ON THE KRULL-SCHMIDT-AZUMAYA THEOREM FOR INTEGRAL GROUP RINGS

PETER HINDMAN, LEE KLINGLER, AND CHARLES J. ODENTHAL

Abstract. We show that the torsion free Krull-Schmidt-Azumaya Theorem holds for the integral group ring ℤD_{8}, where D_{8} is the dihedral group of order 8, but torsion free cancellation (and hence also the torsion free Krull-Schmidt-Azumaya theorem) fails for the integral group ring ℤD_{32}. We summarize the cases for which the integral group ring ℤG is known to satisfy the torsion free Krull-Schmidt-Azumaya Theorem, ℤD_{16} being the only open case.

1. Introduction and Notation

Let R be a Dedekind domain and Λ an order over R. Recall that a Λ-lattice is a Λ-module that is finitely generated projective as an R-module. We say that torsion free cancellation (TFC) holds for Λ if X ⊕ M ≅ X ⊕ N implies M ≅ N for lattices X, M, and N over Λ. We say that the torsion free Krull-Schmidt-Azumaya Theorem (TFKS) holds for Λ if M_{1} ⊕ ... ⊕ M_{m} ≅ N_{1} ⊕ ... ⊕ N_{n} implies that m = n and (reordering if necessary) M_{i} ≅ N_{i} for each index i, for indecomposable lattices M_{1}, ..., M_{m}, N_{1}, ..., N_{n} over Λ. Clearly TFKS implies TFC.

We are interested in the question of whether or not TFKS holds for ℤG, the integral group ring of a finite group G. Let us fix the notation and recall a few needed facts. Throughout this paper, modules are always assumed to be finitely generated left modules, and maps are always written on the right.

If R is a Dedekind domain with quotient field K, Λ an R-order, and M a Λlattice, we define the genus of M to be the set gen_{Λ}(M) consisting of all isomorphism classes of Λ-lattices N such that the localizations M_{P} ≅ N_{P} for all maximal ideals P of R. Fix Γ to be some maximal R-overorder of Λ. Then M embeds in KM = K ⊗_{R} M, and we can form the Γ-lattice ΓM = {Σγ_{i}m_{i} | γ_{i} ∈ Γ, m_{i} ∈ M}. We define the restricted genus of M to be the set D(M) consisting of all isomorphism classes of Λ-lattices N such that N is in the genus of M and ΓN ≅ ΓM. For the maximal overorder Γ of Λ, there is a natural map from gen_{Λ}(M) to gen_{Γ}(ΓM) obtained by sending the isomorphism class of the Λ-lattice N of gen_{Λ}(M) to the isomorphism class of the Γ-lattice ΓN of gen_{Γ}(ΓM). Since this function is a surjection (see [4, Lemma 3.7], for example), we see that gen_{Λ}(M) = D(M) if and only if |gen_{Γ}(ΓM)| = 1.

For the maximal R-overorder Γ of Λ, let C denote the largest two-sided ideal of Γ such that C ⊆ Λ. The ideal C is called the conductor from Γ to Λ. If M is a Λ-lattice, we obtain the conductor square

1991 Mathematics Subject Classification. Primary 20C10; Secondary 16D70.

Key words and phrases. Krull-Schmidt-Azumaya, unique decomposition, integral group ring.

\begin{matrix}M & ⟶ & ΓM \\ ↓ & & ^{ν}↓ \\ M/CM & ⟶^{ι} & ΓM/CM \end{matrix} (1.0.1)

in which the horizontal maps are the natural inclusions and the vertical maps are the natural surjections. The conductor square (1.0.1) defines M as a pullback; that is, M = (M/CM)ιν^{−1} ⊆ ΓM.

