A universal coefficient decomposition for subgroups induced by submodules of group algebras

by Manfred Hartl

Institut de Recherche Mathématique Avancée, ULP et CNRS, 7 rue René-Descartes, 67084 Strasbourg Cedex, FRANCE

It was first observed by Sandling [9] that dimension subgroups over an arbitrary commutative ring of coefficients can be decomposed in terms of the dimension subgroups over the ‘universal’ coefficient rings ZZ and ZZ/p^{e}ZZ, where p and e run through all primes and positive integers, respectively. Borrowing a notion from group cohomology we may conveniently term this property by saying that dimension subgroups satisfy a ‘universal coefficient decomposition’.

This property was conceptually proved and extended to Lie dimension subgroups by Parmenter, Passi and Seghal [7], in developing a theory of polynomial ideals and their induced subgroups for this purpose. But unfortunately important classes of induced subgroups are not covered by this theory, such as dimension subgroups with respect to arbitrary N-series, relative dimension subgroups or Fox subgroups. So in this paper we prove the universal coefficient decomposition for subgroups induced by a very general type of suitable additive subgroups of group algebras, which includes not only all types of induced subgroups mentioned before, but also certain common generalizations of them, such as ‘relative dimension subgroups with respect to N-series’ or ‘relative Fox dimension subgroups’ (we compute some of these explicitly in subsequent work). This result is based on a quite elementary homological lemma which extends the universal coefficient decomposition proved in [7] for polynomial ideals not only to all ideals, but even to all additive subgroups of group rings. So, surprisingly enough, it turns out that this nice property depends only on the ring structure of the coefficient rings, not the one of group algebras.

Throughout in this paper R denotes a commutative ring with identity 1_{R}. As usual, the characteristic of R is the least non-negative integer such that n1_{R} = 0.

Let G be a group, R(G) its group algebra with coefficients in R, I_{R}(G) the augmentation ideal of R(G), I_{R}^{n}(G) its n-th associative and I_{R}^{(n)}(G) its n-th Lie power, see [8, p. 2]. Write i_{R}: ZZ(G) → R(G) for the canonical ring homomorphism extending the identity map on G.

Research supported by successive research fellowships of the Sonderforschungsbereich 343, Bielefeld, and of the Human Capital and Mobility Programme of the European Union.

For an additive subgroup J of ZZ(G) let J_{R} denote the R-submodule of R(G) spanned by i_{R}(J).

Definition 1 Let J ≤ ZZ(G) be an additive subgroup with the property that the set G ∩ (1_{R} + J_{R}) is a subgroup of G for all commutative unitary rings R. Then we say that these subgroups satisfy a universal coefficient decomposition if the following properties hold for all rings R as above.

(i) If characteristic of R is zero, then

G∩(1_{R} +J_{R}) = ∏_{p∈σ(R)}{t_{p}(G mod (G ∩ (1_{Z Z} + J)) ∩(G∩(1_{Z Z/p^{e}Z Z} +J_{Z Z/p^{e}Z Z}))}

as an ordered product of subsets of G. Here σ(R) = {p|p is a prime and p^{n}R = p^{n+1}R for some n ≥ 0}, and for p ∈ σ(R), p^{e} is the smallest power of p for which p^{e}R = p^{e+1}R. (If σ(R) is empty then the right hand side is to be interpreted as G ∩ (1_{Z Z} + J)). By definition,

t_{p}(G mod (G ∩ (1_{Z Z} + J)) = {g ∈ G | g^{p^{k}} ∈ G ∩ (1_{Z Z} + J) for some k ≥ 0}.

(ii) If characteristic of R is r > 0, then

G ∩ (1_{R} + J_{R}) = G ∩ (1_{Z Z/rZ Z} + J_{Z Z/rZ Z}) = ⋂_{k} G ∩ (1_{Z Z/p_{k}^{e_{k}} Z Z} + J_{Z Z/p_{k}^{e_{k}} Z Z}) ,

where r = ∏ p_{k}^{e_{k}} is the prime factorization of r. □

Dimension subgroups D_{n,R}(G) = G ∩ (1_{R} + I_{R}^{n}(G)) and Lie dimension subgroups D_{(n),R}(G) = G ∩ (1_{R} + I_{R}^{(n)}(G)) (for n ≥ 2) are known to have these properties; indeed, the theory of polynomial ideals was invented in [7] to furnish a complete proof of this fact which was originally discovered by Sandling [9]. Using homological methods instead we can generalize these results to the following

Proposition 2 Let J be an additive subgroup of ZZ(G) such that for all commutative unitary rings R the subset G∩(1_{R} +J_{R}) is a subgroup of G and satisfies the following condition: For all g ∈ G ∩ (1_{R} + J_{R}) there is some number n ≥ 1 such that for all k ≥ n, (g − 1)^{k} ∈ J. Then the subgroups G ∩ (1_{R} + J_{R}) satisfy a universal coefficient decomposition. □

This result can be derived from theorem 6 below by a word–by–word copy of the proof of theorem II.2.1 in [8], just throughout replacing I_{R}^{n}(G) by J.

