1 Introduction

Commutator relations involving elementary matrices play a key role in answering questions in K-theory. Steinberg’s celebrated commutator formulae were generalized to the setting of Chevalley-Demazure group schemes over commutative rings by Stein. The commutator formulae were pivotal in obtaining local-global principles for various groups. Localization is one of the most powerful tools in the study of structure of quadratic modules and more generally, of algebraic groups over rings. It helps to reduce many important problems over arbitrary commutative rings to similar problems for semi-local rings. Localization comes in a number of versions such as localisation and patching, proposed by Quillen in [20] and Suslin in [26], and localisation-completion, proposed by Bak (see [8]). Both of these methods rely on the yoga of commutators. This term was coined by Hazrat et al. (see [12]) and stands for a large body of common calculations, known as conjugation calculus and as commutator calculus. Their main objective is to obtain explicit estimates of the modulus of continuity in s-adic topology for conjugation by a specific matrix, and to calculate mutual commutator subgroups, nilpotent filtration etc.

The commutator calculus in the setting of general linear groups, Bak’s unitary groups, Hermitian groups, Chevalley groups and Petrov’s odd unitary groups has been established by authors Bak (see [8]), Tang (see [27]), Hazrat (see [11]), Hazrat and Vavilov (see [14]), Petrov (see [18]), Yu and Tang (see [28]), and Preusser (see [19]). Due to the presence of different roots, and complicated elementary relations compared to the general linear groups, establishing the commutator calculus and showing the corresponding elementary subgroup is perfect in these classical-like groups requires long and challenging calculations.

For more historical background on classical-like groups, we refer the reader to Hazrat and Vavilov [13].

In this article, we consider a group of orthogonal transformations defined and studied by Roy in his Ph.D. thesis (see [22]) generalizing the classical Eichler–Siegel transformations to commutative rings. These elementary transformations are defined for the quadratic spaces \(Q\,{\perp }\, {\mathbb {H}}(P)\) with a hyperbolic summand over a commutative ring in which 2 is invertible and we call the elementary orthogonal group generated by these transformations as Dickson–Siegel–Eichler–Roy or DSER elementary orthogonal group. We establish several commutator relations among these elementary orthogonal transformations. Using these relations, we prove that the DSER elementary orthogonal group \({{\text {EO}}}_A(Q,\mathbb {H}(A)^m)\) is perfect. We also prove an action version of the Quillen’s local-global principle for this group.

The commutator relations obtained here have been used in recent articles [2, 3] to establish a local-global principle for the DSER elementary orthogonal group over a polynomial extension and to show that the DSER elementary orthogonal group is normalised by the orthogonal group of smaller size and under certain stable range conditions, the DSER elementary orthogonal group is normal in the full orthogonal group. Further, in [2], we have proved a stability result for \(K_1\) of the orthogonal group using the commutator formulae proved in this article. Thus, the commutator calculus developed here have proved to be quite useful. This would serve as a starting point for exploring the lower K-theory of the DSER elementary orthogonal groups.

Remark 1.1

Even though we needed only commutator relations in the above applications most of the time, the relations themselves could be found only after computing the commutators explicitly. Thus, we are forced to work out the rather involved expressions for commutators appearing in this article. Our commutator formulae are done by hand although the shape emerged in a few small-dimensional cases using the computer algebra system GAP (see [11]).

2 Preliminaries

Let A be a commutative ring in which 2 is invertible. A quadratic space over A is a pair (M, q), where M is a finitely generated projective A-module and \(q:M\longrightarrow A\) is a non-singular (or non-degenerate) quadratic form. Let \(B_q\) be the symmetric bilinear form associated to q on M, which is given by \(B_q(x,y) = q(x+y)-q(x)-q(y)\) and \(d_{B_q} : M \rightarrow M^*\) be the induced isomorphism given by \(d_{B_q}(x)(y) = B_q(x,y)\), where \(x,y\in M\) and \(M^*\) is the dual of the module M.

A quadraticA-module is a pair (M, q), where M is an A-module and q is a quadratic form on M. Given two quadratic A-modules \((M_1,q_1)\) and \((M_2,q_2)\), their orthogonal sum is defined by \(M=M_1\oplus M_2\) and \(q((x_1,x_2))=q_1(x_1)+q_2(x_2)\) for \(x_1 \in M_1, x_2 \in M_2\). The orthogonal sum (M, q) is denoted by \((M_1,q_1)\!\perp \!(M_2,q_2)\) and the quadratic form q by \(q_1 \!\perp \! q_2\).

Let P be a finitely generated projective A-module. The module \(P \oplus P^*\) has a natural quadratic form given by \(p((x,f)) = f(x)\) for \(x\in P\), \(f\in P^*\). Notice that the associated bilinear form \(B_p\) is given by \(B_p((x_1,f_1),(x_2,f_2)) = f_1(x_2)+f_2(x_1)\) for \(x_1,x_2 \in P\) and \(f_1,f_2 \in P^*\). The quadratic space \((P \oplus P^*, p)\) is called the hyperbolic space of P and is denoted by \(\mathbb {H}(P)\). If P is a free A-module of rank n, then \(\mathbb {H}(P) \cong \mathbb {H}(A)^{n}\), where \(\mathbb {H}(A)\) is the hyperbolic plane.

Let Q be a quadratic A-space and P be a finitely generated projective A-module. Consider the quadratic space \(Q \perp \mathbb {H}(P)\) with the quadratic form \(q\!\perp \! p\). The associated bilinear form on \(Q\perp \mathbb {H}(P)\), denoted by \(\langle \cdot , \cdot \rangle \), is given by \(\langle (a,x),(b,y) \rangle = B_q((a,b)) + B_p((x,y)){\text { for all }} a,b \in Q {\text { and }} \,x,y \in \mathbb {H}(P)\), where \(B_q\) and \(B_p\) are the bilinear forms on Q and P respectively.

Let \({\text {Aut}}(M)\) denote the group of all A-linear automorphisms of M. The orthogonal group of a quadratic module \(M = M(B,q)\) is given by

$$\begin{aligned} {{\text {O}}}_A(M) = \{\sigma \in {\text {Aut}}(M)\mid q(\sigma (x))=q(x) {\text { for all }} x \in M\}. \end{aligned}$$

The dual map \(\alpha ^t\) of an A-linear map \(\alpha : Q \rightarrow P\)(or \(\beta : Q\rightarrow P^*\)), is defined by \(\alpha ^t(\varphi ) = \varphi \circ \alpha \) (or \(\beta ^t(\varphi ^*) = \varphi ^* \circ \beta \)) for \(\varphi \in P^*\) (or \(\varphi ^* \in P^{**}\)). Now the A-linear map \(\alpha ^* : P^*\rightarrow Q\) (\(\beta ^* : P\rightarrow Q\)) is defined by \(\alpha ^* = d_{B_q}^{-1}\circ \alpha ^t\) (\(\beta ^* = d_{B_q}^{-1}\circ \beta ^t \circ \varepsilon \), where \(\varepsilon \) is the natural isomorphism \(P\rightarrow P^{**}\)) and is characterized by the relation

$$\begin{aligned} (f\circ \alpha )(z) = B_q\left( \alpha ^*(f),z \right) {\text { for }} \,f\in P^*, z\in Q. \end{aligned}$$

In [22], A. Roy defined a pair of “elementary” orthogonal transformations \(E_{\alpha }, E_{\beta }^*\) of \(Q\!\perp \! \mathbb {H}(P)\) given by

$$\begin{aligned}\begin{array}{lllll} E_{\alpha }: (z,x,f) &{}\mapsto (z-\alpha ^*(f), x +\alpha (z)-\frac{1}{2}\alpha \alpha ^*(f),f), \\ E_{\beta }^*: (z,x,f) &{}\mapsto (z-\beta ^*(x), x, f-\beta (z)-\frac{1}{2}\beta \beta ^*(x)), \end{array}\end{aligned}$$

for \((z,x,f) \in Q\perp \mathbb {H}(P)\). Denote by \({{\text {EO}}}_{A}(Q, \mathbb {H}(P))\), the subgroup of the orthogonal group \(O_A(Q\perp \mathbb {H}(P))\) generated by the transformations of type \(E_{\alpha }\) and \(E_{\beta }^{*}\). We call this elementary orthogonal group Dickson–Siegel–Eichler–Roy (DSER) elementary orthogonal group.