Following [4], let E^{∗}(ΓM/CM) denote the autmorphism group of the Γ/Cmodule ΓM/CM, let E^{∗}(βM) denote the subgroup consisting of those automorphisms θ ∈ E^{∗}(ΓM/CM) such that (M/CM)θ = M/CM, and let E^{∗}(ΓM) denote the subgroup of automorphisms in E^{∗}(ΓM/CM) which lift (via ν) to automorphisms of ΓM. A means of calculating the size of a restricted genus is provided by the following Double Coset Theorem. (For a proof see [4, Theorem 3.4] or [7, Theorem 4.3].)

Theorem 1.1. With notation as above, there is a one-to-one correspondence between the set of isomorphism classes of lattices in the restricted genus D(M) and the double cosets E^{∗}(βM)\E^{∗}(ΓM/CM)/E^{∗}(ΓM).

The correspondence in the theorem is obtained by sending θ ∈ E^{∗}[ΓM/CM] to the lattice M^{θ} defined by the following pullback.

M^{θ}  ⟶  ΓM
↓  ^{ν}↓
M/CM ⟶^{ι} ΓM/CM ⟶^{θ} ΓM/CM

We shall also need the following theorem, an immediate consequence of [6, Theorem 1.3] and [6, Proposition 2.3].

Theorem 1.2. Let Λ be an order over the Dedekind domain R; let Γ be a maximal R-overorder. Then Λ satisfies TFKS if and only if the following conditions are satisfied.

(a) There is at most one maximal ideal 𝔪 of R such that Λ_{𝔪} ≠ Γ_{𝔪}.

(b) If the exceptional 𝔪 from part (a) exists, then TFKS holds for Λ_{𝔪}.

(c) Every genus of Λ-lattices consists of a single isomorphism class. If Λ is not a prime ring, then condition (b) can be replaced by (b′) If the exceptional 𝔪 from part (a) exists, then primitive idempotents of Γ_{𝔪}

remain primitive modulo the jacobson radical of Γ_{𝔪}.

The above can be contrasted with the following theorem, the proof of which is based on ideas from [12, §2].

Theorem 1.3. Let Λ be an order over the Dedekind domain R; let Γ be a maximal R-overorder. Then Λ satisfies TFC if and only if the following conditions are satisfied.

(a) TFC holds for Γ.

(b) Every restricted genus of Λ-lattices consists of a single isomorphism class.

Proof. (⇒) Suppose TFC holds for Λ. Clearly TFC holds for Γ. Take Λ-lattices M and N in the same restricted genus. We need to show that M and N are isomorphic. We will show that M ⊕ Γ^{(2)} ≅ N ⊕ Γ^{(2)}, then cancellation finishes the proof.

First note that M ⊕ Γ^{(2)} and N ⊕ Γ^{(2)} are in the same restricted genus, so it will suffice to show that the restricted genus of M ⊕Γ^{(2)} consists of a single isomorphism class. We use Theorem 1.1. Consider first E^{∗}(ΓM ⊕Γ^{(2)}) ⊆ E^{∗}(ΓM/CM ⊕Γ/C^{(2)}), where C is the conductor from Γ to Λ.

For the moment we work one prime summand of Γ at a time, and so can assume that Γ is prime. Let H = end_{Γ}(ΓM ⊕ Γ^{(2)}), and I = {θ ∈ H|(ΓM ⊕ Γ^{(2)})θ ⊆ CM ⊕ C^{(2)}}. Then H is Morita equivalent to Γ, and I is a non-zero ideal of H with H/I ≅ end_{Γ/C}(ΓM/CM ⊕ Γ/C^{(2)}). We also have group isomorphisms (H/I)^{∗} ≅ E^{∗}(ΓM/CM ⊕ Γ/C^{(2)}) and (H^{∗} + I)/I ≅ E^{∗}(ΓM ⊕ Γ^{(2)}). Note that H has uniform rank r ≥ 2, and is a prime maximal order [9, Theorem 21.6]. In particular, H is an HNP having no idempotent ideals.