This criterion is very general but has the disadvantage that the condition has to be verified for the groups G ∩ (1_{R} + J_{R}) which one hopes to determine by use of the conclusion. So in the theorem to follow we provide a handy criterion which captures all types of induced subgroups studied frequently in the literature, as well as certain common generalizations of them, see corollary 5 below.

Theorem 3 Let H be a subgroup of G and J ⊂ ZZ(G)I_{Z Z}(H) be a right Hsubmodule, with the property that for all h ∈ H there exists some n ≥ 1 such that (h − 1)^{n} ∈ J. Then the sets G ∩ (1_{R} + J_{R}) are subgroups of H for all R and satisfy a universal coefficient decomposition. Moreover, all factors

t_{p}(G mod (G ∩ (1_{Z Z} + J)) ∩ (G ∩ (1_{Z Z/p^{e}Z Z} + J_{Z Z/p^{e}Z Z})) , p ∈ σ(R) ,

arising in the characteristic zero case are also subgroups of H. □

Before giving the proof we first discuss some

Examples 4 The following additive subgroups J of ZZ(G) satisfy the hypothesis of the theorem:

(i) associative powers I_{Z Z}^{n}(G) for all n ≥ 1 and Lie powers I_{Z Z}^{(n)}(G) for n ≥ 2, taking H = G and H = G′, respectively. Indeed, for n ≥ 2 one has I_{Z Z}^{(n)}(G) ⊂ I_{Z Z}^{(2)}(G) = ZZ(G)I_{Z Z}(G′) and I_{Z Z}^{n−1}(G′) ⊂ I_{Z Z}^{(n)}(G) by Sandling’s formula for I_{R}^{(n)}(G), cf. [8, I.1.8].

(ii) subgroups J = MI_{Z Z}(H), where H ≤ G and M is any right H -submodule of I_{Z Z}(G) with the property that for all h ∈ H there exists some n ≥ 1 such that (h − 1)^{n} ∈ M. In this case we obtain in [1] a homological construction of a right H -submodule J′_{R} ⊂ I_{R}^{2}(H) such that

G ∩ (1 + MI_{R}(H)) = H′ ∩ (1 + (M ∩ I_{R}(H))I_{R}(H) + J′_{R}) .

In the important case that H is free one even has J′_{R} = 0. This fact is further exploited in [5].

Thus we obtain

Corollary 5 The following common generalizations of classical types of induced subgroups satisfy a universal coefficient decomposition:

(1) relative dimension subgroups

D_{n,R}^{N}(G, K) =^{def} G ∩ (1 + I_{R}(K)I_{R}(G) + I_{R,N}^{n} (G))

with respect to a subgroup K ≤ G and an N-series N of G; here {I_{R,N}^{i} (G)} denotes the ideal filtration of R(G) induced by N, cf. [8, III.1.5].

(2) relative Fox dimension subgroups

G ∩ (1 + I_{R}(K)I_{R}(H) + I_{R,N}^{n} (G)I_{R}(H))

with respect to any subgroups K, H ≤ G and N-series N of G.

We remark that such groups with an N-series different from the lower central series naturally arise in the study of classical Fox subgroups, namely when H is free (nilpotent) or is one of the two factors of a semidirect product, see [5] and [4]. In [3] we calculate the groups in (1), (2) above for n = 3, 2, respectively.

Now we turn to the proof of theorem 3. As a key step we first obtain a generalization of the ‘universal coefficient decomposition’ for polynomial ideals [7] to arbitrary additive subgroups of ZZ(G).

Theorem 6 Let J ≤ ZZ(G) be any additive subgroup and R any commutative unitary ring. Then

(i) if characteristic of R is zero,

i_{R}^{−1}J_{R} = ∑_{p∈σ(R)}{t_{p}(ZZ(G) mod J) ∩ (i_{Z Z/p^{e}Z Z}^{−1}J_{Z Z/p^{e}Z Z})} .

If σ(R) is empty then the right hand side is to be interpreted as being J.

