Journal of Algebra 399 (2014) 378–388

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Journal of Algebra

www.elsevier.com/locate/jalgebra

On the non-injectivity of the Vaserstein symbolin dimension three

Dhvanita R. Rao a,∗, Neena Gupta b

a Bhavan’s College, Andheri (W), Munshi Nagar, Mumbai 400058, Indiab Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 September 2013Available online 4 November 2013Communicated by Luchezar L. Avramov

MSC:13C1015A6319B1419G12

Keywords:Unimodular rows and their orbit spacesElementary symplectic Witt groupsVaserstein symbol

L.N. Vaserstein proved over a two dimensional ring that the orbitspace of unimodular rows of length three modulo elementaryaction has a Witt group structure. R.A. Rao and W. van der Kallenshowed that the Vaserstein symbol need not be injective over threedimensional affine algebras over the real field, but is injective overthree dimensional affine algebras over a field of cohomologicaldimension one whose characteristic �= 2,3. R.G. Swan, R.A. Raoand J. Fasel gave another example of a real affine algebra ofdimension three for which the Vaserstein symbol is not injective.We demonstrate an uncountably infinite family of non-isomorphicaffine algebras of dimension three over the real field for which theVaserstein symbol is not injective.

© 2013 Published by Elsevier Inc.

1. Introduction

In [9] L.N. Vaserstein described a beautiful Witt group structure on the orbit space of unimodularrows of length three Um3(A) modulo the action of the elementary subgroup E3(A), when A is acommutative noetherian ring of Krull dimension two. This association between the orbit space andthe elementary symplectic Witt group denoted by V A : Um3(A)/E3(A) −→ W E (A) is defined and canbe studied in any dimension, and is now called the Vaserstein symbol.

In [7] R.A. Rao and W. van der Kallen studied V A when A is an affine threefold over a field k.They proved that if A is an affine threefold over a field of cohomological dimension at most one,and of characteristic �= 2,3, then V A is an isomorphism. However, for the coordinate ring of the real3-sphere Γ (S3) they proved that VΓ (S3) is not injective.

* Corresponding author.E-mail addresses: dhvanita@gmail.com (D.R. Rao), neenag@isical.ac.in (N. Gupta).

0021-8693/$ – see front matter © 2013 Published by Elsevier Inc.http://dx.doi.org/10.1016/j.jalgebra.2013.09.049

D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388 379

In the homepage of R.G. Swan (see [10]) another example has been given of a real algebra A ofdimension three for which V A is not injective.

In this article we show that there is an uncountable family of real threefolds Aλ , λ ∈ R, for whichthe Vaserstein symbol V Aλ is not injective.

2. Alternating matrices and the Witt group W G (A)

Let A be a commutative ring with 1. We shall also assume that A is a noetherian ring.A matrix from Mr(A) is said to be alternating if it has the form ν − νt , where ν ∈ Mr(A) and the

superscript ‘t ’ denotes the transpose, i.e. it is skew-symmetric and its diagonal elements are equal tozero.

For α from Mr(A) and β from Ms(A) we denote by α ⊥ β the matrix in Mr+s(A) given by

(α 00 β

).

The operation ⊥ is obviously associative.We define inductively an alternating matrix ψr in E2r(A), setting

ψ1 =(

0 1−1 0

),

ψr = ψr−1 ⊥ ψ1.

It is well known that there exists a Pfaffian – a polynomial pf in the matrix elements with coefficients±1 such that det(ϕ) = (pf (ϕ))2, for all alternating matrices ϕ .

On matrices of odd order the Pfaffian is identically equal to 0, and on matrices of even order it isdefined up to sign, to fix which we insist that pf (ψr) = 1, for all r.

For any α from Mr(A) and any alternating ϕ from Mr(A) we have pf (αtϕα) = pf (ϕ).det(α). Forany alternating matrices ϕ1, ϕ2, it is easy to check that pf (ϕ1 ⊥ ϕ2) = pf (ϕ1)pf (ϕ2).

