Multiplicative Spectra for Rings and Automorphism Groups of Modules and Hermitian Forms

Anthony Bak

Department of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany e-mail: bak@@ mathematik.uni-bielefeld.de

and

Guoping Tang

Dept. of Applied Math. Northwestern Polytechnical University 710072 Xian, P.R.China e-mail:tanggp@@nwpu.edu.cn

Abstract This article introduces an analog of the maximal ideal spectrum of a commutative ring, for arbitrary associative rings with identity element. A given ring can have many spectra. Using all of the spectra, we define the local stable rank of a ring and the local λ-stable rank of a ring with λ-involution. Upper bounds for both notions of local rank are established in terms of the Gabriel-Rentschler J. Krull dimension at each quotient of the ring modulo a primitive ideal. This upper bound shows that the local rank of a commutative or module finite ring is 1.

Let R be a ring with λ-involution, of local λ-stable rank m. The main result is that the elementary subgroup of the automorphism group of an arbitrary λ-Hermitian form of hyperbolic Witt index ≥ m + 2 is normal. In particular, the elementary subgroup EH_{2n}(R, a_{1}, ⋯, a_{r}) of the general Hermitian group GH_{2n}(R, a_{1}, ⋯, a_{r}) is normal whenever n − r ≥ m + 2. As a corollary, the analogous linear result is deduced for the automorphism group of an arbitrary module of unimodular index ≥ m + 2 over a ring of local stable rank m. This extends theorems of Suslin and Tulenbaev for the general linear group of free modules of finite rank over commutative and module finite rings.

1 Introduction

We explain first the ring theoretic concepts required in formulating our results.

Throughout the article, R denotes an associative ring with identity element or such a ring with additional structures.

A λ-involution on R is an additive, bijective map R → R, a ↦ ā, such that \overline{ab} = b̄ā and λ̄¯āλ = a for all a, b ∈ R. Setting a = 1, it follows that λ̄ = λ^{−1}. If λ ∈ center (A) then ¯ ā = a and the map a ↦ ā is an involution in the usual sense.

The concept of a multiplicative spectrum for associative rings is designed to provide an environment for carrying out local-global arguments that had been restricted to either commutative rings or rings which are module finite over their centers where localization can be carried out effectively with respect to commutative multiplicative sets. The problem for general rings is that one cannot carry out localization effectively by restricting to commutative multiplicative sets and one cannot form a ring of fractions at an arbitrary multiplicative set, but only at ones that are denominator sets (cf. [S] II Prop. 1.4). This means that the natural interplay in the commutative situation between maximal ideals 𝔪 of R and multiplicative sets of R, consisting of all elements which become a unit in R/𝔪, is lost in the general situation, because such multiplicative sets in general are not denominator sets. The idea of a multiplicative spectrum for an associative ring is to replace the set of maximal ideals of a commutative ring by a set of denominator sets, which satisfies two simple properties supporting local-global procedures.

Recall that a right denominator set in R is a multiplicative set S ⫅ R satisfying the following properties.

D1. (Ore condition) If s ∈ S and a ∈ R then sb = at for some t ∈ S and b ∈ R.

D2. If s ∈ S and a ∈ R such that sa = 0 then at = 0 for some t ∈ S.

The ring of right S-fractions (cf. [S] II §1) of R will be denoted by R[S^{−1}]. The notion of a left denominator set is defined similarly. A right multiplicative spectrum on R is a set ∑ of right denominator sets satisfying the following properties.

MS1. (Local condition) For each S ∈ ∑, the canonical homomorphism R → R[S^{−1}]/J(0) is surjective where J(0) denotes the Jacobson radical of R[S^{−1}].

MS2. (Global condition) Any subset of R which meets each S ∈ ∑ nontrivially generates a right ideal of R, which is all of R.

The notion of a left multiplicative spectrum on R is defined similarly. A multiplicative spectrum on R is a right multiplicative spectrum which is simultaneously a left multiplicative spectrum. The set of all multiplicative spectra on R will be denoted by MS(R). A λ-multiplicative spectrum on a ring R with λ-involution is a right multplicative spectrum ∑ such that each S ∈ ∑ is involution invariant. Because f the last condition, it follows that a λ-multiplicative spectrum is a multiplicative spectrum. The set of all λ-multiplicative spectra on R will be denoted by λMS(R).

Next we several concepts of stable rank for rings and rings with λ-involution and then use them to define their local versions.

If T ⫅ R is a subset, let J_{ℓ}(T) denote the intersection of R and all maximal left ideals of R, which contain T. Let R^{n} denote the direct sum of n copies of R. If (a_{1}, ⋯, a_{n}) ∈ R^{n}, let J_{ℓ}(a_{1}, ⋯, a_{n}) = J_{ℓ}{a_{1}, ⋯, a_{n}}. One says that a vector (a_{1}, ⋯, a_{n}) in the right R-module R^{n} can be shortened, if there are elements x_{1}, ⋯, x_{n−1} ∈ R such that J_{ℓ}(a_{1} + x_{1}a_{n}, ⋯, a_{n−1} + x_{n−1}a_{n}) = J_{ℓ}(a_{1}, ⋯, a_{n}). The right absolute stable rank asr(R) [KMV] is the smallest natural number m such that every vector of length m + 1 can be shortened. If no such number exists then by definition asr(R) = ∞. A vector (a_{1}, ⋯, a_{n}) ∈ R^{n} is called right unimodular, if J_{ℓ}(a_{1}, ⋯, a_{n}) = R, i.e. the left ideal of R generated by {a_{1}, ⋯, a_{r}} is all of R. The right stable rank sr(R) is the smallest natural number m such that every right unimodular vector of length m + 1 can be shortened. If no such number exists then by definition sr(R) = ∞. By [MR] 11.3.4, the right stable rank and left stable rank of R are equal. Obviously sr(R) ≦ asr(R).

Suppose R has a λ-involution. In order to have the correct local notion of stable rank for rings with λ-involution, we need to use form parameters in defining the absolute notion. Let min^{λ}(R) = {a−āλ|a ∈ R} and max^{λ}(R) = {a ∈ R|a = −āλ}. A λ-form parameter is an additive subgroup Λ of R such that

FP1. min^{λ}(R) ⫅ Λ ⫅ max^{λ}(R),

FP2. āΛa ⫅ Λ for all a ∈ R.

If α = (a_{ij}) is an m × n matrix with coefficients in R, let ᾱ = (a′_{ℓk}) denote the n × m matrix such that a′_{ℓk} = ā_{ℓk}. the matrix ᾱ is called the conjugate transpose of α. The operation of conjugate transpose defines a λ-involution on the ring 𝕄_{n}(R) of all n × n matrices with coefficients in R. Let 𝕄_{n}(Λ̄) = {γ ∈ 𝕄_{n}(R) | γ = −λ̄γ̄, diagonal coefficients of γ lie in Λ̄}. The right Λ-stable rank Λsr(R) [BT] ist the smallest natural number m such that sr(R) ≦ m and such that given a right unimodular vector (a_{1}, ⋯, a_{m+1}, b_{1}, ⋯, b_{m+1}) ∈ R^{2(m+1)}, there is a 2(m + 1) × 2(m + 1) matrix γ ∈ 𝕄_{2(m+1)}(Λ̄) with the property that

( \begin{matrix}I & 0 \\ γ & I \end{matrix} ) ^{t}(a_{1}, ⋯, a_{m+1}, b_{1}, ⋯, b_{m+1}) = ^{t}(a′_{1}, ⋯, a′_{m+1}, b′_{1}, ⋯, b′_{m+1})

with (b′_{1}, ⋯, b′_{m+1}) right unimodular. If no such number exists then by definition Λsr(R) = ∞. It is not difficult to show that the right and left Λ-stable rank are equal. Obviously sr(R) ≦ Λsr(R) and by [BT] (3.4), Λsr(R) ≦ asr(R).

We define local versions of the notions of rank above as follows. If ∑ ∈ MS(R) or λMS(R) and S ∈ ∑, let Jac(R[S^{−1}]) denote the Jacobson radical of R[S^{−1}] and let J(R, S) denote its preimage in R. By MS1, R/J(R, S) = R[S^{−1}]/Jac(R[S^{−1}]). Let R(S) = R/J(R, S). If R has a λ-involution, let Λ(S) denote the image of max^{λ}(R) in R(S). Λ(S) is a λ-form parameter on R(S), but is not true in general that Λ(R) = max^{λ}(R(S)). This is the reason why we had to introduce form parameters in defining stable rank for rings with λ-involution. Define the

right local absolute stable rank lasr(R) = inf ; ∑∈MS(R)(sup ; S∈∑(asr(R(S)))),

local stable rank lsr(R) = inf ; ∑∈MS(R)(sup ; S∈∑(sr(R(S)))),

and for a ring R with λ-involution, define

local λ- stable rank lλsr(R) = inf ; ∑∈λMS(R)(sup ; S∈∑(Λ(S)sr(R(S)))).

Since sr(R) and Λsr(R) are left-right invariant and since multiplicative spectra are leftright invariant, it follows that lsr(R) and λlsr(R) are left-right invariant. From remarks in the penultimate paragraph, it is obvious that

(1.1) lsr(r) ≦ lasr(R), and

(1.2) for a given λ-multiplicative spectrum ∑, sup ; S∈∑(Λ(S)sr(R(S)) ≦ sup ; S∈∑(asr(R(S)).

But it does not follow from (1.2) that lλsr(R) ≦ lasr(R), because the set MS(R)\λMS(R) is apriori not empty.

Next we state upper bounds on the local ranks above.

