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A Note on Free Groups in the Ring of Fractions of SkewPolynomial RingsJ. Z. Gonçalves a & E. Tengan ba Department of Mathematics , Instituto de Matemática e Estatıstica, University of SãuPaulo , Brazilb Department of Mathematics , Instituto de Ciëncias Matemáticas e de Computação,University of São Paulo , BrazilPublished online: 24 Jun 2009.

To cite this article: J. Z. Gonçalves & E. Tengan (2009) A Note on Free Groups in the Ring of Fractions of Skew PolynomialRings, Communications in Algebra, 37:7, 2477-2484, DOI: 10.1080/00927870802258646

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Communications in Algebra®, 37: 2477–2484, 2009Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870802258646

A NOTE ON FREE GROUPS IN THE RING OF FRACTIONSOF SKEW POLYNOMIAL RINGS

J. Z. Gonçalves1 and E. Tengan21Department of Mathematics, Instituto de Matemática e Estatıstica,University of Sãu Paulo, Brazil2Department of Mathematics, Instituto de Ciëncias Matemáticase de Computação, University of São Paulo, Brazil

Let L be a function field over the rationals and let D denote the skew field of fractionsof L�t� ��, the skew polynomial ring in t, over L, with automorphism �. We prove thatthe multiplicative group D× of D contains a free noncyclic subgroup.

Key Words: Free groups; Skew fields.

2000 Mathematics Subject Classification: Primary 16K40; Secondary 20E05.

1. INTRODUCTION

Let D be a skew field with centre K and multiplicative group of unitsD× = D − �0�. In [13], answering negatively a question posed by Bachmut,Lichtman raised the following conjecture.

Conjecture 1.1 (Lichtman). D× contains a noncyclic free subgroup.

Many instances of this conjecture have been verified, see for example [3, 4,14, 15]. More recently, building on earlier work by Reichstein and Vonessen [17],Chiba [1] also verified the conjecture when K is uncountable. When dimK D = �and K is countable, the answer is still unknown.

In this note we show the following theorem.

Theorem 1.2. Let L be a function field over � and � ∈ Aut�L� be a nontrivialautomorphism. If L�t� �� denotes the ring of fractions of the skew polynomial ringL�t� ��, then L�t� ��× contains a noncyclic free subgroup.

(Recall that a function field L over � is a finitely generated extension of� with tr. deg�L = 1.) The proof of the above theorem combines a classical

Received December 31, 2007; Revised May 30, 2008. Communicated by I. Shestakov.Address correspondence to E. Tengan, Department of Mathematics, Instituto de Ciëncias

Matemáticas e de Computação, University of São Paulo Av. Trabalhador São-carlense, 400, São Paulo13560-970, Brazil; E-mail: etengan@gmail.com

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2478 GONÇALVES AND TENGAN

specialisation argument with some algebro-geometric techniques, and can easily beadapted to other situations, for instance, mutatis mutandis the same proof yields

Theorem 1.3. Let L = ��t1 tn� and � �= id be a linear automorphism of Lcorresponding to an element of PSLn+1���. Then L�t� ��× contains a noncyclic freesubgroup.

For simplicity, we will stick to the situation of Theorem 1.2. Here is anoverview of the main steps of the proof. Throughout, we will use the language andnotation of standard references of Algebraic Geometry such as [5].

1. Since L is a function field over �, there exists a smooth projective curve Xover Spec� with function field L and a nontrivial automorphism � of X whichinduces �;

2. Since X is of finite type over Spec�, both X and � can actually be definedover some localisation B

df= ��1/h� of � for some nonzero h ∈ �, that is, thereis a smooth projective scheme Y of relative dimension 1 over SpecB and aSpecB-automorphism � of Y such that X = Y ×SpecB Spec� and � = �× id(Theorems 3.1 and 3.2);

3. We apply a specialisation argument to Y : for infinitely many primes p ∈ � notdividing h, we may “reduce everything modulo p” in such a way that the fibreYp

df= Y ×SpecB Spec�p of Y over p is a smooth projective curve over Spec�p

and the restriction �p

df= �× id of � to Yp is nontrivial (Lemma 4.1). Hence thegeneric point of Yp corresponds to a codimension 1 point y ∈ Y such that theautomorphism � of L restricts to an automorphism of the dvr (discrete valuationring) �Yy, and the induced automorphism � on the residue field ��y� (which isthe function field of Yp) is not trivial;

4. Since Yp is a nonsingular curve over a finite field, the automorphism � must beof finite order, and hence ��y��t� �� is finite dimensional over its centre. Henceby [4] ��y��t� ��× contains a noncyclic free subgroup, which can be lifted to anoncyclic free subgroup of L�t� ��× (Corollary 2.3).