(ii) If characteristic of R is r > 0, then

i_{R}^{−1}J_{R} = i_{Z Z/rZ Z}J_{Z Z/rZ Z} = ⋂_{k} i_{Z Z/p_{k}^{e_{k}} Z Z}J_{Z Z/p_{k}^{e_{k}} Z Z}

where r = ∏ p_{k}^{e_{k}} is the prime factorization of r. □

Theorem 6 rests on the following crucial homological

Lemma 7 Let A be an abelian group and R be any ring with identity 1_{R}. Consider the homomorphism j_{R}: A → R ⊗ A, a ↦ 1_{R} ⊗ a. Let r = characteristic of R. Then

Ker(j_{R}) = rA + ∑_{p∈σ(R)} p^{e}t_{p}(A)  (1)

for σ(R) and e = e_{p} as in definition 1. If σ(R) is empty then r = 0, whence Ker(j_{R}) = 0. If r > 0 then Ker(j_{R}) = rA.

Proof: The right hand side of (1) is contained in Ker(j_{R}) since for p ∈ σ(R), a ∈ A and some k ≥ e such that p^{k}a = 0 we have 1_{R} ⊗p^{e}a = p^{e}1_{R} ⊗a ∈ p^{k}R⊗a = 0. Conversely, let u_{R} : ZZ/rZZ ⟶^{>} R , i(1) = 1_{R}. Then the map j_{R} factors as

j_{R}: A → → A/rA ≅ ZZ/rZZ ⊗ A ⟶^{u_{R}⊗A} R ⊗ A  (2)

Now consider the following part of a six-term exact sequence,

R/⟨1_{R}⟩ ∗ A ⟶^{τ} ZZ/rZZ ⊗ A ⟶^{u_{R}⊗A} R ⊗ A → R/⟨1_{R}⟩ ⊗ A → 0 ,

where ∗ denotes the torsion product of abelian groups and τ is the appropiate connecting homomorphism. The inclusions of the torsion subgroups induce an isomorphism

t(R/⟨1_{R}⟩) ∗ t(A) ⟶^{≅} R/⟨1_{R}⟩ ∗ A ,

as follows directly from the suitable six-term exact sequences. By the decomposition t(X) = ⨁{t_{p}(X) | p prime} for any abelian group X and by additivity of the torsion product we have

Im(τ) = ∑ ; p prime τ(t_{p}(R/⟨1_{R}⟩) ∗ t_{p}(A)) .

Let ⟨x, p^{k}, a⟩ be a canonical generator of t_{p}(R/⟨1_{R}⟩)∗t_{p}(A), i.e. x ∈ R, a ∈ A such that p^{k}x = n1_{R} for some integer n and p^{k}a = 0, cf. [6, V.6]. Then τ⟨x, p^{k}, a⟩ = n1 ⊗ a = 1 ⊗ na. Write n = p^{l}m with (p, m) = 1. If l ≥ k then τ⟨x, p^{k}, a⟩ = 0, so we need only to consider the case l < k. Let m′, p′ ∈ ZZ such that mm′ + pp′ = 1. Then

p^{l}1_{R} = p^{l}mm′1_{R} + p^{l}pp′1_{R}
= nm′1_{R} + p^{l+1}p′1_{R}
= p^{k}m′x + p^{l+1}p′1_{R}
= p^{l+1}(p^{k−l−1}m′x + p′1_{R}) ,

whence p ∈ σ(R) and l ≥ e. Thus τ⟨x, p^{k}, a⟩ = 1 ⊗ (p^{l}m)a ∈ 1 ⊗ p^{e}t_{p}(A), and Ker(u_{R} ⊗ A) = Im(τ) ⊂ ∑_{p∈σ(R)} 1 ⊗ p^{e}t_{p}(A). By (2) equality (1) is proven.

Now suppose σ(R) = ∅. Then Ker(j_{R}) = rA. But if r > 0 then any prime not dividing r belongs to σ(R), so r = 0.

Finally suppose r > 0. Let p ∈ σ(R). If p does not divide r then t_{p}(A) = rt_{p}(A). If p divides r write r = p^{s}r′ with (p, r′) = 1. Assuming e < s implies r′p^{s−1}1_{R} = r′p^{s−e−1}p^{e}1_{R} ⊂ r′p^{s−e−1}p^{e+1}R = rR = 0, which contradicts the fact that characteristic of R = r. Thus e ≥ s, and p^{e}t_{p}(A) ⊂ p^{s}t_{p}(A) = p^{s}r′t_{p}(A) = rt_{p}(A). Thus ∑_{p∈σ(R)} p^{e}t_{p}(A) ⊂ rA, whence Ker(j_{R}) = rA by (1) which completes the proof. □