Recall that a ring is called right Goldie, if it contains no infinite direct sum of nonzero right ideals and if right annihilators satisfy the ascending chain condition. Clearly right Noetherian rings are right Goldie. A 2-sided ideal 𝔮 ⫅ R is called a right Artin-Rees ideal, if the graded ring R⊕𝔮⊕𝔮^{2}⊕⋯ is right graded Noetherian, i.e. every homogeneous right ideal of R ⊕ 𝔮 ⊕ 𝔮^{2} ⊕ ⋯ is finitely generated or equivalently homogenous right ideals satisfy the ascending chain condition. Obviously the condition forces R to be right Noetherian. If T ⫅ R is a subset, let J(T) denote the intersection of R and all maximal right ideals of R which contain T. A right ideal 𝔮 in R is called a J-ideal if 𝔮 = J(𝔮). A ring is called right J- Noetherian, if its right J-ideals satisfy the ascending chain condition. Let JK(R) denote the J Krull-dimension of R of Gabriel and Rentschler [GR]. Finally recall that a 2-sided ideal 𝔮 ⫅ R is called right primitive, if there is a right faithful simple R/𝔮-module. Every 2-sided maximal ideal is primitive and every primitive ideal is prime. Let prtvspec(R) denote the set of all (right) primitive ideals in R.

Theorem 1.3 For each 𝔪 ∈ prtvspec(R), let 𝔮(𝔪) ⫅ 𝔪 be a 2-sided ideal. Suppose that each quotient R/𝔮(𝔪) is right J-Noetherian and right Goldie (e.g. each R/𝔮(𝔪) is right Noetherian). Suppose that for each 𝔪, there is a right denominator set S_{𝔪} such that for each s ∈ S_{𝔪}, s is a unit in R/𝔮(𝔪) and s + 𝔮(𝔪) ⫅ S_{𝔪} (e.g. S_{𝔪} = 1 + 𝔮(𝔪) and R is a direct limit lim_{⟶} R_{i} of subrings R_{i} such that each ideal 𝔮(𝔪)∩R_{i} is right Artin-Rees in R_{i}). Then lsr(R) ≦ lasr(R) ≦ sup_{𝔪∈prtvspec(R)}JK(R/J(𝔮(𝔪))) + 1 and if R has a λ-involution

then lλsr(R) ≦ sup_{𝔪∈prtvspec(R)}JK(R/J(𝔮(𝔪))) + 1.

Corollary 1.4 Suppose that for each 𝔪 ∈ prtvspec(R), R/𝔪 is right Artinian and R is a direct limit lim_{⟶} R_{i} of subrings R_{i} such that each intersection 𝔪 ∩ R_{i} is a right Artin-Rees ideal in R_{i} (e.g. R is commutative or module finite over its center). Then lsr(R) = lasr(R) = 1 and if R has a λ-involution then lλsr(R) = 1.

We explain next the Hermitian form concepts required in formulating our results.

Let V be a right R-module and let h : V × V → R be a λ-Hermitian on V. By definition, h is a biadditive map such that for all v, w ∈ V and a, b ∈ R, h(va, wb) = āh(v, w)b and h(v, w) = \overline{h(w, v)}λ. The pair (V, h) is called a λ-Hermitian module. If λ ∈ center (R), we get the more familiar identity h(v, w) = λ\overline{h(w, v)}. A morphism f : (V, h) → (V′, h′) of λ-Hermitian modules is by definition an R-linear map f : V → V′ which preserves the λ-Hermitian forms. The automorphism group of (V, h) is denoted by GH(V, h).

To economize notation, we shall frequently abbreviate

h(u, v) by (u, v).

An element v ∈ V is called isotropic if (v, v) = 0. An ordered pair e, e_{−} of elements of V is called a metabolic pair if e_{−} is isotropic and (e, e_{−}) = λ. Thus (e_{−}, e) = 1. An ordered pair e, e_{−} of elements of V is called a hyperbolic pair if it is metabolic and e is isotropic. (V, h) is called nonsingular, if the map h : V → Hom_{A}(V, R), v ↦ h(v,_{−}), is bijective. It is easy to check that a metabolic pair e, e_{−} generates a free submodule eR ⊕ e_{−}R of rank 2 such that (eR ⊕ e_{−}R, h|_{eR⊕e_{−}R}) is nonsingular. If e, e_{−} is metabolic (resp. hyperbolic) pair then (eR ⊕ e_{−}R, h|_{eR⊕e_{−}R}) is called metabolic plane (resp. hyperbolic plane). Since metabolic planes are nonsingular, it follows easily that for any metabolic pair e, e_{−}, V decomposes as an orthogonal sum (eR⊕e_{−}R) ⊥ (eR⊕e_{−}R)^{⊥} where (eR ⊕ e_{−}R)^{⊥} = {v ∈ V |(v, e) = (v, e_{−}) = 0}. Metabolic pairs e, e_{−} and f, f_{−} are called complementary if (e, f) = (e_{−}, f) = (e, f_{−}) = (e_{−}, f_{−}) = 0. The hyperbolic Witt index ind(V, h) is the largest nonnegative integer or infinity such that for any nonnegative integer n ≦ ind(V, h), V contains n mutually complementary hyperbolic pairs. From the discussion above, it follows that n ≤ ind(V, h) ⇔ V has an orthogonal decomposition V = H_{1} ⊥ ⋯ ⊥ H_{n} ⊥ V′ where each H_{i} (1 ≦ i ≦ n) is a hyperbolic plane. (The Witt index i of (V, h) is the largest nonnegative integer or infinity such that for any nonnegative integer n ≦ i, V contains n mutually complementary metabolic pairs).

An element v ∈ V is called even if (v, v) = a + āλ for some a ∈ A. It is obvious that the set of all even elements in V forms an R-submodule. It will be denoted by V_{even}. If v is isotropic then clearly v ∈ V_{even}. (V, h) is called even, if V = V_{even}. An ordered pair u, v ∈ V is called a transvection pair if u is isotropic, v is even, and (u, v) = 0. A transvection pair u, v and an element s ∈ A such that s + s̄λ = (v, v) defines an automorphism τ(u, v, s) of (V, h) called an Eichler transvection as follows:

τ(u, v, s)(x) = x + v(u, x) − uλ̄(v, x) − uλ̄s(u, x).

Let GH(V, H) denote the automorphism group of (V, h). Let T (V, h) denote the subgroup of GH(V, h) generated by all transvections. If τ(u, v, s) is a transvection and σ ∈ GH(V, h) then σu, σv is a transvection pair such that s + s̄λ = (σv, σv). Thus τ(σu, σv, s) is a transvection and one checks straightforward that στ(u, v, s)σ^{−1} = τ(σu, σv, s). thus T (V, h) is a normal subgroup of GH(V, h).

For a hyperbolic pair e, e_{−} ∈ V, let T_{⟨e,e_{−}⟩}(V, h) denote the subgroup of T (V, h) generated by all transvections τ(e, v, s) and τ(e_{−}, w, t).

Theorem 1.5 Suppose R is a ring with λ-involution, of local λ-stable rank m. Let (V, h) be a λ-Hermitian module and let e, e_{−} and f, f_{−} be hyperbolic pairs. If ind(V, h) ≧ m + 2 then T_{⟨e,e_{−}⟩}(V, h) = T_{⟨f,f_{−}⟩}(V, h).

Corollary 1.6 Under the conditions of (1.5), T_{⟨e,e_{−}⟩}(V, h) is normal in GH(V, h).

Proof Let τ(e, v, s) and τ(e_{−}, w, t) be typical generators of T_{⟨e,e_{−}⟩}(V, h) and let σ ∈ GH(V, h). Then σe, σe_{−} is a hyperbolic pair and the elements στ(e, v, s)σ^{−1} = τ(σe, σv, s) and στ(e_{−}, w, t)σ^{−1} = τ(σe_{−}, σw, t) ∈ T_{⟨σe,σe_{−}⟩}(V, h). But by Theorem 1.5, T_{⟨σe,σe_{−}⟩}(V, h) = T_{⟨e,e_{−}⟩}(V, h).□

Suppose that V has a basis e_{1}, ⋯, e_{n}, e_{−1}, ⋯, e_{−n} consisting of mutually complementary metabolic pairs e_{i}, e_{−i}. Let 1 ≦ r ≦ n and suppose that (e_{i}, e_{i}) = 0 for all i > r. thus e_{i}, e_{−i} is a hyperbolic pair for i > r. Thus ind(V, h) ≥ n − r. Let a_{i} = (e_{i}, e_{−i}). Obviously a_{i} = 0 for i > r. Using the basis above for V, we can associate to each element σ ∈ GH(V, h), a 2n × 2n invertible matrix, and identify GH(V, h) with the subgroup of the general linear group GL_{2n}(R), consisting precisely of these matrices. This is called the general Hermitian group [B] p. 40 - 42 and is denoted by GH_{2n}(R, a_{1}, ⋯, a_{r}). We allow r = 0 and interpret this to mean that a_{1} = ⋯ = a_{n} = 0. The Hermitian module (V, h) with the basis e_{1}, ⋯, e_{n}, e_{−1}, ⋯, e_{−n} is by definition the underlying module [B] §1 C and (2.10) - (2.13) of GH_{2n}(R, a_{1}, ⋯, a_{r}). It will be denoted by 𝕄_{2n}(R, a_{1}, ⋯, a_{r}). The elementary subgroup of GH_{2n}(R, a_{1}, ⋯, a_{r}) is denoted by EH_{2n}(R, a_{1}, ⋯, a_{r}) and is called the Hermitian elementary group [T] §4 and §8. Its definition is recalled in §2.