As usual, GLr(A) is the group of all invertible matrices over A, and SLr(A) is the subgroup of GLr(A)

consisting of matrices of determinant one.Let SL(A) denote the infinite linear group

⋃r SLr(A), where SLr(A) is thought of as a subgroup of

SLr+1(A) under the usual identification α �→ (1) ⊥ α.Let E(A) denote the infinite elementary subgroup of SL(A) consisting of

⋃r Er(A), where Er(A)

denotes the usual subgroup of SLr(A) generated by the elementary generators Eij(a), i �= j, a ∈ A.(Of course, here Er(A) is regarded as a subgroup of SLr(A), and so sits inside Er+1(A) by the previousidentification.)

We recall Whitehead’s lemma:

Lemma 2.1 (Whitehead). If α ∈ GLr(A) then α ⊥ α−1 ∈ E2r(A).

Note that by Whitehead’s lemma [SL(A), SL(A)] = E(A).In particular, E(A) is a normal subgroup of SL(A) (and even GL(A)). In fact, Suslin showed in

[8, Corollary 1.4] that Er(A) is a normal subgroup of GLr(A), for r � 3.We fix some subgroup G of SL(A), containing E(A). This G is automatically normal in GL(A) as

GL(A)/E(A) is an abelian group. (In view of Whitehead’s lemma above.)Two alternating matrices α from M2r(A) and β from M2s(A) are said to be equivalent relative

to G (written α ∼ β) if

α ⊥ ψs+p = γ t(β ⊥ ψr+p)γ ,

for some natural number p and some matrix γ from G ∩ SL2(r+s+p)(A).

380 D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388

This relation is reflexive, symmetric and transitive, i.e. an equivalence relation on the set of allalternating matrices (even those of Pfaffian one) since a matrix that is equivalent to an alternatingmatrix is alternating and has the same Pfaffian.

Note: One can see that α ⊥ β ∼ β ⊥ α as the matrix

(0 Is

Ir 0

)∈ Er+s(A),

when r, s is even.Vaserstein showed that the operation ⊥ induces the structure of an abelian group on the set of

equivalence classes relative to G of alternating matrices with Pfaffian 1; this group is denoted byW G(A), and is known as the symplectic Witt group (w.r.t. G).

3. The Vaserstein symbol V A : Um3(A)/E3(A) −→ W E (A)

A row v = (v0, . . . , vr) is called unimodular of length (r + 1) if there is a row w = (w0, . . . , wr)

such that 〈v, w〉 = v · wt = ∑i vi wi = 1.

(This is the case when the ideal generated by the coordinates of v is the unit ideal. Hence a rowcan be checked to be unimodular if it is a non-zero vector over the field A/m, for every maximalideal m of A.)

The set of all unimodular rows of length (r + 1) over a ring A is denoted by Umr+1(A).There is a very interesting association of a unimodular row of length 3 with an alternating matrix,

which was pointed out by L.N. Vaserstein in [9]:Given a pair of unimodular rows v = (a,b, c), w = (a′,b′, c′), with a relation 〈v, w〉 = aa′ + bb′ +

cc′ = 1, one can associate an alternating matrix V (v, w) as follows:

V (v, w) =⎛⎜⎝

0 a b c−a 0 c′ −b′−b −c′ 0 a′−c b′ −a′ 0

⎞⎟⎠ ∈ SL4(A).

It is easily checked that V (v, w) has Pfaffian (aa′ + bb′ + cc′) = 1.Vaserstein considered the map from Um3(A) −→ W E (A) given by v �→ [V (v, w)] ∈ W E (A).He showed that it did not depend on the choice of w . Moreover, if v was replaced by an ele-

mentary transformation vε of v (and w replaced by the corresponding row wεt−1) then [V (v, w)] =

[V (vε, wεt−1)] ∈ W E (A).

Vaserstein studied the map Um3(A)/E3(A) −→ W E (A) given by [v] �→ [V (v, w)]. (We call this theVaserstein symbol today.)

