On the Group Rings of Abelian Minimax Groups

Journal of Algebra 237, 64᎐94 Ž2001. doi:10.1006rjabr.2000.8579, available online at http:rrwww.idealibrary.com on

On the Group Rings of Abelian Minimax Groups Dan Segal All Souls College, Oxford OX1 4AL, United Kingdom E-mail: [email protected] Communicated by Alexander Lubotzky Received June 27, 1999

INTRODUCTION An abelian group G is called minimax if it contains a finitely generated subgroup H such that GrH satisfies the minimal condition for subgroups Žwhich I shall abbreviate to min.. In this case, we may choose H to be free abelian Žby making it smaller if necessary ., or we may choose GrH to be divisible Žby making H bigger if necessary .. Recall that the divisible abelian groups with min are direct products of finitely many quasicyclic groups Žgroups of type C p⬁ , for various primes p ., and that an abelian group with min is the direct product of a divisible one with a finite group. There are two reasons for being interested in the group rings of such groups, ‘‘external’’ and ‘‘internal.’’ The external reason, and original motivation for the present work, comes from the theory of infinite soluble groups. It has become apparent in recent years that among finitely generated groups, the soluble groups of finite rank are singled out by quite a varied range of finiteness conditions, such as polynomial subgroup growth or finite upper rank wLMS, MSx. In proving results of this nature, some sort of induction argument may well reduce the problem to the consideration of a group ⌫ with an abelian normal subgroup A such that ⌫rA is soluble of finite rank; to conclude that ⌫ itself has finite rank one then has to understand the nature of A as a module for the group ring ⺪Ž ⌫rA.. The analogous situation in the case where ⌫rA was polycyclic was studied by P. Hall in the 1950s and J. E. Roseblade in the 1970s; the key idea of their beautiful and highly successful theory was to pick an abelian normal subgroup G in ⌫rA, and consider A as a module for the group ring ⺪G, with ⌫rA as a group of operators. In their situation, ⺪G was a finitely generated commutative ring, and the whole arsenal of Noetherian ring 64 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

ABELIAN GROUP RINGS

65

theory was available for exploitation. In our case, we are only allowed to suppose that G is a minimax group Žas is every abelian subgroup of a finitely generated soluble group of finite rank.: to have any hope of emulating Hall and Roseblade we therefore need to develop some analogous theory for rings like ⺪G. That is what this paper sets out to do. That the results obtained go some way towards fulfilling the stated purpose will be demonstrated elsewhere wSx; a sample of the methodology is provided by the proof of the final theorem in this paper, 8.5 below. The internal reason for pursuing this topic is that in fact a rather satisfying theory emerges. At first glance, one might expect this to be little more than a routine exercise in commutative algebra: after all, ⺪G is an integral extension of its subring ⺪ H, and ⺪ H is a finitely generated commutative ring. However, the main results seem to depend critically on the interplay between the ideals of the ring and the structure of the multiplicative subgroup G; the resulting theory is specific to such group rings. Before summarising the main results let us establish some terminology. Throughout the paper, G will denote an abelian minimax group. The set of primes p such that G has an element of order p is denoted ␲ Ž G .. A subgroup H is dense in G if GrH is periodic Žthis is equivalent to GrH satisfying min.; and H is co-di¨ isible if GrH is divisible. Note that every finitely generated subgroup of G is contained in a finitely generated dense subgroup of G. We write specŽ G . s ␲ Ž GrH ., where H is a finitely generated dense co-divisible subgroup of G; this is the set of primes p such that C p⬁ is a section of G. G is said to be reduced if the torsion subgroup of G is finite; this is equivalent to G being residually finite. We work over a finitely generated commutative ring k. A kG-module M is non-singular if

␲ Ž M . l spec Ž G . s ⭋. An ideal I of kG is regular if the module kGrI is non-singular; in particular, a prime ideal P is regular if and only if charŽ kGrP . f specŽ G .. For an ideal I of kG we write I † s Ž I q 1. l G, and say that I is faithful if I † s 1. The first main result is 1.2 COROLLARY. generated.

E¨ ery faithful regular prime ideal of kG is finitely

This is fundamental for the whole theory. The second important result shows that, under some circumstances, the group ring kG shares another

66

DAN SEGAL

key property of finitely generated rings; the proof, however, depends on both Hilbert’s irreducibility theorem and Chebotarev’s theorem: 4.2 THEOREM. If G is reduced then e¨ ery faithful regular prime ideal of kG is an intersection of maximal ideals of finite index in kG. The last three sections develop a kind of primary decomposition, for modules which satisfy the following DEFINITION. A kG-module M is quasi-residually finite, or qrf, if GrC G Ž a. is reduced for every a g M. This property is a generalisation of residual finiteness, but has the advantage of passing more readily to quotient modules. It is a curious, though not particularly useful, fact that for non-singular modules, being qrf is equivalent to being ‘‘poly-Žlocally-Žpoly-Žresidually finite....’’ We show that if a non-zero non-singular module M is qrf then the set P Ž M . of associated primes of M is non-empty Žthis is the set of prime ideals of kG of the form ann k G Ž a. for some a g M .. I shall say that M is unmixed if, additionally, P Ž M . consists of maximal annihilators, and primary if P Ž M . is a singleton. What then emerges from the theory is that every non-singular qrf module has a finite filtration in which each factor is unmixed; 䢇

every unmixed module has a natural decomposition as a subdirect sum of primary modules. 䢇

This decomposition serves to reduce the study of general Žnon-singular qrf. modules to that of prime modules, which may be more tractable; the point is that it is available in situations that are generally far from Noetherian. This is illustrated in Section 8, where we consider certain finitely generated k⌫-modules where ⌫ is a group in which G sits as a normal subgroup, with ⌫rG polycyclic.

1. PRIME MODULES In this section, F denotes a field, p denotes a prime, and ␨m denotes a primitive mth root of unity when m is odd and a primitive 2 mth root of unity when m is even. Note that ␨m g F entails charŽ F . ¦ m. The group of units of a ring R is denoted R*. The field of fractions of a domain R is denoted ffrŽ R .. An R-module M is said to be prime if ann R Ž M . is a prime ideal and M is torsion-free when considered as an Rrann R Ž M .-module. A kG-module

ABELIAN GROUP RINGS

67

M is induced from a subgroup H of G if there is a kH-submodule U of M such that the natural mapping U

mk H

kG ª M

is an isomorphism. In this case we write M s U ­GH . Note that in this case Ži. if the kG-module M has one of the properties finitely generated, prime, or torsion-free then the kH-module U has the same property; Žii. M is induced from H1 for every subgroup H1 of G containing H. The main result of this section shows how a prime module is built from a module over a Noetherian subring of kG: 1.1 THEOREM. Let M be a finitely generated non-singular prime kG-module. Then M is induced from a dense subgroup H of G containing C G Ž M . such that HrC G Ž M . is finitely generated. Before embarking on the proof we deduce 1.2 COROLLARY. Ži.

Let P be a faithful regular prime ideal of kG. Then

G contains a finitely generated dense subgroup H such that

P s Ž P l kH . ­GH Žii.

and

kGrP s Ž Ž kH q P . rP . ­GH .

P is finitely generated.

Proof. Put M s kGrP. Let H be as in 1.1; since P is faithful, C G Ž M . s 1, so H is finitely generated. Now let  g i 4 be a transversal to the cosets of H in G. Then kG s [kHg i , and M s [Ug i for some kH-submodule U of M. It follows that P s ann k G Ž M . s ann k G Ž U . s

[ann k H Ž U . g i s [ Ž P l kH . g i ,

which implies that kGrP s

Ý Ž kHg i q P . rP s [ Ž kHg i q P . rP.

This gives Ži.. Since H is finitely generated, kH is a Noetherian ring. Hence P l kH is a finitely generated ideal of kH, and Žii. follows. We build up to the proof of Theorem 1.1 with a series of lemmas.

68

DAN SEGAL

1.3 LEMMA. Let F s F Ž y 0 . : F Ž y 1 . : ⭈⭈⭈ : F Ž yn . be a chain of fields, where yip s yiy1 for i s 1, . . . , n. If F contains ␨ p and F Ž yn . s F Ž yny1 . then F Ž yn . s F. Proof. This follows from wKa, Theorem 51x. 1.4 LEMMA. Let S be an integrally closed domain with field of fractions F, and let E s F Ž C . be an extension field of F, where C is a subgroup of E*. Suppose that CrŽ C l S*. is a ␲-group, where ␲ is a finite set of primes such that ␨ p g S for all p g ␲ . Then the ␲-torsion subgroup of E*rS* is CS*rS*. Proof. We may clearly suppose that C G S* and that CrS* is finite. Now we argue by induction on < C : S* <. It will suffice to show that if x g E* and x p g C, where p g ␲ , then x g C. Replacing x by a suitable m power of x, we may assume also that x p g S* for some m. m

Case 1. Suppose C s S*² x p :. Then x g F Ž x p .. Since x p g F, Lemma 1.3 shows that x g F. As S is integrally closed, it follows that x g S, and m hence that x g S* since x p g S*. Case 2. Suppose C ) S*² x p :. Then S*² x p : F D - C where C s D ² y : and y q g D for some q g ␲ . If x g F Ž D . we are done by inductive hypothesis. Suppose x f F Ž D .. Then F Ž D . - F Ž D .Ž x . : E s F Ž D .Ž y .; as Ž F Ž D .Ž y . : F Ž D .. s q it follows that F Ž D .Ž x . s F Ž D .Ž y . and that q s p. Hence there exists z g F Ž D . such that x p s y h p z p for some integer h ŽKummer theory.. Now z p g D, and so by inductive hypothesis z g D. Consequently x s y h z ␨ pj g C for some j. 1.5 LEMMA. Let R s k w A x be a domain, where A is a subgroup of R*. Let B be a subgroup of A with ArB ( C p⬁ , where p / charŽ R .. Then one of the following holds: Ži. Žii.