We consider the case when Q and P are free A-modules of rank \(n \ge 1\) and m respectively. In this case, we fix the bases for Q, P (say, \(\{x_i : 1\le i\le m \}\) for P and \(\{f_i : 1\le i\le m \}\) for \(P^{*}\)) and we define the linear transformations \(\alpha _{i}, \alpha _{ij} \in {\text {Hom}}(Q,P)\) and \(\beta _{i}, \beta _{ij} \in {\text {Hom}}(Q,P^{*})\) for \(\alpha \in {\text {Hom}}(Q,P)\), \(\beta \in {\text {Hom}}(Q,P^*)\), \(1 \le i \le m\) and \(1\le j \le n\), as the maps given by

$$\begin{aligned} \alpha _{i}= & {} \eta _i\circ p_i\circ \alpha , \\ \alpha _{ij}= & {} \eta _i\circ p_i\circ \alpha \circ \eta _j\circ p_j, \\ \beta _{i}= & {} \eta _i\circ p_i\circ \beta \\ {\text {and}}\quad \quad \beta _{ij}= & {} \eta _i\circ p_i\circ \beta \circ \eta _j\circ p_j, \end{aligned}$$

where \(p_i: A^n \longrightarrow A\) be the projection onto the \(i^{th}\) component and \(\eta _i: A \longrightarrow A^n\) be the inclusion into the \(i^{th}\) component.

For \(1 \le i \le m\) and \(1 \le j \le n\), a basis \(\{f_i : 1\le i\le m \}\) for \(P^*\), the maps \(\alpha _i^*\) and \(\alpha _{ij}^*\) are given by

$$\begin{aligned} \begin{array}{llll} \alpha _i^*(f_j) = {\left\{ \begin{array}{ll} w_i &{}\quad {\text {if}}\quad j=i,\\ 0 &{}\quad {\text {if}}\quad j \ne i. \end{array}\right. } &{} &{} &{} \alpha _{ij}^*(f_k) = {\left\{ \begin{array}{ll} w_{ij}&{}\quad {\text {if}}\quad k=i,\\ 0 &{}\quad {\text {if}}\quad k \ne i. \end{array}\right. } \end{array} \end{aligned}$$

Similarly, the maps \(\beta _{i}^{*}\) and \(\beta _{ij}^*\) are given by

$$\begin{aligned} \begin{array}{llll} \beta _i^*(x_j) = {\left\{ \begin{array}{ll} v_i &{}\quad {\text {if}}\quad j=i,\\ 0 &{}\quad {\text {if}}\quad j \ne i. \end{array}\right. } &{} &{} &{} \beta _{ij}^*(x_k) = {\left\{ \begin{array}{ll} v_{ij}&{}\quad {\text {if}}\quad k=i,\\ 0 &{}\quad {\text {if}}\quad k \ne i. \end{array}\right. } \end{array} \end{aligned}$$

Here \(w_{i}\), \(v_{i}\) are elements of Q and \(w_{ij}\), \(v_{ij}\) are given by \(\eta _{j}\circ p_{j} (w_{i})\), \(\eta _{j}\circ p_{j} (v_{i})\) respectively.

Now with these definitions, the DSER elementary orthogonal transformations \(E_{{\alpha }_{ij}}\) and \(E_{{\beta }_{ij}}^*\) on \(Q\!\perp \! \mathbb {H}(P)\) on \(Q\perp \mathbb {H}(P)\) are given by

$$\begin{aligned} E_{\alpha _{ij}}(z,x,f) =&\Bigl (I-\alpha ^*_{ij}+\alpha _{ij}-\frac{1}{2}\alpha _{ij}\alpha ^*_{ij}\Bigr )(z,x,f)\\ =&\Bigl (z-\langle f,x_i \rangle w_{ij},\;x+\langle w_{ij},z\rangle x_i -\langle f,x_i \rangle q(w_{ij})x_i,\;f\Bigr )\\ E_{\beta _{ij}}^*(z,x,f) =&\Bigl (I-\beta _{ij}^*+\beta _{ij}-\frac{1}{2}\beta _{ij}\beta _{ij}^*\Bigr )(z,x,f)\\ =&\Bigl (z-\langle f_i,x \rangle v_{ij},\;x,\;f+\langle v_{ij},z\rangle f_i -\langle x,f_i \rangle q(v_{ij})f_i\;\Bigr ). \end{aligned}$$

Notation 2.1

Let G be a group and \(a,b \in G\). Then [a, b] denotes the commutator \(aba^{-1}b^{-1}\).

3 Commutators of elementary transformations

In this section, we establish various commutator relations among the elementary generators of the DSER elementary orthogonal group. We will carry out the computations in two different ways - one is by choosing bases (which we call the method using coordinates), and the other by just using the formal definition without choosing bases (which we call the coordinate-free method). By the ‘length’of a commutator, we mean the number of words in the commutator expression.

The following is a coordinate-free definition of the elementary generators.

Definition 3.1

For \(\theta \in {\text {Hom}}_A(Q ,P)\) or \({\text {Hom}}_A(Q, P^*)\), define \(\theta ^*\) as \(d_{B_q}^{-1}\circ \theta ^t\) or \(d_{B_q}^{-1}\circ \theta ^t \circ \varepsilon \), where \(\varepsilon \) is the natural isomorphism \(P\rightarrow P^{**}\) according to whether \(\theta \in {\text {Hom}}_A(Q,P)\) or \({\text {Hom}}_A(Q,P^*)\) respectively. Then the elementary transformations \(E_{\theta }\) and \(E_{\theta }^{-1}\) are given by

$$\begin{aligned} E_{\theta }&= I + \theta - \theta ^* - \frac{1}{2} \theta \theta ^*,\\ E_{\theta }^{-1}&= I - \theta + \theta ^* - \frac{1}{2} \theta \theta ^* = E_{(-\theta )}. \end{aligned}$$

The elementary generators are defined below using coordinates.