A final observation: by [16], if the genus of X is greater than 1, then � hasfinite order and thus L�t� �� is finite dimensional over its centre; in this case theconjecture is know to be true (see [4]). Hence we could assume that the genus of Xis either 0 or 1, but we will avoid this simplification in order to keep the generalityof the method.

2. LIFTING FREE SUBGROUPS

Let us begin with some notation and conventions. We write �R� k� for acommutative local ring R with maximal ideal � and residue field k = R/�. For any� ∈ Aut�R�, we have that ���� = � and hence � induces an automorphism of k,that will be denoted by �.

A subset S of nonzero divisors of a ring A is called a left localising set ora left Ore set if (1) it is closed under multiplication; (2) it satisfies the so-calledOre condition: for any a ∈ A and s ∈ S, there are a′ ∈ A and s′ ∈ S such thats′a = a′s. Given a left localising subset S ⊂ A, we can build a new ring S−1A, the

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FREE GROUPS IN FRACTION RING OF SKEW POLYNOMIALS 2479

the left localisation of A with respect to S, whose elements are fractions of theform s−1a with s ∈ S and a ∈ A and two fractions s−1a and t−1b are identifiedif there exists elements u v ∈ A such that us = vt ∈ S and ua = vb. Addition andmultiplication are performed in the usual way, using the Ore condition to finda common denominator. One important particular case occurs when A is a leftnoetherian domain; then S = A− �0� is a left localising set, and we can build S−1A,which is a skew field containing A. We refer to [2] for further details.

Let �R� k� be a dvr, � ∈ Aut�R� and L = FracR. In this section, followingLichtman [14], we show how to lift a free subgroup from k�t� ��× to a free subgroupof L�t� ��×.

Lemma 2.1. Let �R� k� be a dvr, and let � ∈ Aut�R�. Let

Adf= R�t� ��

�df= �A =

{∑rit

i ∈ A∣∣ ri ∈ �

}

Sdf= A−�

Then � is localisable, i.e., S is a left localising subset of A, and S−1A is anoncommutative local ring with maximal ideal S−1�

df= �s−1a ∈ S−1A � s ∈ S a ∈ ��and residue skew field S−1A/S−1� � k�t� ��.

Proof. It is easy to see that � is an ideal of A, and that A/� � k�t� ��, whichis a domain. Hence � is “prime” (in the sense that ab ∈ � ⇒ a ∈ � or b ∈ �),and thus S is closed under multiplication. By Hilbert’s basis theorem, A is aleft noetherian domain, and hence its nonzero elements form a left localising set.Therefore, given nonzero elements a ∈ A and s ∈ S, there are a′ s′ ∈ A− �0� suchthat a′s = s′a. To show that S is also left localising, we must prove that we may takes′ ∈ S in the last equation. For that, using the fact that R is a valuation ring, write

s′ = xs′′

a′ = ya′′ with x y ∈ R and a′′ s′′ ∈ S

Then yx−1 ∈ R since otherwise xy−1 ∈ �, and thus a′′s = �xy−1�s′′a would belongto �, contradicting the fact that S is closed under multiplication. Hence we maywrite

�yx−1a′′�s = s′′a with yx−1a′′ ∈ A and s′′ ∈ S

This completes the proof that S is a left localising set.To show that S−1� is an ideal of S−1A, let s−1

1 a1 ∈ S−1A and s−12 a2 ∈

S−1� with s1 s2 ∈ S and a1 ∈ A, a2 ∈ �. It is easy to check that �s−11 a1��s

−12 a2� ∈

S−1�. On the other hand, to show that �s−12 a2��s

−11 a1� ∈ S−1�, let a′

2 ∈ A ands′1 ∈ S be such that a′

2s1 = s′1a2. Then a′2 ∈ � since s1 ∈ S and � is prime. Hence

�s−12 a2��s

−11 a1� = �s′1s2�

−1�a′2a1� ∈ S−1�, as required.