ArB has a finite subgroup CrB such that R : ffrŽ k w C x., or ArB has a finite subgroup CrB such that R s k w C x­CA .

Proof. A is the ascending union of a chain of subgroups Bi , with B0 s B and BirB cyclic of order p i for each i G 1. Put F s ffrŽ k w B x. and F⬘ s F Ž ␨ p .. Case 1. Where F⬘Ž Bn . s F⬘Ž Bny1 . for infinitely many values of n G 0. Suppose we have F⬘ - F⬘Ž Bm . for some m. Let n ) m, and choose y such ny j that Bn s B² y :; putting y j s y p for j s 0, . . . , n, we have F⬘Ž Bj . s F⬘Ž y j ., and Lemma 1.3 shows that F⬘Ž Bn . ) F⬘Ž Bny1 .. It follows that in fact F⬘ s F⬘Ž Bm . for all m. Since Ž F⬘ : F . is finite, there exists m such

69

ABELIAN GROUP RINGS

that F Ž Bn . s F Ž Bm . for all n G m. Put C s Bm . Then R:

D F Ž Bn . s F Ž C . s ffr Ž k w C x . ,

nG0

and Ži. holds. Case 2. There exists m such that F⬘Ž Bn . ) F⬘Ž Bny1 . for all n ) m. Let j G 0, and choose y such that B m q j s B ² y : . Then Ž F⬘Ž Bmq j . : F⬘Ž Bm .. s p j, so the first p j powers of y are linearly independent over F⬘Ž Bm ., and a fortiori over k w Bm x. As this holds for each j, we see that Žii. holds with C s Bm . 1.6 LEMMA. Let A, B, and R be as in 1.5, and let U be a finitely generated torsion-free R-module. Assume that JacŽ R . s 0. Suppose that U is not induced from any proper subgroup of A containing B. Then there exists a finite subgroup CrB of ArB such that R : ffrŽ k w C x.. Proof. The R-module U contains a maximal R-linearly independent set S, say ŽZorn’s lemma.; then S generates a free R-submodule W, and UrW is a torsion module for R. As U is finitely generated it follows that ann R ŽUrW . / 0, and as R s k w A x is integral over k w B x Žbecause ArB is a periodic group. this implies that ann kw B xŽUrW . / 0. Choose ␭ g ann kw B xŽUrW . _  04 . Since JacŽ R . s 0, there is a maximal ideal L of R with ␭ f L. Then R s L q ␭ R, so U s UL q U␭ s UL q W and UL l W s WL. Therefore WrWL ( UrUL is a finitely generated RrL-module. As W is a free R-module it follows that W is finitely generated; say d

Ws

[ w R. is1

i

Now suppose that the lemma is false. Then 1.5Žii. holds. Say U s and put V s Ý ris1 u i k w C x where C is as in 1.5Žii.. Then ␭V is a finitely generated k w C x-submodule of W; since R s k w A x and ArC is periodic, it follows that ArC contains a finite subgroup C1rC such that d ␭V F [is1 wi k w C1 x. Now let T be a transversal to the cosets of C1 in A. Then Ý ris1 u i R,

Rs

[ k w C x t. 1

tgT

We can therefore write d

Ws

[ [ w kwC x tgT

ž

is1

i

1

/

t.

Ž 1.1.

70

DAN SEGAL

Now U s Vk w A x s

Ý Vk w C1 x t.

Ž 1.2.

tgT

Suppose ݨ j t j s 0 where each ¨ j g Vk w C1 x and the t j are distinct elements d of T. Then Ý ␭¨ j t j s 0; but ␭¨ j g [is1 wi k w C1 x for each j, so Ž1.1. gives ␭¨ j s 0 for each j. Since U is torsion-free it follows that ¨ j s 0 for each j. Thus the sum in Ž1.2. is direct, and so U is induced from C1. This contradicts the hypothesis, and so completes the proof. 1.7 LEMMA. Let S0 be a finitely generated domain with field of fractions F0 , let F be a finite extension field of F0 , and let S be the integral closure of S0 in F. Then S* is a finitely generated group. Proof. By the generalised units theorem wLa, p. 39x, it will suffice to show that S is contained in some finitely generated subring of F. If char S0 / 0 then S0 is a finitely generated algebra over its prime field P, and wZS, Chap. V, Theorem 9x shows that S is a finitely generated algebra over P. Suppose now that char S0 s 0. It follows from Noether’s normalisation lemma that for some integer m / 0, the ring S0 w m1 x is integral over a polynomial subring of the form ⺪w m1 xw T1 , . . . , Tr x s S1 , say. Then F is still a finite extension of ffrŽ S1 ., and S is contained in the integral closure S2 , say, of S1 in F. In this case, wZS, Chap. V, Cor. 1 to Theorem 7x shows that S2 is finitely generated as a module over S1 , and the result follows. We are now ready for the Proof of Theorem 1.1. M is a finitely generated prime kG-module, with ␲ Ž M . l specŽ G . s ⭋. We may assume without loss of generality that ann k Ž M . s 0 and C G Ž M . s 1. Now G has a co-divisible finitely generated dense subgroup G 0 ; then GrG 0 satisfies min, so we may choose a subgroup ArG 0 of GrG 0 minimal subject to M being induced from A. Thus M s U ­GA , where U is a finitely generated prime kA-module. It will suffice to prove that A is finitely generated. Put ␲ s ␲ Ž ArG 0 ., so ␲ : specŽ G .. Assume, then, that A is not finitely generated. Then A has a subgroup B G G 0 such that ArB ( C p⬁ for some p g ␲ . Let R be the image of kA in the endomorphism ring of U. Since Žwe have assumed that. M is faithful for both k and G, we see that U is faithful for both k and A; so we may identify k and A with their images in R, and we can write R s k w A x Žfor any subgroup A1 of A, k w A1 x will then denote the subring of R generated over k by A1 .. Since U is a prime kA-module, R is a domain and U is torsion-free as an R-module, and charŽ R . f ␲ . Since k is a Hilbert ring and G 0 is finitely generated, kG 0 is again a Hilbert ring, so JacŽ k w G 0 x. s 0,

ABELIAN GROUP RINGS

71

and as R is integral over k w G 0 x it follows that JacŽ R . s 0. The minimal choice of A ensures that U is not induced from any proper subgroup of A containing B. Thus we are in the situation of Lemma 1.6. Hence there exists a finite subgroup CrB of ArB such that R : ffr Ž k w C x . s E0 , say. Now put S0 s k w G 0 x and F0 s ffrŽ S0 .. Then F0 Ž C . s E0 . So if Z s  ␨ l < l g ␲ 4 and E s E0 Ž Z ., F s F0 Ž Z ., we have F Ž C . s E. Let S be the integral closure of S0 in F. Then F s ffrŽ S ., since Z : S, and since S* G G 0 , both CrŽ C l S*. and AS*rS* are ␲-groups. It follows by Lemma 1.4 that A F CS*. But S* is a finitely generated group, by Lemma 1.7; this implies that ArC is finitely generated, a contradiction since ArC ( C p⬁ . This completes the proof. Corollary 1.2 says that regular faithful prime ideals are finitely generated. It is easy to see that a non-faithful prime ideal need not be finitely generated; the next result shows that, in a sense, a regular prime ideal is not far from being finitely generated: 1.8 PROPOSITION. Let P be a regular prime ideal of kG. Then P contains a finitely generated ideal Ph such that for e¨ ery positi¨ e integer m, the additi¨ e group PrŽ Ph q P m . is a di¨ isible ␲-group, where ␲ s specŽ G .. Proof. Put K s P †. By Corollary 1.2, there is a finitely generated subgroup HrK of GrK such that P s ᒈ kG q Ž P l kH . kG, where ᒈ s Ž K y 1. kK is the augmentation ideal of K. Now K has a finitely generated subgroup F such that KrF is a divisible ␲-group; then ᒃ s Ž F y 1. kK is a finitely generated ideal of kK. As k Ž HrK . is Noetherian we have P l kH s ᒈ kH q XkH for some finite subset X of P l kH. We define Ph s XkG q ᒃ kG. As P s ᒈ kG q XkG, we see that PrŽ Ph q P m . is an image of ᒈ kGrŽ ᒃ kG q ᒈ m kG .; as ᒈ kG is additively generated by G-translates of ᒈ, it will be enough to show that ᒈrŽ ᒃ q ᒈ m . is a divisible ␲-group.