Theorem 3.1 (Vaserstein, 1973/74). (See [9].) The (Vaserstein) symbol

V (= V A) : Um3(A)/E3(A) −→ W E(A)

is an isomorphism when Krull dimension of A is 2.

3.1. The Vaserstein symbol in dimension three and four

When A is a non-singular affine algebra over a nice field (say an algebraically closed field, saythe complex numbers C, or a field like C(t) which is a function field in one variable over C; moregenerally, a field of cohomological dimension at most one) then Ravi A. Rao and Wilberd van derKallen showed that the injective stability estimate for K1(A) improves by 1, i.e. SLd+1(A) ∩ E(A) =Ed+1(A), if A has Krull dimension d for such algebras A. As a consequence they could prove:

D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388 381

Theorem 3.2 (Ravi Rao–Wilberd van der Kallen). (See [7].) Let A be a non-singular affine threefold over analgebraically closed field of characteristic �= 2,3. Then the Vaserstein symbol V A is an isomorphism.

Recently, there is a theorem of Jean Fasel, Ravi A. Rao, and Richard G. Swan (FRS theorem) whichhas as a consequence of the following theorem:

Theorem 3.3 (Fasel–Rao–Swan). (See [3].) Let A be a non-singular affine algebra of dimension 4 over analgebraically closed field of characteristic �= 2,3. Then the Vaserstein symbol V A is injective.

3.2. Counter examples of Rao and van der Kallen, Swan, Rao and Fasel

Ravi A. Rao and Wilberd van der Kallen also showed that, in general, the Vaserstein symbol is notinjective in dimension 3. (Note: V A is always surjective when A has Krull dimension � 3.)

Theorem 3.4 (Ravi Rao–W. van der Kallen). (See [7].) Let A = R[x, y, z, t]/(x2 + y2 + z2 + t2 − 1) (= Γ (S3))

be the coordinate ring of the real 3-sphere S3 . Then the Vaserstein symbol V A is not injective.

Thus, it was important that one considers algebras over nice fields, if one expects to get positiveresults.

Remark 3.5. The example given by Rao and van der Kallen in [7, §4.5] says that the unimodular rows

vR = (−t2 + x2 + y2 − z2,−2tx + 2yz,2ty + 2xz),

wR = (−t2 + x2 + y2 − z2,−2tx − 2yz,2ty − 2xz)

in Um3(AR), where

AR = R[x, y, z, t]/(x2 + y2 + z2 + t2 − 1)

are not in the same elementary orbit. In particular, we may conclude that the corresponding rowsv L, w L in Um3(AL), where

AL = L[x, y, z, t]/(x2 + y2 + z2 + t2 − 1),

are not in the same elementary orbit, when L is a subfield of R.

In dimension 3 it is known that the Vaserstein symbol V A is always surjective. Its kernel wascomputed by Anuradha Garge and Ravi A. Rao in [1] when A is an affine algebra of dimension threeover a field of cohomological dimension at most 1, and of characteristic �= 2,3, and shown to be{[e1ρ] | ρ ∈ SL3(A) ∩ E5(A)}. (The latter set is always contained in the kernel.)

Recently, Richard G. Swan, Ravi A. Rao and Jean Fasel gave another three dimensional examplewhen the Vaserstein symbol is not injective.

Theorem 3.6 (Swan–Rao–Fasel). (See [10].) If 2n − 1 ≡ 3 mod 8, there is an affine domain A over R ofdimension 2n − 1 and a 2-stably elementary element ρ ∈ SL2n−1(A) such that (1 ⊥ ρ) /∈ E2n(A)Sp2n(A).Moreover, if n = 2, we can choose ρ in such a way that its first row is not completable to an elementary matrixor, equivalently, e1ρ is not elementarily equivalent to e1 .

The last sentence will give the desired example as it shows that the kernel has a non-trivial ele-ment.

382 D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388

4. The main counter-examples

We begin with a slew of easy observations.

