, …, _{jℓ_{j}} are the valuations lying over j, and such that _{j1} _{j}^{h} _{M_{j}}. Since the _{i} have rank 1, they are independent, and hence the jk are pairwise independent by Lemma 4.3, since they all have rank 1 by [EP, (3.2.5)]. By Corollary 4.2, there exists an element x_{j} M_{j} such that _{j1} x_{j} _{j} e and _{jk} x_{j} 0 for

k 2, …, n_{j}. Then by Proposition 4.5, we have that _{j} N_{M_{j} F} x_{j}  _{j1} x_{j}  _{j} e and
i N_{M_{j} F} x_{j}  0 for i j. Let y _{j V_{e}} N_{M_{j} F} x_{j} . We have that _{j} y  _{j} e for j V_{e}

and _{j} y 0 for j V_{e}. It follows that ye F^{2} _{j}^{}for j 1, …, r. Then Lemma 4.6yields that ye F^{2}R^{}. Since σ becomes hyperbolic over M_{j}, invoking Proposition 1.21 (a) and (b) yields that N_{M_{j} F} x_{j} G , σ _{F} . Since G , σ _{F} is a group and F^{2} G , σ _{F} , it follows that e G , σ _{F} R^{}.

4.8 Theorem. Suppose R has finite Krull dimension. Let , σ be an R algebra with involution. Let e F^{}. The following are equivalent.

(i) e G , σ _{F} R^{}.

(ii) For each valuation ring of F containing R, such that e F^{2} ^{}, we have that , σ becomes hyperbolic over the residue field of .

Proof. That (i) implies (ii) follows from Proposition 1.19.

Let _{1}, …, _{r} be the maximal ideals of R. For i 1, …, r, we denote R_{_{i}} by _{i}. Let _{1}, …, _{r} be valuations on F with respective valuation rings _{1}, …, _{r}. Denote their respective value groups by Γ_{1}, …, Γ_{r}. We will show that (ii) implies (i) by induction on the sum of the ranks of the _{i}. We will denote this sum by m.

We have that m r, since _{1}, …, _{r} all have rank at least 1. If r m, then they all have rank 1, and hence R has Krull dimension 1. Proposition 4.7 then yields the statement.

Suppose now r m. Then at least one of the _{i} has rank bigger than 1. Without loss of generality, we may assume that the rank of _{1} is bigger than 1. Let V_{1} be the smallest overring of _{1} in F strictly bigger than _{1} (which exists since _{1} has finite rank). Then rank V_{1} rank O_{1} . Let R^{}V_{1} _{i 2}^{m} _{i} . Since rank V_{1} _{i 2}^{r} rank _{i} m and R V_{1}, by induction we have that e G , σ _{F} u, for some u R^{}. Put v u^{1}, then ev G , σ _{F} . Let g _{F}^{} be such that ev σ_{F} g g. We have that _{i} v 0 for i 2, …, r. If _{1} v 2Γ_{1} we are done, since then, by Lemma 4.6, v R^{}F^{2} and e G , σ _{F} R^{}follows. So suppose _{1} v 2Γ_{1}. Let F_{1}^{h}, _{1}^{h} be a Henselisation of F, _{1} . If _{1} e 2Γ_{1} then _{1} σ_{F} g g 2Γ_{1}. Then σ becomes hyperbolic over F_{1}^{h} by Proposition 1.19. If _{1} e 2Γ_{1} then σ is hyperbolic over the residue field of _{1} by assumption, and hence σ is hyperbolic over F_{h}^{1} by Corollary 1.13. In any case, there exists a finite field extension F M_{1} F_{1}^{h} with the properties as in Proposition 4.5. Let _{11}, …, _{1n} be the set of valuations on M_{1} lying over some _{i}, starting with the ones lying over _{1}, that is, let ℓ 1, …, n be such that _{11}, …, _{1ℓ} are the valuations lying over 1, and such that _{11} _{1}^{h} _{M_{1}}. For k 2, …, n, let V_{1k} be the smallest overring of _{_{11}} and _{_{1k}} and let Δ_{1k} be the corresponding convex subgroup of Γ_{1} Γ_{_{11}} such that the value group of V_{1k} is Γ_{1} Δ_{1k}. We have that V_{12}, …, V_{1n} are linearly ordered. Without loss of generality, we may assume that V_{12} _{k 2}^{n}V_{1k}. It follows that Δ_{12} _{k 2}^{n}Δ_{1k}. Let β be the maximal ideal of V_{12}. Then V_{12} _{_{11}} _{β} and hence β is a prime ideal of _{_{11}} that is not maximal. Let Ṽ V_{12} F. Then Ṽ is a valuation ring of F containing _{1}. We have that β Ṽ is the maximal ideal of Ṽ. Furthermore, it is clear that β Ṽ β _{1}. Hence, Ṽ _{1} _{β _{1}} and since β _{1} is a prime ideal of _{1} that is not maximal by Lemma 4.4, we have that _{1} Ṽ. Let Δ̃ be the convex subgroup of Γ_{1} corresponding to Ṽ, that is, such that the value group of Ṽ is isomorphic to Γ_{1} Δ̃. Then by [EP, (2.3.1)], we have that Δ_{12} Δ̃. By [EP, (3.2.5)], Ṽ and V_{12} have the same rank. This implies that Δ̃ Δ_{12}.