Now an application of [7, Proposition 5.1] shows that there is an artinian PIR A with (H/I)^{∗} ≅ GL_{r}(A). Under this isomorphism (H^{∗} + I)/I ⊇ EL_{r}(A), the subgroup of GL_{r}(A) generated by the elementary matrices I_{r} + αe_{ij} (α ∈ A, i ≠ j), where e_{ij} denotes the matrix with a 1 in the (i, j) position and zeros elsewhere. Again from [7, Proposition 5.1], there are canonical group epimorphisms

(H/I)^{∗} → → \frac{GL_{r}(A)}{EL_{r}(A)} ≅ K_{1}(A) ≅ K_{1}(H/I) ≅ K_{1}(Γ/C)

→ → \frac{(H/I)^{∗}}{(H^{∗} + I)/I} ≅ \frac{E^{∗}(ΓM/CM ⊕ Γ/C^{(2)})}{E^{∗}(ΓM ⊕ Γ^{(2)})}

(1.3.1)

In particular, E^{∗}(ΓM ⊕ Γ^{(2)}) is a normal subgroup of E^{∗}(ΓM/CM ⊕ Γ/C^{(2)}) with abelian quotient, and the space of double cosets

E^{∗}(β(M ⊕ Γ^{(2)}))\E^{∗}(ΓM/CM ⊕ Γ/C^{(2)})/E^{∗}(ΓM ⊕ Γ^{(2)})

is just the factor group

\frac{E^{∗}(ΓM/CM ⊕ Γ/C^{(2)})}{E^{∗}(β(M ⊕ Γ^{(2)}))E^{∗}(ΓM ⊕ Γ^{(2)})}

Since (1.3.1) holds for each prime summand of our original Γ, in the semiprime case we still have the epimorphism

K_{1}(Γ/C) → → \frac{E^{∗}(ΓM/CM ⊕ Γ/C^{(2)})}{E^{∗}(ΓM ⊕ Γ^{(2)})} = Ē (1.3.2)

where the last equality defines Ē.

Return now to the case of semiprime Γ. Note that 1 is in the stable range of the artinian ring Γ/C, so we have the isomorphism of abelian groups

(1.3.3) \frac{GL_{2}(Γ/C)}{EL_{2}(Γ/C)} ≅ K_{1}(Γ/C)

Next observe that

[\begin{matrix}E^{∗}(βM) & 0 \\ 0 & E^{∗}(βΓ^{(2)}) \end{matrix}] ⊆ E^{∗}(β(M ⊕ Γ^{(2)}))

So the image of E^{∗}(β(M ⊕Γ^{(2)})) in Ē contains the image of E^{∗}(βΓ^{(2)}) = GL_{2}(Γ/C). But GL_{2}(Γ/C) maps onto K_{1}(Γ/C) by (1.3.3), so (1.3.2) implies that E^{∗}(β(M ⊕ Γ^{(2)})) maps onto Ē. That is, E^{∗}(β(M ⊕ Γ^{(2)}))E^{∗}(ΓM ⊕ Γ^{(2)}) = E^{∗}(ΓM/CM ⊕ (Γ/C)^{(2)}), and D(M ⊕ Γ^{(2)}) consists of a single isomorphism class by Theorem 1.1.

(⇐) Let conditions (a) and (b) be satisfied, and suppose M ⊕X ≅ N ⊕X for some Λ-lattices M, N, and X. We have ΓM ⊕ΓX ≅ ΓN ⊕ΓX and M_{P} ⊕X_{P} ≅ N_{P} ⊕X_{P} for every maximal ideal P of R. Since TFC holds for Γ as well as for Λ_{P}, we see that ΓM ≅ ΓN and M_{P} ≅ N_{P} for all P. So M and N are in the same restricted genus, and M ≅ N by condition (b), finishing the proof.

Combining these theorems provides us with the tool we need.

Theorem 1.4. Let Λ be an order over the Dedekind domain R; let Γ be a maximal R-overorder. Then Λ satisfies TFKS if and only if the following conditions are satisfied.

