arXiv:1703.04083v1 [] 12 Mar 2017

NORMALITY OF DSER ELEMENTARY ORTHOGONAL GROUP
A.A. AMBILY AND RAVI A. RAO

Abstract. Let (Q, q) be a quadratic space over a commutative ring R in which 2 is invertible,
and consider the Dickson–Siegel–Eichler–Roy’s subgroup EOR (Q, H(R)m ) of the orthogonal group
OR (Q ⊥ H(R)m ), with rank Q = n ≥ 1 and m ≥ 2. We show that EOR (Q, H(R)m ) is a normal
subgroup of OR (Q ⊥ H(R)m ), for all m ≥ 2. We also prove that the DSER group EOR (Q, H(P )) is
a normal subgroup of OR (Q ⊥ H(P )), where Q and H(P ) are quadratic spaces over a commutative
ring R, with rank(Q) ≥ 1 and rank(P ) ≥ 2.

1. Introduction
J-P. Serre’s problem on projective modules (See [19]) played a vital role in the development of
K-theory. It stimulated the investigation of K1 , higher K-groups, the stable and unstable behaviour
of K-groups. The study of K1 began with Bass’s seminal work in 1964 (See [6]). In [9], P.M. Cohn
proved that E2 (R) is not normal in GL2 (R) for a commutative ring R. In 1977, A.A. Suslin proved the
normality of the elementary linear group En (R) in the general linear group GLn (R) for n ≥ 3, over a
commutative ring R (See [20]). Analogues of Suslin’s normality theorem were proved for the elementary
orthogonal group EO2n (R) ⊂ O2n (R) for n ≥ 3 by A.A. Suslin–V.I. Kopeı̆ko in [21] and the failure
of the result in the case n = 2 was proved by V.I. Kopeı̆ko in [12]. Similar results for the elementary
symplectic group ESp2n (R) ⊆ Sp2n (R) for n ≥ 3 was proved by Kopeı̆ko in [11]. The normality results
for the elementary Chevalley groups over local rings was proved by E. Abe in [1] and for elementary
Chevalley–Demazure group schemes over commutative rings was studied by G. Taddei in [23]. In [10],
R. Hazrat–N. Vavilov gave a simpler proof of normality of elementary Chevalley groups. The case of
twisted elementary Chevalley groups was proved by K. Suzuki in [22] and A. Bak–N. Vavilov in [5].
For classical-like groups such as general quadratic groups, general Hermitian groups, odd unitary
groups, isotropic reductive groups, the normal subgroup structure was studied by A. Bak–N. Vavilov
in [5], G. Tang in [24], V. Petrov in [14], V.Petrov–A. Stavrova in [15] and W. Yu in [25] respectively.
Recently, in [26], W. Yu–G. Tang studied the nipotency of odd unitary K1 -functor and in [7], R.
Basu showed that the unstable K1 -groups of general Hermitian groups over module finite rings are
nilpotent-by-abelian.
We study the normality of the elementary Dickson–Siegel–Eichler–Roy’s orthogonal group which
was considered by A. Roy in [18]. We describe this briefly below.
Let R be a commutative ring and let (Q, q) be a quadratic R-space with associated bilinear form
Bq and P be a finitely generated projective R-module. The module P ⊕ P ∗ has a natural quadratic
form given by p((x, f )) = f (x) for x ∈ P , f ∈ P ∗ . The corresponding bilinear form Bp is given by
Bp ((x1 , f1 ), (x2 , f2 )) = f1 (x2 )+f2 (x1 ) for x1 , x2 ∈ P and f1 , f2 ∈ P ∗ . The quadratic space (P ⊕P ∗ , p),
denoted by H(P ), is called the hyperbolic space of P . Given any homomorphism α : Q → P , define
2010 Mathematics Subject Classification 19G99 (primary), 19B99, 20G35, 20H25 (secondary).

−1
α∗ : P ∗ → Q by the formula α∗ = dB
◦ αt , where αt denotes the dual map P ∗ → Q∗ . If β : Q → P ∗ ,
q
t
∗∗
t
then define β ∗ : P → Q by precomposing d−1
Bq ◦ β with ε : P → P , where β denotes the dual map
∗∗
∗
∗
P → Q . The A-linear map α is characterized by the relation

(f ◦ α)(z) = Bq (α∗ (f ), z) for f ∈ P ∗ , z ∈ Q.
Let OR (Q) denote the orthogonal group of the quadratic module (Q, q). That is,
OR (Q) = {α ∈ AutR (Q) | q(α(z)) = q(z) for all z ∈ Q}
In [18], A. Roy defined the “elementary” transformations Eα and Eβ∗ of Q ⊥ H(P ) given by
Eα (z)

= z + α(z)

Eβ∗ (z)

= z + β(z)

Eα (x)