Theorem 1.7 If r < i ≤ n then T_{⟨e_{i},e_{−i}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})) = EH_{2n}(R, a_{1}, ⋯, a_{r}).

We deduce from (1.6) and (1.7) the following normality result.

Theorem 1.8 If R is a ring with λ-involution, of local λ-stable rank m, then EH_{2n} (R, a_{1}, ⋯, a_{n}) is normal in GH_{2n}(R, a_{1}, ⋯, a_{r}) whenever n − r ≥ m + 2.

Proof By Theorem 1.7, EH_{2n}(R, a_{1}, ⋯, a_{r}) = T_{⟨e_{i},e_{−i}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})). Since n − r ≥ sup(m + 1, 3), ind(𝕄_{2n}(R, a_{1}, ⋯, a_{r})) ≥ m + 2. Thus by Corollary 1.6, T_{⟨e_{i},e_{−i}⟩}(𝕄_{2n} (R, a_{1}, ⋯, a_{r})) is normal in GH(𝕄_{2n}(R, a_{1}, ⋯, a_{r})) = GH_{2n}(R, a_{1}, ⋯, a_{r}).□

We explain now the module theoretic concepts required to state the linear analogs of the results above.

Let V be a right R-module. An element v ∈ V is called unimodular, if there is an element f ∈ Hom_{R}(V, R) such that f(v) = 1. Clearly v is unimodular ⇔ the submodule vR of V is a free direct summand isomorphic to R. The unimodular index ind(V) of V is the largest nonnegative integer n such that V contains a free direct summand isomorphic to R^{n}. If no such integer exists then by definition ind(V) = ∞. (It is possible that ind(R) = ∞.)

Let R^{op} denote the opposite ring of R; by definition, the additive group of R^{op} is that of R and multiplication is defined by a ∘ b = ba. Let V be a right R-module. Then V ⊗_{ℤ} Hom_{R}(V, R) is a right R ⊗_{ℤ} R^{op}-module such that (v ⊗ f)(a ⊗ b) = va ⊗ bf and the pairing V × Hom_{r}(V, R) → R, (v, f) ↦ f(v), defines an R ⊗_{ℤ} R^{op}-linear map V ⊗_{ℤ} Hom_{R}(V, R) → R. A pair (v, f) ∈ V × Hom_{r}(V, R) is called isotropic if f(v) = 0. Pairs (v, f) and (v′, f′) are called complementary, if f(v′) = f′(v) = 0. An ordered pair ((v, f), (v′, f′)) of complementary pairs such that (v, f) is isotropic is called a transvection pair. A transvection pair (u, f), (v, g) defines an automorphism τ((u, f), (v, g)) of V called a transvection as follows:

τ((u, f), (v, g))(x) = x + vf(x) − ug(x) − ug(v)f(x).

Let GL(V) denote the automorphism group of V. Let T (V) denote the subgroup of GL(V) generated by all transvections. If τ((u, f), (v, g)) is a transvection and σ ∈ GL(V) then ((σu, σ^{∗}f), (σv, σ^{∗}g)) is a transvection pair where σ^{∗} denotes the dual σ^{∗} : Hom_{R}(V, R) → Hom_{R}(V, R), h ↦ hσ, of σ. Thus τ((σu, σ^{∗}f), (σv, σ^{∗}g)) is a transvection and one checks straightforward that στ((u, f), (v, g))σ^{−1} = τ((σu, σ^{∗}f), (σv, σ^{∗}g)). Thus T (V) is a normal subgroup of GL(V). For a unimodular element e ∈ V, let T_{e}(V) denote the subgroup of T (V) generated by all transvections of the form τ((e, f), (v, g)).

Theorem 1.9 Suppose R has local stable rank m. Let V be a right R-module and let e and e′ be unimodular elements in V. If ind(V) ≧ m + 2 then T_{e}(V) = T_{e′}(V).

Corollary 1.10 Under the conditions of (1.9), T_{e}(V) is normal in GL(V).

Proof Same as that of (1.6) □

Suppose that V has a basis e_{1}, ⋯, e_{n}. Thus ind(V) ≥ n. Using the basis above of V, we can identify GL(V) with the general linear group GL_{n}(R) of all invertible n × n matrices. Let E_{n}(R) denote the elementary subgroup of GL_{n}(R). Let F_{n}(R) denote the free R-module with basis e_{1}, ⋯, e_{n}.

Theorem 1.11 T_{e_{i}}(F_{n}(R)) = E_{n}(R) for any 1 ≦ i ≦ n.

We deduce from (1.10) and (1.11) the following result.

Theorem 1.12 If R is a ring of local stable rank m then E_{n}(R) is normal in GL_{n}(R) whenever n ≧ m + 2.

Proof Same as that of 1.8. Of course, the hyperbolic Witt index is replaced by the unimodular index. □

The theorem above extends Suslin’s pioneering result [Su] that E_{n}(R) is normal in GL_{n}(R) when R is commutative and n ≥ 3 and Tulenbaev’s extension [Tul] of this result to rings that are module finite over their center.

The rest of the paper is organized as follows. §2 begins by recalling basic identities for Eichler transvections. Then the definition of the Hermitian elementary group EH_{2n} (A, a_{1}, ⋯, a_{r}) is given. It is generated by Hermitian elementary matrices and their definition is formulated slightly differently than in [T] §8 because we are not assuming that λ is central. It is shown how to pass back and forth between Hermitian elementary matrices and Eichler transvections. Theorem 1.7 is proved. §3 proves the main result Theorem 1.5. §4 deduces our linear results, namely Theorem 1.9 and Theorem 1.11, as a special case of their Hermitian counterparts, namely Theorem 1.9 and Theorem 1.11, respectively. §5 establishes the upper bounds for lsr, lasr, and lλsr announced in Theorem 1.3 and Corollary 1.4.

2 Eichler Transvections and Hermitian elementary matrices

The goal of the section is to prove Theorem 1.7. This is done following Lemma 2.9. To prepare for the proof, we write each transvection of the form τ(e_{±i}, v, s) as a product of Hermitian elementary matrices and conversely, we write each Hermitian elementary matrix as a product of transvections of this kind. This done, the proof of Theorem 1.7 is not difficult. The section closes by studying certain quadratic subgroups of the Hermitian elementary group and how they act on unimodular vectors. This result will be applied in the next section.

Let R be an associative ring with identity element 1 and λ-involution a ↦ ā. Let (V, h) be a λ-Hermitian module.

Lemma 2.1 Transvections on (V, h) satisfy the following identities.

T1. στ(u, v, s)σ^{−1} = τ(σu, σv, s), ∀ σ ∈ GH(V, h).

T2. τ(ua, v, s) = τ(u, vā, ¯ āsā), ∀ a ∈ R.

T3. τ(u, v, s)τ(u, w, t) = τ(u, v + w, s + t + (v, w)).

T4. τ(u + u′, v, s) = τ(u, v, s)τ(u′, v + uλ̄s̄λ, s), providing (u, u′) = (u, v) = (u′, v) = 0.

T5. τ(u, v, 0) = τ(vλ, −u, 0).

T6. τ(ua, u, 0) = τ(u, 0, λaλ̄ − λā).

The identities above are verified by straightforward computation.

Lemma 2.2 Let e_{1}, e_{−1} and e_{2}, e_{−2} be complementary hyperbolic pairs in V. Then T_{⟨e_{1},e_{−1}⟩}(V, h) = T_{⟨e_{2},e_{−2}⟩}(V, h).

Proof Let τ(e_{1}, v, s) and τ(e_{−1}, w, t) be typical generators of T_{⟨e_{1},e_{−1}⟩}(V, h). Let σ = τ(e_{−2}, e_{−1}(−λ̄), 0)τ(e_{2}, e_{−1}(−1), 0). Then σ(e_{1}) = e_{2} and by (2.1) T1, στ(e_{1}, v, s)σ^{−1} = τ(e_{2}, σv, s). Thus τ(e_{1}, v, s) = σ^{−1}τ(e_{2}, σv, s)σ ∈ T_{⟨e_{2},e_{−2}⟩}(V, h). Similarly τ(e_{−1}, w, t) ∈ T_{⟨e_{2},e_{−2}⟩}(V, h). Thus T_{⟨e_{1},e_{−1}⟩}(V, h) ⫅ T_{⟨e_{2},e_{−2}⟩}(V, h).

Interchanging in the argument above e_{1} with e_{2} and e_{−1} with e_{−2}, we obtain the reverse inclusion. Thus T_{⟨e_{1},e_{−1}⟩}(V, h) = T_{⟨e_{2},e_{−2}⟩}(V, h).□

If α denotes a k × ℓ matrix (α_{ij}) whose (i, j)’th coefficient α_{ij} ∈ A, let ᾱ denote the ℓ × k matrix (α′_{ji}) where α′_{ji} = ᾱ_{ij}. ᾱ is called the conjugate transpose of α.

Suppose for the rest of the section that (V, h) = 𝕄_{2n}(R, a_{1}, ⋯, a_{r}). Recall that 𝕄_{2n}(R, a_{1}, ⋯, a_{r}) is equipped with a basis e_{1}, ⋯, e_{n}, e_{−1}, ⋯, e_{−n} such that the pairs e_{i}, e_{−i} (1 ≦ i ≦ n) are mutually complementary metabolic pairs with the property that (e_{i}, e_{i}) = a_{i} (1 ≦ i ≦ r) and (e_{i}, e_{i}) = 0 (r < i ≦ n). Let

A_{1} = r × r diagonal matrix ( \begin{matrix}a_{1} \\ & . & . & . \\ & & & & a_{r} \end{matrix} ) ,

A = n × n diagonal matrix ( \begin{matrix}a_{1} \\ & . & . & . \\ & & & & a_{r} \\ & & & & & 0 \\ & & & & & & . & . & . \\ & & & & & & & & & 0 \end{matrix} ) ,

I = an identity matrix .