Since V_{1} is the smallest overring of _{1}, we have that V_{1} Ṽ. Let Δ̃ be the convex subgroup of Γ_{1} corresponding to V_{1}, then this means that Δ̃ Δ_{12}. Since v V_{1}^{}, we get that _{1} v Δ̃ Δ_{12}. Then we can apply Theorem 4.1 to find an element x_{1} M_{1} with _{1k} x_{1} _{1} v for k 1 and _{1k} x_{1} 0 otherwise. Proposition 4.5 then implies that _{i} N_{M_{1} F} x_{1} _{1} v for i 1 and _{i} N_{M_{1} F} x_{1} 0 otherwise. It follows that vN_{M_{1} F} x_{1} R^{}. Furthermore, since σ is hyperbolic over M_{1}, Proposition 1.21 (a) yields that x_{1} G _{M_{1}}, σ_{M_{1}} and hence N_{M_{1} F} x_{1} G , σ _{F} by Proposition 1.21 (b). Then evN_{M_{1} F} x_{1} G , σ _{F} , since ev G , σ _{F} and G , σ _{F} is a group. It follows that e G , σ _{F} R^{}, as desired.

4.9 Remarks.

(a) As in the Noetherian case, see Remark 3.6, the properties (i) and (ii) of Theorem 4.8 both hold for trivial reasons if Z is not a domain. The reasoning is the same.

(b) Note that, whereas it suffices to impose the hyperbolicity condition on the localisations of R at its maximal ideals in Proposition 4.7, we need to impose the condition on all valuation rings containing R in Theorem 4.8, for the induction proof to work.

The above theorem implies that if R has finite Krull dimension, algebras with involution that are rationally isomorphic, through a central isomorphism, are isomorphic.

4.10 Proposition. Suppose R has finite Krull dimension. Let , σ , , σ^{}be a pair of R algebras with involution and assume Z is a domain. If , σ _{F} _{F} , σ^{}_{F} through a Z _{F} isomorphism, then , σ _{R} , σ^{}through a Z -isomorphism.

Proof. By Proposition 1.17, there exists an element s ^{}such that σ s s and σ^{}Int s σ. By Proposition 1.16, there exist elements e F^{}and g _{F}^{}such that es σ_{F} g g. Combining Proposition 1.19, Theorem 4.8 and Corollary 1.22 yields that , σ _{R} , σ^{}, through a Z isomorphism.

Note that, in view of Propositions 1.14 and 1.15, the above already implies that if R has finite Krull dimension, then R has the RIII property. We can show more generally that R also has the RIII property if it does not necessarily have finite Krull dimension, by reducing to the case of finite Krull dimension.

4.11 Proposition. Let F_{0} be the prime field of F. Let , σ be an R algebra with involution. Then there exists a finitely generated field extension F̃ F_{0} and an R F̃ algebra with involution ˜ , σ̃ such that R F̃ is a semilocal Bézout domain of finite Krull dimension and , σ _{R} ˜ , σ̃ _{R}.

Proof. Since R is a semilocal Bézout domain, is free as an R module by Proposition 1.6. Let e_{1}, …, e_{n} be an R basis for . Then it is an F basis for _{F}. In fact, by [K, (I.1.3.5)], R is a direct summand of , and hence, we may assume that e_{1} 1. For i, j, k 1, …, n, let ε_{ijk} R be such that e_{i}e_{j} _{k 1}^{n} e_{k}ε_{ijk}. By [K, (III.5.1.2)], since is separable over R, there exists an idempotent x _{R} ^{op} such that under the map m _{R} ^{op} ; a b ab, we have that m x 1 and a 1 x 1 a x, for all a . We write x _{i 1}^{ℓ} x_{i} y_{i} and x_{i} _{k 1}^{n} e_{k}x_{ki} and y _{k 1}^{n} e_{k}y_{ki}. Furthermore, for i, j 1, …, n, there exist elements α_{ij} R