=x

Eβ∗ (x)

= −β ∗ (x) + x − 12 ββ ∗ (x)

Eα (f ) = −α∗ (f ) − 12 αα∗ (f ) + f

Eβ∗ (f ) = f

for z ∈ Q, x ∈ P and f ∈ P ∗ .
The Dickson–Siegel–Eichler–Roy’s subgroup EOR (Q ⊥ H(P )) of the orthogonal group OR (Q ⊥
H(R)m ) is the subgroup generated by the elementary generators Eα , Eβ∗ , where α ∈ Hom(Q, P ) and
β ∈ Hom(Q, P ∗ ). We shall refer to this group by DSER group. For more historical details on this
group we refer the reader to the introduction of [4].
In [8], B. Calmès–J. Fasel have described an elementary subgroup EO2n+1 (R) of the special orthogonal group SO2n+1 (R) generated by a set of 5 generators. By comparing the generators with the
generators of DSER, we can identify two of the elementary generators are of the form Eαij and Eβ∗ij ,
and other three generators are commutators of these type of generators.
The following normality results were proved by the first author in [3].
(i) OR (Q ⊥ H(R)m−1 ) normalizes EOR (Q ⊥ H(R)m ). In particular, EOR is a normal subgroup of
OR .
(ii) If m ≥ dim Max(R) + 2, then OR (Q ⊥ H(R)m ) normalizes EOR (Q ⊥ H(R)m ).
(iii) If m > l, then OR (Q ⊥ H(R)m ) normalizes EOR (Q ⊥ H(R)m ) provided R satisfies the stable
range condition 0-SRl .
All aforementioned results about normality of this elementary orthogonal group in the literature
have assumed stable rank conditions. In this paper, we prove that the normality of the DSER subgroup
in most generality without these restrictions on the hyperbolic rank, except for the case of rank(Q) = 2
and m = 1. We prove the following theorems.
Theorem 1.1. EOR (Q, H(R)m ) is normal in OR (Q ⊥ H(R)m ), where Q and H(R)m are quadratic
spaces over a commutative ring R with rank(Q) ≥ 1 and m ≥ 2.
Theorem 1.2. The DSER group EOR (Q, H(P )) is a normal subgroup of OR (Q ⊥ H(P )), where Q
and H(P ) are quadratic spaces over a commutative ring R, with rank(Q) ≥ 1 and rank(P ) ≥ 2.
In order to prove these theorems, we prove an analogue of Quillen’s Local-Global principle in the
extended module case for the DSER group. In addition to these results, we see that in the free module
case, the elementary DSER group with q to be the standard hyperbolic form coincides with the usual
elementary orthogonal group EOn+2m (R). We also use a decomposition result of R.A. Rao in [17].
2

Notations:
• GLn (R) will denote the general linear
(R), then by
! group. Let α ∈ GLn (R) and β ∈ GLm!
α 0
0 α
α ⊥ β we denote the matrix
and by α⊤β we denote the matrix
.
0 β
β 0
!
f1 = 0 1 , ψ
fr = ψ
f1 ⊥ ψ
]
• ψ
r−1 .
1 0
• The matrix of the bilinear form corresponding to the quadratic form q is denoted by ϕ.
• E(α) denote either Eα or Eα∗ , where α ∈ Hom(Q, P ) or Hom(Q, P ∗ ).
• h denote the hyperbolic plane H(R) and hm = h ⊥ h ⊥ . . . ⊥ h = H(R)m .
• EOR (q, hm ) = EOR (Q, H(R)m ) and OR (q ⊥ hm ) = OR (Q ⊥ H(R)m ).
2. Normality of EOR (q, hm ) in OR (q ⊥ hm )
In this section, we prove the normality of the elementary orthogonal group EOR (q, hm ) of OR (q ⊥
hm ), where R is a commutative ring in which 2 is invertible, (Q, q) a quadratic space, and h denote
the hyperbolic plane H(R).
Now we recall a decomposition result of R.A. Rao from [17].
Lemma 2.1 ([17, Lemma 2.2]). Let (Q, q) be a diagonizable quadratic R-space. Let m ≥ dim Max R+1.
Then
OR (q ⊥ hm ) = EOR (q, hm ) · OR (hm ) = OR (hm ) · EOR (q, hm )),
where EOR (q, hm ) · OR (hm ) denote the subset {σ1 σ2 | σ1 ∈ EOR (q, hm ), σ2 ∈ OR (hm )} of OR (q ⊥ hm ).
In [13] on page 290, R. Parimala remarked that if α ∈ Ok (h), where k is a field of characteristic
6= 2, then there is a nonzero element u ∈ R such that α = [u] ⊥ [u−1 ] or [u]⊤[u−1 ]. We prove that this
also holds over a local ring.
Lemma 2.2. Let R be a local ring with maximal ideal m. Assume that 2R = R. Let α ∈ OR (h). Then
there exists a unit u ∈ R such that α = [u] ⊥ [u−1 ] or [u]⊤[u−1 ].
!
!
a b
0
1
f1 denotes the matrix
. Then one has the relations
∈ OR (h) and ψ
Proof. Let α =
1 0
c d
f1 α = ψ
f1 , αψ
f1 = ψ
f1 α−1 . From this one deduces that det(α)2 = 1, and 2ac = 0 = 2bd, ad + bc = 1,
αt ψ
∆c = −c, ∆b = −b, ∆a = a, ∆d = d, where ∆ = det(α). From these relations it is easy to deduce
Parimala’s result in the case when R is a field. We shall use this below.
Let the α denote α modulo m. From above equations, it follows that
∆ = det(α) = ±1.
If ∆ = 1, then α = [u] ⊥ [u−1 ], for some unit u ∈ R/m. Hence, a, d are units and b, c ∈ m. Since ad = 1
modulo m, let ad = 1 + m, for some m ∈ m. Now 1 = ∆2 = (1 + m − bc)2 = 1 + (m − bc)(2 + m − bc),
whence m − bc = 0. Hence ∆ = 1 + m − bc = 1. But then the equations ∆b = −b, and ∆c = −c give
b = 0 = c, and so α = [a] ⊥ [a−1 ] as required.
f1 ∈ OR (h). Then
If ∆ = −1, then α = [u]⊤[u−1 ], for some unit u ∈ R/m. Consider β = αψ
−1
−1
det(β) = 1; whence β = [b] ⊥ [b ], for some unit b ∈ R. Hence α = [b]⊤[b ], for some unit b in
R.

