POLYCYCLIC-BY-FINITE GROUP ALGEBRAS ARE CATENARY

EDWARD S. LETZTER AND MARTIN LORENZ

Abstract. We show that group algebras kG of polycyclic-by-finite groups G, where k is a field, are catenary: If P = I_{0} ⊊ I_{1} ⊊ ⋯ ⊊ I_{m} = P′ and P = J_{0} ⊊ J_{2} ⊊ ⋯ ⊊ J_{n} = P′ are both saturated chains of prime ideals of kG, then m = n.

Introduction

Group algebras kG of polycyclic-by-finite groups G, over a field k, are among the most prominent special classes of noncommutative noetherian rings, partly because the only known noetherian group algebras (over fields) are of the form kG, and partly because of the intrinsic group theoretic interest in G. Indeed, the investigation of the prime ideals of kG was a major enterprise throughout the 1970’s. Zalesskii’s work [20] initiated a program that was essentially completed in 1980 by [11], [10] and [1]; the most notable contribution was by Roseblade [15]. The aforementioned papers resulted in a detailed, albeit complicated, description of the prime ideals of kG, and the subject has been largely dormant ever since. Disappointingly, the question of whether kG is, or is not, catenary (see below) remained unanswered, despite several attempts in that direction: [17] gave an affirmative answer for f.g. nilpotent groups, [15, Corrigenda] did the same for ℨ-groups, and [10] settled the case of orbitally sound-by-(finite nilpotent) groups. But the intervening almost 20 years have witnessed substantial progress in our understanding of prime ideals in both noetherian rings and their extensions (see, e.g., [4]), and so we can now offer in the present paper a proof of the following

Theorem. Group algebras of polycyclic-by-finite groups are catenary.

Recall that a ring R is called catenary when the following holds: For every inclusion P ⊊ P′ of prime ideals in R, if P = I_{0} ⊊ I_{1} ⊊ ⋯ ⊊ I_{m} = P′ and P = J_{0} ⊊ J_{2} ⊊ ⋯ ⊊ J_{n} = P′ are both chains of prime ideals that cannot be lengthened by the insertion of extra primes, then m = n. A fundamental result of commutative algebra asserts that all affine commutative k-algebras are catenary. Various noncommutative generalizations of this result are known: Affine PI-algebras [16], enveloping algebras of finite dimensional solvable Lie (super-)algebras [2], [8] and, most recently, certain algebras arising from quantum groups [3] have all been proven to be catenary.

1991 Mathematics Subject Classification. 16S34, 16D30, 16P40.

Key words and phrases. group algebra, polycyclic-by-finite group, prime ideal, catenary ring, noetherian ring, second layer condition.

Research of the first author supported in part by NSF Grant DMS-9623579.

Research of the second author supported in part by NSF Grant DMS-9618521.

Notations and Conventions. Throughout this note, k will denote a commutative field. Furthermore, ⊊ denotes strict inclusion, while equality is permitted in ⊂; similarly for ⊋ and ⊃. We will write ann X_{R} when referring to the right annihilator of X in R and ann_{S} Y when referring to the left annihilator of Y in S.

1. Preliminaries

1.1. Catenarity. A ring R is called catenary if, given prime ideals Q ⊊ Q′ of R, any two saturated chains of primes between Q and Q′ have the same length. Here, a chain Q_{0} ⊊ Q_{1} ⊊ ⋯ ⊊ Q_{m} of prime ideals of R is saturated if Q_{i−1} and Q_{i} are adjacent for all 1 ≤ i ≤ m; a pair of prime ideals Q and Q′ in R are adjacent if Q ⊊ Q′ and if there exist no prime ideals Q′′ of R such that Q ⊊ Q′′ ⊊ Q′.