(a) There is at most one maximal ideal 𝔪 of R such that Λ_{𝔪} ≠ Γ_{𝔪}.

(b) If the exceptional 𝔪 from part (a) exists, then TFKS holds for Λ_{𝔪}.

(c) TFC holds for Λ.

(d) TFKS holds for Γ. If Λ is not a prime ring, then condition (b) can be replaced by (b′) If the exceptional 𝔪 from part (a) exists, then primitive idempotents of Γ_{𝔪}

remain primitive modulo the jacobson radical of Γ_{𝔪}. For any Λ (prime or non-prime) condition (c) can be replace by (c′) Every restricted genus of Λ-lattices consists of a single isomorphism class.

Proof. Suppose that Λ satisfies TFKS. Then clearly so does Γ, while Λ satisfies TFC. Conditions (a) , (b), (b′) and (c′) are satisfied by Theorem 1.2.

Conversely, suppose conditions (a)–(d) are satisfied. By Theorem 1.2 it will suffice to show that every genus of Λ-lattices consists of a single isomorphism class. Theorem 1.2 applied to Γ (assumed to satisfy TFKS) implies that |gen_{Γ}(ΓM)| = 1 for every Λ-lattice M. As observed above, this is equivalent to gen_{Λ}(M) = D(M) for every Λ-lattice M. Since Λ satisfies TFC we have, by Theorem 1.3, |gen_{Λ}(M)| = |D(M)| = 1 for every Λ-lattice M.

By Theorem 1.2, condition (b′) can replace condition (b). In the presence of TFKS for Γ, conditions (c) and (c′) are equivalent by Theorem 1.3.

In the situation where Λ is the group ring ℤG, things simplify even further: Condition (a) just says that G is a p-group for some prime p; while, for odd primes p, condition (b), that ℤ_{(p)}G satisfies TFKS, holds by [1, Theorem 36.1]. For (nontrivial) 2-groups we can use condition (b′) of Theorem 1.2 since ℤG is not a prime ring.

Theorem 1.5. Let Γ be a maximal ℤ-overorder of the group ring ℤG. Then ℤG satisfies TFKS if and only if the following conditions are satisfied.

(a) G is a p-group for some prime p.

(b) If G is a 2-group, then primitive idempotents of Γ_{(2)} remain primitive modulo the jacobson radical of Γ_{(2)}.

(c) ℤG satisfies TFC.

(d) Γ satisfies TFKS. Condition (c) can be replaced by (c′) Every restricted genus of Λ-lattices consists of a single isomorphism class.

As noted in [10, §7], Endo and Hironaka [2] have shown that, if TFC holds for ℤG, then G must be either abelian, dihedral, or one of A_{4}, S_{4}, or A_{5}. Now Theorem 1.5 shows that if TFKS holds for ℤG, then G must be either an abelian p-group or a binary dihedral group.

For abelian groups G, Wiegand [12, Proposition 5.1] has shown that, if G is not cyclic of order 8 or 9, then TFC holds for ℤG if and only if |G| is prime, or |G| = 4, or G is cyclic of order 1, 6, 10, or 14. Thus by Theorem 1.5, for abelian groups G we need only consider |G| prime, |G| = 4, and G cyclic of order 1, 8, or 9.