3

Lemma 2.3. Let R be a local ring with maximal ideal m. Assume that 2R = R. Let (Q, q) be a
quadratic space over R of rank n. Then the group OR (h) normalises EOR (q, h).
Proof. The elements of OR (h) are of the form
τu = [u] ⊥ [u−1 ]

or σu = [u]⊤[u−1 ].

The elementary generators of EOR (q, h) are of the form Eα1j or Eβ∗1j for 1 ≤ j ≤ rank Q = n. From
the following set of equations
(In ⊥ τu ) Eα1j (In ⊥ τu−1 ) = Euα1j ,
(In ⊥ τu ) Eβ∗1j (In ⊥ τu−1 )

∗
= Euβ
,
1j

(In ⊥ σ1 ) Eα1j (In ⊥ σ1−1 )

= Eα1j ,

(In ⊥ σ1 )

Eβ∗1j (In

⊥

σ1−1 )

= Eβ∗1j ,

it is clear that OR (h) normalises EOR (q, h).



Corollary 2.4. Let R be a local ring with maximal ideal m. Assume that 2R = R. Let (Q, q) be
a quadratic space of rank n over R. Then the elementary orthogonal group EOR (q, h) is normal in
OR (q ⊥ h).
Proof. By Lemma 2.1, we have OR (q ⊥ h) = EOR (q, h) · OR (h) = OR (h) · EOR (q, h). By Lemma 2.3,
it follows that OR (q ⊥ h) normalises EOR (q, h).

Lemma 2.5. Let R be a local ring with maximal ideal m. Assume that 2R = R. Let (Q, q) be a
quadratic space. Then the group OR (h) normalises EOR (q, hm ).
Proof. The elementary generators of EOR (q, hm ) are either of the form Eαij or Eβ∗ij for α, β ∈
Hom(Q, Rm ), 1 ≤ i ≤ m, and 1 ≤ j ≤ n.
For 1 ≤ i ≤ m, 1 ≤ j ≤ n, we have the set of equations

E
uαij , if i = m
(In+2m−2 ⊥ τu ) Eαij (In+2m−2 ⊥ τu−1 ) =
Eα , if i 6= m.
ij

E ∗ , if i = m
uβij
(In+2m−2 ⊥ τu ) Eβ∗ij (In+2m−2 ⊥ τu−1 ) =
E ∗ , if i 6= m.
βij

σ1−1 )

= Eαij ,

(In+2m−2 ⊥ σ1 ) Eβ∗ij (In+2m−2 ⊥ σ1−1 )

= Eβ∗ij .

(In+2m−2 ⊥ σ1 ) Eαij (In+2m−2 ⊥

From these equations it is clear that OR (h) normalises EOR (q, hm ).