72

DAN SEGAL

˜ One verifies Write K˜ s k m⺪ K, and F˜ for the image of k m⺪ F in K. ˜ ˜ is a divisible ␲-group. The group epimorphism K˜ ª ᒈrᒈ 2 easily that KrF given by ␭ m x ¬ ␭Ž x y 1. q ᒈ 2 maps F˜ onto Ž ᒃ q ᒈ 2 .rᒈ 2 , so ᒈrŽ ᒃ q ᒈ 2 . is ˜ ˜ For each i ) 1, Ž ᒃ q ᒈ i .rŽ ᒃ q ᒈ iq1 . is an image of an image of KrF. iy1 ᒈ m⺪ ᒈrŽ ᒃ q ᒈ 2 .. It follows that ᒈrŽ ᒃ q ᒈ m . is a divisible ␲-group as required. However, the hypothesis that P be regular is definitely necessary in 1.2: 1.9 EXAMPLE. Let k be an integral domain of characteristic p, and let G be the additive group of ⺪w 1p x [ ⺪w 1p x [ ⺪w 1p x, written multiplicatively. Then kG contains a faithful prime ideal which is not finitely generated. Proof. Write G s X = Y = Z, where X s ² x n < n g ⺪: ,

Y s ² yn < n g ⺪: ,

Z s ² z n < n g ⺪:

with x np s x ny1 , ynp s yny1 , and z np s z ny1 for each n. Let S be the group ring k Ž XY ., and for each n put t n s x n q yn . Then t np s t ny1 for each n, x a group so the set  t n < n g ⺪4 generates inside the unit group of Sw ty1 0 1 w x isomorphic to ⺪ p . We may therefore define a homomorphism ␪ : kG ª x by setting z n ␪ s t n for all n, s␪ s s for all s g S. Sw ty1 0 Put P s ker ␪ . This is evidently a prime ideal of kG, and it is faithful, x, which occurs only for since if Ž x ia y jb z lc .␪ s 1 then x ia y jb t lc s 1 in Sw ty1 0 a s b s c s 0. On the other hand, P cannot be finitely generated. For if P s Ž P l S² z m :. kG, then S ty1 s kG␪ s S² z m :␪ ­²Zz m : s S ty1 ­²Zz m : , 0 0 which is plainly impossible.

2. INDUCED MODULES In this section, we consider the relationship between the submodules of finite index in a kG-module M and those in a kH-submodule U, when H is a subgroup of G and M s U ­GH . We write DŽ G . to denote the maximal divisible subgroup of G. A subgroup X of H is said to be open in H if X s H l K for some subgroup K of finite index in G. Note that if H is co-divisible in G, then X is open in H if and only if GrX splits over HrX; in this case, GrX s HrX = D X rX, where D X rX s DŽ GrX . is the unique complement to HrX in GrX. In the following proposition, G can be any abelian group, and k can be any ring with identity.

ABELIAN GROUP RINGS

73

2.1 PROPOSITION. Let M be a kG-module and let H be a co-di¨ isible subgroup of G, and suppose that M s UkG where U is a kH-submodule of M. Put N s  N Fk G M < < G : C G Ž MrN . < is finite4 , V s  V Fk H U < C H Ž UrV . is open in H 4 . There exists a mapping ⌽: N ª V gi¨ en by ⌽ Ž N . s N l U, and the following hold: Ži.

⌽ is a bijection from N onto ⌽ Ž N ., with in¨ erse ⌿ gi¨ en by ⌿ Ž V . s VkG q M Ž D X y 1 . s V q U Ž DX y 1. ,

Ž 2.1.

where X s C H ŽUrV .. Žii. If M s U ­GH then ⌽ Ž N . s V . Both ⌽ and ⌿ preser¨ e inclusions. Moreo¨ er, if N g N then Ur⌽Ž N . and MrN are isomorphic kH-modules, and if X s C H ŽUr⌽Ž N .. then D X s C G Ž MrN .. Proof. Let N g N and put Y s C G Ž MrN ., V s N l U, X s C H ŽUrV .. Then HY s G since GrH has no proper finite quotients. Thus M s UkG s U Ž kH q Ž Y y 1 . kH . : U q N,

Ž 2.2.

so M s U q N; hence MrN ( UrŽ N l U . s UrV as kH-modules. It follows that X s H l Y, so X is open in H and V g V . Since now GrX s HrX = YrX, we also have Y s D X , and this implies that VkG q M Ž D X y 1 . F N.

Ž 2.3.

Moreover, G s HY implies kGŽ Y y 1. s kH Ž Y y 1., so M Ž D X y 1 . s UkG Ž Y y 1 . s U Ž Y y 1 . .

Ž 2.4.

On the other hand, suppose w g N. Then Žas in Ž2.2.. we can write w s u 0 q Ý i G 1 u i Ž yi y 1. with each u i g U and yi g Y; the sum on the right lies in M Ž Y y 1. F N, so u 0 g N l U s V. Thus N F V q M Ž Y y 1.. With Ž2.3. and Ž2.4. this establishes the second equality in Ž2.1., and shows that ⌿ Ž V . s N. It is obvious that ⌽ preserves inclusions. That ⌿ does so follows from Ž2.1., and the easily verified fact that if X 1 F X 2 are open subgroups of H then D X 1 F D X 2 .

74

DAN SEGAL

We have established everything except Žii.. Suppose now that M s U ­GH ; we have to show that ⌽ maps N onto V . So let V g V , and put X s C H ŽUrV ., Y s D X , and N s V q M Ž Y y 1.. I claim that N g N and that N l U s V. Now N is certainly a kG-submodule of M, because G s HY and V is a kH-submodule of M. Since Y has finite index in G and C G Ž MrN . G Y, it follows that N g N . Finally, let T be a transversal to the cosets of X in Y, with 1 g T ; as G s HY, we have kG Ž Y y 1 . s

Ý kH Ž X y 1. t q Ý kH Ž t y 1. . tgT

tgT

Since GrX s HrX = YrX, T is also a transversal to the cosets of H in G, so Ms

[ Ut. tgT

Now M Ž Y y 1. s

Ý UkH Ž X y 1. t q Ý UkH Ž t y 1. tgT

F

tgT

Ý Vt q Ý U Ž t y 1. ; tgT

tgT

so M Ž Y y 1. l U F V Žfor if u s ݨ t t q Ýu t Ž t y 1. g U then ¨ t q u t s 0 for each t / 1, giving u s ¨ 1 q Ý t / 1¨ t g V .. Therefore N l U s V q Ž M Ž Y y 1. l U . s V, as required. 2.2 COROLLARY. Ži. E¨ ery finite kG-module image of M is kH-isomorphic to some kH-module image of U. Žii. Suppose that GrH is a ␲-group and that M s U ­GH . If U is a finite kH-module image of U and HrC H ŽU . is a ␲ ⬘-group, then U is kH-isomorphic to some kG-module image of M. We shall mostly apply Proposition 2.1 with M s kGrP where P is a prime ideal of kG. In this case we can say a little more Žwe revert to the standing assumption that G is a minimax group.: 2.3 PROPOSITION. Let P be an ideal of kG and let H be a co-di¨ isible subgroup of G. Put S s kGrP and R s Ž kH q P .rP, and assume that S s R­GH . Write Nf for the set of all ideals of finite index in S and V f for the set of all ideals J of finite index in R such that J † is open in H. Then Ži. the mapping ⌽: Nf ª V f gi¨ en by I ¬ I l R is a bijection, and Rr⌽ Ž I . ( SrI for each I g Nf ;

ABELIAN GROUP RINGS

75

Žii. let I g Nf and n g ⺞; then I n g Nf , and if < S : I < is coprime to ␲ Ž GrH . then ⌽ Ž I n . s ⌽ Ž I . n. Proof. Ži. We apply Proposition 2.1 with M s S and U s R. Since C G Ž SrI . s I † for each ideal I of S, it is clear that Nf : N . As Rr⌽ Ž I . ( SrI for each such I, it follows that ⌽ Ž Nf . : V f . If J g V f then, similarly, J g V , so we have ⌿ Ž J . g N and Sr⌿ Ž J . ( RrJ; thus Sr⌿ Ž J . is finite and ⌿ Ž J . g Nf . Žii. Put q s < S : I < and Gn s Ž I n . †. Then I †rGn has exponent dividing q ny 1 , and GrI † is finite, so GrGn is finite. Also krŽ k l I n . s k n , say, is a finite ring. It follows that the group ring k nŽ GrGn . is finite, and hence so is its image SrI n. Thus I n g Nf . Suppose now that q is coprime to ␲ Ž GrH .. Put J s ⌽ Ž I .. Arguing as above, with H for G and J for I, we see that J n has finite index in R and that J †rŽ J n . † is a finite group of exponent dividing q ny1. As J † is open in H, it now follows that Ž J n . † is open in H. Thus J n g V f . Put In s ⌿ Ž J n .. Then In F I, since ⌿ is inclusion-preserving, and In q R s S, by Ž2.2. above. Therefore I s In q Ž I l R . s In q J , so I n F In q J n s In and so I n l R F In l R s J n. The reverse inclusion being obvious, we deduce that ⌽ Ž I n . s I n l R s J n s ⌽ Ž I . n.

3. AN INTERSECTION THEOREM Here we establish 3.1 THEOREM. Let M be a non-singular, finitely generated prime kGmodule, and let Irann k G M be an ideal of finite index q ) 1 in kGrann k G M. If q is coprime to specŽ G . then ⬁

F MI n s 0.

ns1

Proof. Put S s kGrann k G M, so M is a finitely generated torsion-free S-module, S is an integral domain, and charŽ S . f specŽ G .. As in the proof of Lemma 1.6, we see that M ␭ F W F M for some free S-module W and 0 / ␭ g S. Now put L s Irann k G M. It will suffice to prove that ⬁

F Ln s 0,

ns1

Ž 3.1.