Making use of [12, Proposition 5.1], if C_{p} is a cyclic group of odd prime order p, we get from Theorem 1.5 that TFKS holds for ℤC_{p} if and only if TFKS holds for the maximal overorder Γ = ℤ × ℤ[ζ_{p}] if and only if ℤ[ζ_{p}] has trivial class group, where ζ_{p} denotes a primitive complex p-th root of unity. If C_{2} is a cyclic group of order 2, we get from Theorem 1.5 that TFKS holds for ℤC_{2} if and only if TFKS holds for the maximal overorder Γ = ℤ×ℤ and primitive idempotents of Γ_{(2)} remain primitive modulo the jacobson radical; thus, clearly TFKS holds for ℤC_{2} as well. (These cases also follows from [5, Theorem 11.5] as well as from [1, Theorem 34.31].) Similarly, if C_{4} is a cyclic group of order 4, we get from Theorem 1.5 that TFKS holds for ℤC_{4} if and only if TFKS holds for the maximal overorder Γ = ℤ×ℤ×ℤ[ζ_{4}] and primitive idempotents of Γ_{(2)} remain primitive modulo the jacobson radical. Since ℤ and ℤ[ζ_{4}] are principal ideal domains and since 2 is totally ramified in ℤ[ζ_{4}], it follows that TFKS holds for ℤC_{4}. Likewise, if G = C_{2} × C_{2} is a non-cyclic group of order 4, we get from Theorem 1.5 that TFKS holds for ℤG if and only if TFKS holds for the maximal overorder Γ = ℤ × ℤ × ℤ × ℤ and primitive idempotents of Γ_{(2)} remain primitive modulo the jacobson radical; so in this case also, TFKS holds for ℤG.

If C_{8} is a cyclic group of order 8, then TFC holds for ℤC_{8} by [10, Theorem 5.1]. Proceeding as above, we have the maximal overorder Γ = ℤ × ℤ × ℤ[ζ_{4}] × ℤ[ζ_{8}], where ℤ, ζ_{4}, and ℤ[ζ_{8}] are principal ideal domains, and 2 is totally ramified in both ℤ[ζ_{4}] and ζ_{8}. Then Γ satisfies TFKS and primitive idempotents of Γ_{(2)} remain primitive modulo the jacobson radical, so TFKS holds for ℤC_{8} by Theorem 1.5.

Finally, if C_{9} is a cyclic group of order 9, then TFC holds for ℤC_{9} by [1, Theorem 34.35]. Again the maximal overorder Γ = ℤ×ℤ[ζ_{3}]×ℤ[ζ_{9}] is a direct sum of principal ideal domains; so Γ satisfies TFKS, and hence TFKS holds for ℤC_{9} by Theorem 1.5. (This also follows from a more detailed reading of [1, Theorem 34.35].)

If C_{1} is a cyclic group of order 1, then obviously ℤC_{1} ≅ ℤ satisfies TFKS. This settles the question for abelian groups G.

For non-abelian groups, we need only consider binary dihedral groups (of order at least 8). Let D_{2^{n}} denote the dihedral group of order 2^{n}. Swan [10, Theorem 6.1] showed that TFC holds for ℤD_{8}, and so it follows easily from Theorem 1.5 that TFKS also holds for ℤD_{8}. We give a different proof of this fact below (Theorem 2.4).

In the other direction, Swan [10, Theorem 8.1] showed that TFC fails for ℤD_{128}. Since D_{2^{n}} is a homomorphic image of D_{2^{n+1}} (and hence ℤD_{2^{n}} is a homomorphic image of ℤD_{2^{n+1}}), it follows that TFC fails for ℤD_{2^{n}} for all integers n ≥ 7. Our second main result below (Theorem 3.2) is that TFC fails for ℤD_{32} and hence for ℤD_{2^{n}} for all integers n ≥ 5. Consequently, TFKS fails for ℤD_{2^{n}} for all integers n ≥ 5. The question of whether or not TFKS holds for ℤD_{16} remains open. (See §4 below.)

We summarize the known cases for which TFKS holds in the following theorem.

Theorem 1.6. Let G be a finite group and suppose that G is not dihedral of order 16. Then TFKS holds for ℤG if and only if one of the following conditions holds

(i) |G| = p, prime, and the cyclotomic ring of integers ℤ[ζ_{p}] has trivial class group;

(ii) |G| = 4; (iii) G is cyclic of order 1, 8, or 9; or (iv) G is dihedral of order 8.

2. The Krull-Schmidt-Azumaya Theorem for ℤD_{8}

In this section, we show that TFKS holds for the integral group ring ℤD_{8}. Since TFKS implies TFC, this yields another proof of [10, Theorem 6.1]. Our proof also serves as an introduction to the more complicated construction in §3, where we show that TFC (and hence TFKS) fails for the integral group ring ℤD_{32}.