Lemma 2.6. Let (Q, q) be a quadratic space of rank n over R. Then the orthogonal group OR (q)
normalises EOR (q, hm ) for m ≥ 2.
Proof. Let α ∈ OR (q). Consider the elementary generator Eβ
Then

In
 −1
−1
(α ⊥ I2m ) Eβ (α ⊥ I2m )
= βα
0
4

∈ EOR (q ⊥ hm ), where β : Q → Rm .
0
Im
0


−αβ ∗
∗
− ββ2  .
Im

T −1
−1 ∗
−1 T
−1 T
Note that d−1
= α◦ d−1
) = d−1
) = d−1
) ◦ βT =
Bq ◦ (α )
Bq . Therefore we get (β ◦ α
Bq ◦ (β ◦ α
Bq ◦ (α
T
∗
α ◦ d−1
Bq ◦ β = α ◦ β . Hence

(α ⊥ I2m ) Eβ (α ⊥ I2m )−1 = Eβ◦α−1 ∈ EOR (q, hm ).

fr . Then EOR (q, hm ) =
Lemma 2.7. Let (Q, q) be a quadratic space of rank n = 2r over R and let ϕ = ψ
EOn+2m (R), where EOR (q, hm ) denote DSER elementary orthogonal group and EOn+2m (R) denote
the usual elementary orthogonal group.
fr . Then q ⊥ hm has the form ψer+m . Let σ denote the permutation matrix associated
Proof. Let ϕ = ψ
to this form where σ is given by

l + 1 if l odd
σ(l) =
,
l − 1 if l even
for 1 ≤ l ≤ n + 2m. The elementary generators for EOn+2m (R) with respect to this form are

(1)

oekl (a) = In+2m + aekl − aeσ(l),σ(k) if k 6= σ(l) and k < l,

for 1 ≤ k 6= l ≤ n + 2m and a ∈ R. The elementary generators for EOR (q, hm ) are
(2)

Eαkl = En+2k−1,l (akl )Eσ(l),n+2k (−akl ),

(3)

Eβ∗kl = En+2k,l (bkl )Eσ(l),n+2k−1 (−bkl ),

for 1 ≤ k ≤ m, 1 ≤ l ≤ n, α = (akl ) and β = (bkl ) for akl , bkl ∈ R.
From Eq.(1), Eq.(2) and Eq.(3), it is clear that for 1 ≤ k ≤ m and 1 ≤ l ≤ n,
Eαkl = oeσ(l),n+2k (−akl ) and Eβ∗kl = oeσ(l),n+2k−1 (−bkl ).
Hence
EOR (q, hm ) ⊆ EOn+2m (R).
To prove the reverse inclusion, it is enough to show that for 1 ≤ k, l ≤ n and n + 1 ≤ k, l ≤ n + 2m,
k 6= σ(l), k < l and a ∈ R, oekl (a) ∈ EOR (q, hm ).
For 1 ≤ k, l ≤ n and for 1 ≤ i ≤ m,
oekl (a) = oeσ(l),σ(k) (−a)
= [oeσ(l),n+2i (−a), oen+2i,σ(k) (1)]
= [oeσ(l),n+2i (−a), oek,n+2i−1 (−1)]
= [Eαil (a) , Eβ∗i,σ(k) (1) ].
For k = n + 2i − 1, l = n + 2j − 1, 1 ≤ i 6= j ≤ m and for 1 ≤ s ≤ n,
oekl (a) = [oen+2i−1,s (a), oes,n+2j−1) (1)]
= [oeσ(s),n+2i (−a), oes,n+2j−1 (1)]
= [Eαis (a) , Eβ∗j,σ(s) (−1) ].
5

For k = n + 2i − 1, l = n + 2j, 1 ≤ i, j ≤ m and for 1 ≤ s ≤ n,
oekl (a) = [oen+2i−1,s (a), oes,n+2j) (1)]
= [oeσ(s),n+2i (−a), oes,n+2j (1)]
= [Eαis (a) , Eαj,σ(s) (−1) ].
For k = n + 2i, l = n + 2j − 1, 1 ≤ i < j ≤ m and for 1 ≤ s ≤ n,
oekl (a) = [oen+2i,s (a), oes,n+2j−1) (1)]
= [oeσ(s),n+2i−1 (−a), oes,n+2j−1 (1)]
= [Eβ∗is (a) , Eβ∗j,σ(s) (−1) ].
For k = n + 2i, l = n + 2j, 1 ≤ i 6= j ≤ m and for 1 ≤ s ≤ n,
oekl (a) = [oen+2i,s (a), oes,n+2j) (1)]
= [oeσ(s),n+2i−1 (−a), oes,n+2j (1)]
= [Eβ∗is (a) , Eαj,σ(s) (−1) ].
Thus we get
EOR (q, hm ) = EOn+2m (R).