Definition 3.2

Let \(\alpha ,\delta \in {\text {Hom}}_A(Q,P)\); \(\beta ,\gamma \in {\text {Hom}}_A(Q,P^*)\) and \(w_i,t_i,v_i,c_i \in Q\) for \(1 \le i \le m\). Then, choosing the bases \(\{x_i\}_{i=1}^m, \{f_i\}_{i=1}^m,\{z_i\}_{i=1}^m\) respectively for \(P,P^*,Q\), one can define the following elements in \({\text {Hom}}_A(Q \perp \mathbb {H}(P))\).

$$\begin{aligned} \begin{array}{llllllll} \alpha _{ij}\left( z,x,f\right) &{}= &{} (0,\langle w_{ij},z\rangle x_i,0), &{} &{} \alpha ^{*}_{ij}\left( z,x,f\right) &{}= &{} (\langle f,x_i \rangle w_{ij},0,0),\\ \delta _{kl}\left( z,x,f\right) &{}= &{} (0,\langle t_{kl},z\rangle x_k,0), &{} &{} \delta ^{*}_{kl}\left( z,x,f\right) &{}= &{} (\langle f,x_k \rangle t_{kl},0,0),\\ \beta _{ij}(z,x,f) &{}= &{}(0,0,\langle v_{ij},z\rangle f_i), &{} &{} \beta ^{*}_{ij}(z,x,f) &{}= &{}(\langle x,f_i \rangle v_{ij},0,0),\\ \gamma _{kl}(z,x,f) &{}= &{}(0,0,\langle c_{kl},z\rangle f_k), &{} &{} \gamma ^{*}_{kl}(z,x,f) &{}= &{}(\langle x,f_k \rangle c_{kl},0,0). \end{array} \end{aligned}$$

Here \(w_{ij}, v_{ij}\) denote the elements \(\eta _j\circ p_j(w_i) , \eta _j\circ p_j(v_i) \) respectively and \(c_{kl}, t_{kl}\) denote the elements \(\eta _l\circ p_l(c_k),\eta _l\circ p_l(t_k)\).

Now, for \(1 \le i,k \le m\) and \(1 \le j,l \le n\), the corresponding DSER elementary orthogonal transformations \(E_{\alpha _{ij}}, E_{\delta _{kl}}, E_{\beta _{ij}}^*, E_{\gamma _{kl}}^*\) and their inverses have the following form.

$$\begin{aligned} E_{\alpha _{ij}}\left( z,x,f\right) =\;&\Bigl (z-\langle f,x_i \rangle w_{ij},\;x+\langle w_{ij},z\rangle x_i - \langle f,x_i \rangle q(w_{ij})x_i,\;f\Bigr ),\\ E_{\delta _{kl}}\left( z,x,f\right) =\;&\Bigl (z-\langle f,x_k \rangle t_{kl},\;x+\langle t_{kl},z\rangle x_k - \langle f,x_k \rangle q(t_{kl})x_k,\;f\Bigr ),\\ E_{\beta _{ij}}^*\left( z,x,f\right) =\;&\Bigl (z-\langle f_i,x \rangle v_{ij},\;x,\;f+\langle v_{ij},z\rangle f_i - \langle x,f_i \rangle q(v_{ij})f_i\Bigr ),\\ E_{\gamma _{kl}}^*\left( z,x,f\right) =\;&\Bigl (z-\langle f_k,x \rangle c_{kl},\;x,f+\langle c_{kl},z\rangle f_k - \langle x,f_k \rangle q(c_{kl})f_k\Bigr ),\\ E_{\alpha _{ij}}^{-1}\left( z,x,f\right) =\;&\Bigl (z+\langle f,x_i \rangle w_{ij} , x-\langle w_{ij},z\rangle x_i -\langle f,x_i \rangle q(w_{ij})x_i , f\Bigr ),\\ E_{\delta _{kl}}^{-1}\left( z,x,f\right) =\;&\Bigl (\;z+\langle f,x_k \rangle t_{kl} , x-\langle t_{kl},z\rangle x_k -\langle f,x_k \rangle q(t_{kl})x_k , f\Bigr ),\\ E_{\beta _{ij}}^{*^{-1}}\left( z,x,f\right) =\;&\Bigl (z+\langle f_i,x \rangle v_{ij},\;x,\;f-\langle v_{ij},z\rangle f_i - \langle x,f_i \rangle q(v_{ij})f_i\Bigr ),\\ E_{\gamma _{kl}}^{*^{-1}}\left( z,x,f\right) =\;&\Bigl (z+\langle f_k,x \rangle c_{kl},\;x,f-\langle c_{kl},z\rangle f_k - \langle x,f_k \rangle q(c_{kl})f_k\Bigr ). \end{aligned}$$

The first (and the simplest) set of commutators which we compute is between the DSER elementary generators corresponding to two elements of \({\text {Hom}}_A(Q,P)\); this is given in the following lemma.

Lemma 3.3

Let \(\alpha , \delta \in {\text {Hom}}_A(Q,P)\). Then, for i, j, k, l with \(1 \le i,k \le m\) and \({1 \le j,l \le n}\), the commutator of the type \([ E_{\alpha _{ij}}, E_{\delta _{kl}}]\) is given by

$$\begin{aligned} \Bigl [ E_{\alpha _{ij}}, E_{\delta _{kl}} \Bigr ](z,x,f) =&\Bigl ( I + \delta _{kl}\alpha ^{*}_{ij} - \alpha _{ij} \delta ^{*}_{kl} \Bigr )\left( z,x,f \right) \\ =&\Bigl (z,\;x + \langle f,x_i \rangle \langle t_{kl},w_{ij} \rangle x_k - \langle f,x_k \rangle \langle w_{ij}, t_{kl}\rangle x_i,\;f\Bigr ). \end{aligned}$$

In particular, if \(i=k\), then \(\Bigl [ E_{\alpha _{ij}}, E_{\delta _{kl}}\Bigr ] = I.\)

Proof

For \(\alpha , \delta \in {\text {Hom}}_A(Q,P)\) and for any i, j, k, l with \(1 \le i,k \le m\) and \(1 \le j,l \le n\), using the coordinate-free definition of the elementary generators, we have

$$\begin{aligned}&\Bigl [ E_{\alpha _{ij}}, E_{\delta _{kl}} \Bigr ](z,x,f)\\&\quad =E_{\alpha _{ij}} E_{\delta _{kl}}E^{-1}_{\alpha _{ij}} E^{-1}_{\delta _{kl}}(z,x,f)\\&\quad =E_{\alpha _{ij}} E_{\delta _{kl}}E^{-1}_{\alpha _{ij}}\Bigl (\Bigl ( I - \delta _{kl} + \delta _{kl}^* - \frac{1}{2} \delta _{kl}\delta _{kl}^* \Bigr )(z,x,f)\Bigr )\\&\quad = E_{\alpha _{ij}} E_{\delta _{kl}} \Bigl (\Bigl ( I - \delta _{kl} + \delta _{kl}^* - \frac{1}{2} \delta _{kl}\delta _{kl}^* - \alpha _{ij} + \alpha _{ij}^* - \frac{1}{2} \alpha _{ij}\alpha _{ij}^* - \alpha _{ij} \delta _{kl}^*\Bigr )(z,x,f)\Bigr )\\&\quad = E_{\alpha _{ij}}\Bigl (\Bigl ( I - \alpha _{ij} + \alpha _{ij}^* - \frac{1}{2} \alpha _{ij} \alpha _{ij}^* - \alpha _{ij} \delta _{kl}^* + \delta _{kl} \alpha _{ij}^*\Bigr )(z,x,f) \Bigr )\\&\quad = \Bigl ( I - \alpha _{ij} \delta _{kl}^* + \delta _{kl} \alpha _{ij}^* \Bigr )(z,x,f). \end{aligned}$$