76

DAN SEGAL

for if this holds then also F ⬁ns 1WLn s 0, giving the result since M ( M ␭ F W. Assume without loss of generality that M is faithful for G. Corollary 1.2 shows that G has a finitely generated dense subgroup H such that S s R­GH , where R denotes the image of kH in S; we can choose H to be co-divisible in G. Proposition 2.3Žii. shows that then Ln l R s Ž L l R .

n

Ž 3.2.

for each n. Since R is a finitely generated domain and L l R is a proper ideal of R, Krull’s intersection theorem tells us that F ⬁ns 1Ž L l R . n s 0. Then Ž3.2. shows that R l F ⬁ns 1 Ln s 0, and Ž3.1. follows since S is integral over R.

4. IDEALS OF FINITE INDEX We begin with a result about finitely generated rings: 4.1 PROPOSITION. Let R be a finitely generated integral domain, and let H be a torsion-free subgroup of R*. Let ␲ be a finite set of primes. Then the maximal ideals M of finite index in R satisfying < H : Ž 1 q M . l H < is a ␲ ⬘-number intersect in zero. If charŽ R . s 0, then the same holds of the maximal ideals of prime index in R. Proof. If charŽ R . s l / 0, we shall assume that l f ␲ ; this will not affect the conclusion, since if M is a maximal ideal of finite index in R then < H : Ž1 q M . l H < is a divisor of <Ž RrM .* <, which is coprime to l. We start with some reductions. By wGS1, Theorem Ax, there exist homomorphisms ␪ from R into global fields of characteristic l such that ␪ < H is injective, and the kernels of all such ␪ intersect in zero. Replacing R by its image under such a ␪ , we may therefore assume that R is a subring of a global field k. We may assume that ␨ p g k for each p g ␲ . As R is finitely generated, there is a finite set S of primes of k such that R consists of S-integers; enlarging R if necessary, we may suppose that R is the full ring of S-integers in k. Now R* is finitely generated, so we have R*rH s GrH = TrH, where GrH is free abelian and TrH is finite, of exponent Ł p eŽ p., say. Let e be the maximum of all the eŽ p ., and denote by ␮ Ž p . the group of p-power roots of unity in k. Since G is torsion-free we then have e

k* p l G␮ Ž p . F G

Ž 4.1.

77

ABELIAN GROUP RINGS

for each prime p. Note also that, since H is torsion-free and ␮ Ž p . is cyclic, < ␮ Ž p .< < p e for each prime p. Put m s Ł p g ␲ p e, for each p g ␲ put ␩p s ␨ p eq 1 , and put K s k Ž ␨m , G 1r m . ,

L s K Ž ␩p < p g ␲ . ,

E p s k Ž ␩p .

for p g ␲ .

Below I shall prove that

␩p f K Ž ␩q < q g ␲ _  p 4 . s L p ,

Ž 4.2.

say, for each p g ␲ . This implies that GalŽ LrK . is the direct product of its subgroups GalŽ LrL p ., each of which has order p; it follows that GalŽ LrK . is not contained in D p g ␲ GalŽ LrE p .. Let

␴ g Gal Ž LrK . _

D Gal Ž LrEp . .

pg ␲

Chebotarev’s theorem wFJ, Chap. 5x tells us that there exist infinitely many primes ᑪ of L such that ᑪ l k s ᒍ f S, and

Lrk

ž / ᑪ

ᒍ is unramified in L,

s␴;

in the number-field case, when charŽ k . s 0, infinitely many of these ᒍ have absolute residue-class degree 1. Now choose such a ᑪ, and put ᑪ 0 s ᑪ l K. Write ␪ : R ª Rrᒍ R s ⺖q , say, for the residue-class map, and R for the integral closure of R in K ; note that ᒍ R s R l ᑪ 0 R. Now the choice of ᑪ ensures that ᒍ splits completely in K but does not split completely in E p for any p g ␲ . The first statement implies that R q ᑪ 0 R s R = G 1r m , and hence that G␪ consists of mth powers in Ž R ␪ .*. The second statement implies that q k 1 Žmod p eq 1 . for each p g ␲ , and hence that the ␲-part of <Ž R ␪ .* < divides m. It follows that G␪ is a ␲ ⬘-group. Now we take M s ᒍ R. Then < H : Ž1 q M . l H < divides < G␪ <, a ␲ ⬘-number, and RrM s ⺖q . ŽWhen charŽ k . s 0, we also have that q s < R : ᒍ R < is prime.. As there are infinitely many different possible choices for ᑪ, the intersection of all such M is zero. This completes the proof, modulo the Proof of Ž4.2.. Fix p g ␲ , and put E s k Ž ␨m , ␩q < q g ␲ _  p4., so we have L p s E Ž G 1r m .. Suppose that ␩p g L p . Then E Ž G 1r m . s E Ž G ² ␨ p : .

ž

1rm

/,

78

DAN SEGAL

since ␩pm is a primitive pth root of unity if p is odd and a primitive fourth root of unity if p s 2. It follows by Kummer theory that ␨ p g E* m ⭈ G; e thus ␨ p s y p g for some y g E* and g g G. e pe Now E* l k* : k* p ␮ Ž p ., by wGS2, Lemmas 5.2 and 5.3x. Since ␨ p and e e g are in k* we therefore have y p s x p ␭, where x g k* and ␭ g ␮ Ž p .; then Ž4.1. gives x p s y p ␭y1 s gy1␨ p ␭y1 g k* p l G␮ Ž p . : G. e

e

e

As G is torsion-free this implies that g s xyp and hence that ␨ p s e Ž yxy1 . p . It follows that ␩p g E. Putting E0 s k Ž␩q < q g ␲ _  p4., and noting that E s E0 Ž ␨ p e ., we deduce from Lemma 1.3 that in fact E s E0 . As Ž E0 : k . is prime to p and ␨ p g k this implies that ␩p g k, contradicting our observation that < ␮ Ž p .< < p e. This contradiction completes the proof. e

Now we state the main result of this section: 4.2 THEOREM. Let P be a regular prime ideal of kG such that GrP † is reduced. Let S s kGrP. Ži. The maximal ideals of finite index in S intersect in 0. If charŽ S . s 0 then the same holds for the maximal ideals of prime index in S. Žii. Assume that S is infinite. Then for infinitely many maximal ideals L of finite index in S we ha¨ e F ⬁ns 1 Ln s 0; if charŽ S . s 0 then the same holds for infinitely many maximal ideals of prime index in S. Proof. We may clearly assume that P † s 1. Then G has a torsion-free subgroup G 1 of finite index, and, by Corollary 1.2, G has a finitely generated dense subgroup H such that P s Ž P l kH . kG; we choose H so that GrH is divisible, and write H1 s H l G 1. Then HG 1 s G and G 1rH1 ( GrH. So putting R s Ž kH q P .rP ( kHrŽ kH l P . we have S s R­GH s R­GH11 . Now put ␲ s ␲ Ž GrH .. Suppose M is a maximal ideal of finite index in R such that < H1 : Ž1 q M . l H1 < is a ␲ ⬘-number. Then Corollary 2.2 shows ˜ such that R q M˜ s S and R l M˜ s M, that S has a kG 1-submodule M ˜ is in and it is clear from the construction given in Proposition 2.1 that M ˜ is an ideal of fact an R-submodule of S. Since S s Rw G 1 x it follows that M ˜ ( RrM is a finite field. S, and SrM Proposition 4.1 says that the intersection of all ideals M as above is zero Žcounting only those of prime index in the case where charŽ S . s 0.. It ˜ of S follows that if I is the intersection of all the corresponding ideals M then I l R s 0. But the integral domain S is an integral extension of its

79

ABELIAN GROUP RINGS

subring R, since GrH is periodic. Consequently I s 0. Part Ži. of the theorem follows. Part Žii. follows from Ži. and Theorem 3.1. Remark. The hypothesis that GrP † be reduced is necessary in Theorem 4.2. If G is not finitely generated, then G has C p⬁ as a homomorphic image, for some prime p; provided only that charŽ k . / p, it is then easy to see that kG can be mapped onto an infinite field of prime characteristic. The kernel P of such a map will be a maximal ideal of infinite index in kG. An application of Theorem 4.2 is the following: 4.3 COROLLARY. Let M be a finitely generated non-singular prime kGmodule. Then the following are equi¨ alent: Ža. GrC G Ž M . is reduced; Žb. M is residually finite as kG-module; Žc. there exist a subgroup K of finite index in G and a maximal ideal J of finite index in k such that ⬁

F M Ž JkG q Ž K y 1. kG .

n

s 0.