Lemma 4.1. Let G be a group and H be a normal subgroup of G. Let a,b, c,d ∈ G, with c,d ∈ H. Then [ac,bd] ∈[a,b]H.

Lemma 4.2. Let ϕ ∈ SL2r(A) be an alternating matrix, whose first row is (0, v), for some v ∈ Um2r−1(A). Then

(1 ⊥ α)tϕ(1 ⊥ α)

is an alternating matrix whose first row is (0, vα).

Lemma 4.3. Let α ∈ SLr(A), r � 2. Then the following statements are equivalent:

(i) α ∈ SLr−1(A)Er(A).(ii) e1α can be completed to an elementary matrix, i.e. [e1α] = [e1].

(iii) [α] = [1 ⊥ β] in SLr(A)/Er(A), for some β ∈ SLr−1(A).

Proof. (i) �⇒ (ii): Clearly, e1SLr−1(A)Er(A) = e1 Er(A).(ii) �⇒ (iii): Suppose that e1α = e1ε, for some ε ∈ Er(A). Then

αε−1 =(

1 0vt δ

),

for some v ∈ M1,r−1(A), δ ∈ SLr−1(A). But then

α =(

1 00 δ

)(1 0

δ−1 vt Ir−1

)ε ∈ SLr−1(A)Er(A).

(iii) �⇒ (i): Obvious. �We now input the main conclusion when V A is assumed to be injective:

Lemma 4.4. Let A be a commutative ring for which the Vaserstein symbol V A : Um3(A)/E3(A) −→ W E (A)

is an injective map. Let δ ∈ SL2(A). For any v ∈ Um3(A) one has [v(1 ⊥ δ)] = [v].

Proof. Since V A is an injective map it suffices to show that [V A(v)] = [V A(v(1 ⊥ δ))] in W E (A).Choose a w such that 〈v, w〉 = 1. Then V A(v) = [V (v, w)], and V A(v(1 ⊥ δ)) = [V (v(1 ⊥ δ), w(1 ⊥

δ)t−1)]. Now

[V

(v(1 ⊥ δ), w

(1 ⊥ δt−1))] = [

(I2 ⊥ δ)t V (v, w)(I2 ⊥ δ)]

= [{(I2 ⊥ δt) ⊥ (I2 ⊥ δ)t−1}{

V (v, w) ⊥ ψ2}{

(I2 ⊥ δ) ⊥ (I2 ⊥ δ)−1}]= [

V (v, w) ⊥ ψ2]

= [V (v, w)

].

(The first equality is via Lemma 4.2 and [9, Lemma 5.1]; the second one is because for any δ ∈ SL2(A),δtψ1δ = ψ1; the third equality is due to Whitehead’s lemma.) �

We now list some observations of W. van der Kallen in [5,6] below:

D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388 383

Lemma 4.5. Let A be a commutative ring of dimension d � 2. Let σ ,τ ∈ SLd+1(A). Then e1[σ ,τ ] can becompleted to an elementary matrix.

Proof. In [5] it is shown that the orbit space Umd+1(A)/Ed+1(A) has an abelian group structure. Itis also shown in [5, §3.26] that the first row map σ −→ [e1σ ] is a homomorphism from SLd+1(A) tothe abelian group Umd+1(A)/Ed+1(A). Therefore, [e1[σ ,τ ]] = 1. �

In [6, Theorem 4.1] W. van der Kallen showed that the orbit space Umd(A)/Ed(A), d � 4, has anabelian group structure, when A has dimension d. However, he gave an example to show that thefirst row map SLd(A) −→ Umd(A)/Ed(A) need not be a homomorphism. We recall this example next.

The following observation (attributed to Whitehead by van der Kallen) is proved in [6, Proposi-tion 7.10]:

Proposition 4.6 (W. van der Kallen’s example). Let

A = R[X1, X2, X3, X4, X5, X6]/(

X21 + X2

2 − 1, X23 + X2

4 + X25 + X2

6 − 1).