Using coordinates, we can compute the above commutator as

$$\begin{aligned}&\Bigl [ E_{\alpha _{ij}}, E_{\delta _{kl}} \Bigr ](z,x,f)\\&\quad = E_{\alpha _{ij}} E_{\delta _{kl}}E^{-1}_{\alpha _{ij}} \Bigl (z + \langle f,x_k \rangle t_{kl} , \; x-\langle t_{kl},z\rangle x_k -\langle f,x_k \rangle q(t_{kl})x_k ,\; f \Bigr )\\&\quad = E_{\alpha _{ij}} E_{\delta _{kl}} \Bigl (z+ \langle f , x_i\rangle w_{ij} +\langle f,x_k \rangle t_{kl},\; x-\Bigl \{ \langle w_{ij}, z \rangle + \langle f , x_i \rangle q\left( w_{ij}\right) \Bigr .\Bigr .\\&\qquad \quad \Bigl .\Bigl . +\langle f,x_k \rangle \langle w_{ij},t_{kl} \rangle \Bigr \} x_i -\Bigl \{\langle t_{kl},z\rangle + \langle f,x_k \rangle q(t_{kl})\Bigr \}x_k,\; f \Bigr )\\&\quad = E_{\alpha _{ij}} \Bigl (\,z+\langle f,x_i \rangle w_{ij},\;x- \Bigl \{\langle w_{ij},z\rangle + q(w_{ij})\langle f , x_i \rangle +\langle f,x_k \rangle \langle w_{ij},t_{kl} \rangle \Bigr \}x_i \Bigr .\Bigr .\\&\qquad \quad \Bigl .\Bigl .+ \langle f,x_i \rangle \langle t_{kl},w_{ij} \rangle x_k , f \Bigr )\\&\quad = \Bigl (\, z,\,x+\langle f,x_i \rangle \langle t_{kl},w_{ij} \rangle x_k - \langle f,x_k \rangle \langle w_{ij}, t_{kl}\rangle x_i,\, f \Bigr ). \end{aligned}$$

If \(i = k\), then we have

$$\begin{aligned} \delta _{kl} \alpha ^{*}_{ij}(z,x,f) = \Bigl (0,\langle f,x_i \rangle \langle t_{il},w_{ij} \rangle x_i,0 \Bigr ) = \alpha _{ij} \delta ^{*}_{kl}(z,x,f). \end{aligned}$$

Hence \(\Bigl [ E_{\alpha _{ij}}, E_{\delta _{il}} \Bigr ] = I.\)\(\square \)

As a consequence of this lemma, we have the following commutator relations.

Corollary 3.4

For any i, j, k, l with \(1 \le i,k \le m\), \(1 \le j,l \le n\) and for \(a,b,c,d \in A\) with \(ab=cd\), the following equation holds.

$$\begin{aligned} \Bigl [E_{a\alpha _{ij}}, E_{b\delta _{kl}}\Bigr ] = \Bigl [E_{c\alpha _{ij}}, E_{d\delta _{kl}}\Bigr ]. \end{aligned}$$

Proof

For \(\alpha ,\delta \in {\text {Hom}}_A(Q,P)\) and for any i, j, k, l with \(1 \le i,k \le m\), \(1 \le j,l \le n\) and \(a,b,c,d \in A\) with \(ab = cd\), we have

$$\begin{aligned} \Bigl [E_{a\alpha _{ij}}, E_{b\delta _{kl}}\Bigr ] =\;&I - ab \alpha _{ij} \delta _{kl}^* + ab\delta _{kl} \alpha _{ij}^* \quad \quad (\mathrm{{by\ Lemma}}~3.3)\\ =\;&I - cd\alpha _{ij} \delta _{kl}^* + cd \delta _{kl} \alpha _{ij}^* = \Bigl [E_{c\alpha _{ij}}, E_{d\delta _{kl}}\Bigr ]. \end{aligned}$$

\(\square \)

Remark 3.5

Since we will be using similar calculations to find the commutators in the rest of the article, we will give only the final expression for the commutators and for the proofs we refer the reader to the preprint [1] placed in arXiv.

We now compute the ‘mixed commutator’ of the DSER elementary orthogonal generators corresponding to the elements of \({\text {Hom}}_A(Q,P)\) and \({\text {Hom}}_A(Q,P^*)\). The general expression for the commutator is complicated, but we need only its special case \(i \ne k\).

Lemma 3.6

Let \(\alpha \in {\text {Hom}}_A(Q,P)\) and \(\beta \in {\text {Hom}}_A(Q,P^*)\). Then, for i, j, k, l with \(1 \le i,k \le m\) and \(1 \le j,l \le n\) with \(i \ne k\),

$$\begin{aligned} \Bigl [E_{\alpha _{ij}},E_{\beta _{kl}}^{*}\Bigr ](z,x,f) =\;&\Bigl (I-\alpha _{ij} \beta ^{*}_{kl}+ \beta _{kl}\alpha ^{*}_{ij}\Bigr )(z,x,f)\\ =\;&\Bigl (\;z,\; x-\langle x,f_k\rangle \langle w_{ij},v_{kl}\rangle x_i,\; f+\langle f,x_i\rangle \langle v_{kl},w_{ij}\rangle f_k\;\Bigr ). \end{aligned}$$

Proof

Similar to Lemma 3.3 (see [1] for details). \(\square \)

The following corollary lists the resultant commutator relations from the above lemma.

Corollary 3.7

For any i, j, k, l with \(1 \le i,k \le m\), \(1 \le j,l \le n\), \(i \ne k\) and for \(a,b,c,d \in A\) with \(ab=cd\), the following equation holds.

$$\begin{aligned} \Bigl [ E_{a\alpha _{ij}},E_{b\beta _{kl}}^{*} \Bigr ] = \Bigl [ E_{c\alpha _{ij}},E_{d\beta _{kl}}^{*}\Bigr ]. \end{aligned}$$

The lemma below computes the commutator of the DSER elementary orthogonal generators corresponding to two elements of \({\text {Hom}}_A(Q,P^*)\).

Lemma 3.8

Let \(\beta , \gamma \in {\text {Hom}}_A(Q,P^*)\). Then, for i, j, k, l with \(1 \le i,k \le m\) and \(1 \le j,l \le n\), the commutator \([E_{\beta _{ij}}^*, E_{\gamma _{kl}}^*]\) is given by

$$\begin{aligned} \Bigl [E_{\beta _{ij}}^*, E_{\gamma _{kl}}^* \Bigr ](z,x,f) =\;&\Bigl ( I + \gamma _{kl} \beta ^{*}_{ij} - \beta _{ij} \gamma ^{*}_{kl} \Bigr )(z,x,f)\\ =\;&\Bigl (z,\; x,\; f + \langle x,f_i \rangle \langle c_{kl},v_{ij} \rangle f_k - \langle x,f_k \rangle \langle v_{ij}, c_{kl}\rangle f_i \Bigr ). \end{aligned}$$

In particular, if \(i=k\), then \([E_{\beta _{ij}}^*, E_{\gamma _{kl}}^*] = I.\)

Proof

Similar to Lemma 3.3 (see [1] for details). \(\square \)

Immediately, we can deduce the following commutator relations.

Corollary 3.9

For any i, j, k, l with \(1 \le i,k \le m\), \(1 \le j,l \le n\) and for \(a,b,c,d \in A\) with \(ab=cd\), the following equation holds.

$$\begin{aligned} \Bigl [E_{a\beta _{ij}}^*, E_{b\gamma _{kl}}^* \Bigr ]= \Bigl [ E_{c\beta _{ij}}^*, E_{d\gamma _{kl}}^* \Bigr ]. \end{aligned}$$

Remark 3.10

Observe that all the lemmas in this section are valid for the transformations \(E_{\alpha _{i}}\), \(E_{\beta _{i}}^{*}\) also.

