on the equality between homological and cohomological dimension of groups

J. reine angew. Math. 664 (2012), 55—70

DOI 10.1515/CRELLE.2011.106

Journal fur die reine undangewandte Mathematik( Walter de Gruyter

Berlin � Boston 2012

On the equality between homological andcohomological dimension of groups

By Ioannis Emmanouil and Olympia Talelli at Athens

Abstract. In this paper, we study criteria for the equality between the homologicaland the cohomological dimension of a group. In particular, we obtain criteria for the free-ness of finitely generated groups of homological dimension one. To that end, we examinetwo types of conditions under which flat modules are projective over a ring R.

0. Introduction

Stallings has shown in [24] that a finitely generated group is free if and only if it hascohomological dimension one. This result was extended to all (not necessarily finitely gen-erated) groups by Swan in [25]. A related question, which is attributed to Bieri by Dicksand Linnell in [3], asks whether a finitely generated group of homological dimension oneis free. Equivalently, the latter question asks whether a group of homological dimensionone is locally free. (The example of the additive group Q of rational numbers shows thatthis result is the best that one could hope for.) This problem has been studied in [3] and [14],by using the embedding of the group ring of a group G into certain algebras of operatorswhich are associated with G, namely the von Neumann algebra NG and the algebra UG ofunbounded operators which are a‰liated to the latter.

If G is a finitely generated group of homological dimension one, then the augmenta-tion ideal IG is a finitely generated flat ZG-module. In view of the above mentioned resultof Stallings, the freeness of G is equivalent to the projectivity of IG as a ZG-module. If theZG-module IG is, in addition, finitely presented (this is the case if, for example, the group G

is finitely presented), then IG can be easily seen to be projective. Having that particularproblem in mind, we shall present in this paper two criteria for a flat module over a ring R

to be projective.

The first one of these is based upon the work [22] of Raynaud and Gruson on therelation between flatness and projectivity. Even though they were mainly interested inmodules over commutative rings (and, more generally, quasi-coherent sheaves on schemes),they established therein a criterion for a countably presented flat left module M over anot necessarily commutative ring R to be projective. As shown by Lazard [17], such a mod-ule M can be expressed as the direct limit of a direct system ðMnÞn of finitely generated free

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left R-modules. With this notation, Raynaud and Gruson proved that M is projective if andonly if the inverse system of abelian groups

�HomRðMn;RÞ

�n

satisfies the Mittag–Le¿ercondition. Using Gray’s characterization [7] of the Mittag–Le¿er condition for inverse sys-tems of countable abelian groups in terms of the vanishing of lim �

1, we obtain the followingresult (which was also proved independently by Jensen in [12]):

Theorem A. Let R be a countable ring and M a countably generated flat left

R-module. Then, the following conditions are equivalent:

(i) M is projective.

(ii) Ext1RðM;RÞ ¼ 0.

(iii) Ext1RðM;RÞ is a countable group.

If G is a group of type FPy, i.e. if there exists a resolution of the trivial ZG-module Z

by finitely generated projective modules, then the homological and the cohomologicaldimension of G are equal; in particular, this is the case if G admits a KðG; 1Þ of finitetype. As an application of Theorem A, we obtain a criterion for the equality betweenhdZ G and cdZ G, where G is any countable group. Our result complements the structuretheorem for soluble groups with hdZ G ¼ cdZ G < y (cf. [13]):

Theorem B. Let G be a torsion-free soluble group of finite Hirsch number n. Then, the

following conditions are equivalent:

(i) hdZ G ¼ cdZ G ¼ n.

(ii) G is of type FP.

(iii) G is a duality group.

(iv) G is constructible.

(v) H nþ1ðG;ZGÞ ¼ 0.

(vi) H nþ1ðG;ZGÞ is countable.

As another application of our criterion, we show (cf. Theorem 3.1) that a countablegroup G of homological dimension one is free if and only if the cohomology groupH 2ðG;ZGÞ is countable. Using the latter result, we exhibit some properties that any groupof homological dimension one shares with locally free groups.

The main idea behind the approach to the study of groups of homological dimensionone, which is presented by Kropholler, Linnell, Luck and Dicks in [3] and [14], consists inembedding the group ring of a group G into a bigger ring, which possesses a dimensiontheory that enables one to show that a certain projective ZG-module P (the relation moduleassociated with a free presentation of G) is finitely generated. One may also try to provethat the projective ZG-module P is finitely generated by passing to the quotient ringZG=IG ¼ Z. Working in that direction, we obtain the following result:

56 Emmanouil and Talell i, On the equality between homological and cohomological dimension of groups



Theorem C. Let R be a ring and assume that I LR is an ideal, such that

(i) the quotient ring R=I is left Noetherian and

(ii)Ty

n¼1

I n ¼ 0.