Ž 4.3.

ns1

If charŽ M . s 0, then these conditions are equi¨ alent to Žd. for infinitely many primes p, there exist a subgroup K of finite index in G and a maximal ideal J of index p in k such that Ž4.3. holds. Proof. If Ž4.3. holds, for some J and K of finite index, then M is residually finite, by Proposition 2.3Žii., so Žd. « Žc. « Žb.. It is easy to see that Žb. implies Ža.. Suppose now that GrC G Ž M . is reduced, and put P s ann k G Ž M .. Then P is a regular prime ideal and GrP † is reduced, so Theorem 4.2Ži. shows that kGrP has a maximal ideal LrP of finite index. Theorem 3.1 shows that F ⬁ns 1 MLn s 0, provided that charŽ kGrL. f specŽ G .. Taking K s L† and J s L l k, we see that Ž4.3. holds. If charŽ kGrP . s l / 0 then also charŽ kGrL. s l f specŽ G ., and Žc. follows. If charŽ M . s 0, we may choose L to have prime index p in kG, and clearly there are infinitely many choices for p. As specŽ G . is finite, this gives Žd..

5. MINIMAL PRIMES Since kG is integral over a finitely generated subring kH, where H is a finitely generated dense subgroup of G, we see that kG has finite Krull dimension: in fact DimŽ kG . s DimŽ k . q hŽ H ., where hŽ H . is the Hirsch

80

DAN SEGAL

length Žtorsion-free rank. of H. It follows that if I is a proper ideal of kG, then the set of prime ideals containing I has minimal members. I shall call these the minimal primes of I. 5.1 PROPOSITION. prime of I.

Let I be a proper ideal of kG and let P be a minimal

Ži. G has a finitely generated dense subgroup H0 such that P l kH is a minimal prime of I l kH for e¨ ery finitely generated subgroup H of G containing H0 . Žii. For each such H, there exists r g kH such that P l kH s ann k H Ž r mod I .. Žiii. If I is regular then P is regular. Živ. If P is regular and P †rI † is finitely generated, then there exists r g kG such that P s ann k G Ž r mod I .. Proof. We assume without loss of generality that I † s 1. Put S s  kH < H is a finitely generated dense subgroup of G 4 . For S g S , define

␮ Ž S . s min  ht Ž Q . < Q a prime ideal of S with I l S : Q : P l S 4 . Now if S : T g S then

␮ Ž S . F ␮ Ž T . F Dim Ž S . s Dim Ž kG . ; so if we choose S0 s kH0 so that ␮ Ž S0 . is maximal and write S 0 s  S g S < S = S0 4 we have S g S0

«

␮ Ž S . s ␮ Ž S0 . s ␮ ,

say. For each S g S 0 let ᑪ Ž S . s  Q < Q a prime ideal of S with I l S : Q : P l S and ht Ž Q . s ␮ 4 .

Then ᑪ Ž S . is non-empty, it consists of minimal primes of I l S, and hence is finite since S is Noetherian. If T = S g S 0 then Q ¬ Q l S maps ᑪ ŽT . into ᑪ Ž S .; the family  ᑪ Ž S . < S g S 0 4 with these maps then forms an inverse system whose inverse limit is non-empty. Let Ž PS .S g S be an element of this inverse limit, and put P˜ s

D

Sg S 0

PS .

ABELIAN GROUP RINGS

81

As kG is the union of its subrings S g S 0 , it is clear that I : P˜ : P and that P˜ is a prime ideal of kG. It follows that P l S s PS is a minimal prime of I l S for each S g S 0 . This proves Ži.. Claim Žii. follows from the fact that S is Noetherian, and Žiii. follows from Žii.. Finally, suppose that P is regular and that P † is finitely generated. Corollary 1.2 shows that then P s Ž P l S . kG for some S g S 0 . There exists r g S such that P l S s ann S Ž r mod I .. Now put K s ann k G Ž r mod I .. Then K l S s P l S, and so K = Ž P l S . kG s P. Since kG is integral over S this implies that K s P, and Živ. follows. Remark. We shall see in the next section that when kGrI is residually finite, the minimal primes of I behave just as in the Noetherian situation ŽCorollary 6.6..

6. QUASI-RESIDUALLY FINITE MODULES A subgroup C of G is closed if GrC is reduced Ži.e., if C is closed in the profinite topology on G .. A kG-module M is qrf Ž quasi-residually finite. if C G Ž a. is closed in G for every element a g M. Since the intersection of any family of closed subgroups is closed, we see that M is qrf if and only if C G Ž S . is closed in G for e¨ ery subset S of M. It is easy to see that any module which is locally residually finite for kG is necessarily qrf. As we shall see, the converse is not far from being true also; however, the qrf hypothesis is more convenient to handle. For an ideal X of kG, we write *X s  a g M < aX s 04 . For a non-empty set X of ideals, we write ² X : to denote the set of all finite products of ideals in X , and MŽ X . s

D  *X < X g ² X :4 .

We also put M Ž⭋. s 0. For a subset S of M, we put S* s ann k G S. We write P Ž M . s  a* < 0 / a g M and a* is prime4 , and denote by M Ž M . the set of all maximal members of P Ž M .. It is well known, and easy to see, that M Ž M . contains all maximal annihilators of non-empty, non-zero subsets of M; it will follow from 6.3, below, that when M is non-singular and qrf then M Ž M . consists of such maximal annihilators. We shall also be interested in the set Q Ž M . of all minimal members of P Ž M ..

82

DAN SEGAL

We put ␲ s specŽ G .. I shall say that a module N has reduced ␲-torsion if the ␲-torsion subgroup ␶␲ Ž N . of N is reduced as an abelian group. We start with some elementary observations; here M is an arbitrary kG-module. 6.1 LEMMA. The set of all closed subgroups of G satisfies the ascending chain condition. Proof. If C is a closed subgroup of G then GrC is finite-by-torsion free; if C1 - C2 then either hŽ GrC2 . - hŽ GrC1 . or <␶ Ž GrC 2 .< <␶ Ž GrC1 .<. 6.2 LEMMA. Ži. Let Ž M␣ . be a chain of submodules of M with union U. If MrM␣ is qrf for each ␣ then MrU is qrf. Žii. Let Ž M␣ . be a family of submodules of M with intersection V. If MrM␣ is qrf for each ␣ then MrV is qrf. Žiii. Let N be a submodule of M which has reduced ␲-torsion. If N and MrN are qrf then M is qrf. Živ. If M is qrf then so are Mr *X, for any ideal X, and Mr␶␴ Ž M ., for any set of primes ␴ . Proof. Ži. Let a g M, and put C␣ s C G Ž a mod M␣ . for each ␣ . Then C G Ž a mod U . s D ␣ C␣ . Now the C␣ form a chain of closed subgroups in G; it follows from 6.1 that D ␣ C␣ s C␤ for some ␤ . Žii. Using the notation above, we have C G Ž a mod V . s FC␣ , which is closed. Žiii. Let a g M and put A s C G Ž a mod N ., B s C G Ž akG l N .. Then A and B are closed, so GrŽ A l B . is reduced. The mapping x ¬ aŽ x y 1. is a homomorphism from A l B into N with kernel equal to C G Ž a., so Ž A l B .rC G Ž a. is a section of G with reduced ␲-torsion. It follows that GrC G Ž a. is reduced. Živ. If a g M then C G Ž a modŽ*X .. s C G Ž aX ., which is closed in G. Now ␶␴ Ž M . is the ascending union of a family of submodules of the form *n, where n ranges over a sequence of ␴-numbers, each dividing the next; the second claim therefore follows by Ži.. Remark. The hypothesis that N have reduced ␲-torsion is necessary in Žiii.. Without that hypothesis, it becomes false: consider G s C p⬁ acting on M s C p⬁ [ ⺪ via x ¬ 1x 10 . From now on, we shall assume that M is a non-singular qrf kG-module. Note that every member of P Ž M . is then a regular prime ideal. The next, key, lemma shows the usefulness of the qrf hypothesis: it implies that every annihilator is contained in a maximal one.

ž /

6.3 LEMMA.

If S ®  04 is a subset of M then S* : P for some P g M Ž M ..

ABELIAN GROUP RINGS

83

Proof. Let X s  ¨ g M _  04 < ¨ * = S*4 , and let Y s  ¨ g X < C G Ž ¨ . is maximal for ¨ g X 4 . Then Y is non-empty by 6.1. Let ¨ g Y, put I s ¨ *, and let P be a minimal prime of I. By Proposition 5.1, there is a finitely generated subgroup H0 of G such that P l kH is a minimal prime of I l kH whenever H0 F H F G and H is finitely generated; and for any such H, there exists r g kH such that P l kH s ann k H Ž r mod I . s ann k H Ž ¨ r .. This implies, in particular, that P is regular. Let H and r be such a pair. Then ¨ r g X and C G Ž ¨ r . G C G Ž ¨ ., so P † l H s C H Ž ¨r . F CG Ž ¨r . s CG Ž ¨ . s I †. As G is covered by subgroups like H it follows that P † s I †. The last part of Proposition 5.1 now shows that there exists r g kG such that P s ann k G Ž r mod I . s Ž ¨ r . *, and ¨ r g Y since now C G Ž ¨ r . s C G Ž ¨ .. Thus the set Z s  w g Y < w* is prime4 is non-empty. Now kG has the ascending chain condition on prime ideals since DimŽ kG . is finite, so the set  w* < w g Z 4 has a maximal member Q. Since Z : X we have S* : Q. I claim that Q g M Ž M .. To see this, suppose Q : ¨ * for some ¨ / 0. Then ¨ g X and C G Ž ¨ . G C G Ž w . where w g Y with Q s w*, so ¨ g Y. Let P be a minimal prime of ¨ *. By the preceding paragraphs we see that P s wU1 for some w 1 g Z; so from Q : ¨ * : P it follows that Q s P s ¨ *. This establishes the claim and completes the proof. 6.4 COROLLARY.