Let e_{n+i} = e_{−i} (1 ≦ i ≦ n). Then e_{1}, ⋯, e_{2n} is the basis above of 𝕄_{2n}(R, a_{1}, ⋯, a_{r}). In this basis, the matrix of h is by definition the 2n × 2n matrix

((e_{i}, e_{j})) = ( \begin{matrix}A & λI \\ I & 0 \end{matrix} )

and

GH_{2n}(R, a_{1}, …, a_{r}) = \{σ ∈ GL_{2n}(R) | σ̄ ( \begin{matrix}A & λI \\ I & 0 \end{matrix} ) σ = ( \begin{matrix}A & λI \\ I & 0 \end{matrix} )\} .

A typical element of this group is denoted by a 2n × 2n matrix

( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ,

where α, β, γ and δ are n × n block matrices.

Let min_{λ}(R) = {a + āλ|a ∈ R}. Let

C = {^{t}(x_{1}, …, x_{r}) ∈ ^{t}(R^{r}) | ∑_{i=1}^{r} x̄_{i}a_{i}x_{i} ∈ min_{λ}(R)}.

In order to deal effectively with technical difficulties caused by the elements a_{1}, …, a_{r}, we shall partition a typical matrix

( \begin{matrix}α & β \\ γ & δ \end{matrix} )

of GH_{2n}(R, a_{1}, …, a_{r}) into the form

(2.3)  ( \begin{matrix}α_{11} & α_{12} & β_{11} & β_{12} \\ α_{21} & α_{22} & β_{21} & β_{22} \\ γ_{11} & γ_{12} & δ_{11} & δ_{12} \\ γ_{21} & γ_{22} & δ_{21} & δ_{22} \end{matrix} )

where α_{11}, β_{11}, γ_{11}, δ_{11} are r×r matrices, α_{12}, β_{12}, γ_{12}, δ_{12} are r×(n−r) matrices, α_{21}, β_{21}, γ_{21}, δ_{21} are (n − r) × r matrices, and α_{22}, β_{22}, γ_{22}, δ_{22} are (n − r) × (n − r) matrices. By modifying routinely the proof of [T] (3.4) which is carried out under the assumption that λ ∈ center(R), we obtain

(2.4) the columns of α_{11} − I, α_{12}, β_{11}, β_{12}, β̄_{11}, β̄_{21}, δ̄_{11} − I and δ̄_{21} belong to C.

We define now Hermitian elementary matrices as in [T] §8 but with small adjustments to account for the fact that λ is not necessarily central in R. Let

Hϵ_{ij}(a) (a ∈ R and r < i ≤ n, 1 ≤ j ≤ n, i ≠ j)

denote the 2n × 2n matrix with 1 along the diagonal, a in the (i, j)’th position, −λ̄āλ in the (n + j, n + i)’th position, and 0 elsewhere. Let

r_{ij}(a) (a ∈ R and r < i, j ≤ n)

denote the 2n × 2n matrix with 1 along the diagonal, a in the (i, n + j)’th position, −āλ in the (j, n + i)’th position, and 0 elsewhere. If i = j, this forces of course that a = −āλ. Let

l_{ij}(a) (a ∈ R and 1 ≤ i, j ≤ n)

denote the 2n × 2n matrix with 1 along the diagonal, a in the (n + i, j)’th position, −λ̄ā in the (n + j, i)’th position, and 0 elsewhere. If i = j, this forces of course that a = −λ̄ā.

For ζ = ^{t}(x_{1}, ⋯, x_{r}) ∈ C, let

ζ_{f} ∈ R such that ζ_{f} + ζ̄_{f}λ = ∑_{i=1}^{r} x̄_{i}a_{i}x_{i}.

The element ζ_{f} is not in general unique. Define

Hm_{i}(ζ) = ( \begin{matrix}I & α_{12} & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & −λ̄A_{1}α_{12} & I & 0 \\ 0 & γ_{22} & −λ̄ᾱ_{12}λ & I \end{matrix} ) (ζ ∈ C and r < i ≤ n)

to be the 2n × 2n matrix such that α_{12} is the r × (n − r) matrix with ζ as its (i − r)’th column and all other column’s zero, and γ_{22} is the (n − r) × (n − r) matrix with ζ̄_{f} in its (i − r, i − r)’th position and 0 elsewhere. Define

r_{i}(ζ) = ( \begin{matrix}I & 0 & 0 & β_{12} \\ 0 & I & −β̄_{12}λ & β_{22} \\ 0 & 0 & I & −λ̄A_{1}β_{12} \\ 0 & 0 & 0 & I \end{matrix} ) (ζ ∈ C and r < i ≤ n)

to be the 2n × 2n matrix such that β_{12} is the r × (n − r) matrix with ζ as its (i − r)’th column and all other columns 0, and β_{22} is the (n − r) × (n − r) matrix with ζ̄_{f}λ in its (i − r, i − r)’th position and 0 elsewhere.

Each of the matrices above is called a Hermitian elementary matrix for the elements a_{1}, …, a_{r}.

One can show by direct computation as in [T] §4 that each Hermitian elementary matrix is in GH_{2n}(R, a_{1}, ⋯, a_{r}).

Define the Hermitian elementary group

EH_{2n}(R, a_{1}, …, a_{r})

of the elements a_{1}, …, a_{r} to be the subgroup of GH_{2n}(R, a_{1}, …, a_{r}) generated by all Hermitian elementary matrices.

If λ ∈ center(R), the definitions above of Hermitian elementary matrix and Hermitian elementary group agree with those in [T] §4 and §8.

Lemma 2.5 Let r < i ≤ n. Let σ ∈ GL_{2n}(R) be a matrix whose diagonal coefficients are 1 and whose only other nonzero coefficients are restricted to the i’th row and n + i’th column. Let (x_{1}, ⋯, x_{2n}) and ^{t}(y_{1}, ⋯, y_{2n}) denote respectively the i’th row and n + i’th column of σ. (Thus x_{i} = y_{n+i} = 1 and x_{n+i} = y_{i}.) Then σ ∈ GH_{2n}(R, a_{i}, ⋯, a_{r}) ⇔ σ =

( ∏_{j=1_{j≠i}}^{n} Hϵ_{ij}(x_{i})) ( ∏_{j=r+1_{j≠i}}^{n} r_{ji}(y_{j})) r_{i}(ζ) where ζ = ^{t}(y_{1}, ⋯, y_{r}) and ζ_{f} =

(\overline{x_{n+i} − ∑n ; j=1_{j≠i} x_{j}y_{j}}) λ. In particular σ ∈ GH_{2n}(R, a_{1}, ⋯, a_{r}) ⇔ σ ∈ EH_{2n}(R, a_{1}, ⋯, a_{r}).

Proof Reduce to the case i = n as follows. Suppose i ≠ n. Set π = Hϵ_{in}(1)Hϵ_{ni}(−1)Hϵ_{in} (1). By definition, π ∈ EH_{2n}(R, a_{1}, ⋯, a_{r}). Moreover πσπ^{−1} satisfies the hypotheses of the lemma for i = n, and πσπ^{−1} satisfies the conclusion of the lemma ⇔ σ does. This reduces the proof to the case i = n.

Set ρ = ( ∏_{j=1}^{n−1} Hϵ_{nj}(−x_{j})) σ ( ∏_{j=r+1}^{n} r_{jn}(−y_{j})) =

( \begin{matrix}1 & & & & & & & & & & & & & y_{1} \\ & · & & & & & & & & & & & & ⋮ \\ & & · & & & & & & & & & & & y_{r} \\ & & & · & & & & & & & & & & 0 \\ & & & & · & & & & & & & & & ⋮ \\ & & & & & · & & & & & & & & 0 \\ & & & & & & 1 & x_{n+1} & ⋯ & x_{n+r} & x′_{n+r+1} & ⋯ & x′_{2n−1} & z \\ & & & & & & & 1 & & & & & & y′_{n+1} \\ & & & & & & & & · & & & & & ⋮ \\ & & & & & & & & & · \\ & & & & & & & & & & · \\ & & & & & & & & & & & · & & ⋮ \\ & & & & & & & & & & & & · & y′_{2n−1} \\ & & & & & & & & & & & & & 1 \end{matrix} )

where x′_{n+j} = x_{n+j} + ȳ_{j}λ (r < j < n), y′_{n+j} = y_{n+j} + λ̄x̄_{j}λ (1 ≤ j < n), and z = x_{2n} −

∑_{j=1}^{n−1} x_{j}y_{j}. Clearly σ ∈ GH_{2n}(R, a_{1}, ⋯, a_{r}) ⇔ ρ ∈ GH_{2n}(R, a_{1}, ⋯, a_{r}) ⇔ ρ̄( \begin{matrix}A & λI \\ I & 0 \end{matrix} ) ρ =

( \begin{matrix}A & λI \\ I & 0 \end{matrix} ) ⇔ (using (2.4), see [T] p. 217 - 218) x_{n+j} = −ȳ_{j}λ (1 ≦ j ≦ r), x′_{n+j} =

0 (r < j < n), y′_{n+j} = −λ̄ā_{j}y_{j} (1 ≦ j ≦ r), y′_{n+j} = 0 (r < j < n), ^{t}(y_{1}, ⋯, y_{r}) ∈