such that σ e_{i}  _{j 1}^{n} e_{j}α_{ij}. Suppose that Z  is a free separable quadratic R algebra. By
Proposition 1.1, Z  R z , for some z Z  satisfying z^{2} az b, with a, b R. If the

polynomial f x x^{2} ax b R x is reducible, let y be a root of f x in R. Let F̃ be the field obtained from F_{0} by adjoining the elements α_{ij}, ε_{ijk}, x_{ki}, y_{ki}, and if Z R by also adjoining a, b, and furthermore if f x is reducible, we also adjoin y. Then F̃ is finitely generated over F_{0}. Let R̃ R F̃and R_{0} R F_{0}. Since R is a semilocal Bézout domain, it is the intersection of finitely many valuation rings of F. Then R̃is the intersection of finitely many valuation rings of F̃ and is therefore a semilocal Bézout domain with fraction field F̃. Similarly, R_{0} is a semilocal Bézout domain with fraction field F_{0}.

We define an algebra ˜ over R̃as follows. We let ˜ be the free R̃ module with basis e_{1}, …, e_{n} . Then ˜ is multiplicatively closed in , since all ε_{ijk} R̃ by construction. Furthermore, σ restricts to an involution on ˜ , which we denote by σ̃. By construction, we have that x ˜R̃ ^{ ˜ op}, and hence ˜ is separable over R̃ by [K, (III.5.1.2)]. Furthermore, we have that Z ˜ Z ˜ . Suppose first Z R and let c R Ã. Then there exist c_{1}, …, c_{n} R̃such that c _{i 1}^{n} e_{i}c_{i}. Considering c as element of and using that e_{1} 1, it follows that c_{1} c and it follows that c R̃. Hence, Z ˜ R̃. Suppose Z R z . Since z ˜ by construction of F̃, it is clear that R̃ z Z ˜ . Suppose c dz ˜ , with c, d R. By construction of F̃, there exist z_{1}, …, z_{n} R̃such that z _{i 1}^{n} e_{i}z_{i}. Since z R, at least one of z_{2}, …, z_{n} is nonzero, say z_{2}. Then c dz c dz_{1} _{i 2}^{n} e_{i} dz_{i} , and it follows that c dz_{1}, dz_{2}, …, dz_{n} R̃. Since z_{2} R̃, it follows that d F̃, and then also c F̃. Hence, c dz F̃ z ˜ . Since ˜ is finite–dimensional over R̃, it is integral over R̃. So, F̃ z ˜ is contained in the integral closure of R̃in F̃ z . Suppose R̃ z is a domain. Since 2 and the discriminant of f x are units in R̃by construction of F̃ and R̃is integrally closed in F̃, a standard argument shows that R̃ z is the integral closure of R̃in F̃ z . Suppose R̃ z is not a domain. By construction of F̃, f x is then reducible in R̃ x (and separable), which implies that F̃ z F̃ F̃ and R̃ z R̃ R̃, and R̃ R̃is the integral closure of R̃in F̃ F̃. So, we get Z Ã R̃ z .

The above shows that ˜ , σ̃ is a R̃ algebra with involution and it is clear that , σ _{R} ˜ , σ̃ _{R}.

It remains to show that R̃ has finite Krull dimension. By Proposition 1.7, it suffices to show that R̃ is an intersection of valuation rings of F̃ of finite rank. If F_{0} is a finite field, then F_{0} only has one valuation ring, namely itself, and hence R_{0} F_{0}. If F_{0} then [EP, 2.1.4] yields that any non–trivial valuation ring of is discrete. Since R_{0} is the intersection of finitely many valuation rings of F_{0}, it therefore has finite Krull dimension . We have that R̃is the intersection of valuation rings of F̃ lying over valuation rings of F_{0}. Let ˜ be a valuation ring of F̃ and ˜ F_{0}. Then, by [EP, 3.4.4],

rank ˜ tr. deg F̃ F_{0} rank ^{}.

Since the right hand side is finite, it follows that ˜ has finite rank. Hence, R̃has finite Krull dimension.

4.12 Proposition. Let , σ , , σ^{}be a pair of R algebras with involution and assume Z is a domain. If , σ _{F} _{F} , σ^{}_{F} through a Z _{F} isomorphism, then , σ _{R} , σ^{}through a Z -isomorphism.