If M / 0 then P Ž M . is non-empty, and

␲ Ž M . s  char Ž kGrP . < P g P Ž M . 4 _  0 4 s  char Ž kGrP . < P g M Ž M . 4 _  0 4 . 6.5 LEMMA. Then

Let X be a set of prime ideals of kG, and put N s M Ž X ..

Ži. MrN is qrf, and ␲ Ž MrN . : ␲ Ž M . Ž so in particular MrN is non-singular .. Žii. P Ž MrN . : P Ž M . _ X and X l P Ž M . : P Ž N .. Žiii. Suppose that X : P Ž M .; if P Ž M . _ X : Q Ž M . then P Ž MrN . s P Ž M . _ X and P Ž N . s X . Živ. M s M Ž P Ž M .. s M Ž Q Ž M ...

84

DAN SEGAL

Proof. Ži. Let w g M _ N. By 6.1 we may choose D 0 g ² X : so as to maximise C G Ž wD 0 .. I claim that then C G Ž wD 0 . s C G Ž w mod N .. Indeed, x g C G Ž wD 0 .

«

w Ž x y 1 . D 0 s wD 0 Ž x y 1 . s 0

«

w Ž x y 1. g N

«

x g C G Ž w mod N . .

On the other hand, if x g C G Ž w mod N . then w Ž x y 1. D s 0 for some D g ² X :, so x g C G Ž wD . F C G Ž wDD 0 . s C G Ž wD 0 . by the maximal choice of the latter. It follows that C G Ž w mod N . s C G Ž wD 0 . is closed; thus MrN is qrf. Suppose now that qw g N where q is a prime number. Then qwD s 0 for some D g ² X :, and wD / 0 since w f N. Hence q g ␲ Ž M .. This shows that ␲ Ž MrN . : ␲ Ž M .. Žii. Suppose now that Q s annŽ w mod N . is prime, where w g M _ N as above; it follows from Ži. that Q is regular. Putting I s annŽ wD 0 ., we have Q † s C G Ž w mod N . s C G Ž wD 0 . s I †. It follows by Corollary 1.2 that QrI is finitely generated, so we have Q s I q J for some finitely genrated ideal J. Then wJD 1 s 0 for some D 1 g ² X :, since now wJ is a finitely generated submodule of N. Since wD 0 I s 0 by definition it follows that wQD 1 D 0 s 0. Thus putting D s D 1 D 0 we have D g ² X : and Q : annŽ wD .. Now wD ­ N; for if wD : N then D : Q and then wD 2 s 0, which is false since w f N. Hence there exists x g D such that wx f N. Put Y s Ž wx .*. Then xY : Q since wxY s 0, but x f Q since wx f N; consequently Y : Q. Thus Q : Ž wD . * : Ž wx . * s Y : Q, showing that Q s Y g P Ž M .. On the other hand, Q f X since wDQ s 0 and w f N. To complete the proof of Žii., note that if X g X l P Ž M . then X s a* for some x g M, and then a g M Ž X 4. F N, so in fact X g P Ž N .. Žiii. Assume now that X : P Ž M . and that P Ž M . _ X : Q Ž M .. Let P g P Ž M . _ X . Then P s a* for some a g M. If a g N then a* G D for some D g ² X :. But as P is prime and minimal in P Ž M . this would force P to be equal to some member of X , so a f N, and hence, by Lemma 6.3, P F Q for some Q g M Ž MrN . : P Ž M . _ X : Q Ž M .. Then P s Q, and the first claim of Žiii. follows. For the second claim, suppose that Y g P Ž N .. Then Y G D for some D g ² X :, so Y G X for some X g X ; if Y f X then Y g Q Ž M ., forcing Y s X g X , a contradiction, so we must have Y g X . Thus P Ž N . : X , and the result follows from Žii..

ABELIAN GROUP RINGS

85

Finally, putting X s P Ž M . and N s M Ž X ., we see from Žii. that P Ž MrN . s ⭋, and hence by 6.4 that M s M Ž P Ž M ... Since DimŽ KG . is finite, each member of P Ž M . contains a member of Q Ž M ., so Živ. follows. 6.6 COROLLARY. Let I be a non-singular ideal of kG, and suppose that kGrI is qrf as a kG-module. Then I has only finitely many minimal primes, P1 , . . . , Pm , say, and there exist natural numbers n i such that m

Ł Pin : I. i

is1

For each i there exists x i g kG such that Pi s ann k G Ž x i mod I .. Proof. Applying 6.5Živ. to the module M s kGrI, we see that the element 1 q I is annihilated by a product P1n1 . . . Pmn m , with each Pi g Q Ž M .. It follows that P1n1 . . . Pmn m : I : Pi for each i, and this implies that  P1 , . . . , Pm 4 is exactly the set of minimal primes of I. The final sentence follows from the definition of Q Ž M .. Recall that M is said to be unmixed if P Ž M . s M Ž M .. We show next that an arbitrary non-singular qrf module has a finite filtration, of bounded length, with each factor unmixed. First we introduce some notation. For X g P Ž M ., let h#Ž X . denote the maximal length of a chain X s X 0 ) X 1 ) ⭈⭈⭈ ) X n with X i g P Ž M . for each i. We define ␭Ž M . to be the maximal length of any chain in P Ž M .; note that

␭Ž M . s

sup

h# Ž X . F Dim Ž kGrM* . ,

Xg P Ž M .

and that X g Q Ž M . if and only if X g P Ž M . and h#Ž X . s 0. 6.7 PROPOSITION.

For 0 F i F ␭Ž M ., let

Xi s  X g P Ž M . < h# Ž X . - i 4 ,

M i s M Ž P Ž M . _ Xi . .

Then Ži. M s M0 ) M1 ) ⭈⭈⭈ ) Mk s 0, where k s ␭Ž M . q 1; Žii. for each i, we ha¨ e P Ž Miy1rMi . s M Ž Miy1rMi . s Xi _ Xiy1. Proof. Note to begin with that M0 s M by 6.5Živ., because X 0 is empty, and that Mk s 0 because P Ž M . _ Xk is empty. Next, we prove Žii.. Soince X1 s Q Ž M ., we see from 6.5Žiii. that P Ž MrM1 . s X1 s X1 _ X 0 ; thus Žii. holds for i s 1, as the members of X1 are pairwise incomparable.

86

DAN SEGAL

The second part of 6.5Žiii. shows that P Ž M1 . s P Ž M . _ X1; that Žii. holds for i ) 1 now follows by induction on ␭Ž M ., on replacing M by M1. To complete the proof of Ži., note now that if 1 F i F k then Xi _ Xiy1 is non-empty, so Corollary 6.4 shows that Miy1rMi / 0. 6.8 COROLLARY. A non-singular kG-module is qrf if and only if it is PLPL-residually finite for kG. PLPL means ‘‘poly locally-Žpoly locally-..’’ We shall see in Section 8 that in fact ‘‘PLPL’’ can be replaced by ‘‘PLP.’’ Proof. Let M be a non-singular qrf kG-module. In the notation of Proposition 6.7, it will suffice to show that if U is a finitely generated submodule of Miy1rMi , for some i F k, then U is poly locally-Žresidually finite.. Now 6.7Žii. implies that P ŽU . s M ŽU ., and as U is finitely generated, it follows from 6.5Živ. that UP1 . . . Ps s 0 for some P1 , . . . , Ps g M ŽU .. Taking s to be minimal, put U0 s 0 and Ui s *Ž P1 . . . Pi . for i s 1, . . . , s. For each i, UrU i iy1 is easily seen to be prime kG-module with annihilator Pi ; so if W is a non-zero finitely generated submodule of UrU i iy1 then W is prime and C G ŽW . s Pi† is closed in G. It follows by Corollary 4.3 that W is residually finite. Thus each UrU i iy1 is locally residually finite for kG. The converse follows from Lemma 6.2Žiii., since every residually finite kG-module is qrf, and every locally qrf module is qrf. Indeed, this argument shows, more generally, that every module with reduced ␲-torsion which is ŽPL. n-Žresidually finite. for kG for some n is a qrf module.

7. DECOMPOSITION OF UNMIXED MODULES With a view to applications, we consider in this section kG-modules with operators. We shall suppose that G is contained as a normal subgroup in another group ⌫, and that M is a k⌫-module; adjectives such as ‘‘qrf’’ applied to M will always refer to the kG-structure of M. It is clear that ⌫ permutes the sets P Ž M . and M Ž M .. If also M is non-singular and qrf, then the subsets Xi of P Ž M . and the submodules Mi of M given in Proposition 6.7 are all ⌫-invariant. Throughout this section, the k⌫-module M is supposed to be non-singular and qrf as a kG-module. We assume also that M is unmixed, which means that ␭Ž M . s 0; that is, PŽ M . s M Ž M . s QŽ M . . The main result, 7.3 below, shows that if M is finitely generated as a k⌫-module, then M can be embedded in a finite direct sum of modules that are induced from primary kG-modules.