Then there exist σ ∈ SL4(A), δ ∈ SL2(A) such that [σ , (I2 ⊥ δ)] /∈ SL3(A)E4(A). Equivalently, [e1[σ , (I2 ⊥δ)]] �= [e1].

The above example will be the key input we shall use to get a set of counter-examples of alge-bras A of dimension three for which the Vaserstein symbol V A is not injective.

Theorem 4.7. There exists an uncountable family of affine real algebras Aλ , λ ∈ R, of dimension three forwhich V Aλ is not injective.

Proof. Consider the four dimensional real algebra A, σ , δ as in Proposition 4.6.(It is actually a domain. This follows from the fact that it is a subring of the complex algebra

C[X1, X2, X3, X4, X5, X6]/(

X21 + X2

2 − 1, X23 + X2

4 + X25 + X2

6 − 1)

= C[X1, X2](X2

1 + X22 − 1)

⊗C

C[X3, X4, X5, X6](X2

3 + X24 + X2

5 + X26 − 1)

,

which is a domain, as the product of two irreducible varieties is an irreducible variety.)Consider e1[σ , (I2 ⊥ δ)] ∈ Um4(A). By Proposition 4.6 [σ , (I2 ⊥ δ)] /∈ SL3(A)E4(A).Therefore, by Lemma 4.3, e1[σ , (I2 ⊥ δ)] /∈ e1 E4(A).Let e1σ = (a,b, c,d) ∈ Um4(A). We may assume a �= 0 or a unit, whence is a non-zero-divisor in A.Let R = A/(a). Then A is a three dimensional affine algebra over R. We prove that V R is not

injective.Suppose V R is injective. Let the overline denote modulo (a).By Lemma 4.4 one has that SL2(A) acts trivially on Um3(A)/E3(A). In particular, one has

[(b, (c,d)δ)] = [(b, c,d)].Therefore, there is an ε ∈ E3(A) such that

(b, (c,d)δ

)ε = (b, c,d).

Let ε ∈ E3(A) be a lift of ε. Then,

(a,b, c,d)(I2 ⊥ δ)(1 ⊥ ε) = (a,b + ax2, c + ax3,d + ax4),

for some x2, x3, x4 ∈ A. But then one has

384 D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388

[e1σ(I2 ⊥ δ)

] = [e1σ(I2 ⊥ δ)(1 ⊥ ε)

]= [

(a,b + ax2, c + ax3,d + ax4)]

= [(a,b, c,d)

]= [e1σ ].

Therefore,

[e1

[σ , (I2 ⊥ δ)

]] = [e1

{σ(I2 ⊥ δ)

}σ−1(I2 ⊥ δ)−1]

= [e1σσ−1(I2 ⊥ δ)−1]

= [e1].A contradiction to Proposition 4.6.

We explain in the next subsection how the argument above can be used to construct a class ofthreefolds for which the Vaserstein symbol is not injective. �A family of counterexamples. We now show how the argument in the proof of Theorem 4.7 can be usedto construct several examples of three dimensional algebras A for which V A is not injective.

Let us begin by restating Proposition 4.6 in a generic way:

Proposition 4.8 (W. van der Kallen’s example). Let

A = R[X1, X2, X3, X4, X5, X6]/(

X21 + X2

2 − 1, X23 + X2

4 + X25 + X2

6 − 1).

Then for δ = ( X1 X2−X2 X1

) ∈ SL2(AR) and

σ =⎛⎜⎝

X3 X4 X5 X6−X4 X3 −X6 X5−X5 X6 X3 −X4−X6 −X5 X4 X3

⎞⎟⎠ ∈ SL4(AR),

one has [σ , (I2 ⊥ δ)] /∈ SL3(AR)E4(AR). In particular, [e1[σ , (I2 ⊥ δ)]] �= [e1].