Proposition 2.8 ([3, Proposition 4.5]). OR (q ⊥ hm−1 ) normalizes EOR (q, hm ), where q and hm are
quadratic spaces over a commutative ring R for m ≥ 2.
To be self-contained, we sketch a different proof of this proposition.
Lemma 2.9. Let (Q, q) be a diagonalizable quadratic R-space. The elementary orthogonal group
EO4 (R) normalizes the DSER elementary orthogonal group EOR (q, hm ) for m ≥ 2.
Proof. The elementary orthogonal group EO4 (R) has elementary generators of the form oe13 (a),
oe14 (b), oe23 (c), oe24 (d), where oekl (x) = In+2m + xekl − xeσ(l),σ(k) if k 6= σ(l) and k < l, for
1 ≤ k 6= l ≤ n + 2m and x ∈ R. The elementary generators of EOR (q, hm ) are of the (matrix)
form

1
Eαij (x) = In+2m + xen+2i−1,j − dj xej,n+2i − dj x2 en+2i−1,n+2i ,
2
1
Eβ∗ij (y) = In+2m + yen+2i,j − dj yej,n+2i−1 − dj y 2 en+2i,n+2i−1 ,
2
where n = rank(Q), x, y ∈ R, dj ∈ R is the (j, j)th entry of the diagonal matrix Bq−1 , 1 ≤ i ≤ m,
1 ≤ j ≤ n; αij (x) is an m × n matrix with (i, j)-th entry is x and all other entries 0 and βij (y) is an
m × n matrix with (i, j)-th entry is y and all other entries 0.

We need to prove that each oekl normalizes both Eαij and Eβ∗ij for all 1 ≤ i ≤ m and 1 ≤ j ≤ n.
We illustrate one example here:
Let n = 2 and m = 2. Then the elementary orthogonal group EOR (q, h2 ) has elementary generators
Eα11 (a11 ) , Eα12 (a12 ) , Eα21 (a21 ) , Eα22 (a22 ) , Eβ∗11 (b11 ) , Eβ∗12 (b12 ) , Eβ∗21 (b21 ) , Eβ∗22 (b22 ) , where a11 , a12 , a21 ,
a22 , b11 , b12 , b21 , b22 ∈ R.
6

oe13 (a)Eα11 (a11 ) oe13 (−a) = Eα11 (a11 ) ,
oe13 (a)Eα12 (a12 ) oe13 (−a) = Eα12 (a12 ) ,
oe13 (a)Eα21 (a21 ) oe13 (−a) = Eα11 ( 12 aa21 ) Eα21 (a21 ) Eα11 ( 12 aa21 ) = Eα11 (aa21 )+α21 (a21 ) ,
oe13 (a)Eα22 (a22 ) oe13 (−a) = Eα12 ( 21 aa22 ) Eα22 (a22 ) Eα12 ( 12 aa22 ) = Eα12 (aa22 )+α22 (a22 ) ,
oe14 (b)Eα11 (a11 ) oe14 (−b) = Eα11 (a11 ) ,
oe14 (b)Eα12 (a12 ) oe14 (−b) = Eα12 (a12 ) ,
oe14 (b)Eα21 (a21 ) oe14 (−b) = Eα21 (a21 ) ,
oe14 (b)Eα22 (a22 ) oe14 (−b) = Eα22 (a22 ) ,
i
h
oe23 (c)Eα11 (a11 ) oe23 (−c) = Eβ∗21 ( 1 ca11 ) , Eα11 (a11 ) Eβ∗21 (−ca11 ) Eα11 (a11 ) ,
2
h
i
∗
oe23 (c)Eα12 (a12 ) oe23 (−c) = Eβ22 ( 1 ca12 ) , Eα12 (a12 ) Eβ∗22 (−ca12 ) Eα12 (a12 ) ,
2
i
h
∗
oe23 (c)Eα21 (a21 ) oe23 (−c) = Eβ11 ( 1 ca21 ) , Eα21 (a21 ) Eβ∗11 (ca21 ) Eα21 (a21 ) ,
2
h
i
oe23 (c)Eα22 (a22 ) oe23 (−c) = Eα22 (a22 ) , Eβ∗12 ( 1 ca22 ) Eβ∗12 (ca22 ) Eα22 (a22 ) ,
2
h
i
oe24 (d)Eα11 (a11 ) oe24 (−d) = Eα21 ( 21 da11 ) , Eα11 (a11 ) Eα21 (−da11 ) Eα11 (a11 ) ,
i
h
oe24 (d)Eα12 (a12 ) oe24 (−d) = Eα22 ( 21 da12 ) , Eα12 (a12 ) Eα22 (−da12 ) Eα12 (a12 ) ,
oe24 (d)Eα21 (a21 ) oe24 (−d) = Eα21 (a21 ) ,

oe24 (d)Eα22 (a22 ) oe24 (−d) = Eα22 (a22 ) .
We will get similar relations for Eβ∗11 (b11 ) , Eβ∗12 (b12 ) , Eβ∗21 (b21 ) , and Eβ∗22 (b22 ) . Hence it follows that the
group EO4 (R) normalizes the DSER elementary orthogonal group EOR (q, h2 ).