C, and z + z̄λ = ∑_{j=1}^{r} ȳ_{j}a_{j}y_{j}. Thus σ ∈ GH_{2n}(R, a_{1}, ⋯, a_{r}) ⇔ σ = ( ∏_{j=1}^{n−1} Hϵ_{nj}(x_{j}))

r_{n}(ζ) ( ∏_{j=r+1}^{n−1} r_{jn}(y_{j})) = ( ∏_{j=1}^{n−1} Hϵ_{nj}(x_{j})) ( ∏_{j=r+1}^{n−1} r_{jn}(y_{j})) r_{n}(ζ) where ζ = ^{t}(y_{1}, ⋯, y_{r})

and ζ_{f} = z̄λ.□

Corollary 2.6 Let r < i ≤ n. Let τ(e_{i}, v, s) be a transvection on 𝕄_{2n}(R, a_{1}, ⋯, a_{r}). Suppose under the basis of 𝕄_{2n}(R, a_{1}, ⋯, a_{r}) that v = ^{t}(v_{1}, ⋯, v_{2n}). Then v_{n+i} = 0 and

τ(e_{i}, v, s) = ( ∏_{j=1}^{r} Hϵ_{ij}(−λ̄v̄_{j}a_{i} − λ̄v̄_{n+j})) ( ∏_{j=r+1_{j≠i}}^{n} Hϵ_{ij}(−λ̄v̄_{n+j})) ( ∏_{j=r+1_{j≠i}}^{n} r_{ji}(v_{j}λ)) r_{i}(ζ)

where ζ = ^{t}(v_{1}λ, ⋯, v_{r}λ) and ζ_{f} =

(\overline{(−λ̄v̄_{i}λ + v_{i}λ − λ̄sλ) − ∑_{j=1}^{r}(−λ̄v̄_{j}a_{j}v_{j}λ) − ∑_{j=1_{j≠i}}^{n}(−λ̄v̄_{n+j}v_{j}λ)}) λ.

Proof The matrix of τ(e_{i}, v, s) has 1 along the diagonal and all other nonzero coefficients are restricted to the i’th row and (n + i)’th column. Since (e_{i}, v) = 0, it follows that v_{n+i} = 0 and an easy computation shows that the i’th row of τ(e_{i}, v, s) is (−λ̄v̄_{1}a_{1} − λ̄v̄_{n+1}, ⋯, −λ̄v̄_{r}a_{r} − λ̄v̄_{n+r}, −λ̄v̄_{n+r+1}, ⋯, −λ̄v̄_{2n}, −λ̄v̄_{1}λ, ⋯, −λ̄v̄_{n}λ) + (v_{i}λ − λ̄v̄_{i}λ − λ̄sλ)_{n+i} + (1)_{i} where (a)_{j} (a ∈ R, 1 ≦ j ≦ 2n) denotes the row vector whose j’th coefficient is a and all other coefficients are 0; the (n + i)’th column of τ(e_{i}, v, s) is t(v_{1}λ, ⋯, v_{2n}λ) +^{t} (−λ̄v̄_{i}λ − λ̄sλ)_{i} +^{t} (1)_{n+i}. The conclusion of the corollary follows now from Lemma 2.5. □

Lemma 2.7 Let r < i ≤ n. Let σ ∈ GL_{2n}(R) be a matrix whose diagonal coefficients are 1 and whose only other nonzero coefficients are restricted to the (n + i)’th row and i’th column. Let (x_{1}, ⋯, x_{2n}) and ^{t}(y_{1}, ⋯, y_{2n}) denote respectively the (n + i)’th row and i’th column of σ. (Thus x_{n+i} = y_{i} = 1 and x_{i} = y_{n+i}.) Then σ ∈ GH_{2n}(R, a_{1}, ⋯, a_{r}) ⇔

σ = ( ∏_{j=1_{j≠i}}^{n} ℓ_{ij}(x_{j})) ( ∏_{j=r+1_{j≠i}}^{n} Hϵ_{ji}(−x̄_{n+j})) Hm_{i}(ζ) where ζ = ^{t}(y_{1}, ⋯, y_{r}) and ζ_{f} =

λ̄(\overline{y_{n+i} − ∑_{j=1_{j≠i}}^{n} x_{j}y_{j}})λ.

Proof Reduce as in the proof of Lemma 2.5 to the case i = n.

Set ρ = ( ∏_{j=1}^{n−1} ℓ_{nj}(−x_{j})) σ ( ∏_{j=r+1}^{n−1} Hϵ_{jn}(x̄_{n+j})) =

( \begin{matrix}1 & & & & & & y_{1} \\ & · & & & & & ⋮ \\ & & · & & & & y_{r} \\ & & & · & & & y′_{r+1} \\ & & & & · & & ⋮ \\ & & & & & · & y′_{n−1} \\ & & & & & & · \\ & & & & & & 1 \\ & & & & & & y′_{n+1} & 1 \\ & & & & & & ⋮ & & · \\ & & & & & & & & & · \\ & & & & & & ⋮ & & & & · \\ & & & & & & y′_{2n−1} & & & & & · \\ & & & & & & z & x_{n+1} & ⋯ & x_{n+r} & 0 ⋯ & 0 1 \end{matrix} )

where y′_{j} = y_{j}+x̄_{n+j} (r < j < n), y′_{n+j} = y_{n+j}+λ̄x̄_{j} (1 ≦ j < n), and z = y_{2n}− ∑_{j=1}^{n−1} x_{j}y_{j}.

Clearly σ ∈ GH_{2n}(R, a_{1}, ⋯, a_{r}) ⇔ ρ ∈ GH_{2n}(R, a_{1}, ⋯, a_{r}) ⇔ ρ̄( \begin{matrix}A & λI \\ I & 0 \end{matrix} ) ρ =

( \begin{matrix}A & λI \\ I & 0 \end{matrix} ) ⇔ (using (2.4), see [T], p. 217 – 218) y′_{j} = 0 (r < j < n), y′_{n+j} = 0 (r <

j < n), y′_{n+j} = −ā_{j}y_{j} (1 ≦ j ≦ r), ^{t}(y_{1}, ⋯, y_{r}) ∈ C, and λ̄z̄λ + zλ = ∑_{j=1}^{r} ȳ_{j}a_{j}y_{j}.

Thus σ ∈ GH_{2n}(R, a_{1}, ⋯, a_{r}) ⇔ σ = ( ∏_{j=1}^{n−1} ℓ_{nj}(x_{j})) Hm_{n}(ζ) ( ∏_{j=r+1}^{n−1} Hϵ_{jn}(−x̄_{n+j})) =

( ∏_{j=1}^{n−1} ℓ_{nj}(x_{j})) ( ∏_{j=r+1}^{n−1} Hϵ_{jn}(−x̄_{n+j})) Hm_{n}(ζ) where ζ = ^{t}(y_{1}, ⋯, y_{r}) and ζ_{f} = λ̄z̄λ.□

Corollary 2.8 Let 1 ≦ i ≦ n. Let τ(e_{−i}, v, s) be a transvection on 𝕄_{2n}(R, a_{1}, ⋯, a_{r}). Suppose under the basis of 𝕄_{2n}(R, a_{1}, ⋯, a_{r}) that v = ^{t}(v_{1}, ⋯, v_{2n}). Then v_{i} = 0 and

τ(e_{−i}, v, s,) = ( ∏_{j=1}^{r} ℓ_{ij}(−λ̄v̄_{j}a_{j} − λ̄v̄_{n+j})) ( ∏_{j=r+1_{j≠i}}^{n} ℓ_{ij}(−λ̄v̄_{n+j})) ∏_{j=r+1_{j≠i}}^{n} Hϵ_{ji}(v_{j})Hm_{i}(ζ)

where ζ = ^{t}(v_{1}, ⋯, v_{r}) and ζ_{f} = λ̄ (\overline{(v_{n+i} − λ̄s) − ∑_{j=1}^{r}(−λ̄v̄_{j}a_{j}v_{j}) − ∑_{j=1_{j≠i}}^{n}(−λ̄v̄_{n+j}v_{j})}) λ.

Proof The matrix of τ(e_{−i}, v, s) has 1 along the diagonal and all other nonzero coefficients are restricted to the (n+i)’th row and i’th column. Since (e_{i}, v) = 0, it follows that v_{i} = 0 and an easy computation shows that the (n + i)’th row of τ(e_{−i}, v, s) is (−λ̄v̄_{1}a_{1} − λ̄v̄_{n+1}, ⋯, −λ̄v̄_{r}a_{r} − λ̄v̄_{n+r}, −λ̄v̄_{n+r+1}, ⋯, −λ̄v̄_{2n}, −λ̄v̄_{1}λ, ⋯, −λ̄v̄_{n}λ) + (v_{n+i} − λ̄s)_{i} + (1)_{n+i} where (a)_{j} (a ∈ R, 1 ≦ j ≦ n) denotes the row vector whose j’th coefficient is a and all other coefficients are 0; the i’th column of τ(e_{−i}, v, s) is ^{t}(v_{1}, ⋯, v_{2n})+ t(−λ̄s)_{n+i}+ ^{t}(1)_{i}. The conclusion of the corollary follows now from Lemma 2.7. □

The next lemma expresses each Hermitian elementary matrix as a product of at most 2 transvections. Its proof is a straightforward computation.

Lemma 2.9 The following identities hold.