Proof. That this works is clear from the proof of [6, Proposition 7.10]. �Corollary 4.9. Let L be a subfield of R and let

AL = L[X1, X2, X3, X4, X5, X6]/(

X21 + X2

2 − 1, X23 + X2

4 + X25 + X2

6 − 1).

Then there exist σL ∈ SL4(AL), δL ∈ SL2(AL) such that [σL, (I2 ⊥ δL)] /∈ SL3(AL)E4(AL). In particular,[e1[σL, (I2 ⊥ δL)]] �= [e1].

We first give an uncountable family of non-isomorphic affine algebras AL (over subfields L of R)for which the Vaserstein symbol V AL is not injective. Before that we interject a calculation of unitsfor some algebras. For any ring R , R∗ will denote the group of units of R .

Lemma 4.10. Let

B(k) := C[X3, . . . , X6]/(

X23 + · · · + X2

6 − 1, X3 + Xk4

)

D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388 385

and

R L = L[X1, X2, X4, X5, X6]/(

X21 + X2

2 − 1, X24 + X2

5 + X26 − 1

),

where L is a subfield of R. Then (i) B(k)∗ = C∗ and (ii) R L

∗ = L∗ .

Proof. (i) We note that B(k) ∼= C[X4, X5, X6]/(X2k4 + X2

4 + X25 + X2

6 − 1). Setting U = X5 + i X6 andV = X5 − i X6, where i2 = −1, we see that

B(k) ∼= C[X4, U , V ]/(X2k4 + X2

4 + U V − 1).

Let u, v, x4 denote, respectively, the images of U , V , X4 in B(k). Since u, v form a regular sequencein B(k), we have B(k) = B(k)[1/u] ∩ B(k)[1/v]. Now, B(k)[1/u] = C[u,1/u, x4] and B(k)[1/v] =C[v,1/v, x4]. Hence, B(k)[1/u]∗ = {λu j | λ ∈ C

∗, j ∈ Z} and B(k)[1/v]∗ = {λv j | λ ∈ C∗, j ∈ Z}. Since

B(k) = B(k)[1/u] ∩ B(k)[1/v], we have B(k)∗ = B(k)[1/u]∗ ∩ B(k)[1/v]∗ . Thus, since u and v are alge-braically independent over C, it follows that B(k)∗ = C

∗ .(ii) Let F := L[i] (a subfield of C) and D := F [X4, X5, X6]/(X2

4 + X25 + X2

6 − 1). As in (i), one can seethat D∗ = F ∗ . Note that R L ⊗L F = D[X1, X2]/(X2

1 + X22 − 1). Setting X := X1 + i X2 and Y := X1 − i X2,

we see that R L ⊗L F = D[X, Y ]/(XY − 1). Hence, since D∗ = F ∗ , we have (R L ⊗L F )∗ = {λx j | λ ∈ F ∗,j ∈ Z}, where x is the image of X in R L ⊗L F . Since R L ⊗L F is an integral extension over R L , we haveR L

∗ = {λx j | λ ∈ F ∗, j ∈ Z} ∩ R L . Thus R L∗ = L∗ . �

We thank Amartya Dutta for drawing our attention to the following result in [4].

Theorem 4.11. There exist uncountably many non-isomorphic subfields of R.

Theorem 4.12. For each subfield L of R, consider AL and σL as in Corollary 4.9; e1σL = (aL,bL, cL,dL) andR L := AL/(aL). Suppose that L1 and L2 are two non-isomorphic subfields of R. Then R L1 and R L2 are non-isomorphic rings. Thus, there exist uncountably many non-isomorphic three dimensional affine algebras forwhich the Vaserstein symbol V BL is not injective.

Proof. Note that in the generic example (aL,bL, cL,dL) = e1σL = (X3, X4, X5, X6). Hence

R L = AL/(aL) � L[X1, X2, X4, X5, X6]/(

X21 + X2

2 − 1, X24 + X2

5 + X26 − 1

).