The first step is to embed ℤD_{8} in a maximal order. Let us write

D_{8} = ⟨x, y | x^{4} = y^{2} = 1, yxy = x^{−1}⟩.

It is easy to get four homomorphisms from ℤD_{8} onto ℤ induced by x ↦ ±1 and y ↦ ±1. Additionally, there is a homomorphism from ℤD_{8} into M_{2×2}(ℤ) (the ring of 2 × 2 matrices over ℤ), induced by

x ↦ [\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}] and y ↦ [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}]

If we set Γ = ℤ × ℤ × ℤ × ℤ × M_{2×2}(ℤ) (a maximal order, by [9, Theorem 8.7]), then it is routine to check that the ring homomorphism induced by

x ↦ (1, 1, −1, −1, [\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}]) and y ↦ (1, −1, 1, −1, [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}])

is an embedding of ℤD_{8} into Γ. We identify ℤD_{8} with its image in Γ.

Following [10], let N = ∑_{g∈D_{8}} g in ℤD_{8}, and set Δ = ℤD_{8} + ℤ · \frac{N}{2} ⊆ Γ. We need the following special case of [10, Lemma 5.6].

Lemma 2.1. Any ℤD_{8}-lattice has the form P ⊕ M, where P is a projective ℤD_{8}- lattice, and M is a Δ-lattice.

The next step is to show that every genus of indecomposable ℤD_{8}-lattices consists of a single isomorphism class. Given Lemma 2.1, we need only consider projective ℤD_{8}-lattices and Δ-lattices.

Lemma 2.2. If M is an indecomposable projective ℤD_{8}-lattice, then M ≅ ℤD_{8}. In particular, the genus of ℤD_{8} consists of a single isomorphism class.

Proof. By [1, Corollary 32.12], M is in the genus of the free module ℤD_{8}, so it suffices to show that the genus of ℤD_{8} consists of a single isomorphism class. But |D(ℤD_{8})| = 1 by [3, Theorem 3], while |gen_{Γ}(Γ)| = 1 by [1, Theorem 31.26], since ℤ is a principal ideal domain; therefore, gen_{ℤD_{8}}(ℤD_{8}) = 1 as well.

The harder case is that of an indecomposable Δ-lattice. The idea is to compute the conductor of Γ into Δ, and then use Theorem 1.1.

Lemma 2.3. If M is an indecomposable Δ-lattice, then the genus of M (as a ℤD_{8}-lattice) consists of a single isomorphism class.

Proof. Clearly it suffices to show that gen_{Δ}(M) is trivial.

Under the above identification of ℤD_{8} as a suborder of Γ = ℤ × ℤ × ℤ × ℤ × M_{2×2}(ℤ), we get the element

\frac{N}{2} = \frac{1}{2} ∑_{g∈D_{8}} g = (4, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}])

in Δ. It is then straightforward but tedious to compute that 4Γ ⊆ Δ, and indeed 4Γ is the conductor from Γ into Δ.

The conductor square for the order Δ becomes

Δ ⟶ Γ
↓  ↓
Δ/4Γ ⟶ Γ/4Γ

where Γ/4Γ = ℤ̄_{4} × ℤ̄_{4} × ℤ̄_{4} × ℤ̄_{4} × M_{2×2}(ℤ̄_{4}) (writing ℤ/4ℤ = ℤ̄_{4}).