(1) Hϵ_{ij}(a) = τ(e_{i}, e_{−j}, ā, 0).
(2) Hm_{i}(ζ) = τ(e_{−i}, ζ, −λζ̄_{f}).
(3) r_{ij}(a) = τ(e_{i}, e_{−j}ā, 0), for i ≠ j.
(4) r_{ii}(a) = τ(e_{i}, e_{−j}, −¯ā)τ(e_{i}, e_{−j}, 0), for j ≠ i.
(5) r_{i}(ζ) = τ(e_{i}, ζ′, −λ^{2}ζ̄_{f} λ̄), where ζ =^{t}(ζ_{1}, ⋯, ζ_{r}) and ζ′ =^{t}(ζ̄_{1}λ, ⋯, ζ̄_{r}λ).
(6) ℓ_{ij}(a) = τ(e_{−i}, e_{−j}ā, 0), for i ≠ j.
(7) ℓ_{ii}(a) = τ(e_{−i}, e_{−j}, λa)τ(e_{−i}, e_{−j}(−1), 0), for j ≠ i.

Proof of Theorem 1.7 Let r < i ≤ n. By (2.6) and (2.8), T_{⟨e_{i},e_{−i}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})) ⊆ EH_{2n}(R, a_{1}, ⋯, a_{r}). To prove the converse, we note that by (2.2), T_{⟨e_{i},e_{−i}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})) = T_{⟨e_{k},e_{−k}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})) for any k such that r < k ≤ n. Each Hermitian elementary matrix of the kind Hϵ_{ℓm}(a), Hm_{ℓ}(ζ), r_{ℓm}(a), r_{ℓℓ}(a), and r_{ℓ}(ζ) has the property that r < ℓ ≤ n and by (2.9) (1) - (5), each of these matrices lies in T_{⟨e_{ℓ},e_{−ℓ}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})). The remaining Hermitian elementary matrices are of the kind ℓ_{km}(a). By (2.9) (7) - (8), Hermitian elementary matrices of this kind lie in T_{⟨e_{k},e_{−k}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})) and so we are okay if r < k ≤ n. If r < m ≤ n then we are also okay, because ℓ_{km}(a) = ℓ_{mk}(−λ̄ā). Suppose 1 ≦ k, m ≦ r. If k ≠ m then ℓ_{km}(a) = Hϵ_{nk}(1)ℓ_{nm}(a)Hϵ_{nk}(1)^{−1}ℓ_{nm}(a)^{−1} ∈ T_{⟨e_{n},e_{−n}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})). If k = m then ℓ_{kk}(a) = Hϵ_{nk}(1)ℓ_{nn}(a)Hϵ_{nk}(1)^{−1}ℓ_{nn}(a)^{−1} ∈ T_{⟨e_{n},e_{−n}⟩}(𝕄_{2n}(R, a_{1}, ⋯, a_{r})).□

If a_{1} = a_{2} = ⋯ = a_{r} = 0, we set

GH_{2n}(R, 0) = GH_{2n}(R, a_{1}, ⋯, a_{r}), and
EH_{2n}(R, 0) = EH_{2n}(R, a_{1}, ⋯, a_{r}).

It is obvious that the definition of GH_{2n}(R, 0) is independent of r and the same is true of EH_{2n}(R, 0), thanks to the following lemma.

Lemma 2.10 Define the 2n × 2n matrices

Hϵ_{ij}(a) (a ∈ R and 1 ≦ i, j ≦ n, i ≠ j)
r_{ij}(a) (a ∈ R and 1 ≦ i, j ≦ n)
ℓ_{ij}(a) (a ∈ R and 1 ≦ i, j ≦ n)

as after (2.4), except replace r by 1. We require that if i = j then r_{ij}(a) has the property that a = −āλ and ℓ_{ij}(a) the property that a = −λ̄ā. Then each of the matrices above is in EH_{2n}(R, 0) and EH_{2n}(R, 0) is generated by these matrices.

Proof Since A_{1} = 0, it is easy to see that each Hermitian elementary matrix defined after (2.4) is either one of the matrices in (2.10) or a product of such matrices. Conversely, since A_{1} = 0 and C = ^{t}(R^{r}), it is easy to check that each matrix in (2.10) is some Hermitian elementary matrix. □

We describe next certain subgroups of GH_{2n}(R, 0) and EH_{2n}(R, 0), which arise in the theory of quadratic forms. Let Λ be a λ-form parameter on R, as defined in §1. Let 𝕄_{n}(Λ) denote the additive subgroup of the ring 𝕄_{n}(R) of n×n matrices with coefficients in R, consisting of all β such that β = −β̄λ and the diagonal coefficients of β lie in Λ. Define the general quadratic group [B] §3

GQ_{2n}(R, Λ) = \{( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ GL_{2n}(R) | ( \overline{\begin{matrix}α & β \\ γ & δ \end{matrix}} ) ( \begin{matrix}0 & λ I \\ I & 0 \end{matrix} ) ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) =

( \begin{matrix}0 & λ I \\ I & 0 \end{matrix} ) , γ̄α and δ̄β ∈ 𝕄_{n}(Λ)\} .

If Λ = max^{λ}(R) then it follows from the equation ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ( \begin{matrix}0 & λ I \\ I & 0 \end{matrix} ) ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) =

( \begin{matrix}0 & λ I \\ I & 0 \end{matrix} ) that γ̄α and δ̄β ∈ 𝕄_{n}(max^{λ}(R)). Thus

GQ_{2n}(R, max^{λ}(R)) = GH_{2n}(R, 0).  (2.11)

Since each λ-form parameter Λ ⫅ max^{λ}(R), it is obvious that each group GQ_{2n}(R, Λ) ⫅ GQ_{2n}(R, max^{λ}(R)). Define the 2n × 2n matrices

Hϵ_{ij}(a) (a ∈ R and 1 ≦ i, j ≦ n, i ≠ j)  (2.12)
r_{ij}(a) (a ∈ R and 1 ≦ i, j ≦ n)
ℓ_{ij}(a) (a ∈ R and 1 ≦ i, j ≦ n)

as in Lemma 2.10, except if i = j, we insist that r_{ij}(a) has the property that a ∈ Λ and ℓ_{ij}(a) the property that a ∈ Λ̄. These matrices are called Λ-quadratic elementary matrices. One checks easily that they are members of GQ_{2n}(R, Λ). They generate by definition the Λ- quadratic elementary group EQ_{2n}(R, Λ) [B] §3. From Lemma 2.10 and the definition of EQ_{2n}(R, max^{λ}(R)), it follows that

EQ_{2n}(R, max^{λ}(R)) = EH_{2n}(R, 0).  (2.13)

Obviously each group EQ_{2n}(R, Λ) ⫅ EQ_{2n}(R, max^{λ}(R)), because each λ-form parameter Λ ⫅ max^{λ}(R).

For technical reasons, we give names to several kinds of Λ-quadratic elementary matrices. Let m be a natural number such that 1 ≦ m ≦ n − 2. We define 5 types of matrices as follows.

(2.14) Type 1. Hϵ_{ni} (n − m − 1 ≦ i < n)
Type 2. ℓ_{in} (n − m − 1 ≦ i < n)
Type 3. ℓ_{ij}  (1 ≦ i, j < n)
Type 4. Hϵ_{ij} (1 ≦ i ≦ n − m − 2, n − m − 2 < j < n)
Type 5. Hϵ_{ij} (n − m − 2 ≦ i, j < n, i ≠ j).

Let

EQ_{2n}^{(m)}(R, Λ) = ⟨ type k Λ-quadratic elementary matrices | 1 ≦ k ≦ 5⟩.

Proposition 2.15 Let 1 ≤ m ≦ n − 2. Let ϵ ∈ EQ_{2n}^{(m)}(R, Λ), v = (b_{1}, ⋯, b_{n}, c_{1}, ⋯, c_{n}) ∈ R^{n}, and ϵv = (b′_{1}, ⋯, b′_{n}, c′_{1}, ⋯, c′_{n}). If b_{n−m−1} = b_{n−m} = ⋯ = b_{n} = c_{n} = 0 then b′_{n−m−1} = b′_{n−m} = ⋯ = b′_{n} = c′_{n} = 0. Furthermore if v is right unimodular and m ≥ Λsr(R) then there is an ϵ such that c′_{n−m−1} = c′_{n−m} = ⋯ = c′_{n} = 1.

Proof The first assertion is trivial to check. We prove the second assertion. If k is a natural number and ϵ^{(1)}, ϵ^{(2)}, ⋯, ϵ^{(k)} ∈ EQ_{2m}^{(m)}(R, Λ), let ϵ^{(k)}ϵ^{(k−1)} ⋯ ϵ^{(1)}v = (b_{1}^{(k)}, ⋯, b_{n}^{(k)}, c_{1}^{(k)}, ⋯, c_{n}^{(k)}). Suppose m ≥ Λsr(R) and v is unimodular. By [K] VI (1.5.1), there is a product ϵ^{(1)} of type 1 matrices such that (b_{1}^{(1)}, ⋯, b_{n}^{(1)}, c_{1}^{(1)}, ⋯, c_{n−1}^{(1)}) is unimodular. Again by [K] VI (1.5.1), there is a product ϵ^{(2)} of type 2 matrices such that (b_{1}^{(2)}, ⋯, b_{n−1}^{(2)}, c_{1}^{(2)}, ⋯, c_{n−1}^{(2)}) is unimodular. By [BT] (3.3), there is a product ϵ^{(3)} of type 3 matrices such that (c_{1}^{(3)}, ⋯, c_{n−1}^{(3)}) is unimodular. By [K] VI (1.5.1), there is a product ϵ^{(4)} of type 4 matrices such that (c_{m−n−1}^{(4)}, ⋯, c_{n−1}^{(4)}) is unimodular. By [K] VI (1.5.2), there is a product ϵ^{(5)} of type 5 matrices such that (c_{m−n−1}^{(5)}, ⋯, c_{n−1}^{(5)}) = (1, ⋯, 1).□

3 Normality of elementary subgroups of general Hermitian groups

The goal of this section is to prove Theorem 1.5. This is done at the end of the section.