Recall that the height of a prime ideal P of R is the supremum of the lengths n of all chains P = P_{0} ⊋ P_{1} ⊋ ⋯ ⊋ P_{n} of prime ideals descending from P. In all known noetherian rings the prime ideals have finite height; for group algebras of polycyclic-by-finite groups, see [15, Theorem H1]. Assuming finiteness of heights, R is certainly catenary if the following condition is satisfied:

(1) height P′ = height P + 1 holds whenever P ⊊ P′ are adjacent prime ideals.

If every prime of R contains exactly one minimal prime, then (1) is actually equivalent with catenarity.

For group algebras kG of polycyclic-by-finite groups G, the latter condition on minimal primes holds. Indeed, if F denotes the finite radical of G, that is, the largest finite normal subgroup of G, then the unique minimal prime of kG contained in a given prime ideal P of kG is (P ∩ kF)kG; see [15, Corollary 22]. Thus, our objective will be to verify (1) for kG.

1.2. Heights. Let kG be the group algebra of a polycyclic-by-finite group G. Following Roseblade [15] we put, for any ideal I of kG,

I^{†} = G ∩ (1 + I) .

Thus I^{†} is the kernel of the natural homomorphism of G into the group of units of kG/I, and I contains the kernel of the natural homomorphism kG → k[G/I^{†}]. The ideal I is said to be faithful iff I^{†} = 1. The image of I in k[G/I^{†}] is always faithful.

The following Lemma will be used to reduce the proof of (1) to the case where P is faithful. Its essence is in [15, 2.4], at least for Roseblade’s ℨ-groups; the general case follows rather routinely. The term

p_{G}(N)

in the formula below denotes the G-plinth length of the normal subgroup N of G. For the definition and basic properties of plinth lengths, the reader is referred to [15, 2.3]. We remark that [15, 2.3] assumes N to be polycyclic, but the notion of plinth length is insensitive to finite factors and applies to polycyclic-by-finite groups as well.

Lemma. Let kG be the group algebra of a polycyclic-by-finite group G, let P be a prime ideal of kG, and let N be a normal subgroup of G that is contained in P^{†}. Then

height P = height \overline{P} + p_{G}(N) ,

where \overline{P} denotes the image of P in k[G/N].

Proof. Let G_{0} denote a normal ℨ-group of finite index in G (see [15, 1.3]) and put N_{0} = N ∩ G_{0}. Further, write

P ∩ kG_{0} = ⋂_{g∈G} P_{0}^{g} (2)

for some prime ideal P_{0} of kG_{0}; cf. [15, 4.1] or [11, Lemmas 1.3(i), 1.4(i)]. Then P^{†} ∩ G_{0} ⊂ P_{0}^{†}, and so N_{0} ⊂ P_{0}^{†}. Letting denote images in k[G/N], it follows that \overline{P} ∩ k[\overline{G_{0}}] = ⋂_{x∈\overline{G}} \overline{P_{0}}^{x} and \overline{P_{0}} is prime in k[\overline{G_{0}}]. Thus, by [15, 8.1] or [11, Lemma 2.5(i)], height P = height P_{0} and height \overline{P} = height \overline{P_{0}}. Finally, p_{G}(N) = p_{G_{0}}(N_{0}), since G_{0} has finite index in G. Therefore, replacing G by G_{0} etc., we may assume that G is a ℨ-group. In this case, the desired formula follows from Roseblade’s height formula in [15, 2.4], noting that Roseblade’s λ equals height for ℨ-groups and that p_{G}( .) is additive on short exact sequences of G-groups.

1.3. Second layer theory. We review some relevant results and background information from noetherian ring theory.

(i) Definitions of the right and left strong second layer conditions of Jategaonkar may be found in [4, p. 183]; also see [5]. When we refer to noetherian rings having the strong second layer condition, abbreviated sslc, we mean rings satisfying both the right and left strong second layer conditions.

(ii) Group rings of polycyclic-by-finite groups over commutative noetherian coefficient rings satisfy the sslc; see [5, A.4.6].