In the following sections, we will prove more complicated commutator relations of lengths 10 and 16; we will take the indices such that the commutator is non-trivial.

4 Triple commutators

In this section, we state certain triple commutator relations among the elementary generators of the DSER elementary orthogonal group. We start with a commutator of length 10 which involves a commutator of the DSER elementary orthogonal generators corresponding to two elements of \({\text {Hom}}_A(Q,P)\) and a triple commutator which involves a mixed commutator. We refer the reader to [1] in arXiv for the proofs.

Lemma 4.1

Let \(\alpha , \delta \in {\text {Hom}}_A(Q,P)\) and \(\beta , \gamma \in {\text {Hom}}_A(Q,P^*)\). Then, for i, j, k, l, p, q with \(1 \le i,k,p \le m\), \(1 \le j,l,q \le n\) and \(k \ne p\);

  1. (i)

    the triple commutator \(\left[ E_{\beta _{ij}}^*,\left[ E_{\alpha _{kl}},E_{\delta _{pq}} \right] \right] \) is given by

    $$\begin{aligned} \left[ E_{\beta _{ij}}^*,\left[ E_{\alpha _{kl}},E_{\delta _{pq}} \right] \right] = {\left\{ \begin{array}{ll} E_{\lambda _{kj}}\left[ E_{\beta _{ij}}^*,E_\frac{\lambda _{kj}}{2} \right] &{}\quad {\text {if}}\quad i = p,\\ E_{\xi _{pj}}\left[ E_{\beta _{ij}}^*,E_\frac{\xi _{pj}}{2} \right] &{}\quad {\text {if}}\quad i = k,\\ I &{} \quad {\text {if}}\quad i \ne p {\text { and }} i \ne k, \end{array}\right. } \end{aligned}$$

    where \(\lambda _{kj}\;=\;\alpha _{kl}\delta _{pq}^*\beta _{ij}\) and \(\xi _{pj}\;=\;-\delta _{pq}\alpha _{kl}^*\beta _{ij}\),

  2. (ii)

    the triple commutator \([ E_{\alpha _{ij}},[ E_{\delta _{kl}},E_{\beta _{pq}}^*] ]\) is given by

    $$\begin{aligned} \left[ E_{\alpha _{ij}},\left[ E_{\delta _{kl}},E_{\beta _{pq}}^* \right] \right] = {\left\{ \begin{array}{ll} E_{\mu _{kj}}\left[ E_{\alpha _{ij}},E_\frac{\mu _{kj}}{2} \right] , &{}{\text { if }}\quad i = p,\\ I &{}{\text { if }}\quad i = k \quad {\text { or }}\quad i \ne p, \end{array}\right. } \end{aligned}$$

    where \(\mu _{kj}=\delta _{kl}\beta _{pq}^*\alpha _{ij}\),

  3. (iii)

    the triple commutator \([E_{\beta _{ij}}^*,[ E_{\alpha _{kl}}, E_{\gamma _{pq}}^* ] ]\) is given by

    $$\begin{aligned} \left[ E_{\beta _{ij}}^*,\left[ E_{\alpha _{kl}}, E_{\gamma _{pq}}^* \right] \right] = {\left\{ \begin{array}{ll} E_{\nu _{pj}}^* \left[ E_{\beta _{ij}}^*,E_\frac{\nu _{pj}}{2}^* \right] , &{}{\text { if }}\quad i = p,\\ I &{}{\text { if }}\quad i = k \quad {\text { or }}\quad i \ne p, \end{array}\right. } \end{aligned}$$

    where \(\nu _{pj}= -\gamma _{pq}\alpha _{kl}^*\beta _{ij}\), and

  4. (iv)

    the triple commutator \([ E_{\alpha _{ij}},[ E_{\beta _{kl}}^*,E_{\gamma _{pq}}^* ] ]\) is given by

    $$\begin{aligned} \left[ E_{\alpha _{ij}},\left[ E_{\beta _{kl}}^*,E_{\gamma _{pq}}^* \right] \right] = {\left\{ \begin{array}{ll} E_{\eta _{kj}}^* \left[ E_{\alpha _{ij}},E_\frac{\eta _{kj}}{2}^*\right] &{}\quad {\text {if}}\quad i = p,\\ E_{\vartheta _{pj}}^*\left[ E_{\alpha _{ij}},E_\frac{\vartheta _{pj}}{2}^* \right] &{}\quad {\text {if}}\quad i = k,\\ I &{}\quad {\text {if}}\quad i \ne p {\text { and }} i \ne k, \end{array}\right. } \end{aligned}$$

    where \(\eta _{kj}=\beta _{kl}\gamma _{pq}^*\alpha _{ij}\) and \(\vartheta _{pj} = \gamma _{pq}\beta _{kl}^*\alpha _{ij}\).

As a consequence of the above lemma on triple commutators, we observe the following commutator relations.

Corollary 4.2

For any i, j, k, l, p, q with \(1 \le i,k,p \le m\), \(1 \le j,l,q \le n\), \(i \ne k\) and \(k \ne p\) and \(a,b,c,d,e,f \in A\) with \(abc=def\) and \(a^2bc=d^2ef\), the following equations hold.

  1. (i)

    \(\left[ E_{a\beta _{ij}}^*, \left[ E_{b\alpha _{kl}},E_{c\delta _{pq}} \right] \right] = \left[ E_{d\beta _{ij}}^*,\left[ E_{e\alpha _{kl}}, E_{f\delta _{pq}} \right] \right] ,\)

  2. (ii)

    \(\left[ E_{a\alpha _{ij}}, \left[ E_{b\delta _{kl}},E_{c\beta _{pq}}^* \right] \right] = \left[ E_{d\alpha _{ij}}, \left[ E_{e\delta _{kl}}, E_{f\beta _{pq}}^* \right] \right] ,\)

  3. (iii)

    \(\left[ E_{a\beta _{ij}}^*,\left[ E_{b\gamma _{kl}}^*,E_{c\alpha _{pq}} \right] \right] = \left[ E_{d\beta _{ij}}^*,\left[ E_{e\gamma _{kl}}^*,E_{f\alpha _{pq}} \right] \right] ,\)

  4. (iv)

    \(\left[ E_{a\alpha _{ij}},\left[ E_{b\beta _{kl}}^*,E_{c\gamma _{pq}}^*\right] \right] =\left[ E_{d\alpha _{ij}},\left[ E_{e\beta _{kl}}^*,E_{f\gamma _{pq}}^*\right] \right] .\)

5 Multiple commutators

In this section, we state some four-fold commutator formulae. For the proofs, we refer to [1].