Let R be an associative ring with identity element and λ-involution a ↦ ā. Let (V, h) be a λ-Hermitian module. We fix the following notation throughout the section. Let n be a natural number and let e_{1}, e_{−1}, ⋯, e_{n}, e_{−n} be a set of mutually complementary hyperbolic pairs e_{i}, e_{−i} in (V, h). The submodule, say X, of V generated by these pairs is nonsingular and V splits as an orthogonal sum X ⊥ X^{⊥} where X^{⊥} = {v ∈ V | h(v, x) = 0 ∀ x ∈ X}. The module X is free on the elements of the hyperbolic pairs above and we let X have the ordered basis e_{1}, ⋯, e_{n}, e_{−1}, ⋯, e_{−n}. Let ℍ_{2n}(R) = (X, h|_{X}) with the choice of basis above. Thus ℍ_{2n}(R) is the λ-Hermitian module 𝕄_{2n}(R, a_{1}, ⋯, a_{r}) defined in §2, with a_{1} = a_{2} = ⋯ = a_{r} = 0 and with basis as in §2. Let V′ = X^{⊥}. Thus V = ℍ_{2n}(R) ⊥ V′. we identify GH(ℍ_{2n}(R)) with its image in GH(V, h) under the map σ ↦ σ ⊥ 1_{V′}. By definition GH_{2n}(R, 0) = GH(ℍ_{2n}(R)), by (2.11) GQ_{2n}(R, max^{λ}(R)) = GH_{2n}(R, 0), and by (2.13) EQ_{2n}(R, max^{λ}(R)) = EH_{2n}(R, 0). For any λ-form parameter Λ, we have a sequence of inclusions EQ_{2n}(R, Λ) ⫅ EQ_{2n}(R, max^{λ}(R)) = EH_{2n}(R, 0) ⫅ GH(V, h) which allow us to view EQ_{2n}(R, Λ) as a subgroup GH(V, h). Moreover from Theorem 1.7 applied to ℍ_{2n}(R), it follows that EQ_{2n}(R, Λ) ⫅ T_{⟨e_{i},e_{−i}⟩}(V, H) for any i (1 ≦ i ≦ n). If v ∈ V then we can write uniquely

v = ∑_{i=−n}^{n} e_{i}v_{i} + v′

where v′ ∈ V′ and we make this a notational convention. In vector notation, ∑_{i=−n}^{n} e_{i}v_{i}

becomes identified with the vector ^{t}(v_{1}, ⋯, v_{n}, v_{−1}, ⋯, v_{−n}) because of our choice of orderd basis. Each element ϵ ∈ EQ_{2n}(R, Λ) acts on ^{t}(v_{1}, ⋯, v_{n}, v_{−1}, ⋯, v_{−n}), but leaves v′ fixed.

Lemma 3.1 Let (τ(u, v, a) be a transvection on (V, h).

(3.1.1) If for some i (1 ≦ i ≦ n), u = e_{±i}u_{i} or v = e_{±i}v_{i} and a = 0 then τ(u, v, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

(3.1.2) If for some i (1 ≦ i ≦ n), u_{i} = u_{−i} = v_{i} = v_{−i} = 0 then τ(u, v, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

(3.1.3) If for some i (1 ≦ i ≦ n), u_{i} = u_{−i} = 0 then τ(u, v, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

(3.1.4) If for some i (1 ≦ i ≦ n), u_{i} = v_{i} = 0 then τ(u, v, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

(3.1.5) Suppose n ≥ 3. Suppose for some i ≠ j (1 ≦ i, j ≦ n), v_{−j} = u_{i} = u_{−i} = 0. Let s ∈ R such that ū_{−i}u_{i}s ∈ ū_{−j}R. Then τ(us, v, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

Proof (3.1.1) Suppose u = e_{i}u_{i}. By (2.1) T 2, τ(e_{i}u_{i}, v, a) = τ(e_{i}, vū_{i}, ¯ ū_{i}au_{i}) ∈ (by (2.2)) T_{⟨e_{1},e_{−1}⟩}(V, h). Suppose v = e_{i}v_{i} and a = 0. By (2.1) T5 and T2, τ(u, e_{i}v_{i}, 0) = τ(e_{i}, −uλ̄v_{i}, 0) ∈ (by (2.2)) T_{⟨e_{1},e_{−1}⟩}(V, h). The remaining cases are proved similarly.

(3.1.2) The identity (2.1) T2 suggests looking for an element σ ∈ T_{⟨e_{1},e_{−1}⟩}(V, h) such that σu = u and σe_{−i} = v + e_{i}b + e_{−i}c for suitable b, c ∈ R. This done, we could try to relate τ(u, v, a) to τ(v + e_{i}b + e_{−i}c, u, 0) = στ(e_{−i}, u, 0)σ^{−1} and finish the proof with (2.2).

By (2.1) T 1, τ(e_{i}, v, a)τ(e_{−i}, −uλ̄, 0)τ(e_{i}, v, a)^{−1} = τ(τ(e_{i}, v, a)e_{−i}, τ(e_{i}, v, a)(−uλ̄), 0) = τ(e_{−i} + v − e_{i}λ̄a, −uλ̄, 0) = (by (2.1) T5 and T2) τ(u, e_{−i} + v − e_{i}λ̄a, 0) = (by (2.1) T3) τ(u, e_{−i}, 0)τ(u, v, a)τ(u, −e_{i}λ̄a, 0) = (by (2.1) T 5 and T 2) τ(e_{−i}, −uλ̄, 0)τ(u, v, a) τ(e_{i}, u\overline{λ̄aλ}, 0). Thus τ(u, v, a) = [τ(e_{−i}, −uλ̄, 0)^{−1}, τ(e_{i}, v, a)]τ(e_{−i}, u\overline{λ̄aλ}, 0)^{−1} ∈ (by (2.2)) T_{⟨e_{1},e_{−1}⟩}(V, h).

(3.1.3) Let v = e_{i}v_{i} + e_{−i}v_{−i} + w where w is orthogonal to e_{i} and e_{−i}. By (2.1) T3, τ(u, v, a) = τ(u, w, a − v̄_{i}v_{−i})τ(u, e_{i}v_{i}, 0)τ(u, e_{−i}v_{−i}, 0). By (3.1.2), τ(u, w, a − v̄_{i}v_{−i}) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h) and by (3.1.1), τ(u, e_{i}v_{i}, 0) and τ(u, e_{−i}v_{−i}, 0) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

(3.1.4) Let u = e_{−i}u_{−i} + w where w is orthogonal to e_{i} and e_{−i}. By (2.1) T4, τ(u, v, a) = τ(e_{−i}u_{−i}, v, a)τ(w, v + e_{−i}u_{−i}λ̄āλ, a). By (3.1.1), τ(e_{−i}u_{−i}, v, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h) and by (3.1.3), τ(w, v + e_{−i}u_{−i}λ̄āλ, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

(3.1.5) Let c ∈ R such that ū_{−i}u_{i}s = ū_{−j}c. Then u is orthogonal to e_{i}u_{i}s − e_{j}c. Thus us is orthogonal to e_{i}u_{i}s − e_{j}c. Define w by the equation us = w + (e_{i}u_{i}s − e_{j}c). Since us and e_{i}u_{i}s − e_{j}c are totally isotropic and e_{i}u_{i}s − e_{j}c is orthogonal to us, it follows that w is totally isotropic and orthogonal to e_{i}u_{i}s − e_{j}c. Thus by (2.1) T4, τ(us, v, a) = τ(w, v, a)τ(e_{i}u_{i}s − e_{j}c, v + wλ̄aλ, a). By (3.1.4), τ(w, v, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h) and because n ≥ 3, it follows from (3.1.3) that τ(e_{i}u_{i}s − e_{j}c, v + wλ̄aλ, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).□

Lemma 3.2 Let τ(u, v+w, a) be a transvection such that v and w are even and orthogonal to u. Let b, c ∈ R such that b + b̄λ = v and c + c̄λ = w. Then the element d = a − (b + c + (v, w)) has the property that d = −d̄λ and τ(u, v + w, a) = τ(u, v, b)τ(u, w, c + d).

Proof Since a + āλ = (v + w, v + w) = (b + c + (v, w)) + (\overline{b + c + (v, w)})λ, it follows that d + d̄λ = 0. Thus d = −d̄λ. The second assertion in the lemma is a direct consequence of (2.1) T3. □

The next lemma is an easy exercise.

Lemma 3.3 Let S be a right denominator set in R. Let x_{1}, ⋯, x_{k}, y_{1}, ⋯, y_{ℓ}, u, v ∈ R such that u and v are units in R[S^{−1}]. Then there is an element s ∈ S and elements b_{1}, ⋯, b_{k}, c_{1}, ⋯, c_{ℓ} ∈ R such that x_{i}s = ub_{i} (1 ≦ i ≦ k) and y_{j}s = vc_{j} (1 ≦ j ≦ ℓ).

Lemma 3.4 Suppose m = lλsr(R) and n ≥ m + 2. If τ(u, v, a) is a transvection on (V, h) such that u can be completed to a hyperbolic pair u, u_{−} and v_{−n} = v_{n} = v_{n−1} = ⋯ = v_{n−m−1} = 0 then τ(u, v, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

Proof Call an element w ∈ V good, if w_{−n} = w_{n} = w_{n−1} = ⋯ = w_{n−m−1} = 0. Thus v is good. Let ∑ be a λ-multiplicative spectrum on R such that m = sup ; S∈∑(Λ(S)sr(R(S))).