Then, any finitely generated flat left R-module is projective.

In fact, Theorem C is a special case of a general lifting property, concerning rings allof whose finitely generated flat modules are projective. As an application of Theorem C, itfollows (cf. Theorem 5.5) that a group G of homological dimension one is locally free if

the augmentation ideal IG is residually nilpotent (i.e. ifTy

n¼1

I nG ¼ 0). We note that the resid-

ual nilpotency of the augmentation ideal has been extensively studied in the theory of grouprings (cf. [20]). The corresponding class of groups has been characterized by Lichtman in

[18]; in particular, it is known thatTy

n¼1

I nG ¼ 0 if G is residually a torsion-free nilpotent

group.

The contents of the paper are as follows: In Section 1, we present the projectivity cri-terion of Raynaud and Gruson and use Gray’s criterion, in order to characterize the pro-jectivity of countably generated flat modules over a countable ring R, in terms of the van-ishing of the functor Ext1

Rð_;RÞ. This result is then applied in Section 2, where we obtain acriterion for the equality between the homological and the cohomological dimension of acountable group. In the following section, we consider groups of homological dimensionone and show that these have certain properties in common with a locally free group. InSection 4, we study rings all of whose finitely generated flat modules are projective andestablish the lifting of that property modulo residually nilpotent ideals. This lifting resultis then applied to the case of group rings in the last section, where we also examine certainclosure properties of the resulting class of groups.

Acknowledgments. It is a pleasure to thank I. B. S. Passi for his helpful commentsand suggestions and C. U. Jensen for bringing to our attention [12].

1. The projectivity criterion of Raynaud and Gruson

Raynaud and Gruson established in [22], §I.3.1 and §II.3.1, Grothendieck’s conjecture[9], Remarque 9.5.8, on the local nature of projectivity for modules over a commutativering. Their proof is based on a result, which describes the precise relation between projec-tivity and flatness, and is itself valid for general (not necessarily commutative) rings.

Before stating the projectivity criterion of Raynaud and Gruson, we shall recall thedefinition of the Mittag–Le¿er condition on inverse systems of abelian groups. All directand inverse systems in the sequel will be indexed by the ordered set N of natural numbers.Let ðAnÞn be an inverse system of abelian groups with structural maps sn;m : Am ! An,nem. Then, we say that ðAnÞn satisfies the Mittag–Le¿er condition if the decreasing fil-tration of An by the images of the sn;m’s is eventually constant for all n.

57Emmanouil and Talell i, On the equality between homological and cohomological dimension of groups



We shall restrict our attention below to countably presented modules, since that willbe su‰cient for the applications that we have in mind; for the general case, the readeris referred to [4], §2. Recall that a left R-module is called countably presented if it is thecokernel of a linear map between countably generated free left R-modules. It is easily seenthat the class of countably presented left R-modules coincides with the class of those mod-ules that may be expressed as direct limits of a direct system of finitely presented leftR-modules. As shown by Lazard in [17], §I.3, the subclass of countably presented flat leftR-modules coincides with the class of those modules that may be expressed as direct limitsof a direct system of finitely generated free left R-modules. In particular, it follows that anycountably presented flat left R-module has projective dimensione 1.

We note that for any direct system ðMnÞn of left R-modules a contravariant functor Ffrom the category of left R-modules to that of abelian groups induces an inverse system ofabelian groups ðFMnÞn, whose structural maps FMm ! FMn are induced by the structuralmaps Mn !Mm of the direct system ðMnÞn for all nem.

Proposition 1.1 ([22], I.3.1.3). Let R be a ring and M a countably presented flat left



(ii) If ðMnÞn is a direct system of finitely generated free left R-modules, such that

M ¼ lim�!n

Mn, then the inverse system of abelian groups�HomRðMn;RÞ

�n

satisfies the

Mittag–Le¿er condition. r

The right derived functors of the inverse limit functor were introduced by Roos [23],who showed that the higher inverse limits lim �

i vanish for all if 2; the assumption thatthe inverse systems are indexed by N is crucial here. The Mittag–Le¿er condition foran inverse system of abelian groups ðAnÞn was then defined by Grothendieck in [8],Chapter 0III, 13.1.2, as a su‰cient condition for the vanishing of the group lim �

n

1 An.

Even though the vanishing of lim �1 does not always imply the Mittag–Le¿er condition,

Gray showed in [7] that these conditions are equivalent for inverse systems of countableabelian groups. More precisely, Gray proved the following result:

Proposition 1.2 ([7]). Let ðAnÞn be an inverse system of countable abelian groups.