Finally, in the canonical surjective map A � k�t� �� with kernel �, the set S ismapped onto k�t� ��− �0�, which consists of invertible elements in k�t� ��. Hence we

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2480 GONÇALVES AND TENGAN

obtain an induced surjective map S−1A � k�t� �� with kernel S−1�. Also S−1A−S−1� = �S−1A�×. This shows that S−1A is a local ring with maximal ideal S−1� andresidue skew field S−1A/S−1� � k�t� ��. �

Remark 2.2. Let L = FracR and v L× → � be the associated valuation to thedvr R. Then v extends to a valuation v L�t� �� → � ∪ ��� given by

v

(∑i

riti

)= min

iv�ri�

for all elements∑

i riti ∈ R�t� ��. In fact, given � � ∈ R�t� ��, it is clear that this

definition satisfies v��+ �� ≥ min�v��� v����. Now, as in the above proof, we maywrite � = x�0 and � = y�0 with x y ∈ R and �0 �0 � � so that v��� = v�x�,v��� = v�y�. Finally, using the fact that v���r�� = v�r� for all r ∈ R, we concludethat �� = x�0y�0 = xy�′0�0 with �′0 � �. Hence �′0�0 � � either and thus v���� =v�xy� = v�x�+ v�y� = v���+ v���, as required.

Corollary 2.3. With the above notation, if k�t� ��× contains a noncyclic free subgroup,so does �S−1A�×. In particular, if � is a nontrivial automorphism of k of finite order,then L�t� ��× contains a noncyclic free subgroup, where L = FracR.

Proof. Suppose that a b ∈ k�t� ��× are two elements that generate a free subgroupof rank 2. Since S−1A− S−1� = �S−1A�×, we can lift a and b to elements a b ∈�S−1A�×, which must necessarily generate a free group of rank 2 since any nontrivialrelation between them would entail a similar relation for a and b. Finally, if� �= 1 has finite order, then by [4] we know that k�t� ��× contains a noncyclic freesubgroup, hence so does L�t� ��× ⊃ �S−1A�×. �

3. DIFFERENCE KERNEL AND LIMITS

In this section, we review a few algebro-geometric facts that will be used in thenext section. Let S be a scheme, and let X and Y be two S-schemes with Y separatedover S. Given two S-morphisms f g X → Y , we define their difference kernel to bethe S-scheme Z given by the cartesian square

Here � is the diagonal map. Since Y is separated over S, � is a closed immersion andhence so is the left vertical arrow. Hence Z can be identified with a closed subschemeof X, which is the largest subscheme on which f and g agree in the following sense:if h T → X is a morphism of S-schemes such that f � h = g � h then h factors

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FREE GROUPS IN FRACTION RING OF SKEW POLYNOMIALS 2481

uniquely through Z. In particular, denoting by ix Spec��x� → X the inclusionmorphism of x ∈ X, we have that

�x ∈ X � f � ix = g � ix�

is a closed subset of X (for more details, see for instance [11]).Next we recall a series of results that allows us to pass from a “limit” situation

to a “finite” one. The setup is the following. Let I be an inductive set with asmallest element 0, and let �Ai�i∈I be a direct system of commutative rings withA� = lim−→ Ai. Write Si = SpecAi and S� = SpecA�. Let X0 and Y0 be schemes overS0, and write Xi = X0 ×S0

Si, Yi = Y0 ×S0Si, and X� = X0 ×S0

S�, Y� = Y0 ×S0S�.

Then �Si�i∈I , �Xi�i∈I and �Yi�i∈I are projective systems of schemes having projectivelimits S�, X� and Y� in the category of schemes. For the proof of the next result,we refer to [8, Théorème 8.8.2, p. 28].

Theorem 3.1. In the above setup:

1. Assume that X0 is quasi-compact and separated over S0 and that Y0 is of finitepresentation over S0 (for instance, S0 is noetherian, and Y0 is of finite type over S0).Then there is a canonical bijection

lim−→ HomSi�Xi Yi� = HomS��X� Y��

2. Let Z� be a scheme that is of finite presentation over S�. Then there exists i ∈ I , afinitely presented scheme Zi over Si, and an S�-isomorphism Z� = Zi ×Si

S�.

For the next result, see [8, Théorème 8.10.5, p. 37, Théorème 11.2.6, p. 123],and [9, Proposition 17.7.8, p. 75].

Theorem 3.2. In the above setup, assume that X0 and Y0 are finitely presentedover S0. Let f� ∈ HomS��X� Y�� and �fi�i∈I , fi ∈ HomSi

�Xi Yi�, be correspondingelements in the bijection of the previous theorem. Then f� is an isomorphism(respectively, a closed immersion, a finite, projective, flat or smooth morphism) if andonly if there exists i ∈ I such that fi has the corresponding property.

As a simple application of the above, let us prove the following well-knownresult, for which we could not find a suitable reference.