Proof. By Proposition 1.17, there exists an element s ^{}such that σ s s and σ^{}Int s σ. By Proposition 1.16, the isomorphism , σ _{F} _{F} , σ^{}_{F} yields the existence of elements e F^{}and g _{F}^{}such that es σ_{F} g g. Let F_{0} be the prime field of F and let F̃, ˜ , σ̃ be as in Proposition 4.11. Let e_{1}, …, e_{n} be an R basis for . There exist s_{1}, …, s_{n} R and g_{1}, …, g_{n} F such that s _{i 1}^{n} e_{i}s_{i} and g _{i 1}^{n} e_{i}g_{i}. We adjoin s_{1}, …, s_{n}, g_{1}, …, g_{n} to F̃. This field we obtain is still a finitely generated field extension of F_{0}. By abuse of notation, we denote it again by F̃. Let R̃ R F̃. Then R̃is a semilocal Bézout domain with fraction field F̃.

By construction of F̃, the equation es σ_{F} g g is already defined in ˜F̃. Furthermore, σ^{} restricts to an involution on ˜ , which we denote by σ̃^{}. Then, by Proposition 1.16, we have that ˜ , σ̃ F̃ F̃ ˜ , σ̃^{}F̃, through a Z _{ ˜ F} isomorphism. Since R̃has finite Krull dimension by Proposition 4.11, Proposition 4.10yields that ˜ , σ̃ R̃ ˜ , σ̃^{}, through a Z ˜ isomorphism, and scalar extension to R then yields that , σ and , σ^{}are Z isomorphic.

4.13 Theorem. R has the RIII property.

Proof. Let , σ , ^{}, σ^{}be a pair of R algebras with involution. Suppose , σ _{F} _{F} , σ^{}_{F}. By Proposition 1.14 we may assume that ^{}and that , σ _{F} _{F} ^{}, σ^{}_{F} via a Z _{F} isomorphism. Furthermore, we may assume Z is a domain by Proposition 1.15. The theorem then follows from Proposition 4.12

4.14 Corollary. Let Δ, θ be an R algebra with involution without zero divisors. Let V, h and V^{}, h^{}be ε hermitian spaces over Δ, θ . Suppose there exists e F^{}such that h_{F}^{}_{F} eh_{F}. Then there exists u R^{}such that h^{}_{R} uh.

Proof. Since by assumption, h_{F}^{}and h_{F} are similar, it follows from Propostion 1.18, applied in the case R F, that Ad h_{F} _{F} Ad h_{F}^{}through a Z Δ_{F} isomorphism. It follows from Proposition 4.12 that Ad h _{R} Ad h^{}, through a Z Δ isomorphism. Applying the other implication given by Proposition 1.18, we obtain that there exists u R^{}such that h^{}_{R} uh.

4.15 Remark. This corollary states that “rational similarity implies similarity” for ε hermitian spaces over Δ, θ . As a special case of the above corollary, namely taking Δ R and θ id, we obtain that rationally similar non–singular symmetric or skew–symmetric bilinear spaces are similar.

Acknowledgements We would like to thank M. Ojanguren for many pleasant discussions, and K. J. Becher for suggesting how to reduce the case of general semilocal Bézout domains to the case of finite rank.

References

[B] Beke S., Specialisation and good reduction for algebras with involution, preprint 2012.

[EP] Engler A.J., Prestel A., Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.

[FrJ] Fried M., Jarden M., Field arithmetic, Third edition. Revised by Jarden. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 11. Springer-Verlag, Berlin, 2008.

[FT] Fröhlich, A., Taylor, M. J., Algebraic number theory. Cambridge Studies in Advanced Mathematics, 27. Cambridge University Press, Cambridge, 1993.

[H] Harder G., Eine Bemerkung zum schwachen Approximationssatz, Arch. Math. (Basel) 19 (1968) 465–471.

[Ka] Kaplansky I., Elementary Divisors and Modules, Trans. Amer. Math. Soc. 66 (1949), 464-491.

[K] Knus M.–A., Quadratic and hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, 294. Springer-Verlag, Berlin, 1991.

[KMRT] Knus M.–A., Merkurjev A.S., Rost M., Tignol J.–P. The book of involutions, American Mathematical Society Colloquium Publications, 44. American Mathematical Society, Providence, RI, 1998.

[L1] Lang S., Algebra, Second edition. Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002.

[L2] Lang S., Algebraic number theory. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont. 1970.

[N] Nisnevich Y., Rationally trivial principal homogeneous spaces, purity and arithmetic of reductive group schemes over extensions of two-dimensional regular local rings. C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 10, 651–655.

[Pa] Panin I.A., Purity for multipliers, Algebra and number theory, Hindustan Book Agency (2005), Delhi, 66–89.

[R] Ribenboim P., Le théorème d’approximation pour les valuations de Krull., Math. Z. 68 (1957), 1–18.

[T] Tignol J.–P., A Cassels Pfister theorem for involutions on central simple algebras, J. Algebra 181 (1996), no. 3, 857–875.