Then, the following conditions are equivalent:

(i) The inverse system ðAnÞn satisfies the Mittag–Le¿er condition.

(ii) lim �n

1 An ¼ 0.

(iii) lim �n

1 An is countable. r

Let R be a countable ring. Then, it is clear that any countably generated free leftR-module is also countable. We conclude that a left R-module M is countably presentedif and only if it is countably generated.

We are now ready to state and prove the main result of this section, which was alsoproved independently by Jensen in [12], Theorem A of the Introduction.




Theorem 1.3. Let R be a countable ring and M a countably generated flat left



(ii) Ext1RðM;RÞ ¼ 0.

(iii) Ext1RðM;RÞ is a countable group.

Proof. It is clear that (i)! (ii) and (ii)! (iii); hence, it only remains to show that(iii)! (i). As we have already observed above, the countably generated left R-module M isnecessarily countably presented. Therefore, using Lazard’s characterization (cf. [17], §I.3),the flat left R-module M can be expressed as the direct limit of a direct system ðMnÞn offinitely generated free left R-modules. It is well-known that one may express the Ext-groupsof M in terms of the Ext-groups of the Mn’s. Since the Mn’s are projective, it follows thatthere is an isomorphism

Ext1RðM;RÞF lim �

n

1 HomRðMn;RÞ:

Then, our assumption implies that the abelian group lim �n

1 HomRðMn;RÞ is countable.

Since the left R-module Mn is finitely generated, the abelian group HomRðMn;RÞ is alsocountable; indeed, if Mn can be generated by an elements, then HomRðMn;RÞ embeds intoRan . This being the case for all n, the countability of the group lim �

n

1 HomRðMn;RÞ implies

that the inverse system�HomRðMn;RÞ

�n

satisfies the Mittag–Le¿er condition (cf. Proposi-tion 1.2). Then, the projectivity of M follows from Proposition 1.1. r

Remarks 1.4. (i) The category of inverse systems of abelian groups has arbitrary di-rect sums and direct products. If ðAnÞn is an inverse system of abelian groups which satisfiesthe Mittag–Le¿er condition and L is any set, then the inverse systems ðAðLÞn Þn and ðAL

n Þnsatisfy the Mittag–Le¿er condition as well. Furthermore, if ðAnÞn and ðA 0nÞn are two in-verse systems of abelian groups and the direct sum ðAn lA 0nÞn satisfies the Mittag–Le¿ercondition, then this is also the case for the inverse systems ðAnÞn and ðA 0nÞn. Using theseelementary observations and arguing as in the proof of Theorem 1.3, we may prove that acountably generated flat left module M over a countable ring R is projective if and only ifthere exists a countably generated left R-module N, such that

(i1) Ext1RðM;NÞ ¼ 0 (or, equivalently, Ext1

RðM;NÞ is a countable group) and

(i2) the regular left R-module R is a direct summand of the direct sum N ðLÞ or thedirect product NL, for a suitable set L.

(ii) Let R be a countable ring and M a countably generated left R-module. In a sub-sequent paper, we shall strengthen the result of Theorem 1.3, by proving that M is projec-tive if and only if the following two conditions are satisfied:

(ii1) M has projective dimensione 1 and

(ii2) Ext1RðM;RÞ ¼ 0.




2. Homological and cohomological dimension of countable groups

Let G be a group and ZG the associated integral group ring. Since projective modulesare flat, it is clear that hdZ G e cdZ G. In the special case where the group G is count-able, we also have cdZ G e 1þ hdZ G; this is a consequence of the previously mentionedresult of Lazard, which states that any countably presented flat module has projectivedimensione 1 (cf. [17], Theoreme I.3.2). We may apply Theorem 1.3 and obtain a criterionfor the equality between the homological and the cohomological dimension of G, asfollows:

Theorem 2.1. Let G be a countable group of homological dimension n < y. Then, the



(ii) H nþ1ðG;ZGÞ ¼ 0.

(iii) H nþ1ðG;ZGÞ is a countable group.