Lemma 5.1

Let \(\alpha ,\delta , \xi \in {\text {Hom}}_A(Q,P)\) and \(\beta ,\gamma , \mu \in {\text {Hom}}_A(Q,P^*)\). Then, for i, j, k, l, r, s, p, q with \(1 \le i,k,r,p \le m\), \(1 \le j,l,s,q \le n\), \(i \ne k\) and \(r \ne p\),

  1. (i)

    the four-fold commutator \([[ E_{\beta _{ij}}^*,E_{\gamma _{kl}}^* ], [ E_{\alpha _{rs}}, E_{\mu _{pq}}^* ]]\) is given by

    $$\begin{aligned} \left[ [E_{\beta _{ij}}^*,E_{\gamma _{kl}}^*], [E_{\alpha _{rs}}, E_{\mu _{pq}}^*]\right] = {\left\{ \begin{array}{ll} \left[ E_{\mu _{pq}\alpha _{rs}^*}^*, E_{\beta _{ij}\gamma _{kl}^*}^* \right] &{}\quad {\text {if}}\quad k=r,\\ \left[ E_{\gamma _{kl}\beta _{ij}^*}^*,E_{\mu _{pq}\alpha _{rs}^*}^* \right] &{}\quad {\text {if}}\quad i = r,\\ I &{} {\text { otherwise }}. \\ \end{array}\right. } \end{aligned}$$
  2. (ii)

    the four-fold commutator \([[E_{\alpha _{ij}},E_{\delta _{kl}}],[E_{\xi _{rs}}, E_{\beta _{pq}}^* ]]\) is given by

    $$\begin{aligned} \left[ [E_{\alpha _{ij}},E_{\delta _{kl}}], [E_{\xi _{rs}}, E_{\beta _{pq}}^* ]\right] = {\left\{ \begin{array}{ll} \left[ E_{\delta _{kl}\alpha _{ij}^*}, E_{\xi _{rs}\beta _{pq}^*} \right] &{}{\text { if }}\quad i = p,\\ \left[ E_{\alpha _{ij}\delta _{kl}^*}, E_{\xi _{rs}\beta _{pq}^*} \right] &{}{\text { if }}\quad k = p,\\ I &{} {\text { otherwise }}. \end{array}\right. } \end{aligned}$$

Lemma 5.2

Let \(\alpha , \delta \in {\text {Hom}}_A(Q,P)\) and \(\beta ,\gamma \in {\text {Hom}}_A(Q,P^*)\). Then, for any i, j, k, l, r,  s, p, q with \(1 \le i,k,r,p \le m\), \(1 \le j,l,s,q \le n\), \(i \ne k\) and \(r \ne p\), the four-fold commutator \([ [E_{\alpha _{ij}}, E_{\beta _{kl}}^*], [ E_{\delta _{rs}}, E_{\gamma _{pq}}^*] ]\)is given by

$$\begin{aligned} \left[ [E_{\alpha _{ij}}, E_{\beta _{kl}}^*], [ E_{\delta _{rs}},E_{\gamma _{pq}}^*] \right] = {\left\{ \begin{array}{ll} {\left[ E_{\alpha _{ij}\beta _{kl}^*},E_{\gamma _{pq}\delta _{rs}^*}^*\right] }^{-1} &{} {\text { if }}\quad k=r {\text { and }} i \ne p, \\ \left[ E_{\delta _{rs}\gamma _{pq}^*}, E_{\beta _{kl}\alpha _{ij}^*}^* \right] &{} {\text { if }}\quad i = p {\text { and }} k \ne r,\\ I &{} {\text { if }}\quad k \ne r {\text { and }} i \ne p. \end{array}\right. } \end{aligned}$$

Lemma 5.3

Let \(\alpha , \delta , \xi , \mu \in {\text {Hom}}_A(Q,P)\) and \(\beta , \gamma , \eta , \nu \in {\text {Hom}}_A(Q,P^{*})\). Then, for i, j, k, l, r, s, p, q with \({1 \le i,k,r,p \le m}\), \(1 \le j,l,s,q \le n\), \(i \ne k\) and \(r \ne p\),

  1. (i)

    the four-fold commutator \([ [E_{\alpha _{ij}},E_{\delta _{kl}}], [ E_{\xi _{rs}}, E_{\mu _{pq}}] ]\) is given by

    $$\begin{aligned} \left[ [E_{\alpha _{ij}},E_{\delta _{kl}}], [ E_{\xi _{rs}}, E_{\mu _{pq}}] \right] = I . \end{aligned}$$
  2. (ii)

    the four-fold commutator \([ [E_{\beta _{ij}}^*,E_{\gamma _{kl}}^*], [E_{\eta _{rs}}^*, E_{\nu _{pq}}^*]]\) is given by

    $$\begin{aligned} \left[ [E_{\beta _{ij}}^*,E_{\gamma _{kl}}^*], [ E_{\eta _{rs}}^*, E_{\nu _{pq}}^*] \right] = I. \end{aligned}$$

Lemma 5.4

Let \(\alpha , \delta \in {\text {Hom}}_A(Q,P)\) and \(\beta ,\gamma \in {\text {Hom}}_A(Q,P^*)\). Then, for any i, j, k, l, r,  s, p, q with \(1 \le i,k,r,p \le m\), \(1 \le j,l,s,q \le n\), \(i \ne k\) and \(r \ne p\), the four-fold commutator \([ [E_{\alpha _{ij}}, E_{\delta _{kl}}], [ E_{\beta _{rs}}^*,E_{\gamma _{pq}}^*] ]\) is given by

$$\begin{aligned} \left[ [E_{\alpha _{ij}}, E_{\delta _{kl}}], [ E_{\beta _{rs}}^*,E_{\gamma _{pq}}^*] \right] = {\left\{ \begin{array}{ll} {\left[ E_{\alpha _{ij}\beta _{kl}^*},E_{\gamma _{pq}\delta _{rs}^*}^*\right] }^{-1} &{} {\text { if }}\quad k=r {\text { and }} i \ne p, \\ \left[ E_{\delta _{rs}\gamma _{pq}^*}, E_{\beta _{kl}\alpha _{ij}^*}^* \right] &{} {\text { if }}\quad i = p {\text { and }} k \ne r,\\ I &{} {\text { if }}\quad k \ne r {\text { and }} i \ne p. \end{array}\right. } \end{aligned}$$

6 Applications of the commutator calculus

In this section, we will give some applications of the commutator calculus which we have established.

6.1 The perfectness of the DSER elementary orthogonal group \({{\text {EO}}}_A(Q, \mathbb {H}(A)^m)\)

We observe that the DSER elementary orthogonal \({{\text {EO}}}_A(Q, {\mathbb {H}}(A)^m)\) is perfect. That is, \({{\text {EO}}}_A(Q, {\mathbb {H}}(A)^m)\) coincides with its own commutator subgroup. First, we recall a lemma from [25].

Lemma 6.1

[25, Lemma 1.4] The group \({{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \) is generated by \(E_i{(\alpha )}\)\((1\le i\le m)\), where \(\alpha \in {\text {Hom}}_A(Q,A)\).

Lemma 6.2

[3, Lemma 3.4] The group \({{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \) is generated by \(E({\alpha _{ij}})\)\((1\le i \le m\) and \(1\le j \le n)\) with \(\alpha \in {\text {Hom}}_A(Q,A^{m})\) or \({\text {Hom}}_A\left( Q,({A}^{m})^{*}\right) \).

Theorem 6.3

If \(m\ge 2\), then the group \({{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \) is perfect.

Proof

To prove \([{{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) , {{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) ] = {{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \), we need to prove that any element in \({{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \) can be written as a commutator. This follows from the commutator relation proved in the previous sections.

Since \({{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \) is generated by elementary transformations of the type \(E_{\alpha _{ij}}\) and \(E_{\beta _{ij}}^*\) by Lemma 6.2, it is enough to show that these transformations can be written as commutators of elements of \({{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \). By triple commutator relations in Sect. 4, we can write the transformations \(E_{\alpha _{ij}}\) and \(E_{\beta _{ij}}^*\) as products of commutators of elements of the group \({{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \).