(iii) Let U and V be noetherian rings satisfying the sslc. Suppose that B is a U-V - bimodule finitely generated and faithful on each side. It was proved by Jategaonkar that U and V have the same classical Krull dimension; see, for example, [4, 12.5] or [5, 8.2.8]. We will use clKdim( .) to denote classical Krull dimension.

(iv) Let U be a prime noetherian ring, and let M be a nonzero left U-module. Recall that u ∈ U is regular provided uv and vu are nonzero for all nonzero v ∈ U. Further recall that M is torsion free provided u.m ≠ 0 for every regular element u ∈ U and every 0 ≠ m ∈ M. It follows from Goldie’s Theorem, when _{U}M is torsion free, that every nonzero submodule of M is faithful. If N is a U-module for which every nonzero submodule is faithful, then we say that N is fully faithful over U.

(v) Let P and Q be prime ideals in a noetherian ring U satisfying the sslc. We say that P is linked to Q, and write P ⇝ Q, if there exists a U-U-bimodule factor B of (P ∩ Q)/P Q with the following properties: ann_{U} B = P, ann B_{U} = Q, and B is torsion free on each side as a U/P -U/Q-bimodule. A graph structure is thus imposed on the set of prime ideals, and if X is a subset thereof then the link closure of X will be defined to be the union of the connected components of the prime ideals in X. Note, by (iii), that if P ⇝ Q then clKdim(U/P) = clKdim(U/Q).

(vi) Continue to let U be a noetherian ring satisfying the sslc, and let M be a finitely generated left U-module. A prime ideal P of U is said to be an associated prime of M if there exists a submodule N of M such that ann_{U} N = P and such that N is a fully faithful U/P -module. By noetherianity, the set of associated primes of M is not empty. Also, if M′ is an essential extension of M, then the sets of associated primes of M and M′ are the same. Let X denote the link closure of the set of associated primes of M. In [4, 11.4], for example, it is proved that M is annihilated by a (finite) product of prime ideals in X. (This last assertion follows from an iterated application of the Main Lemma of Jategaonkar; see [5].)

1.4. Krull dimension. Here we collect some facts on Krull dimension, in the sense of Gabriel and Rentschler, that will be needed later. The Krull dimension of a module will be denoted Kdim(M); for the definition and basic facts concerning Kdim, we refer the reader to [4, Chapter 13]. All rings and modules discussed below are assumed noetherian, and so Kdim(M) will always exist. Further, Kdim(U), for a ring U, will stand for Kdim(U_{U}).

(i) A nonzero module M is called homogeneous (or α-homogeneous, if Kdim(M) = α) provided all nonzero submodules of M have the same Krull dimension as M. Prime rings are homogeneous; see [4, p. 229]. Further, finite subdirect products and extensions of α-homogeneous modules are easily seen to be α-homogeneous.

(ii) Assume the ring U is (right) homogeneous. If the element u ∈ U satisfies Kdim(U/uU) < Kdim(U) then u is regular. This assertion can be proved as follows: First, the left regularity of u follows from [14, 6.8.12]. As to right regularity (which is due to Stafford [19]), observe that (U/uU)_{U} maps onto (u^{n−1}U/u^{n}U)_{U} for all n > 0. Hence, Kdim(u^{n−1}U/u^{n}U)_{U} ≤ Kdim(U/uU)_{U} < Kdim(U) and, consequently, Kdim(U/u^{n}U)_{U} < Kdim(U). But, by “Fitting’s Lemma” [14, 2.3.2(ii)], there is an n > 0 such that the right annihilator, ann(u^{n})_{U}, of u^{n} in U embeds into U/u^{n}U. Therefore, Kdim(ann(u^{n})_{U}) < Kdim(U), whence ann(u^{n})_{U} = 0, by homogeneity. This proves that u is right regular.