ABELIAN GROUP RINGS

87

7.1 LEMMA. Let Xi Ž i g I . be a family of pairwise disjoint subsets of P Ž M ., and put X s D i g I Xi , M i s M Ž Xi . ,

M i s M Ž X _ Xi .

for each i. Then Ži. Ý i g I Mi s [i g I Mi ; Žii. if X s P Ž M . then the natural map M ª Ł i g I MrM i is injecti¨ e and maps M into [i g I MrM i. Proof. Ži. Suppose 0 / a g M1 l Ž M2 q ⭈⭈⭈ qMn ., where 1, 2, . . . , n are distinct indices in I . Then there exist X g ² X1 : and Y g ² X 2 j ⭈⭈⭈ j Xn : such that aŽ X q Y . s 0. Then X q Y : P for some P g M Ž M ., forcing P s X 1 s Yj for some X 1 g X1 and Yj g Xj , where j G 2. This is impossible since X1 l Xj s ⭋. Žii. Let 0 / a g M, and put I s a*. Then kGrI ( akG is non-singular and qrf; by Corollary 6.6, I has finitely many minimal primes nj P1 , . . . , Pm , say, and these satisfy Ł m js1 Pj : I. Also each Pj is the annihilator of some element of akG, and since M s M Ž X ., by 6.5Živ., each Pj contains some member of X . It follows that Pj g X for each j. Now a g M i if and only if I contains some member of ² X _ Xi :; this holds whenever Xi l  P1 , . . . , Pm 4 s ⭋, which is true for all but finitely many values of i. It follows that the image of a in Ł MrM i lies in [MrM i, as claimed. On the other hand, if Pj g XiŽ j. then a f M iŽ j., for j s 1, . . . , m: for if I contains some member of ² X _ XiŽ j. : then Pj = Q for some Q g X _ XiŽ j. , forcing Pj s Q g X _ XiŽ j. . Thus the image of a in Ł MrM i is not zero, showing that the map M ª Ł MrM i is injective. 7.2 COROLLARY. Suppose that X s M Ž M ., and that the action of ⌫ on X is transiti¨ e. Fix P g X , put ⌬ s N⌫ Ž P ., and let U s MrM Ž X _  P 4 . . Ži. M has a natural embedding into the k⌫-module U ­⌫⌬ . Žii. If M is finitely generated as a k⌫-module then U is finitely generated as a k⌬-module. Proof. Ži. Let T be a transversal to the right cosets of ⌬ in ⌫. Then X s  P t < t g T 4 . We apply 7.1 with I s T, Xt s  P t 4 , and M t s M Ž X _  P t 4. for each t. Supposing that 1 g T, we thus have U s MrM 1 , a

88

DAN SEGAL

k⌬-module since clearly M 1 is ⌬-invariant, and M t s M 1 t for each t g T. The operation of t g T therefore induces a k-module isomorphism ␪ t : U s MrM 1 ª MrM t. Writing U ­⌫⌬ s [t g T U m t, we may define a kmodule isomorphism ␪ : U ­⌬⌫ ª [t g T MrM t by putting Ž u m t .␪ s u␪ t . Now let ⌿ s ⌸ ( ␪y1 : M ª U ­⌫⌬ , where ⌸: M ª [t g T MrM t is the natural injective mapping given by 7.1Žii.. As G F ⌫, we only have to verify that ⌿ respects the action of ⌫. So let ␥ g ⌫, and suppose that t␥ s ␣ t st , where ␣ t g ⌬ and st g T, for each t g T. Write ᎏ: M ª U for the quotient mapping. Then for a g M we have

Ž a⌿ . ␥ s ž Ý aty1 m t / ␥ s

Ý aty1␣ t m st

s

m st s Ž a␥ . ⌿, Ý a␥ sy1 t

as required. Žii. Let S be a finite subset of M. Then S⌿ is contained in V ­⌫⌬ for some finitely generated k⌬-submodule V of U. Writing ␲ 1 for the projection U ­⌫⌬ ª U m 1 s U, we have Sk ⌫⌿␲ 1 s S⌿k⌫␲ 1 F Ž V ­⌬⌫ . ␲ 1 s V . Hence if Sk ⌫ s M then U s M ⌿␲ 1 F V. Thus U s V and the result follows. 7.3 PROPOSITION. Let  Pj < j g I 4 be a set of representati¨ es for the orbits of ⌫ in X s M Ž M .. For each j let ⌬ j s N⌫ Ž Pj . and put Uj s MrM Ž X _  Pj 4.. Then P ŽUj . s  Pj 4 for each j, and M has a natural embedding as a k⌫-module into

[ Uj ­⌫⌬ jg I

j

.

If M is finitely generated as a k⌫-module, then I is finite, and for each j g I , Uj is finitely generated as a k⌬ j-module and satisfies Uj Pje j s 0 for some e j g ⺞. Proof. Denote by Xj the ⌫-orbit of Pj , and put M j s M Ž X _ Xj .. Lemma 7.1 shows that M embeds as a subdirect sum in [j g I MrM j; each M j is ⌫-invariant, and it is clear that this embedding is a k⌫-homomorphism. Now fix j g I . By Lemma 6.5Žiii. we have P Ž MrM j . s Xj ; it

ABELIAN GROUP RINGS

89

follows that M Ž MrM j . s P Ž MrM j . s Xj is permuted transitively by ⌫. This also shows, by 6.5Živ., that MrM j s Ž MrM j .Ž Xj ., and by 6.4 that M / Mj. That P ŽUj . s  Pj 4 also follows from 6.5. Put M j>rM j s Ž MrM j .Ž Xj _  Pj 4.. The preceding corollary gives a natural embedding of k⌫-modules MrM j ª MrM j> ­⌫⌬ j .

ž

/

So to establish the first part it suffices now to show that M j> s M Ž X _  Pj 4.. It is clear that M j F M Ž X _  Pj 4. F M j>; consequently, by 6.5Žiii., P M j>rM Ž X _  Pj 4 . :  Pj 4 l Ž Xj _  Pj 4 . s ⭋.

ž

/

Therefore M j> s M Ž X _  Pj 4. as required. Suppose now that M is finitely generated as a k⌫-module. Then so is MrM j, for each j; and 7.2Žii. shows that Uj s MrM j> is then finitely generated as a k⌬ j-module. Since Uj s UŽ Pj 4., by 6.5, and Pj is ⌬ j-invariant, it follows that some finite power of Pj annihilates Uj . Now let S be a finite generating set for M as a k⌫-module. Lemma 7.1 shows that S : M j for all but finitely many indices j g I . As each M j is ⌫-invariant, it follows that M s M j for all but finitely many indices j g I . But we have seen above that M / M j for all j g I , so I must in fact be finite.

8. A FILTRATION As in Section 7, G is supposed to be a normal subgroup of a group ⌫. We saw in Corollary 4.3 that if M is a finitely generated non-singular prime kG-module and M is residually finite, then there is a subgroup of finite index in G that acts ‘‘residually nilpotently’’ on M. We are now in a position to generalise this to the case where M is finitely generated as a k⌫-module, provided that ⌫rG is a polycyclic group. Throughout this section, P will denote a ⌫-invariant regular prime ideal of kG. We write l s char Ž kGrP . ,

␲ s spec Ž G . .

8.1 LEMMA. Let M be a finitely generated k⌫-module such that MP m s 0 for some m. Then MP contains a finitely generated k⌫-submodule N such that the additi¨ e group MPrN is a di¨ isible ␲-group. If l / 0 then MP is finitely generated as a k⌫-module.

90

DAN SEGAL

Proof. By Proposition 1.8, P contains a finitely generated ideal Ph such that PrŽ Ph q P m . is a divisible ␲-group. Let U be a finitely generated kG-submodule of M such that M s U ⌫, and put N s UPhk⌫. Now MPrN is additively generated by ⌫-translates of its subgroups Ž uP q N .rN, with u g U. Since uP l N G uŽ Ph q P m ., each of these subgroups is an image of PrŽ Ph q P m ., and hence a divisible ␲-group. It follows that MPrN is likewise a divisible ␲-group. Finally, if l / 0 then MPrN has finite exponent dividing l my 1, as well as being divisible; so MP s N and the last claim follows. Let ␭ g kG. We say that a kG-module B is ² ␭⌫ :-torsion if every element of B is annihilated by a product of the form ␭␥ 1 . . . ␭␥ n with ␥ 1 , . . . , ␥n g ⌫, that is, if B s B Ž I ⌫ ., where I s ␭ kG and I ⌫ s  I ␥ < ␥ g ⌫4 . The pair Ž kGrP, ⌫ . has the Hall᎐Roseblade property if for every finitely generated k⌫-module A satisfying AP s 0, there exists a free kGrP-submodule F of A such that ArF is ² ␭⌫ :-torsion for some ␭ g kG _ P. It is easy to see that this holds when ⌫ s G; a theorem of Brown wBnx shows that it holds whenever ⌫rG is polycyclic. For any module M and ideal J, we write MJ ⬁ s

⬁

F MJ n .

ns1

8.2 LEMMA. Let A G B be kG-modules and let I, J be ideals of kG. Suppose that ArB s Ž ArB .Ž I ⌫ . and that J q I ␥ s kG for e¨ ery ␥ g ⌫. Then AJ ⬁ rBJ ⬁ s Ž AJ ⬁ rBJ ⬁ .Ž I ⌫ ., and ArAJ n ( BrBJ n for each n. Proof. Suppose a s Ýa j y j g AJ n l B, where each a j g A and each y j g J n. There exists X g ² I ⌫ : such that a j X : B for each j. Then J n q X s kG, so a g aJ n q

Ý a j Xyj : BJ n .