Lemma 2.10. Let R be a local ring. The orthogonal group OR (hm−1 ) normalizes the DSER elementary
orthogonal group EOR (q, hm ) for m ≥ 2.
Proof. For m = 2, this lemma reduces to Lemma 2.3. For m > 2, we have OR (hm−1 ) = EOR (hm−2 , h)·
OR (h) by Lemma 2.1. The orthogonal group OR (h) normalizes EOR (q, hm ) by Lemma 2.3. The
elementary orthogonal group EOR (hm−2 , h) ∼
= EO2m−2 (R). Any elementary generator in EO2m−2 (R)
can be considered as an element in some EO4 (R). Hence by Lemma 2.9, EO2m−2 (R) normalizes
EOR (q, hm ).

of Proposition 2.8. Let α ∈ OR (q ⊥ hm−1 ). It is enough to prove that α normalises the elementary
generators Eβ , Eγ∗ in EOR (q, hm ) for β, γ ∈ HomR (Q, Rm ). Let θ(T ) = αET β α−1 ∈ EOR[T ] (q ⊗
R[T ], hm ⊗ R[T ]). Then θ(0) = I and (θ(T ))m = αm (ET β )m (α−1 )m .
If (θ(T ))m ∈ EO(R[T ])m (q ⊗ (R[T ])m , hm ⊗ (R[T ])m ) for all m ∈ Max(R), then θ(T ) ∈ EOR[T ] (q ⊗
R[T ], hm ⊗ R[T ]) by Theorem 3.3. Hence we have θ(1) = αEβ α−1 ∈ EOR (q, hm ). So we can assume R to be a local ring. Assume that R is a local ring. Then by Lemma 2.1, we have OR (q ⊥
hm−1 ) = EOR (q, hm−1 ) · OR (hm−1 ). By Lemma 2.10, we have OR (hm−1 ) normalises EOR (q, hm ) and
EOR (q, hm−1 ) ⊂ EOR (q, hm ). Hence OR (q ⊥ hm−1 ) normalizes EOR (q, hm ).

Now we prove the normality of the DSER group EOR (Q, H(P )) when P is free R-module.
7

Theorem 2.11. EOR (q, hm ) ⊳ OR (q ⊥ hm ), where q and hm are quadratic spaces over a commutative
ring R and m ≥ 2.
Proof. Let α ∈ OR (q ⊥ hm ) and Eβ , Eγ∗ , elementary generators in EOR (q, hm ). Let
θ(T ) = αET β α−1 ∈ EOR[T ] (q ⊗ R[T ], hm ⊗ R[T ]).
Then θ(0) = I and (θ(T ))m = αm (ET β )m (α−1 )m . If (θ(T ))m ∈ EO(R[T ])m (q ⊗ (R[T ])m , hm ⊗ (R[T ])m )
for all m ∈ Max(R), then by Theorem 3.3,
θ(T ) ∈ EOR[T ] (q ⊗ R[T ], hm ⊗ R[T ]).
Hence we have θ(1) = αEβ α−1 ∈ EOR (q, hm ). So we can assume R to be a local ring.
Assume that R is a local ring. Then q is diagonalizable. By Lemma 2.1, we have the decomposition
OR (q ⊥ hm ) = EOR (q, hm ) · OR (hm ) = EOR (q, hm ) · EOR (hm−1 , h) · OR (h).
Note that by Lemma 2.5, OR (h) normalises EOR (q, hm ). By Lemma 2.7, we conclude that EOR (hm−1 , h) =
EO2m (R). Since any elementary generator of EO2m (R) is an element of some EO2m−2 (R) for m > 2,
S
it will be contained in OR (hm−1 ) ⊂ OR (q ⊥ hm−1 ). But by Proposition 2.8, we have OR (q, hm−1 )
normalises EOR (q, hm ). Hence the group OR (q ⊥ hm ) normalises EOR (q, hm ).