(iii) For group algebras kG of polycyclic-by-finite groups G, one has Kdim(kG) = h(G), the Hirsch number of G; see [18]. If H is a subgroup of finite index in G, then for any (right) kG-module M, one has

Kdim(M_{kG}) = Kdim(M_{kH}) .

If H is normal, this is covered by a more general result on finite normalizing extensions; see [7, Théorème 5.3]. The general case follows by passing to the normal core H_{G} = ⋂_{g∈G} H^{g}, a normal subgroup of finite index in G (and in H).

(iv) Let G, H, and M be as above and suppose that M = W kG for some submodule W of M_{kH}. Then

Kdim(M) = Kdim(W) .

Indeed, by (iii), it suffices to prove Kdim(M_{kN}) = Kdim(W_{kN}), where N = H_{G} is the normal core of H. But, as kN-module, M = W kG is a finite sum of G-conjugates of W_{kN} all of which have isomorphic submodule lattices, and hence the same Kdim. Thus, Kdim(M_{kN}) = Kdim(W_{kN}), as required.

(v) Keeping the above notations G and H, a kG-module M is α-homogeneous if and only if it is so as kH-module. To verify this claim, first note that α-homogeneity as kH-module certainly entails α-homogeneity over kG, by (iii). Conversely, if M_{kG} is α-homogeneous and W is a nonzero submodule of M_{kH}, then W kG is a nonzero kG-submodule of M, and so α = Kdim(W kG) = Kdim(W), by (iv). Finally, if M = W kG for some submodule W of M_{kH}, then M is α-homogeneous if and only if W is. Indeed, if M = W kG is α-homogeneous and X is a nonzero submodule of W, then α = Kdim(XkG) = Kdim(X), by (iv). Conversely, assume that W is αhomogeneous. It suffices to show that M = W kG is α-homogeneous as kN-module, where N = H_{G}. But M_{kN} is a finite sum of G-conjugates of W_{kN}, and all these are α-homogeneous. Hence M_{kN} is α-homogeneous.

2. Induced Ideals

Throughout this section, R ⊂ S is an extension of rings. Additional assumptions will be required later.

2.1. Definition and basic properties. For any ideal I of R, we put

I^{S} = ann(S/IS)_{S} = {s ∈ S | Ss ⊂ IS} .

In other words, I^{S} is the largest ideal of S that is contained in IS. We refer to I^{S} as the induced ideal of I, and we will say that ideals in S of this general form are induced from R. In the case of group algebras R = kH ⊂ S = kG, for some subgroup H of G, one also writes I^{G} in place of I^{S} and calls these ideals induced from H.

The induction operator

( .)^{S} : Ideals of R → Ideals of S

obviously preserves inclusions. Furthermore,

I_{1}^{S}I_{2}^{S} ⊂ (I_{1}I_{2})^{S} (3)

holds for any two ideals I_{1}, I_{2} of R. Indeed, I_{1}^{S}I_{2}^{S} ⊂ I_{1}SI_{2}^{S} ⊂ I_{1}I_{2}S.

2.2. Minimal covering primes. Equation (3) has the following consequence.

Lemma. Let P be a prime ideal of S, and let Q be an ideal of R maximal among ideals whose induced ideals are contained in P. Then Q is prime.

Proof. Let I_{1} and I_{2} be ideals of R properly containing Q. Then I_{1}^{S} and I_{2}^{S} are not contained in P, and hence neither is their product. By (3), (I_{1}I_{2})^{S} is not contained in P, whence I_{1}I_{2} is not contained in Q.

In the situation of the above Lemma, P will often actually be minimal over Q^{S}. A sufficient condition for this to happen is given in the following

Proposition. Suppose that S is finitely generated and free as a left R-module. Assume further that R and S are noetherian and satisfy the sslc. Let Q be a prime ideal of R, and let P be a prime ideal of S containing Q^{S}. Then the following are equivalent:

i. P is minimal over Q^{S}; ii. S/P and R/Q have the same classical Krull dimension; iii. Q is maximal among ideals in R whose induced ideal is contained in P.