By Lemma 4.10, R L∗ = L∗ . Suppose that Φ : R L1 → R L2 be isomorphism of rings. Then Φ(R L1

∗) = R L2∗

and so, Φ(L1∗) = L2

∗ and hence Φ(L1) = L2. This contradicts the fact L1 and L2 are non-isomorphic.The result now follows from Theorem 4.7, Corollary 4.9 and Theorem 4.11. �Remark 4.13. Suppose we want to construct a family of threefolds over R for which the Vasersteinsymbol is not injective. The proof of Theorem 4.7 tells us how we can go about this.

The argument in the proof of Theorem 4.7 works, for instance, over A/(b) also. In fact, it worksfor such quotients for any choice of coordinate for e1σε, as ε varies in E4(A). This observation canbe used to construct more examples.

Recall that in the generic examples (a,b, c,d) = e1σ = (X3, X4, X5, X6). Let us say we factor out by

a+λ(b −b2). For λ =√

112 + 5

√5

2 the algebra A/(X3 +λX4 −λX24) is singular by the Jacobian criterion,

and so is not isomorphic to A/(X3). (The singular locus is (X21 + X2

2 − 1, X3 +λ(√

5 − 2), X4 − 12 (

√5 −

1), X5, X6).)

In a similar vein one can construct a countable family of real threefolds A which are not isomor-phic but for which the Vaserstein symbol V A is not injective. Here is a set of examples:

For any ring A, Cl(A) denotes the Weil divisor class group of A.

386 D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388

Lemma 4.14. For any integer k ∈N, let

A(k) := C[X1, X2, . . . , X6]/(

X21 + X2

2 − 1, X23 + · · · + X2

6 − 1, X3 + Xk4

).

Then A(k) ∼= A(k′) if and only if k = k′ . Thus, if

R(k) = R[X1, X2, . . . , X6]/(

X21 + X2

2 − 1, X23 + · · · + X2

6 − 1, X3 + Xk4

),

then R(k) ∼= R(k′) if and only if k = k′ .

Proof. Let

B(k) := C[X3, . . . , X6]/(

X23 + · · · + X2

6 − 1, X3 + Xk4

).

Then A(k) = B(k)[X1, X2]/(X21 + X2

2 − 1). Setting X := X1 + i X2 and Y := X1 − i X2, we see that A(k) =B(k)[X, Y ]/(XY − 1) ∼= B(k)[X, X−1]. Suppose that A(k) ∼= A(k′) for some integers k,k′ ∈ Z. ThenCl(A(k)) ∼= Cl(A(k′)). Since A(k) ∼= B(k)[X, X−1], we have Cl(A(k)) = Cl(B(k)). Therefore, Cl(B(k)) ∼=Cl(B(k′)). We now show that Cl(B(k)) ∼= Z

2k−1. This will prove that k = k′ .We note that B(k) ∼= C[X4, X5, X6]/(X2k

4 + X24 + X2

5 + X26 − 1). Setting U := X5 + i X6 and V :=

X5 − i X6, where i2 = −1, we see that

B(k) ∼= C[X4, U , V ]/(X2k4 + X2

4 + U V − 1).

Let u, v, x4 denote, respectively, the images of U , V , X4 in B(k). Since the polynomial X2k4 + X2

4 − 1has no multiple roots in C[X4], B(k) is smooth and Cl(B(k)) is generated by the height one primeideals

p1 := (u, x4 − a1), . . . , p2k := (u, x4 − a2k),

where a1, . . . ,a2k are distinct roots of X2k4 + X2

4 − 1 and having one relation among the generators,namely, p1 + · · · + p2k = 0. Thus, Cl(B(k)) ∼= Z

2k−1.Since R(k) ⊗R C = A(k), using the above, we have R(k) ∼= R(k′) if and only if k = k′ . �Thus we have exhibited a countable family of non-isomorphic threefolds over R, as required.

An uncountable family of counter examples over the reals. We now exhibit an uncountable family of non-isomorphic R-algebras for which the Vaserstein symbol is not injective.

The following result follows from a result of D. Daigle [2, Lemma 2.10].