3. Local-Global principle for the DSER group
In this section, we prove an analogue of Quillen’s Local-Global principle(extended case) for the
DSER elementary orthogonal group. We take M = Q ⊥ H(P ) and M [X] will denote (Q ⊥ H(P ))[X].
We assume that the rank of the projective module Q is ≥ 1 and rank P ≥ 2.
We recall lemmas from [4].
Lemma 3.1 ([4, Lemma 5.5]). Let Q, P be free R-modules of rank n and m respectively. The kernel of the evaluation map EOA[X] (Q[X], H(P [X])) → EOA (Q, H(P )) at X = 0 is generated by the
elements of the type γE (Xαij (X)) γ −1 , where γ ∈ EOA (Q, H(P )), αij (X) ∈ Hom(Q[X], P [X]) or
Hom(Q[X], P ∗ [X]).
Lemma 3.2 ((Dilation Lemma)[4, Lemma 5.7]). Let Q, P be free modules of rank n and m respectively.
Let s be a non-nilpotent element of R and M = Q ⊥ H(P ). Let θ(X) ∈ OR[X] (M [X]) with θ(0) = I. Let
Y, Z ∈ Hom(Q, P ) or Hom(Q, P ∗ ). If θs (X) := (θ(X))s is an element of EORs [X] (Q ⊗ Rs [X], H(R)m ⊗ Rs [X]),
then for N ≫ 0 and for all b ∈ (sN )R, we have θ(bX) ∈ EOR[X] (Q ⊗ R[X], H(R)m ⊗ R[X]).
Theorem 3.3 ((Local-Global Principle)[4, Theorem 5.3]). Let θ(X) ∈ OR[X] (M [X]) with θ(0) = I.
If for all maximal ideals m of R, θ(X)m ∈ EORm [X] (Q ⊗ Rm [X], H(R)m ⊗ Rm [X]), then θ(X) ∈
EOR[X] (Q ⊗ R[X], H(R)m ⊗ R[X]).
We state here some useful facts.
(1) Let R be a ring and S be a multiplicative subset of R. Let M be a finitely presented R-module
and N be any R-module. Then we have a natural isomorphism,
η : S −1 (HomR (M, N )) → HomS −1 R (S −1 M, S −1 N ).
(2) Let R be a ring and M be a finitely presented R-module and let N be any R module. Then
we have a natural isomorphism,
γ : HomR (M, N )[X] → HomR[X] (M [X], N [X]).
8

We state a variation of [4, Lemma 5.6] which can be proved by the similar arguments of [4,
Lemma 5.6]. we also refer the reader to [2] for the various commutator relations involved in proving this lemma.
Lemma 3.4. Let Q be a quadratic R-space and P be a finitely generated projective R-module. Let s
be a non-nilpotent element of R. Fix r ∈ N. Given an integer N ∈ N and x ∈ R, then for integers
Nt ≥ N and xt ∈ R, there exists a product decomposition
µ
Y


E (Wij ) E sN xYkl E (−Wij ) =
E sNt xt Zpt qt ,
t=1

where W, Y, Z ∈ HomRs (Qs , Ps ) or W, Y, Z ∈ HomRs (Qs , Ps∗ ), i, k, pt ∈ {1, 2, ..., m} and j, l, qt ∈
{1, 2, ..., n} for every integer 1 ≤ t ≤ µ.
Proof. Use the similar arguments as in [4, Lemma 5.6] which involves various commutator relations

among the elementary generators of EORs (Qs , Ps ).
Corollary 3.5. Let Q be a quadratic R-space and P be a finitely generated projective R-module. If ε =
ε1 ε2 . . . εk , where each εj is an elementary generator of the type Eαij or Eβ∗ij with α ∈ HomRs (Qs , Ps )
or β ∈ HomRs (Qs , Ps∗ ). For l > 0, 1 ≤ r ≤ m, 1 ≤ p ≤ n and x ∈ R, there is a product decomposition
εE(s

2k l

xWrp )ε

−1

=

µk
Y

t=1


E sl xt Yrt pt ,

where W, Y ∈ Hom(Qs , Ps ) or Hom(Qs , Ps∗ ), 1 ≤ rt ≤ m, 1 ≤ pt ≤ n and xt ∈ R chosen suitably.
Proof. Apply Lemma 3.4 repeatedly.



Proposition 3.6 ((Dilation Principle)). Let (Q, q) be a quadratic R-space of rank n ≥ 1 and let P be
a finitely generated projective R-module of rank m ≥ 2. Let M = Q ⊥ H(P ). Let s be a non-nilpotent
element in R such that Ps and Qs are free. Let θ(X) ∈ OR[X] (M [X]) with θ(0) = I. Suppose that
b
θs (X) ∈ EORs [X] (Qs [X], Rs [X]m ). Then there exists θ(X)
∈ EOR[X] (Q[X], P [X]) and ℓ ≫ 0 such
ℓ
b
b = I.
that θ(X)
localizes to θ(bX) for some b ∈ (s ) and θ(0)

Q
Proof. Since θ(0) = I, we can write θs (X) = νt=1 γt E (Xht (X)Wit jt ) γt −1 , where ht (X) ∈ Rs [X] and
γt ∈ EORs (Qs , Ps ). For l > 0, we can write θs (sl X) as

Q
θs (sl X) = νk=1 γk E sl Xhk (sl X)Wik jk γk −1 .
Let γk = ε1 ε2 . . . εrk and l = 2rk −1 . Using Corollary 3.5, we get
2l

θs (s X) =

νrk
ν Y
Y

k=1 t=1


E sl gt (X)Wit jt , where Wit jt ∈ Hom(Qs , Ps ).