Proof. Set M = S/QS and J = Q^{S} = ann M_{S}. By hypothesis on _{R}S, M is free of finite rank as left R/Q-module. In particular, it follows from (1.3 iii) that clKdim(R/Q) = clKdim(S/J), and so clKdim(S/P) ≤ clKdim(R/Q).

(i ⇒ ii) Note that M is nonzero and torsion free as a left R/Q-module. Therefore, by [4, 7.7], there exist prime ideals P_{1}, …, P_{m} of S, and a series of R-S-bimodules

0 = M_{0} ⊊ M_{1} ⊊ ⋯ ⊊ M_{m} = M,

with the following properties: For each 1 ≤ i ≤ n, the right annihilator in S of M_{i}/M_{i−1} is equal to P_{i}, and M_{i}/M_{i−1} is faithful on each side as an R/Q-S/P_{i}- bimodule. Note that all of the prime ideals in S minimal over J, and in particular P, are included among P_{1}, …, P_{m}. Therefore, clKdim(S/P) = clKdim(R/Q), by (1.3 iii).

(ii ⇒ i) If P is not minimal over J, then

clKdim(S/P) < clKdim(S/J) = clKdim(R/Q).

(ii ⇒ iii) Suppose that I is an ideal of R, properly containing Q, such that I^{S} ⊂ P. Since S is free as a left R-module, ann_{R}(S/IS) = I. Therefore, by (1.3 iii),

clKdim(R/Q) > clKdim(R/I) = clKdim(S/I^{S}) ≥ clKdim(S/P).

(iii ⇒ ii) Suppose that clKdim(S/P) ≠ clKdim(R/Q), and so clKdim(S/P) < clKdim(R/Q). We will show that there exists an ideal I of R, strictly containing Q, such that I^{S} ⊂ P.

By [4, 12.3], there exists a series of R-S-bimodules

0 ⊂ L_{1} ⊊ L_{2} ⊂ M

such that ann(L_{2}/L_{1})_{S} = P, and such that L_{2}/L_{1} is torsion free as a right S/P - module. Let K be an R-S-sub-bimodule of M, containing L_{1}, and maximal with respect to K ∩ L_{2} ⊂ L_{1}. Set N = M/K. Note that L_{2}/L_{1} embeds as an R-S-bimodule into N; we will let L denote the image of this embedding. Observe that every nonzero R-S-sub-bimodule of N intersects L nontrivially. Also, by (1.3 iv), L is a fully faithful right S/P -module.

Let Q′ be an associated prime of _{R}L, and let L′ = ann Q′_{L}. Note that L′ is a nonzero R-S-sub-bimodule of L, and that Q′ = ann_{R} L′. In particular, L′ is an R/Q′-S/P -bimodule finitely generated and faithful on each side. Consequently, by (1.3 iii), clKdim(R/Q′) = clKdim(S/P). Now let X denote the link closure of the associated primes of _{R}L. By (1.3 v), if Q′′ ∈ X, then clKdim(R/Q′′) = clKdim(S/P) < clKdim(R/Q).

Now let T be a left R-submodule of N maximal with respect to T ∩ L = 0. Set V = N/T and I = ann_{R} V. Note that I ⊃ Q and that L embeds into _{R}V as an essential submodule. It now follows from (1.3 vi) that V is annihilated on the left by a product Q_{1} ⋯ Q_{n} of prime ideals Q_{1}, …, Q_{n} ∈ X. However, the classical Krull codimension of each Q_{i} is strictly smaller than clKdim(R/Q), and so I properly contains Q.

Since IV = 0, we have IN ⊂ T, and so IN ∩ L = 0. However, IN is an R-S-sub-bimodule of N, and so IN = 0 because each nonzero R-S-sub-bimodule of N intersects L nontrivially. Therefore, IM ⊂ K, and so IM ∩L_{2} ⊂ L_{1}. Hence, L_{2}/L_{1} is an R-S-bimodule subfactor of M/IM. Since ann(L_{2}/L_{1})_{S} = P and M/IM ≅ S/IS, we conclude that

P ⊃ ann(M/IM)_{S} = ann(S/IS)_{S} = I^{S}.