Proof. Since G is countable, the group ring ZG is also countable. Hence, there is aresolution of the trivial ZG-module Z by countably generated free modules. In particular,there is an exact sequence of ZG-modules of the form

0!M ! Fn�1 ! � � � ! F1 ! F0 ! Z! 0;

where Fi is free for all i ¼ 0; 1; . . . ; n� 1 and M is countably presented. Since hdZ G ¼ n,we conclude that M is flat. Then, the result is an immediate consequence of Theorem 1.3, inview of the natural isomorphism Ext1

ZGðM; _ÞFExtnþ1ZG ðZ; _Þ ¼ H nþ1ðG; _Þ. r

The structure theorem of soluble groups G with hdZ G ¼ cdZ G < y has a longhistory (cf. [6]); it was finally completed by Kropholler in [13]. We recall that the class ofconstructible groups is the smallest class of groups, which contains the trivial group and isclosed under

(i) finite extensions and

(ii) HNN extensions with base group and associated subgroups already constructible.

Using Theorem 2.1, we may supplement Kropholler’s result, as follows (Theorem Bof the Introduction):

Corollary 2.2. Let G be a torsion-free soluble group of finite Hirsch number n. Then,the following conditions are equivalent:










Proof. As shown in [1], Lemma 7.9 and Theorem 7.10, a torsion-free soluble groupG with finite Hirsch number n is necessarily countable and has homological dimensionequal to n. Then, the result follows from the main theorem of [13] and Theorem 2.1. r

The notion of Hirsch length for soluble groups has been extended by Hillman [10] tothe class of all elementary amenable groups. We recall that the latter is the smallest class ofgroups, which contains all abelian and all finite groups and is closed under extensions anddirected unions. Combining Corollary 2.2 with the structure theorem for elementaryamenable groups of finite Hirsch number (cf. [11] and [26]), we obtain the following result:

Corollary 2.3. Let G be a torsion-free elementary amenable group of finite Hirsch

number n. Then, the following conditions are equivalent:







Proof. As shown in [11] and [26], a torsion-free elementary amenable group of finiteHirsch length is soluble by finite. Let K LG be a soluble subgroup of finite index. Then, K

has finite Hirsch length n and the equalities hdZ G ¼ cdZ G ¼ n are equivalent to the equal-ities hdZ K ¼ cdZ K ¼ n, in view of Serre’s theorem [1], Theorem 5.4. Therefore, the resultis an immediate consequence of Corollary 2.2, since the group G is of type FP (resp.a duality group, resp. constructible) if and only if the same is true for K (cf. [1], Proposi-tion 2.5 and Theorem 9.9), whereas H nþ1ðG;ZGÞ ¼ H nþ1ðK;ZKÞ, since K has finite indexin G. r

3. Groups of homological dimension one, I

Our goal in this section is to obtain some information about groups of homologicaldimension one. We shall begin by stating explicitly the following result, which follows fromthe Stallings–Swan theorem characterizing free groups and Theorem 2.1:

Theorem 3.1. Let G be a countable group of homological dimension one. Then, the





(i) G is free.

(ii) cdZ G ¼ 1.

(iii) H 2ðG;ZGÞ ¼ 0.

(iv) H 2ðG;ZGÞ is a countable group. r

As an application of Theorem 3.1, we obtain the following result:

Proposition 3.2. Let G be a non-trivial finitely generated group of homological dimen-

sion one. Then, the following conditions are equivalent:

(i) G is a finitely generated free group.

(ii) G contains a non-trivial finitely generated free group N as a normal subgroup.

Proof. Since the implication (i)! (ii) is obvious, we only have to show that(ii)! (i). To that end, we assume that N is a non-trivial normal subgroup of G, which isfinitely generated and free. We shall prove that the cohomology group H 2ðG;ZGÞ is count-able; then, in view of Theorem 3.1, it will follow that G is free. We may compute the coho-mology groups of G with coe‰cients in ZG by using the Lyndon–Hochschild–Serre spec-tral sequence which is associated with the extension

1! N ! G ! Q! 1;

where Q ¼ G=N. The spectral sequence has the form

Epq2 ¼ H p

�Q;H qðN;ZGÞ

�) H pþqðG;ZGÞ:

Since N is a free group, it follows that Epq2 ¼ 0 if qf 2. On the other hand, we also have

H 0ðN;ZGÞ ¼ 0, since N is an infinite group and ZG a projective ZN-module; hence,E

p02 ¼ 0. Therefore, we conclude that

H 2ðG;ZGÞ ¼ E112 ¼ H 1

�Q;H 1ðN;ZGÞ

�:

The countability of H 2ðG;ZGÞ then results from the following simple lemma. r

Lemma 3.3. Let G be a finitely generated group and A a countable ZG-module. Then,the cohomology group H 1ðG;AÞ is also countable.