It follows that AJ n l B s BJ n, for each n. Since ArŽ AJ n q B . is both ² I ⌫ :-torsion and ² J :-torsion, we also have AJ n q B s A. The second claim follows. For the first claim, note that AJ ⬁ l B s

F AJ n l B s F BJ n s BJ ⬁ . n

n

91

ABELIAN GROUP RINGS

Now we make an ad hoc definition: an ideal J of kG is I ⌫-admissible if < kG : J < is a finite ␲ ⬘-number,

P F J - kG,

␥

J q I s kG

and

for every ␥ g ⌫

This depends on P, but the relevant P will always be clear from the context. 8.3 PROPOSITION. Assume that the pair Ž kGrP, ⌫ . has the Hall᎐Roseblade property. Let M be a finitely generated k⌫-module such that MP m s 0, where m G 1. Then there exist an ideal I of kG, strictly containing P, and a chain M s M0 G M1 G ⭈⭈⭈ G Mm of k⌫-submodules in M such that Ži. ideal J; Žii.

for i s 0, . . . , m y 1, Ž MirMiq1 . J ⬁ s 0 for e¨ ery I ⌫-admissible Mm F M Ž I ⌫ ..

M also contains a chain of finitely generated k⌫-submodules Ž Ni . such that Mir Ž Mi J n q Miq1 . ( Nir Ž Ni J n q Niq1 . for each i and J as in Ži. and all n. Proof. We start by constructing the chain Ž Ni .. Put N0 s M0 s M. Now let i G 0. Having specified the finitely generated k⌫-submodule Ni of MP i, we use Lemma 8.1 to find a finitely generated k⌫-submodule Niq1 of Ni P such that Ni PrNiq1 is a ␲-group. In this way we obtain a chain M s N0 G N0 P G N1 G N1 P G N2 G ⭈⭈⭈ G Nmy1 G Nmy1 P s Nm s 0. For each i, the Hall᎐Roseblade property ensures that NirNi P contains a free kGrP-submodule FirNi P such that NirFi is ² ␭⌫i :-torsion for some ␭ i g kG _ P. Put

␭ s q ␭0 . . . ␭ my1

and

I s P q ␭ kG,

where q s Ł p g ␲ p Žnote that q f P as P is regular.. The chain Ž Mi . is now defined recursively by setting Miq1rNiq1 s Ž MirNiq1 . Ž I ⌫ . for i G 0. Note that Ni P F Miq1 and that MirFi is ² I ⌫ :-torsion, for each i.

92

DAN SEGAL

Certainly Žii. holds, as Nm s 0. To establish Ži., fix i g  0, . . . , m y 14 and let J be an I ⌫-admissible ideal. Write ᎏ: M ª MrNiq1. Now Theorem 3.1 shows that Fi J ⬁ F Ni P F Miq1, and Lemma 8.2 shows that Mi J ⬁ rFi J ⬁ is ² ␭⌫ :-torsion. It follows that Mi J ⬁ F Miq1; but the ² ␭⌫ :-torsion module Miq1 has no non-zero image annihilated by J, so in fact Mi J ⬁ s Miq1. This implies Ži.. For the final claim, put Y s Miq1 q Ni and T s Miq1 l Ni . Then Ni J n q Niq1 G T, since TrNiq1 is ² I ⌫ :-torsion; so we have Nir Ž Ni J n q Niq1 . s Nir Ž Ni J n q T . ( Yr Ž YJ n q Miq1 . ( Mir Ž Mi J n q Miq1 . by 8.2. 8.4 COROLLARY. Let A be a finitely generated non-singular qrf kG-module. If A is unmixed then A is poly-Ž residually finite. for kG. Proof. By Proposition 6.5Živ., each element of A is annihilated by some member of M Ž A.. As A is finitely generated, it follows that M Ž A. s  P1 , . . . , Pt 4 , say, is finite, and that AP1m . . . Ptm s 0 for some m. Lemma t 7.1 shows that A embeds in [is1 ArAi, where now Ai s AŽ Pj < j / i4.. It will suffice to show that each ArAi is poly-Žresidually finite.. Fix i g  1, . . . , t 4 and put P s Pi , M s ArAi. Then M is finitely generated and MP m s 0. It follows from 6.5Žiii. that M Ž M . s  P 4 . We now apply 8.3, taking ⌫ s G. The existence of an I-admissible ideal J is assured by Theorem 4.2Ži.; note that our hypotheses imply that P is regular and that GrP † is reduced. Proposition 2.3Žii. shows that NirŽ Ni J n q Niq1 . is finite for each i and all n, so the same holds for MirŽ Mi J n q Miq1 .. Hence MirMiq1 is residually finite as a kG-module. Finally, Mm s 0 because M Ž M . s  P 4 . Thus M is poly-Žresidually finite.. Inserting Corollary 8.4 into the proof of Corollary 6.8, we see now that every non-singular qrf kG-module is poly-Žlocally-Žpoly-Žresidually finite.... 8.5 THEOREM. Suppose that ⌫rG is polycyclic. Let M be a finitely generated k⌫-module which is unmixed as a kG-module. Then there exist an ideal J of finite index in kG and a finite chain M s M0 G M1 G ⭈⭈⭈ G Mm s 0 of k⌫-submodules in M such that Ž Miy1rMi . J ⬁ s 0 for 1 s 1, . . . , m. In group-theoretic terms, this result shows that if E is any extension of the module M by G, then E has a subgroup E1 of finite index Žnamely the inverse image of J † . such that the lower central series of E1 , continued transfinitely, terminates at the identity.

ABELIAN GROUP RINGS

93

Proof. According to Proposition 7.3, we can embed M into a direct sum

[I U ­ j

jg

⌫ ⌬j

;

here, I is a finite set of indices such that P Ž M . s  Pj␥ < j g I , ␥ g ⌫4 , and for each j g I , ⌬ j s N⌫ Ž Pj . and U j is a finitely generated k⌬ j-module which is Pj-primary as a kG-module. If the result holds for each of the finitely many summands, it holds for M; so, changing notation, we may as well suppose that M s U ­⌫⌬ , where U s U j and ⌬ s ⌬ j for some j. Thus U is P-primary where P s Pj . Now we apply Proposition 8.3 Žwith U in place of M and ⌬ in place of ⌫ .. This provides an ideal I of kG, strictly containing P, and a chain U s U0 G U1 G ⭈⭈⭈ G Um of k⌬-submodules in U such that Ži. for each i, ŽUiy1rUi . J ⬁ s 0 for every I ⌬ -admissible ideal J of kG, and Žii. Um s UmŽ I ⌬ .. As U is P-primary and P is ⌬-invariant, it follows from Žii. that in fact Um s 0. Now one of the main theorems of wSx Žan application of Theorem 4.2 above. shows that the maximal ideals LrP of finite index in kGrP which satisfy L q I ␥ s kG for all ␥ g ⌬ intersect in P. If charŽ kGrP . s 0 then charŽ kGrL. must take infinitely many distinct values as L ranges over all such maximal ideals, so we may choose one such L with charŽ kGrL. f specŽ G .. If charŽ kGrP . s l / 0, then l f specŽ G ., since M is non-singular and P g P Ž M .; so for any such ideal L we have charŽ kGrL. s l f specŽ G .. Thus in either case we find an I ⌬ -admissible ideal L of kG, containing lkG for some prime l. There exists e g ⺞ such that G e F L†, and we put J s Ž L l k . kG q Ž G e y 1 . kG. Then J is a ⌫-invariant ideal of finite index in kG and J F L, so ŽUiy1rUi . J ⬁ s 0 for each i. Putting Mi s Ui ­⌫⌬ for i s 0, 1, . . . , m we now get Ž Miy1rMi . J ⬁ s 0, as required. REFERENCES wBnx

K. A. Brown, The Nullstellensatz for certain group rings, J. London Math. Soc. (2) 26 Ž1982., 425᎐434.

94 wFJx wGS1x wGS2x wKax wLax wLMSx wMSx wNx wSx wZSx

DAN SEGAL

M. D. Fried and M. Jarden, ‘‘Field Arithmetic,’’ Springer-Verlag, BerlinrNew York, 1986. F. J. Grunewald and D. Segal, Remarks on injective specialisations, J. Algebra 61 Ž1979., 538᎐547. F. J. Grunewald and D. Segal, On congruence topologies in number fields, J. Reine Angew. Math. 311 Ž1979., 389᎐396. I. Kaplansky, ‘‘Fields and Rings,’’ Univ. of Chicago Press, Chicago and London, 1972. S. Lang, ‘‘Diophantine Geometry,’’ Interscience, New York, 1962. A. Lubotzky, A. Mann, and D. Segal, Finitely generated groups of polynomial subgroup growth, Israel J. Math. 82 Ž1993., 363᎐371. A. Mann and D. Segal, Uniform finiteness conditions in residually finite groups, Proc. London Math. Soc. (3) 61 Ž1990., 529᎐545. M. Nagata, ‘‘Local Rings,’’ Interscience, New York, 1962. D. Segal, On modules of finite upper rank, Trans. Amer. Math. Soc. 353 Ž2001., 391᎐410. O. Zariski and P. Samuel, ‘‘Commutative Algebra,’’ Vol. I, Springer-Verlag, New York, 1975.