The proposition follows. Remark. The argument establishing (iii ⇒ ii) above is similar to the proof in [9, 4.6] of “lying over” for finite extensions of noetherian rings with the second layer condition. 2.3. The case of group rings. We will broaden the list of equivalent properties in Proposition 2.2 by adding another condition concerning heights. This strengthening of the proposition seems, however, to require further restrictions on the ring embedding R ⊂ S. Therefore, we will concentrate here on extensions of group algebras,

R = kH ⊂ S = kG ,

where G is a polycyclic-by-finite group and H is a subgroup of finite index in G.

We start with a lemma on (Krull) homogeneity whose essence is gleaned from [1, Lemma 4.2]. Lemma. Let G be a polycyclic-by-finite group, let H be a subgroup of finite index in G, and put N = H_{G}, the normal core of H. Further, let Q be a prime ideal of kH and put I = Q^{G} ∩ kN. Then

kH/Q, kG/Q^{G}, and kG/IkG

are all (right) α-homogeneous rings, where α = Kdim(kH/Q).

Proof. First, kH/Q is prime and therefore α-homogeneous (1.4 i). Next, by [11, Lemma 1.2(i)],

Q^{G} = ⋂_{g∈G} Q^{g}kG , (4)

and this intersection is finite. Thus, as right kG-module, kG/Q^{G} is a finite subdirect product of the modules kG/Q^{g}kG. Furthermore, each kG/Q^{g}kG is generated, as kG-module, by the k[H^{g}]-submodule k[H^{g}]/Q^{g}. Now k[H^{g}]/Q^{g} is isomorphic to kH/Q (as ring) and is therefore α-homogeneous. Consequently, each kG/Q^{g}kG is α-homogeneous, by (1.4 v), and hence so is kG/Q^{G}, by (1.4 i).

Finally, using (4), I = ⋂_{g∈G}(Q ∩ kN)^{g} and, by [1, Lemma 4.2(i)], kN/Q ∩ kN is α-homogeneous. Therefore, so are its G-conjugates kN/(Q ∩ kN)^{g} and consequently kN/I as well, by the usual (finite) subdirect product argument (1.4 i). Since kN/I generates the kG-module kG/IkG, the latter is α-homogeneous, by (1.4 v), and the proof is complete.

Proposition. Let G, H, N, and Q be as in the above Lemma. Further, let P be a prime ideal of kG with P ⊃ Q^{G}. Then the following are equivalent:

i. P is minimal over Q^{G}; ii. P ∩ kN = Q^{G} ∩ kN; iii. height P = height Q.

Proof. (i ⇒ ii) Put I = Q^{G} ∩kN = ⋂_{g∈G}(Q∩kN)^{g}, as in the Lemma (and its proof), and observe that I is a G-prime ideal of kN. Indeed, as in (2), Q∩kN = ⋂_{h∈H} Q_{0}^{n} for some prime Q_{0} of kN, and so I = ⋂_{g∈G}(Q ∩ kN)^{g} = ⋂_{g∈G} Q_{0}^{g}. Clearly, I ⊂ P ∩ kN. Suppose, by way of contradiction, that this inclusion is strict. Then, by [13, Lemma 3.4(i)], there exists an element c ∈ P ∩ kN such that c is regular modulo I. Since kG/IkG is free as left and right kN/I-module, c is also regular in kG modulo IkG. Therefore, Kdim(kG/IkG + ckG)_{kG} < Kdim(kG/IkG); see [4, 13.6]. Further, by the above Lemma, Kdim(kG/IkG) = Kdim(kG/Q^{G}). Since IkG + ckG ⊂ Q^{G} + ckG, we obtain altogether that

Kdim(kG/Q^{G} + ckG)_{kG} < Kdim(kG/Q^{G}) .