By the triple commutators, we have

$$\begin{aligned} E_{\alpha _{ij}}= & {} E_{\delta _{il}\beta _{pq}^{*}\mu _{pj}} = \left[ E_{\mu _{pj}},\left[ E_{\delta _{il}},E_{\beta _{pq}}^* \right] \right] \left[ E_{\frac{\delta _{il}\beta _{pq}^{*}\mu _{pj}}{2}},E_{\mu _{pj}} \right] , \end{aligned}$$
(1)
$$\begin{aligned} E_{\beta _{ij}}^{*}= & {} E_{\eta _{il}\gamma _{pq}^{*}\alpha _{pj}}^{*} = \left[ E_{\alpha _{pj}},\left[ E_{\eta _{il}}^{*},E_{\gamma _{pq}}^* \right] \right] \left[ E_{\frac{\eta _{il}\gamma _{pq}^{*}\alpha _{pj}}{2}}^{*},E_{\alpha _{pj}} \right] , \end{aligned}$$
(2)

where \(\alpha ,\delta ,\mu \in {\text {Hom}}_A(Q,A^{m})\) and \(\beta ,\gamma ,\eta \in {\text {Hom}}_A\left( Q,({A}^{m})^{*}\right) \). Thus the elements \(E_{\alpha _{ij}}\) and \(E_{\beta _{ij}}^*\) belong to the commutator subgroup \(\left[ \,{{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) , {{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \,\right] .\) Hence \({{\text {EO}}}_A\left( Q, {\mathbb {H}}(A)^m\right) \) is perfect. \(\square \)

Remark 6.4

The condition \(m \ge 2\) in the above theorem is necessary in order to have non-trivial commutator relations. Note that we can choose different triple commutators from Lemma 4.1 to represent the transformations \(E_{\alpha _{ij}}\) and \(E_{\beta _{ij}}^{*}\).

6.2 Action version of local-global principle

Here, we prove an “action version” of analogue of Quillen’s local-global principle for the DSER elementary orthogonal group \({{\text {EO}}}_A(Q, {\mathbb {H}}(A)^m)\). We begin by recalling some known results in this direction. In a letter to Bass, Vaserstein proved the following action version of Quillen’s well-known local-global principle.

Theorem 6.5

[16, Chapter III, Theorem 2.5] Let \(n \ge 3\) and \(\nu \left( X \right) \in {{\text {Um}}}_n\left( A[X] \right) \). If \(\nu \left( X \right) \in {{\text {GL}}}_n\left( A_{\mathfrak {m}}[X] \right) \), for all maximal ideals \(\mathfrak {m}\) of A, then \(\nu \left( X \right) \in \nu \left( 0 \right) {{\text {GL}}}_n\left( A[X] \right) \).

A result similar to the one above was proved for the elementary linear group by Rao which is the following.

Theorem 6.6

[21, Theorem 2.3] Let \(\nu \left( X \right) \in {{\text {Um}}}_n\left( A[X] \right) , n \ge 3\). Suppose, for all maximal ideals \(\mathfrak {m}\) in A, \(\nu \left( X \right) \in \nu \left( 0 \right) {{\text {E}}}_n\left( A_{\mathfrak {m}}[X] \right) \). Then \(\nu \left( X \right) \in \nu \left( 0\right) {{\text {E}}}_n\left( A[X] \right) \).

Similar results are also proved in [5, 7, 9]. More generalized results of the action version of local-global principle for Chevalley groups are established in [6, 23].

In [22], A. Roy proved the following result.

Theorem 6.7

[22] Let A be a commutative Noetherian ring and \(d = \dim \; {\text {Max}}A < \infty \). Let P be a finitely generated projective A-module of \({\text {rank}}\ge d+1\), and Q a quadratic A-space. Let \(\mathfrak {a}\) be an ideal of A and \(w \in Q\!\perp \!\mathbb {H}(P)\) such that \(Aq(w) + \mathfrak {a} = A\). Then there exist A-linear maps \(\alpha _1, \ldots , \alpha _n : Q \rightarrow P\) such that

$$\begin{aligned} o(\text{ P-component } \text{ of } E_{\alpha _n}\circ \cdots \circ E_{\alpha _1} (w)) + \mathfrak {a} = A \end{aligned}$$

In [17], R. Parimala extended this result for generalised dimension.

Theorem 6.8

[17, Theorem 3.1] Let A be a commutative ring and d be a generalised dimension function on \({\text {Spec}}A\). Let \(\left( Q_0,q_0 \right) \) be a quadratic A-space and let \({Q\!=\! Q_0\!\perp \!\mathbb {H}(P)}\), where P is a finitely generated projective A-module of \({\text {rank}}\ge d\left( A \right) + 1\). Let \(w = \left( z,x,f \right) \) be an element in \(Q_0\!\perp \!\mathbb {H}(P)\) such that \(q\left( w \right) = q_0\left( z \right) + f\left( x \right) \) is a unit in A. Then there exists \(\eta = E_{\alpha _1} \circ E_{\alpha _2} \circ \cdots \circ E_{\alpha _n} \in {{\text {EO}}}_A\left( Q_0, H\left( P \right) \right) \) such that \(\eta \left( z,x,f \right) = \left( z',x',f' \right) \) with \(x'\) unimodular in P.

The above result states that elements of unit norm in a quadratic space of sufficiently large Witt index can be brought into general position by elementary orthogonal transformations. This can be considered as a quadratic analogue of a stability theorem of Eisenbud-Evans [10, Theorem A (ii)b].

Rao, in his Ph.D. thesis (1984), raised the following question.

Question 6.9

Is there a “local- global” principle for the action of the elementary orthogonal group \({{\text {EO}}}_{A[T]}\left( Q\otimes A[T], \mathbb {H}(A[T])^n \right) \) on non-singular elements? Explicitly, let \(\left( Q,q \right) \) be a quadratic A-space and let w be a non-singular element in \(\left( Q\!\perp \mathbb {H}(A)^n \right) \otimes A[T]\). Assume that, for all \(\mathfrak {m}\!\in \!{\text {Max}}\left( A \right) \), there exists an element \(\sigma _{\mathfrak {m}}\in {{\text {EO}}}_{A_{\mathfrak {m}}[T]}\left( Q\otimes A_{\mathfrak {m}}[T], \mathbb {H}(A_{\mathfrak {m}}[T])^n \right) \) such that

$$\begin{aligned} \sigma _{\mathfrak {m}}w = w\left( 0 \right) {{\text {EO}}}_{A[T]}\left( Q\otimes A[T], \mathbb {H}(A[T])^n \right) . \end{aligned}$$

Does there exist an element \(\sigma \) in \({{\text {EO}}}_{A[T]}\left( Q\otimes A[T], \mathbb {H}(A[T])^n \right) \) with \(\sigma w = w\left( 0 \right) \)?

In this paper, we give an affirmative answer to this question.

Let Q and P be free A-modules of rank n and m respectively. The main theorem of this section is:

Theorem 6.10

Let A be a commutative ring and d be a generalized dimension function on \({\text {Spec}}A\). Let \(\left( Q,q \right) \) be a quadratic A-space and let \(M = Q\!\perp \!\mathbb {H}(A)^m\), where m is at least \(d(A)+1\). Let \(\left. w \in \left( Q\!\perp \!\mathbb {H}(A)^m \right) \otimes A[T] \right. \) be non-singular. Assume that, for all \(\mathfrak {m} \in {\text {Max}}\left( A \right) \), there exists an element \({{\text {EO}}}_{A_{\mathfrak {m}}[T]} (Q\otimes A_{\mathfrak {m}}[T], \mathbb {H}(A_{\mathfrak {m}}[T])^m )\) such that \(\left. \sigma _{\mathfrak {m}}w = w\left( 0 \right) {{\text {EO}}}_{A[T]}\left( Q\otimes A[T],\mathbb {H}(A[T])^m \right) \right. \). Then there exists an element \(\sigma \) in the elementary group \({{\text {EO}}}_{A[T]}\left( Q\otimes A[T], \mathbb {H}(A[T])^m \right) \)  with  \(\sigma w = w(0)\).