Lemma 3.3. Let k be a field and let X be a variety over k (i.e., an integralseparated scheme of finite type over Spec k). Let � X → X be a nontrivial (Spec k)-automorphism of X. Then there exists a finite field extension k′ of k and a k′-valued pointf Spec k′ → X of X such that � � f �= f .

Proof. First we show that if � � f = f for all f kalg → X, then � = id (here kalg

denotes the algebraic closure of k). Let Z be the difference kernel of � and id.Since � � f = f for all f kalg → X any such f factors through Z. By Hilbert’sNullstellensatz, a point x ∈ X is closed if and only if its residue field ��x� is a finiteextension of k. Hence any closed point belongs to the image of some kalg-valued

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2482 GONÇALVES AND TENGAN

point f , and the set of these closed points is dense in X. But Z is a closed subschemeof the integral scheme X, hence Z = X and � = id on X, as was to be shown.

Now suppose that � �= id. Then there exists a kalg-point f such that � � f �= f .Since kalg = lim−→ k′, where k′ runs over all finite extensions of k, we conclude by the

proof of Theorem 3.1 (see [8, Lemme 8.8.2.3, p. 29]) that f factors through a k′-valuedpoint g Speck′ → X for some finite extension k′ such that � � g �= g, as required. �

Since � = lim−→ ��1/h�, where h runs over all nonzero integers, the above limittheorems allow us to show that Spec�-schemes of finite type that are smooth,projective, and so on, can be defined over some “finite” localisation ��1/h� of � sothat the corresponding properties are preserved. This is done in the next section.

4. PROOF OF THEOREM 1.2

First we deal with a small technical issue. We may assume that � is of infiniteorder. Let k be the algebraic closure of � in L. Since L is finitely generated over �,we have that k is a finite extension of � (see for instance [12], Chap. VIII, Ex. 4,p. 374). Hence � restricts to an automorphism of k of finite order, say n, and wehave that �n ∈ AutkL is a nontrivial automorphism. But since L�t� �� ⊃ L�tn� �n� �L�t� �n� it is enough to show that L�t� �n�× contains a noncyclic free subgroup.Hence we may assume without loss of generality that � ∈ AutkL, where k is a finiteextension of � which is algebraically closed in L.

By the correspondence between function fields and smooth projective curves(see for instance [6, Proposition 7.4.18, p. 152]), there is a smooth projective curve Xover Spec k and a nontrivial �Spec k�-automorphism � of X such that the functionfield of X is L and � induces �. Moreover, the fact that k is algebraically closed inL implies that X is geometrically integral, i.e., X ×Spec k Spec k

alg is integral (see [7,Corollaire 4.5.10, p. 63]).

Now we apply the results of the previous section. Let A be the integral closureof � in k. By Lemma 3.3 there is a finite extension k′ of k and a k′-valued point f Speck′ → X such that � � f �= f . Since k = lim−→ A�1/h�, where h runs over all nonzero

elements ofA, byTheorems 3.1 and 3.2wemayfind a localisationB=A�1/h�ofA and:

1. A geometrically integral scheme Y which is smooth and projective overS

df= SpecB of relative dimension 1;2. An S-automorphism � Y → Y ;3. A finite étale extension B′ of B and a B′-valued point g SpecB′ → Y such that

� � g �= g;

such that the original data is obtained from the new one by base change −×S

Spec k, namely, X = Y ×S Spec k, � = �× id, k′ = B′ ⊗B k, and f = g × id.

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FREE GROUPS IN FRACTION RING OF SKEW POLYNOMIALS 2483

Observe that both B and B′ are Dedekind domains with infinitely many maximalideals whose residue fields are finite fields. Next we want to “reduce modulo �” forsome convenient maximal ideal � ∈ SpecB.

Lemma 4.1. With the above notation and hypotheses, there exist infinitely manyclosed points s ∈ S = SpecB such that the fibre Ys

df= Y ×S Spec��s� of s is a smooth

projective geometrically integral curve over Spec��s� and such that �s

df= �× id is nottrivial.

Proof. Since the proper closed sets of SpecB are its finite subsets, it is enough toshow that the set of points s ∈ S for which Ys and �s satisfy the above conditionsis open and nonempty. The nonemptiness is clear, since the generic point � of Sbelongs to this set: X = Y� is a geometrically integral projective smooth curve overSpec k, and � = �� is nontrivial by hypothesis.

Observe that the properties of being smooth and projective are stable underarbitrary base change. On the other hand, since Y is proper and flat over S, the setof points s ∈ S for which Ys is geometrically integral is open by [8, Théorème 12.2.1,p. 179].