By the above Lemma again, we know that kG/Q^{G} is homogeneous. So the last inequality forces c to be regular modulo Q^{G} in kG; see (1.4 ii). Inasmuch as c ∈ P, we deduce from [4, 10.8] that P cannot be minimal over Q^{G}, contrary to our hypothesis.

(ii ⇒ iii) Recall from the previous part of the proof that Q ∩ kN = ⋂_{h∈H} Q_{0}^{n} for some prime Q_{0} of kN, and Q^{G}∩kN = ⋂_{g∈G} Q_{0}^{g}. Therefore, (ii) implies that P ∩kN = ⋂_{g∈G} Q_{0}^{g}. By [11, Lemma 2.5(i)], we conclude that height Q = height Q_{0} = height P, thereby proving (iii).

(iii ⇒ i) Assume height Q = height P. If P is not minimal over Q^{G} then P contains a prime, P′, that is minimal over Q^{G}. Since we already have seen that (i) implies (iii), we know that height P′ = height Q and, consequently, height P = height P′. But this last equality is impossible, as P properly contains P′. The proof of the proposition is now complete.

3. The proof

In this section, G will always denote a polycyclic-by-finite group. Thus, the group algebra kG is left and right noetherian and satisfies the strong second layer condition.

3.1. The Roseblade subgroup. Following Roseblade [15], a subgroup H of G is called orbital if H has only finitely many G-conjugates. An orbital subgroup H is isolated if there is no orbital subgroup H′ of G with H ⊊ H′ and [H′ : H] finite. Finally, a polycyclic-by-finite group is said to be orbitally sound if all its isolated orbital subgroups are normal. As in [15], we shall write

nio(G)

for the intersection of the normalizers of all isolated orbital subgroups of G. The crucial properties of nio(G) for our purposes are the following:

• nio(G) is an orbitally sound normal subgroup of finite index in G. Moreover, nio(G) is the unique maximal such subgroup and nio(G) contains every finiteby-nilpotent normal subgroup of G (see [15, 1.3]).

• The group algebra of nio(G) is catenary [15, Corrigenda]. Since images of orbitally sound groups are clearly orbitally sound as well, the maximality property of nio(G) stated above implies in particular that, for every epimorphism φ : G ↠ \overline{G}, one has φ(nio(G)) ⊂ nio(\overline{G}).

The proof of the following lemma assumes some familiarity with [11], [10].

Lemma. Assume that G ≠ nio(G). Then every faithful prime ideal of kG is induced from some proper subgroup of G containing nio(G).

Proof. Let P be the prime in question, and let H denote the normalizer of its vertex. Then H contains nio(G) and P is induced from kH; see [10, (2.2) and Lemma 2.5]. If H ≠ G we are done; so assume that H = G. In this case, [10, Lemma 2.3] implies that P is induced from kΔ, where Δ = Δ(G) denotes the FC-center of G. Since Δ is finite-by-abelian, Δ is contained in nio(G). Therefore, by transitivity of induction [11, Lemma 1.2(iii)], P is induced from k[nio(G)] as well and the proof is complete.

3.2. A special case. The following Lemma proves the requisite condition (1) in the special case where the lower prime is induced from a subgroup of finite index in G whose group algebra is catenary.

Lemma. Let H be a subgroup of finite index in G such that kH is catenary. Then, for any pair P ⊊ P′ of adjacent primes of kG such that P is induced from kH, one has height P′ = height P + 1.