In the following lemma, we use a standard argument of L.N. Vaserstein (see [16, Chapter III, Proposition 2.3]).

Lemma 6.11

Let S be a multiplicatively closed set in A and let \(n+2m \ge 6\). Let \(w(X) \in {{\text {Um}}}_{n+2m} (A[X])\) and let \(w(X) \in w(0)\;{{\text {EO}}}_{A[X]} (Q\otimes A[X], \mathbb {H}(A[X])^m )\). Then there is an element s in S such that, for any a in A,

$$\begin{aligned} w\left( X + asT \right) \in w(X)\;{{\text {EO}}}_{A[X,T]} (Q\otimes A[X,T], \mathbb {H}(A[X,T])^m ). \end{aligned}$$

Proof

Let \(\vartheta (X) \in {{\text {EO}}}_{A_{S}[X]} (Q\otimes A_{S}[X], \mathbb {H}(A_{S}[X])^m )\) such that \(w(X)\vartheta (X) = w(0).\) Let

$$\begin{aligned} \theta (X,T) \;=\; \vartheta (X +T)\vartheta (X)^{-1} \;\in \;{{\text {EO}}}_{A_{S}[X,T]} (Q\otimes A_{S}[X,T], \mathbb {H}(A_{S}[X,T])^m ). \end{aligned}$$

Then

$$\begin{aligned} w(X+T)\theta (X,T)&= w(X+T)\vartheta (X+T)\vartheta (X)^{-1}\\&= w(0)\vartheta (X)^{-1} \\&= w(X) \in A_S [X,T ]^{n+2m}. \end{aligned}$$

Since \(\theta (X,0) = I\), we can find \(\theta ^*(X,T) \in {{\text {EO}}}_{A[X,T]} (Q\otimes A[X,T], \mathbb {H}(A[X,T])^m )\) which localizes to \(\theta (X, sT )\) for some \(s \in S\) with \(\theta ^* (X, 0) = I\) (by applying Dilation Lemma to the base ring A[X] ). Then in \(A[X, T ]^{n}\), we have

$$\begin{aligned} w(X + sT){\theta }^*(X,T) w(X) = T v(X,T) \end{aligned}$$

for some v(X, T) which localizes to 0. Thus, for some \(s^* \in S\) and for all \(a \in A\), we get

$$\begin{aligned} w(X + ass^* T ){\theta }^* (X, as^* T ) - w(X) = T as^* v(X, as^* T ) = 0. \end{aligned}$$

\(\square \)

Proof of Theorem 6.10

Let w be a non-singular element in \(\left( Q\!\perp \!\mathbb {H}(A)^m \right) \otimes A[T]\). By Theorem 6.8, there exists an element \(\eta \in {{\text {EO}}}_{A}\left( Q, \mathbb {H}(A)^m \right) \) such that \(\eta \left( w \right) \) has its P-component unimodular in P. This implies that the order ideal

$$\begin{aligned} o\left( P\text{- } \text{ component } \left( \eta \left( w \right) \right) \right) = A. \end{aligned}$$

which in turn implies that \(o\left( \eta \left( w \right) \right) = A.\) Hence \(\eta \left( w \right) \) is unimodular in \(Q\!\perp \!\mathbb {H}(A)^m\).

Let \(n+2m \ge 6\). Let \(w(X) \in {{\text {Um}}}_{n+2m}(A[X])\). If, for all maximal ideals \(\mathfrak {m}\) of A, \(w(X)_{\mathfrak {m}} \in {w(0)_\mathfrak {m}}\, {{\text {EO}}}_{A_{\mathfrak {m}}[X]}(Q \otimes A_{\mathfrak {m}}[X], {\mathbb {H}}(A_{\mathfrak {m}}[X])^m)\). Using Lemma 6.11 it follows that, for each maximal ideal \(\mathfrak {m}\) of A, there exists \(s_k \in A \backslash \mathfrak {m}\) such that, for all \(a \in A\),

$$\begin{aligned} w(X + as_kT ) \in w(X)\; {{\text {EO}}}_{A[X,T]} (Q\otimes A[X,T], \mathbb {H}(A[X,T])^m ). \end{aligned}$$
(3)

We note that the ideal generated by \(s_k's\) is the whole ring A. Therefore there exist elements \(s_{k_1} ,\ldots , s_{k_r}\) in \(A\backslash \mathfrak {m}\) such that \(a_1s_{k_1} + \cdots + a_r s_{k_r} = 1\), where \(a_i \in A\) for \(1 \le i \le r.\) In Eq. (3), replacing X by \(a_2 s_{k_2} X + \cdots + a_r s_{k_r} X\) and \(a_{s_k} T\) by \(a_1 s_{k_1}X\), we get

$$\begin{aligned} w(X)&= w(a_1 s_{k_1} X + a_2 s_{k_2} X + \cdots + a_r s_{k_r} X) \\&\in w(a_2 s_{k_2} X + \cdots + a_r s_{k_r} X)\; {{\text {EO}}}_{A[X]} (Q\otimes A[X], \mathbb {H}(A[X])^m ). \end{aligned}$$

Again in Eq. (3), replacing X by \(a_3 s_{k_3} X + \cdots + a_r s_{k_r} X\) and \(a_{s_k} T\) by \(a_2 s_{k_2}X\), we get

$$\begin{aligned}&w(a_2 s_{k_2} X + \cdots + a_r s_{k_r} X) \\&\quad \in w(a_3 s_{k_3} X + \cdots + a_r s_{k_r} X)\; {{\text {EO}}}_{A[X]} (Q\otimes A[X], \mathbb {H}(A[X])^m ).\end{aligned}$$

Continuing in this way, we have

$$\begin{aligned} w(a_r s_{k_r} X + 0) \in w(0){{\text {EO}}}_{A[X]} (Q\otimes A[X], \mathbb {H}(A[X])^m ). \end{aligned}$$

Combining all of these, we get

$$\begin{aligned} w(X) \in w(0) {{\text {EO}}}_{A[X]} (Q\otimes A[X], \mathbb {H}(A[X])^m ) \end{aligned}$$

and hence the result is proved. \(\square \)

6.3 Other applications

Apart from the applications illustrated above, we could use the commutator calculus to prove a local-global principle for the DSER elementary orthogonal group \({{\text{ EO }}}_A(Q, \mathbb {H}(A)^m)\) which in turn used to prove the extendability of quadratic modules (see [3]). Also, we have used these commutators to get the normality of the DSER elementary orthogonal group \({{\text{ EO }}}_A(Q, \mathbb {H}(A)^m)\) in the orthogonal group \({{\text{ EO }}}_A(Q,\mathbb {H}(A)^m)\) (see [2] and [4]).

Remark 6.12

In [15], Hazrat–Zhang established a generalized commutator formula. By analyzing all possible multi-commutator formulas in the formulas in the DSER elementary orthogonal group, it will be interesting to explore similar results for this group.