Since Qs and Ps are free Rs -modules, we have

Ps [X, Z] ∼
= Rs2m [X, Z] ∼
= Ps [X, Z]∗ and Qs [X, Z] ∼
= Rsn [X, Z].
Using this isomorphism, the polynomials in Ps [X, Z] can be regarded as the bilinear form as follows:
For x = (z1 , z2 , . . . , zn , x1 , x2 , . . . , x2m ), y = (w1 , w2 , . . . , wn , y1 , y2 , . . . , y2m ) ∈ Rsn+2m [X, Z];
T
hx, yi = (z1 , z2 , . . . , zn )ϕ(w1 , w2 , . . . , wn )T ⊥ (x1 , x2 , . . . , x2m )ψf
m (y1 , y2 , . . . , y2m ) ,
9

where ϕ is a diagonal invertible matrix and ψf
m denotes the standard hyperbolic form
2m
X

e2i−1,i +

2m
X

e2i,2i−1 .

i=1

i=1

Let e∗i be the standard basis for Rs2m . Consider the element Zgt (X)e∗i ∈ Rs2m [X, Z] as an element
in (Ps )[X, Z]. By Corollary 3.5, we can choose Nt > 0 such that Nt is the maximum power of s
occurring in the denominator of Zgt (X)e∗i . Now using fact 2, we can consider the element Zgt (X)e∗i
as a polynomial in Z. Applying the homomorphism Z 7→ sNt Z, it follows that θ(bXZ 2l ) is defined
over R[X]. That is, we have
θ(bXZ 2l ) =

Qν

k=1

Qνrk

t=1


E sNt Zgt (X)Wit jt ∈ EOR[X] (Q[X], P [X]) for all b ∈ (sℓ )R, (ℓ ≫ 0).

b
Now substitute Z = 1, then by Lemma 3.2, there exists ℓ ≫ 0 such that θ(X)
∈ EOR[X] (Q[X], P [X]))
ℓ
b
localizes to θ(bX) for some b ∈ (s ) and θ(0) = I.

Theorem 3.7 ((Local-Global Principle)). Let R be a commutative ring with identity in which 2 is
invertible. Let (Q, q) be a quadratic R-space of rank n ≥ 1 and let P be a projective R-module of
rank m ≥ 2. Let M = Q ⊥ H(P ). Assume that for every maximal ideal m of R, the module Mm is
∼

n+2m
with the bilinear form ϕ ⊥ ψ m .
isomorphic to Rm
Let θ(X) ∈ OR[X] (M [X]) with θ(0) = I and θm (X) ∈ EORm [X] (Qm [X], Rm [X]m ) for all m ∈
Max(R). Then θ(X) ∈ EOR (Q, H(P )).

Proof. Use the Dilation Principle (Lemma 3.2) and D. Quillen’s arguments in [16] to deduce the LocalGlobal Principle from the Dilation Principle.


4. Normality of EOR (Q, H(P )) in OR (Q ⊥ H(P ))
In this section, we prove the main result of this paper. Let P be a finitely generated projective
module.
Theorem 4.1. EOR (Q, H(P )) is a normal subgroup of OR (Q ⊥ H(P )), where Q and H(P ) are
quadratic spaces over a commutative ring R, where rank(Q) ≥ 1 and rank(P ) ≥ 2.
Proof. Let η ∈ OR (Q ⊥ H(P )) and Eβ , Eγ∗ , elementary generators in EOR (Q, H(P )). For any maximal
ideal m of R, the Rm -modules Qm and H(P )m are free and H(P )m ∼
= hm .
Let θ(T ) = ηET β η −1 ∈ EOR[T ] (Q ⊗ R[T ], H(P ) ⊗ R[T ]). Then θ(0) = I and
(θ(T ))m = ηm (ET β )m (η −1 )m ∈ EORm [T ] (Q ⊗ Rm [T ], H(P ) ⊗ Rm [T ])
for all m ∈ Max(R). By Theorem 3.7, θ(T ) ∈ EOR[T ] (Q ⊗ R[T ], H(P ) ⊗ R[T ]) . Therefore, we have
θ(1) = ηEβ η −1 ∈ EOR (Q, H(P )).

Acknowledgements. The first author thanks the School of Mathematics, Tata Institute of Fundamental Research, Mumbai for its hospitality during this work.
10

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Department of Mathematics

School of Mathematics

Cochin University of Science and Technology
Cochin 682022, Kerala

Tata Institute of Fundamental Research
1, Dr. Homi Bhabha Road, Mumbai 400 005

India

India

E-mail address: ambily@cusat.ac.in, aaambily@gmail.com
E-mail address: ravi@math.tifr.res.in

11