Proof. Write P = Q^{G} for some ideal Q of kH which we may assume prime, by Lemma 2.2. By Proposition 2.3, height Q = height P. Choose an ideal Q′ of kH containing Q and maximal with respect to (Q′)^{G} ⊂ P′. Then, by (2.2), Q′ is prime and P′ is minimal over (Q′)^{G}. Moreover, Proposition 2.3 gives height Q′ = height P′. Therefore, it suffices to show that height Q′ = height Q + 1 or, since kH is catenary, that Q and Q′ are adjacent. Suppose otherwise and fix a prime I of kH properly between Q and Q′. Then height Q < height I < height Q′. Further, P = Q^{G} ⊂ I^{G} ⊂ (Q′)^{G} ⊂ P′, and so P′ contains a prime, J, of kG minimal over I^{G}. By Proposition 2.3 again, height J = height I. Consequently, all inclusions in P = Q^{G} ⊂ J ⊂ P′ are strict, contradicting adjacency of P and P′.

3.3. Proof of the Theorem. To show that kG is catenary, we argue by induction on the index [G : nio(G)]. The case G = nio(G) is known [15]. So assume G ≠ nio(G), and let P ⊊ P′ be adjacent primes of kG. We have to show that height P′ = height P + 1. In view of Lemma 1.2, we may pass to G/P^{†} and thus reduce to the case where P is faithful. Note also that [G : nio(G)] ≥ [G/P^{†} : nio(G/P^{†})]; see (3.1). Thus, by Lemma 3.1, P is induced from kH for some proper subgroup H of G containing nio(G). Since nio(G) is contained in nio(H), by the maximality property of nio, we have [H : nio(H)] < [G : nio(G)] and so kH is catenary, by induction. Lemma 3.2 now completes the proof.

4. Concluding remarks

G will continue to denote a polycyclic-by-finite group.

Roseblade’s work [15] establishes catenarity of kG up to a finite index: G_{0} = nio(G) is a normal subgroup of finite index in G such that kG_{0} is catenary. Putting R = kG_{0} and S = kG, we have

S = R∗Γ ,

a crossed product of the finite group Γ = G/G_{0} over R. It is tempting to approach the catenary problem for kG from a more general angle by establishing a positive answer to the following

Question. Suppose that S = R∗Γ is a crossed product with R an affine noetherian k-algebra and Γ a finite group. If R satisfies (1), will S do so as well?

Recall that (without any hypotheses on R) the ideals of R of the form P ∩ R, for some prime ideal P of S = R ∗Γ, are precisely the Γ-primes of R and that these ideals are exactly those of the form Q_{Γ} = ⋂_{x∈Γ} Q^{x} for some prime Q of R. Moreover, if Q_{Γ} ⊂ Q′_{Γ} then we may assume, after replacing Q by a suitable Γ-conjugate, that Q ⊂ Q′. The latter will be adjacent if Q_{Γ} and Q′_{Γ} are adjacent Γ-primes. Finally, P ∩ R = Q_{Γ} entails height P = height Q. See [12] for details. A positive answer to the above question would therefore result if the following were true:

Question′. Do adjacent primes of S = R∗Γ contract to adjacent Γ-primes of R?

Here is what appears to be known about Question′:

• Without noetherian hypotheses, the answer is negative in general. Indeed, Kaplansky [6] constructs an extension of commutative (non-noetherian) rings R ⊂ S, with S = R ∗ Γ and Γ the group of order 2, so that Question′ fails for S.

• Question′ has a positive answer if R is affine PI. This is an easy application of Schelter’s catenarity result [16, Theorem 4] for affine PI-algebras.

It would be interesting to have more definitive results about this topic.

Acknowledgment. The authors would like to thank K. A. Brown for reading a preliminary draft of this article and for his valuable comments.

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[16] W. Schelter, Non-commutative affine P.I. rings are catenary, J. Algebra 51 (1978), 12–18.

[17] D. Segal, unpublished manuscript, ca. 1976.

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Department of Mathematics, Texas A&M University, College Station, TX 77843

  E-mail address: letzter@math.tamu.edu

Department of Mathematics, Temple University, Philadelphia, PA 19122-6094

  E-mail address: lorenz@math.temple.edu