Let Z′ be the difference kernel of � � g and g. Then Z′ is a proper closedsubscheme of SpecB′, and hence its underlying topological space consists of finitelymany prime ideals. Let Z ⊂ SpecB be the image of Z′, which is also a finite set,and write gs = g × id for the restriction of g to the fibres over s. Then for anys ∈ S − Z, we have that �s � gs �= gs, and thus �s is nontrivial. Since S − Z is open,we are done. �

Let s ∈ S be one of the infinitely many closed points above. Since s is closed,in the cartesian diagram

both horizontal arrows are closed immersions. We have that the closed subschemeYs ⊂ Y corresponds to the Weil divisor of Y obtained by flat base change −×S Yfrom the divisor s of S (see [9, Proposition 21.10.6, p. 292]). Let y ∈ Y be the genericpoint of Ys; it is a codimension 1 point of Y and R

df= �Yy is a dvr with FracR = L,the function field of Y . Notice that the residue field ��y� of R is just the functionfield of Ys.

Since �s takes the generic point of Ys to itself, we have that ��y� = y andhence � restricts to an automorphism of R. Moreover, since �s is not trivial on Ysthe induced automorphism � of ��y� is not trivial either. The result now followsfrom Corollary 2.3, since Ys is a smooth projective curve over a finite field, hence� must have finite order (see [16] or [10], Exercise A.4.15, p. 90; observe that by[7], Corollaire 2.2.16, p. 13, it is enough to consider the case when the base field isalgebraically closed).

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ACKNOWLEDGMENTS

The first author has been supported by CNPq, Brazil, grant 303.756/82-5,and FAPESP, Brazil, Projeto Temático 2004/15.319-3. The second author has beensupported by FAPESP, Brazil, processo 06/59613-8.

REFERENCES

[1] Chiba, K. (1996). Free subgroups and free subsemigroups of division rings. J. Algebra184(2):570–574.

[2] Cohn, P. M. (1995). Skew Fields. Encyclopedia of Mathematics and its Applications,Vol. 57. Cambridge: Cambridge University Press, Theory of general division rings.

[3] Figueiredo, L. M. V., Gonçalves, J. Z., Shirvani, M. (1996). Free group algebras incertain division rings. J. Algebra 185(2):298–313.

[4] Gonçalves, J. Z. (1984). Free subgroups of units in group rings. Canad. Math. Bull.27(3):309–312.

[5] Grothendieck, A. (1960). Éléments de géométrie algébrique. I. Le langage des schémas.Inst. Hautes Études Sci. Publ. Math. (4):228.

[6] Grothendieck, A. Éléments de géométrie algébrique. II. Étude globale élémentaire dequelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. (8):222.

[7] Grothendieck, A. (1965). Éléments de géométrie algébrique. IV. Étude locale desschémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. (24):231.

[8] Grothendieck, A. (1966). Éléments de géométrie algébrique. IV. Étude locale desschémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math.(28):255.

[9] Grothendieck, A. (1967). Éléments de géométrie algébrique. IV. Étude locale desschémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. (32):361.

[10] Hindry, M., Silverman, J. H. (2000). Diophantine Geometry. Graduate Texts inMathematics, Vol. 201. New York: Springer-Verlag, An introduction.

[11] Iitaka, S. (1982). Algebraic Geometry. Graduate Texts in Mathematics, Vol. 76.New York: Springer-Verlag, An introduction to birational geometry of algebraicvarieties, North-Holland Mathematical Library, 24.

[12] Lang, S. (2002). Algebra. 3rd ed. Graduate Texts in Mathematics, Vol. 211. New York:Springer-Verlag.

[13] Lichtman, A. I. (1977). On subgroups of the multiplicative group of skew fields. Proc.Amer. Math. Soc. 63(1):15–16.

[14] Lichtman, A. I. (1978). Free subgroups of normal subgroups of the multiplicativegroup of skew fields. Proc. Amer. Math. Soc. 71(2):174–178.

[15] Lichtman, A. I. (1982). On normal subgroups of multiplicative group of skew fieldsgenerated by a polycyclic-by-finite group. J. Algebra 78(2):548–577.

[16] Rosenlicht, M. (1955). Automorphisms of function fields. Trans. Amer. Math. Soc.79:1–11.

[17] Reichstein, Z., Vonessen, N. (1995). Free subgroups of division algebras. Comm.Algebra 23(6):2181–2185.

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