Proof. If g1; . . . ; gn are generators of the group G, then the elements

g1 � 1; . . . ; gn � 1

generate the augmentation ideal IG as a ZG-module. In particular, there is a free resolutionof the trivial ZG-module Z, which begins as follows:

ZGn ! ZG ! Z! 0:




Therefore, the cohomology group H 1ðG;AÞ is a certain subquotient of the countableabelian group HomZGðZGn;AÞFAn. r

As an application of Proposition 3.2, we shall now exhibit some properties that anygroup of homological dimension one must necessarily share with a locally free group.

Corollary 3.4. Let G be a group of homological dimension one and consider a non-

trivial subgroup H LG, which is finitely generated and free. Then, the normalizer NGðHÞof H is locally free and the quotient group NGðHÞ=H is locally finite.

Proof. Let K be a finitely generated subgroup of the normalizer NGðHÞ, such thatH LK. The result will follow if we show that

(i) K is free and

(ii) the index ½K : H� is finite.

We note that, being a subgroup of G, the group K has homological dimension one.The non-trivial finitely generated free group H is normal in K and hence we may applyProposition 3.2, in order to conclude that K is free. Since H is a finitely generated freegroup, which is normal in K , the finiteness of the index ½K : H� follows from [19], Proposi-tion 3.11. r

Corollary 3.5. Let G be a group of homological dimension one and consider an infinite

cyclic subgroup H LG. Then:

(i) The normalizer NGðHÞ coincides with the centralizer CGðHÞ.

(ii) The centralizer CGðHÞ is locally infinite cyclic.

Proof. Let h be a generator of H.

(i) If g A NGðHÞ then the subgroup hh; gi generated by h and g is free, being a finitelygenerated subgroup of the normalizer NGðHÞ (cf. Corollary 3.4), and soluble, since thecyclic subgroup hgi normalizes H ¼ hhi. Therefore, hh; gi is infinite cyclic and hence h

and g commute. It follows that NGðHÞ ¼ CGðhÞ ¼ CGðHÞ.

(ii) It follows from (i) above and Corollary 3.4 that the centralizer CGðHÞ is locallyfree. If g1; . . . ; gn A CGðHÞ then the free group hh; g1; . . . ; gni contains h in its center; hence,the latter group must be infinite cyclic. It follows that CGðHÞ is locally infinite cyclic. r

As an application of the above, we may state the following result, which appears alsoin [2] (with a di¤erent proof).

Corollary 3.6. Let G be a group of homological dimension one.

(i) If G is abelian, then G embeds as a subgroup of the additive group Q of rational

numbers.

(ii) If G is non-abelian, then the center of G is trivial.




Proof. (i) It is easily seen that the subgroups of Q are precisely the locally infinitecyclic groups. Then, the result follows, since any finitely generated abelian group of homo-logical dimension one must be infinite cyclic.

(ii) Since the group G has finite homological dimension, it is torsion-free. Assumethat G is non-abelian and has a central element g A G with g3 1. Then, the centralizer ofthe infinite cyclic subgroup generated by g is the (non-abelian) group G. This is a contra-diction, in view of Corollary 3.5(ii). r

4. Residually nilpotent ideals

It is well-known that properties of the class of flat modules over a ring R very oftenreflect properties of (one-sided) ideals of R and vice versa. For example, it is known that:

� The ring R is von Neumann regular (i.e. any finitely generated left ideal of R is adirect summand of the left regular module R) if and only if all left R-modules are flat; for aproof, see [16], Theorem 4.21.

� The ring R is right coherent (i.e. any finitely generated right ideal of R is finitelypresented as a right R-module) if and only if any direct product of flat left R-modules isflat. This result is due to Chase; for a proof, see [16], Theorem 4.47.

� The ring R is left perfect (i.e. it satisfies the descending chain condition on principalright ideals) if and only if any flat left R-module is projective. This result is due to Bass; fora proof, see [15], Theorem 24.25.