Proof of theorem 6: The map i_{R} induces a homomorphism i′_{R}: ZZ(G)/J → R(G)/J_{R}, so we can write i_{R}^{−1}J_{R}/J = Ker(i′_{R}) . But R(G) ≅ R ⊗ ZZ(G) and J_{R} = Im(R ⊗ J → R ⊗ ZZ(G)), so by right exactness of the tensor product R(G)/J_{R} ≅ R ⊗ (ZZ(G)/J). Thus Ker(i′_{R}) = Ker(j_{R}: ZZ(G)/J → R ⊗ (ZZ(G)/J)) , so we can apply lemma 7 for A = ZZ(G)/J. Just note that

p^{e}t_{p}(A) = t_{p}(A) ∩ p^{e}A
= t_{p}(A) ∩ Ker(j_{Z Z/p^{e}Z Z})
= t_{p}(ZZ(G)/J) ∩ (i_{Z Z/p^{e}Z Z}^{−1}J_{Z Z/p^{e}Z Z}/J)
= {t_{p}(ZZ(G) mod J) ∩ (i_{Z Z/p^{e}Z Z}^{−1}J_{Z Z/p^{e}Z Z})}/J ,

since J ⊂ t_{p}(ZZ(G) mod J) ∩ (i_{Z Z/p^{e}Z Z}^{−1}J_{Z Z/p^{e}Z Z}). □

We still need the following useful little lemma which is well-known for J ⊂ I_{Z Z}(H), see [10].

Lemma 8 Let H be a subgroup of a group G and J ⊂ R(G)I_{R}(H) be any subset. Then G ∩ (1_{R} + J) ⊂ H.

Proof: Let T be a right transversal of H in G. Since R(G) is a free right H -module with basis {[t], t ∈ T } we have a composite isomorphism

ψ: R(G)/R(G)I_{R}(H) ≅ R(G) ⊗_{R(H)} R ≅ ⨁_{t∈T} R · [t] .

Now suppose th ∈ (1_{R} + J) ⊂ (1_{R} + R(G)I_{R}(H)) for some t ∈ T, h ∈ H. Then 0 = ψ(th − 1_{R}) = ψ((t − 1_{R}) + (h − 1_{R}) + (t − 1_{R})(h − 1_{R})) = ψ(t − 1_{R}) = [t] − [1], whence t = 1 as was to be shown. □

Proof of theorem 3: We have G ∩ (1_{R} + J_{R}) ⊂ H by lemma 8. This implies that G ∩ (1_{R} + J_{R}) is a subgroup of H since for g, h ∈ G ∩ (1_{R} + J_{R}),

gh^{−1} − 1 = (g − 1) − (h − 1)h^{−1} + (g − 1)(h^{−1} − 1) ∈ J_{R}

since J_{R} is a H -submodule of R(G). Then the universal coefficient decomposition follows from proposition 2.

It remains to show that all factors of the decomposition in the characteristic zero case are subgroups. This actually follows by just applying the universal coefficient decomposition just proved to a ring R which satisfies σ(R) = {p} and e = f for any given prime p and nonnegative number f. Such a ring can be obtained, for example, as a quotient of the monoid ring of the additive monoid of nonnegative integers, ZZ(IN_{o}) = ⨁_{n≥0} ZZ · [n], modulo the ideal generated by the element p^{f}[0] − p^{f+1}[1]. Thus the theorem is proved. □

References

[1] M. Hartl, The second Fox quotient of arbitrary group rings, submitted.

[2] M. Hartl, The third Fox quotient and Fox subgroup of arbitrary groups, in preparation.

[3] M. Hartl, The second Fox and third relative dimension subgroup, in preparation.

[4] M. Hartl, On Fox quotients of semidirect products, in preparation.

[5] M. Hartl, Fox subgroups with respect to free nilpotent subgroups, in preparation.

[6] S. Mac Lane, “Homology”, Springer Grundlehren, Vol. 114, Springer-Verlag Berlin-Göttingen-Heidelberg, 1963.

[7] M.M. Parmenter, I.B.S. Passi and S.K. Seghal, Polynomial ideals in group rings, Canad. J. Math. 25 (1973), 1174-1182.

[8] I.B.S. Passi, “Group Rings and Their Augmentation Ideals”, Lecture Notes in Math., Vol. 715, Springer-Verlag, Berlin, Heidelberg, New York, 1979.

[9] R. Sandling, Dimension subgroups over arbitrary coefficient rings, J. Algebra 21 (1972), 250-265.

[10] R. Sandling, Note on the integral group ring problem, Math. Z. 124 (1972), 255-258.