Let w ∈ V such that u, w is a hyperbolic pair. Let S ∈ ∑. (Keep in mind the notational conventions.) Since (w, u) = 1, it follows that (u_{1}, ⋯, u_{n}, u_{−1}, ⋯, u_{−n}, (w′, u′)) is a unimodular vector in R^{2n+1} and hence also in R(S)^{2n+1}. Since m ≥ sr(R(S)), there are elements x_{−1}, ⋯, x_{−m} ∈ R such that the vector (u_{1}, ⋯, u_{n}, u_{−1} + x_{−1}(w′, u′), ⋯, u_{−m} + x_{−m}(w′, u′), u_{−m−1}, ⋯, u_{−n}) is unimodular in R(S)^{2n}. Trivially for any choice of elements a_{−1}, ⋯, a_{−m} ∈ R, the vector (u_{1}, ⋯, u_{n}, u_{−1} + x_{−1}(w′, u′) + a_{−1}u_{−1}, ⋯, u_{−m} + x_{−m}(w′, u′) + a_{−m}u_{m}, u_{−m−1}, ⋯, u_{−n}) is unimodular in R(S)^{2n}. Let u(a_{−1}, ⋯, a_{−n}) denote this vector as a member of R^{2n} and using the ordered basis of ℍ_{2n}(R), identify

R^{2n} with ℍ_{2n}(R). Since w′ = w − ∑_{i=−n}^{n} e_{i}w_{i} and since w and each e_{i} are even, it fol-

lows that w′ is even. Choose elements b_{i} ∈ R (1 ≦ i ≦ m) such that b_{−i} + b̄_{−i}λ =

(−w′λ̄x̄_{−i}, −w′λ̄x̄_{−i}). Set σ_{1} = ∏_{i=1}^{m} τ(e_{−i}, −w′λ̄x̄_{−i}, b_{−i}). A straightforward computation

shows that σ_{1}u = u(−λ̄b_{−1}, ⋯, −λ̄b_{−m}) + (σ_{1}u)′ where (σ_{1}u)′ is orthogonal to ℍ_{2n}(R). Moreover σ_{1}v is good. By (2.15), there is an element σ_{2} ∈ EQ_{2n}^{(m)}(R, max^{λ}(R)) such that (σ_{2}σ_{1}u)_{−(n−1)} ≡ 1 in R(S) and σ_{2}σ_{1}v is good. By MS1 in §1, (σ_{2}σ_{1}u)_{−(n−1)} must be a unit in R[S^{−1}]. Let z′_{n−1} denote its inverse in R[S^{−1}]. Since (\overline{σ_{2}σ_{1}u})_{−n}λ(σ_{2}σ_{1}u)_{n} and (σ_{2}σ_{1}u)_{−(n−1)}z′_{n−1}(\overline{σ_{2}σ_{1}u})_{−n}λ(σ_{2}σ_{1}u)_{n} are equal in R[S^{−1}], there are elements t_{S} ∈ S and z_{n−1} ∈ R such that (\overline{σ_{2}σ_{1}u})_{−n}λ(σ_{2}σ_{1}u)_{n}t_{S} = (σ_{2}σ_{1}u)_{−(n−1)}z_{n−1} as elements of R. An easy computation shows that for any element c_{S} ∈ R, σ_{2}σ_{1}u is orthogonal to e_{n}(σ_{2}σ_{1}u)_{n}t_{S}c_{S} − e_{n−1}z_{n−1}c_{S}. Thus σ_{2}σ_{1}ut_{S}c_{S} is orthogonal to f_{S} = e_{n}(σ_{2}σ_{1}u)_{n}t_{S}c_{S} − e_{n−1}z_{n−1}c_{S}. By (2.1) T4, τ(σ_{2}σ_{1}ut_{S}c_{S}, σ_{2}σ_{1}v, a) = τ(σ_{2}σ_{1}ut_{S}c_{S}−f_{S}, σ_{2}σ_{1}v, a)τ(f_{S}, σ_{2}σ_{1}v− f_{S}λ̄āλ, a). By (3.1.4), the first factor above lies in T_{⟨e_{1},e_{−1}⟩}(V, h) and by (3.1.3), the seconds factor lies in T_{⟨e_{1},e_{−1}⟩}(V, h). Thus for any c_{S} ∈ R, τ(u, vt_{S}c_{S}, \overline{t_{S}c_{S}}at_{S}c_{S}) = (by (2.1) T2) τ(ut_{S}c_{S}, v, a) = (σ_{2}σ_{1})^{−1}τ(σ_{2}σ_{1}ut_{S}c_{S}, σ_{2}σ_{1}v, a)(σ_{2}σ_{1}) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h). Moreover t_{S} is independent of the choice of the element a in τ(u, v, a). This will be important. By MS 2 in §1, there are a finite number of denominator sets S_{1}, ⋯, S_{k} ∈ ∑ and elements

c_{1}, ⋯, c_{k} ∈ R such that t_{S_{1}}c_{1} + ⋯ + t_{S_{k}}c_{k} = 1. Thus τ(u, v, a) = τ(u, ∑_{i=1}^{k} vt_{S_{i}}c_{i}, a) = (by

(3.2), for suitably chosen a_{i}’s) ∏_{i=1}^{k} τ(u, vt_{S_{i}}c_{i}, a_{i}) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).□

Proof of Theorem 1.5 Let m = lλsr(R). Let ∑ be a λ-multiplicative spectrum on R such that m = sup ; S∈∑(Λ(S)sr(R(S))). Suppose u ∈ V, which can be completed to a

hyperbolic pair u, u_{−}. We shall show that any transvection τ(u, v, a) on (V, h) lies in T_{⟨e_{1},e_{−1}⟩}(V, h). Let S ∈ ∑. The proof of (3.4) demonstrates that we can find a σ ∈ T_{⟨e_{1},e_{−1}⟩}(V, h) such that (σu)_{n−1} and (σu)_{n−2} ≡ 1 in R(S). By MS 1 in §1, (σu)_{n−1} and (σu)_{n−2} are units in R[S^{−1}]. Let N = {−n, n, n − 1, ⋯, n − m − 1}. Since (σu)_{n−1} and (σu)_{n−2} are units in R[S^{−1}], there is by (3.3) an element t_{S} ∈ S and elements z_{i} ∈ R (i ∈ N) such that

(\overline{σu})_{−i}(σv)_{i}t_{S} = (\overline{σu})_{n−1}λz_{i},  for i ≠ n − 1 and − n,
(\overline{σu})_{n−i}(σv)_{n−i}t_{S} = (\overline{σu})_{n−2}λz_{n−1}, for i = n − 1,
(\overline{σu})_{n}λ(σv)_{−n}t_{S} = (\overline{σu})_{n−1}λz_{−n},  for i = n.

It is important to observe that t_{S} is independent of the choice of the element a in τ(u, v, a). For i ∈ N, define f_{i} such that

f_{i} = e_{i}(σv)_{i}t_{S} − e_{−(n−1)}z_{i},  for i ≠ n − 1 and − n,
f_{n−1} = e_{n−1}(σv)_{n−1}t_{S} − e_{−(n−2)}z_{n−1}, for i = n − 1,
f_{−n} = e_{−n}(σv)_{−n}t_{S} − e_{−(n−1)}z_{−n},  for i = n.

An easy computation shows that for any element c_{S} ∈ R, u is orthogonal to f_{i}c_{S} (i ∈ N).

Let c_{S} ∈ R. Let w = ∑_{i∈N} f_{i}. Then στ(u, vt_{S}c_{S}, a)σ^{−1} = (by (2.1) T1) τ(σu, σvt_{S}c_{S} −

wc_{S} +wc_{S}, a) = (by (3.2), for suitable a′ and a_{i}) τ(σu, σvt_{S}c_{S} −wc_{S}, a′) ∏_{i∈N} τ(σu, f_{i}c_{S}, a_{i}).

By (3.4), τ(σu, σvt_{S}c_{S} − wc_{S}, a′) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h) and by (3.1.3), each τ(σu, f_{i}c_{S}, a_{i}) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h). Thus τ(u, v, t_{S}c_{S}, a) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h). Keep in mind that t_{S} is independent of the choice of the element a in τ(u, v, a). By MS 2 in §1, there are finitely many denominator sets S_{1}, ⋯, S_{k} ∈ ∑ and elements

c_{1}, ⋯, c_{k} ∈ R such that t_{S_{1}}c_{1} + ⋯ + t_{S_{k}}c_{k} = 1. Thus τ(u, v, a) = τ(u, ∑_{i=1}^{k} vt_{S_{i}}c_{i}, a) =

(by (3.2), for suitable a_{i})

∏_{i=1}^{k} τ(u, vt_{S_{i}}c_{i}, a_{i}) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).

Finally let u ∈ V such that u can be completed to a hyperbolic pair w, u. Then τ(u, v, a) = τ(u, vλ̄λ, a) = (by (2.1) T2) τ(uλ̄, vλ̄, λ̄aλ). But uλ̄ can be completed to the hyperbolic pair uλ̄, w and so by the paragraph above, τ(uλ̄, vλ̄, λ̄aλ) ∈ T_{⟨e_{1},e_{−1}⟩}(V, h).□

References

[B] [BT] [BV] [GR] [K] [KMV] [MR] [T] [Tu] [Tul] [S] [Su]

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