In the same spirit, the class of those rings R over which any finitely generated flat leftmodule is projective has been studied in [5] and [21], where it is shown that this property isequivalent to the ascending chain condition on certain sequences of left ideals in matrixrings over R. More precisely, it is shown that all finitely generated flat left R-modules areprojective if and only if the following condition holds: For any nf 1 and any sequenceof n� n matrices ðAtÞt A MnðRÞ, for which AtAtþ1 ¼ At for all t, there exists t0 g 0 suchthat MnðRÞAt0

¼MnðRÞAt0þ1 ¼MnðRÞAt0þ2 ¼ � � � . In that case, we say (following the ter-minology of [21]) that R is a left S-ring. As an example, we note that any left Noetherianring is a left S-ring. Indeed, if R is left Noetherian, then any finitely generated left R-moduleis finitely presented and finitely presented flat modules are well-known to be projective.

Our goal in this section is to show that the left S-property can be lifted modulo certaintypes of ideals. For other lifting results of the S-property, the reader may consult [21], §5.We shall begin with the following basic observation:

Lemma 4.1. Let P be a projective left R-module and assume that P0 is a finitely gen-

erated submodule of it, such that HomRðP=P0;RÞ ¼ 0. Then, P is finitely generated.

Proof. Since P is projective, it can be embedded as a direct summand into a free leftR-module F ¼ RðLÞ ¼

L

l ALRel. Since P0 is finitely generated, there is a finite subset

L0 LL, such that P0 is contained in the submodule F0 LF , where F0 ¼ RðL0Þ ¼L

l AL0

Rel.




Let l A LnL0 and define fl : P! R as the restriction to P of the l-coordinate projectionmap F ¼

L

l 0 AL

Rel 0 ! R. Since P0 is a submodule of F0 and l B L0, it follows that fl

vanishes on P0. In view of our assumption, we conclude that fl ¼ 0. This being the case forany index l A LnL0, it follows that the submodule P of F is actually a submodule of F0

and, in fact, a retract of it. In particular, P is finitely generated. r

For any ideal I LR we define I o as the intersection of the powers I n, nf 1. We shallalso consider the ideals I om

, mf 1, which are defined inductively by letting I omþ1 ¼ ðI omÞofor all mf 1. It is easily seen that ðI omÞo

n

¼ I omþn

for all m; nf 1. We say that I is resid-ually nilpotent if I o ¼ 0.

Proposition 4.2. Let R be a ring, P a projective left R-module and I LR an ideal. If

the quotient module P=IP is finitely generated, then the quotient module P=I om

P is also

finitely generated for all mf 1.

Proof. In view of the inductive definition of I om

, it su‰ces to prove the result for thecase where m ¼ 1.

Our assumption that P=IP is finitely generated implies the existence of a finitely gen-erated R-submodule P0 LP, which is such that P ¼ P0 þ IP. Then, P ¼ P=P0 is a leftR-module, which is such that P ¼ IP and hence R=I nR P ¼ P=IP ¼ 0. In particular, forany left R-module M with IM ¼ 0 (i.e. for any left R=I -module) we have

HomRðP;MÞ ¼ HomR=I ðR=I nR P;MÞ ¼ 0:ð1Þ

We shall prove that HomRðP;R=I nÞ ¼ 0 for all nf 1, using induction. In the casewhere n ¼ 1, this follows from (1). Assuming that nf 1 and HomRðP;R=I nÞ ¼ 0, we con-sider the short exact sequence of left R-modules

0! I n=I nþ1 ! R=I nþ1 ! R=I n ! 0

and the induced exact sequence of abelian groups

0! HomRðP; I n=I nþ1Þ ! HomRðP;R=I nþ1Þ ! HomRðP;R=I nÞ:

Since the group HomRðP; I n=I nþ1Þ is trivial, in view of (1), the inductive hypothesis impliesthat HomRðP;R=I nþ1Þ ¼ 0, as needed.

Then, the natural embedding of R=I o into the direct productQ

n

R=I n identifies theabelian group HomRðP;R=I oÞ as a subgroup of the abelian group

HomRðP;Q

n

R=I nÞ ¼Q

n

HomRðP;R=I nÞ ¼ 0:

In particular, it follows that HomRðP;R=I oÞ ¼ 0. We now consider the projective leftR=I o-module Q ¼ P=I oP ¼ R=I o nR P and its finitely generated submodule

Q0 ¼ ðP0 þ I oPÞ=I oP.