Proposition 4.15. Let k be a field of characteristic zero and f (Z), g(Z) ∈ k[Z ]. Suppose that k[X, Y , Z ]/(XY −f (Z)) ∼=k k[X, Y , Z ]/(XY − g(Z)). Then there exist α,μ ∈ k∗ and β ∈ k, such that μ f (Z) = g(αZ + β).

Daigle’s result enables us to exhibit an uncountable class of affine algebras over the reals whichare not isomorphic; and this class is the natural choice to the question raised at the beginning of thissection.

Theorem 4.16. For any integer λ ∈R, let

A(λ) := C[X1, X2, . . . , X6]/(

X21 + X2

2 − 1, X23 + · · · + X2

6 − 1, X3 + λX24

).

D.R. Rao, N. Gupta / Journal of Algebra 399 (2014) 378–388 387

Then A(λ) ∼= A(λ′) if and only if λ = ±λ′ . Thus, if

R(λ) = R[X1, X2, . . . , X6]/(

X21 + X2

2 − 1, X23 + · · · + X2

6 − 1, X3 + λX24

),

then R(λ) ∼= R(λ) if and only if λ = ±λ′ .

Proof. Let

B(λ) := C[X3, . . . , X6]/(

X23 + · · · + X2

6 − 1, X3 + λX24

).

Then A(λ) = B(λ)[X1, X2]/(X21 + X2

2 − 1). Setting X := X1 + i X2 and Y := X1 − i X2, we see thatA(λ) = B(λ)[X, Y ]/(XY − 1) ∼= B(λ)[X, X−1]. Suppose that Φ : A(λ) → A(λ′) be an isomorphism forsome λ,λ′ ∈ R. Then, Φ(A(λ)∗) = A(λ′)∗ . Arguing as in Lemma 4.10(i), we see that B(λ)∗ = C

∗ andB(λ′)∗ = C

∗ . Let x denote the image of X in A(λ) and x′ denote the image of X in A(λ′). Thus,A(λ)∗ = {θxi | θ ∈ C

∗, i ∈ Z} and A(λ′)∗ = {θx′ i | θ ∈ C∗, i ∈ Z}. Therefore, Φ(C[x, x−1]) = C[x′, x′ −1].

Set D := C[x, x−1]. Identify C[x′, x′ −1] with D by Φ , and let F denote the field of fractions of D .Then, Φ extends to an isomorphism

Φ̃ : A(λ) ⊗D F → A(λ′) ⊗D F .

For λ ∈ R, set E(λ) := A(λ) ⊗D F . We note that E(λ) ∼= F [X4, X5, X6]/(λ2 X44 + X2

4 + X25 + X2

6 − 1).Setting U := X5 + i X6 and V := X5 − i X6, where i2 = −1, we see that

E(λ) ∼= F [X4, U , V ]/(λ2 X44 + X2

4 + U V − 1).

Thus, since E(λ) ∼= E(λ′), by Lemma 4.15, there exist α,μ ∈ F ∗ and β ∈ F , such that

μ(λ2 X4

4 + X24 − 1

) = λ′ 2(αX4 + β)4 + (αX4 + β)2 − 1.

Note that λ = 0 if and only if λ′ = 0. So we assume that λ �= 0 and λ′ �= 0. Equating the coefficientof X3

4 , we obtain that β = 0 and then equating the constant term, we get μ = 1. Now, equating thecoefficient of X2

4 , we get α2 = 1 and hence equating the coefficient of X44 , we get λ2 = λ′ 2. This proves

the result.Since R(λ) ⊗R C = A(λ), using the above, we have R(λ) ∼= R(λ) if and only if λ = ±λ′ . �As a consequence of the proof of 4.7, Remark 4.13 and Theorem 4.16 we have exhibited an un-

countable family of non-isomorphic threefolds over R, as required to establish Theorem 4.7.

Remark 4.17. We expect that, in some sense, for most threefolds over R the Vaserstein symbol is notinjective.

References

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