Since Q=Q0 ¼ P=ðP0 þ I oPÞ ¼ R=I o nR P, we have

HomR=I oðQ=Q0;R=I oÞ ¼ HomR=I oðR=I onR P;R=I oÞ ¼ HomRðP;R=I oÞ ¼ 0:

Therefore, we may use Lemma 4.1 and conclude that the projective left R=I o-module Q isfinitely generated, as needed. r

We can now state and prove the main result of this section:

Theorem 4.3. Let R be a ring and assume that I LR is an ideal, such that

(i) the quotient ring R=I is a left S-ring and

(ii) I om ¼ 0 for some mg 0.

Then, R is a left S-ring.

Proof. In view of [5], Proposition 3.5, it su‰ces to show that any finitely generatedand countably presented flat left R-module is projective. To that end, let M be a finitelygenerated and countably presented flat left R-module. Then, being countably presentedand flat, M has projective dimensione 1 (cf. [17], Theoreme I.3.2). Therefore, it followsthat there is a short exact sequence of left R-modules of the form

0! P! Rn !M ! 0;

with P projective. Since M is flat, we may apply the functor R=I nR _ to the exact sequenceabove and obtain the exact sequence of left R=I -modules

0! P=IP! ðR=IÞn !M=IM ! 0:

The left R=I -module M=IM ¼ R=I nR M is finitely generated and flat. Since R=I is a leftS-ring, we conclude that M=IM is projective over R=I and hence the above short exact se-quence splits as an exact sequence of R=I -modules. In particular, it follows that P=IP is afinitely generated left R=I -module and hence a finitely generated left R-module. UsingProposition 4.2, we conclude that the left R-module P=I om

P is finitely generated for allmf 1. In view of our assumption (ii), it follows that the left R-module P is finitely gener-ated and hence the left R-module M is finitely presented. Then, being finitely presented andflat, the left R-module M is projective, as needed. r

Since the class of left S-rings includes all left Noetherian rings, the following result(which is a generalization of Theorem C of the Introduction) is a special case of Theorem 4.3.

Corollary 4.4. Let R be a ring and assume that I LR is an ideal, such that

(i) the quotient ring R=I is left Noetherian and

(ii) I om ¼ 0 for some mg 0.

Then, any finitely generated flat left R-module is projective. r




5. Groups of homological dimension one, II

We shall now apply the results of the previous section, in the special case where R isthe integral group ring ZG of a group G and I the associated augmentation ideal IG LZG.Then, the quotient ring R=I ¼ ZG=IG is identified with the ring Z of integers, a commuta-tive Noetherian ring.

We consider for all mf 1 the class Xm, which consists of those groups G withI om

G ¼ 0. The class X1, which contains all groups G for which the augmentation ideal IG isresidually nilpotent, has been extensively studied in the literature; for example, see [20]. Acharacterization of groups in X1 has been obtained by Lichtman [18], where it is shown thata su‰cient condition for a group G to be in X1 is that G be residually a torsion-free nilpo-tent group. In particular, X1 contains all free groups (cf. [19], §I.10) and all torsion-free abe-lian groups. We also define X ¼

S

m

Xm.

We shall need the following lemma, whose easy proof is omitted.

Lemma 5.1. (i) Let j : G ! H be a group homomorphism and consider the induced

ring homomorphism F : ZG ! ZH. Then, FðI om

G ÞL I om

H for all mf 1.

(ii) Let N be a normal subgroup of a group G and consider the ideal J ¼ INZG LZG.

Then, Jom ¼ I om

N ZG for all mf 1. r

Proposition 5.2. The class X is closed under

(i) subgroups and

(ii) extensions.

Proof. (i) In order to show that X is closed under subgroups, it su‰ces to show thatthe same is true for Xm for all mf 1. To that end, let H be a subgroup of a group G A Xm.Then, the group ring ZH is a subring of ZG and the ideal I ok

H is contained in I ok

G for allk f 1. In particular, it follows that I om

H L I om

G ¼ 0 and hence H A Xm, as needed.

(ii) Let N be a normal subgroup of a group G and consider the quotient group

Q ¼ G=N. If N A Xm and Q A Xl , then we shall prove that G A Xmþl , i.e. that I omþl

G ¼ 0.To that end, we note that the canonical projection p : G ! Q induces a surjective ringhomomorphism P : ZG ! ZQ with kernel kerP ¼ INZG LZG. Since PðI o l

G ÞL I o l

Q ¼ 0(cf. Lemma 5.1(i)), it follows that I o l

G L INZG. We conclude that

I o lþk

G ¼ ðI o l

G Þok

L ðINZGÞok

¼ I ok

N ZG for all k f 1

(cf. Lemma 5.1(ii)) and hence I o lþm

G L I om

N ZG ¼ 0, as needed. r

Let S be the class of those groups, which are such that the integral group ring ZG is a(left) S-ring. In other words, a group G is contained in S if and only if any finitely gener-ated flat ZG-module is projective.




Proposition 5.3. (i) The class S is closed under subgroups.

(ii) If N LG is a normal subgroup with N A X and G=N A S, then G A S.

Proof. (i) This follows since the class of S-rings is closed under taking subrings, asshown in [21], Lemma 3.1.

(ii) Assume that N A Xm and consider the quotient group Q ¼ G=N. Then, the canon-ical projection map p : G ! Q induces a surjective ring homomorphism P : ZG ! ZQ,which enables us to identify the group ring ZQ as the quotient ring ZG=J, whereJ ¼ INZG. Since ZQ is, by hypothesis, an S-ring and Jom ¼ I om

N ZG ¼ 0 (cf. Lemma5.1(ii)), we may invoke Theorem 4.3, in order to conclude that ZG is an S-ring, as needed.

r

Corollary 5.4. The class X is a subclass of S.

Proof. The Noetherian ring Z is an S-ring and hence the trivial group is containedin S. Then, the inclusion XLS is an immediate consequence of Proposition 5.3(ii). r

Our interest in the classes X and S stems from the following result.

Theorem 5.5. Let G be a finitely generated group of homological dimension one.

Then, the following conditions are equivalent:

(i) G is a finitely generated free group.

(ii) G is residually a torsion-free nilpotent group.

(iii) G A X1.

(iv) G A X.

(v) G A S.

Proof. As we noted above, any (finitely generated) free group is residually torsion-free nilpotent, whereas any residually torsion-free nilpotent group is contained in X1. Sincethe implication (iii)! (iv) is clear, whereas the implication (iv)! (v) is precisely Corollary5.4, it only remains to show that (v)! (i). To that end, we note that our assumption on G

implies that IG is a finitely generated flat ZG-module. Since ZG is an S-ring, we concludethat IG is actually a projective ZG-module. Then, G has cohomological dimensione 1; thefreeness of G therefore follows from Stallings’ theorem [24]. r

Residually torsion-free nilpotent groups are known to admit a left order. As shownby Dicks and Linnell in [3], Corollary 6.12, any 2-generated subgroup of a left orderablegroup of homological dimension one is free. Using Theorem 5.5, we conclude that thesame result holds for any finitely generated subgroup of a residually torsion-free nilpotentgroup of homological dimension one.




Corollary 5.6. Let Y be the smallest class of groups, which is such that

(i) Y contains all residually torsion-free nilpotent groups and

(ii) Y is closed under extensions.

If G A Y is a group of homological dimension one, then G is locally free.

Proof. Since any residually torsion-free nilpotent group is contained in X1 (andhence in X), Proposition 5.2(ii) implies that YLX. Hence, if G A Y is a group of homolog-ical dimension one and H is a finitely generated subgroup of it, then H is contained in X

(cf. Proposition 5.2(i)) and has homological dimension one. Therefore, Theorem 5.5 impliesthat H is free. r

Remark 5.7. Kropholler, Linnell and Luck have shown in [14] that a group ofhomological dimension one, which satisfies Atiyah’s conjecture, is necessarily locally free.Let Afin be the class consisting of those groups that satisfy Atiyah’s conjecture (over Z) andhave a finite upper bound on the orders of their finite subgroups. Then, the argumentswhich are provided in [loc. cit.], in order to prove Lemma 4, Lemma 5 and Theorem 2therein, show that if G is a group in Afin and M a finitely generated flat ZG-module ofprojective dimensione 1, then M is finitely presented and hence projective. Invoking [5],Proposition 3.5, we may conclude that G A S and hence that Afin LS. It follows that anyextension of a group in X by a group in Afin is contained in S (cf. Proposition 5.3(ii)). If,in addition, such an extension has homological dimension one, then it is locally free (cf.Proposition 5.3(i) and Theorem 5.5).

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Department of Mathematics, University of Athens, Athens 15784, Greece

e-mail: [email protected]

Department of Mathematics, University of Athens, Athens 15784, Greece

e-mail: [email protected]

Eingegangen 15. April 2009




on the equality between homological and cohomological dimension of groups

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