Thus, given the (indecomposable) Δ-lattice M, the conductor square (1.0.1) becomes

M ⟶  ΓM = ℤ^{(e_{1})} × ℤ^{(e_{2})} × ℤ^{(e_{3})} × ℤ^{(e_{4})} × M_{2×1}(ℤ)^{(e_{5})}
↓  ↓
M/4ΓM ⟶ ΓM/4ΓM = ℤ̄_{4}^{(e_{1})} × ℤ̄_{4}^{(e_{2})} × ℤ̄_{4}^{(e_{3})} × ℤ̄_{4}^{(e_{4})} × M_{2×1}(ℤ̄_{4})^{(e_{5})}

for some non-negative integers e_{1}, e_{2}, e_{3}, e_{4}, and e_{5}, where M_{2×1}(ℤ) (respectively M_{2×1}(ℤ̄_{4})) denotes the indecomposable M_{2×2}(ℤ)-lattice (respectively M_{2×2}(ℤ̄_{4})- module) of 2 × 1 column vectors over ℤ (respectively ℤ̄_{4}). Now the automorphism group of the lower right-hand corner becomes

E^{∗}(ΓM/4ΓM) = GL(e_{1}, ℤ̄_{4}) × GL(e_{2}, ℤ̄_{4}) × GL(e_{3}, ℤ̄_{4}) × GL(e_{4}, ℤ̄_{4}) × GL(e_{5}, ℤ̄_{4}),

where GL(n, ℤ̄_{4}) denotes the group of invertible n × n matrices over the ring ℤ̄_{4}. (Note that the endomorphism ring of the indecomposable M_{2×2}(ℤ̄_{4})-module M_{2×1}(ℤ̄_{4}) is just ℤ̄_{4}.) Similarly, the automorphism group of the upper right-hand corner becomes

E^{∗}(ΓM) = GL(e_{1}, ℤ) × GL(e_{2}, ℤ) × GL(e_{3}, ℤ) × GL(e_{4}, ℤ) × GL(e_{5}, ℤ),

where the endomorphism ring of the indecomposable M_{2×2}(ℤ)-lattice M_{2×1}(ℤ) is just ℤ.

We claim that every invertible matrix of GL(n, ℤ̄_{4}) lifts to an invertible matrix of GL(n, ℤ). Note that elementary matrices in GL(n, ℤ̄_{4}) lift to elementary matrices in GL(n, ℤ), and, because units in ℤ̄_{4} lift to units in ℤ, diagonal matrices in GL(n, ℤ̄_{4}) lift to diagonal matrices in GL(n, ℤ). Since the elementary matrices and the diagonal matrices together generate GL(n, ℤ̄_{4}), the claim follows.

Therefore, E^{∗}(ΓM) maps onto E^{∗}(ΓM/4ΓM), and hence, by Theorem 1.1, the restricted genus D(M) consists of a single isomorphism class. But ΓM = ℤ^{(e_{1})} × ℤ^{(e_{2})} × ℤ^{(e_{3})} × ℤ^{(e_{4})} × M_{2×1}(ℤ)^{(e_{5})}, where ℤ is a principal ideal domain. As above, from [1, Theorem 31.26] it follows that |gen_{Γ}(ΓM)| = 1 also. Therefore, gen_{Δ}(M), the genus of M (as a Δ-lattice), must consist of a single isomorphism class.

It is now easy to show that TFKS holds for ℤD_{8}.

Theorem 2.4. The torsion free Krull-Schmidt-Azumaya Theorem holds for ℤD_{8}.

Proof. If M is a ℤD_{8}-lattice, write M as a direct sum of indecomposable lattices. By Lemmas 2.1, 2.2, and 2.3, each of these indecomposable summands has genus of size one, from which it follows that gen_{ℤD_{8}}(M) consists of a single isomorphism class. First, we can conclude that every genus of Γ-lattices consists of a single isomorphism class, and hence Γ satisfies TFKS by Theorem 1.2. Second, we can also conclude that the restricted genus of M also has size one. Since it is clear that primitive idempotents of Γ_{(2)} remain primitive modulo the jacobson radical of Γ_{(2)}, it follows from Theorem 1.5 that TFKS holds for ℤD_{8}.

Corollary 2.5. Torsion free cancellation holds for ℤD_{8}.

3. Failure of Torsion free Cancellation for ℤD_{32}

In this section, we show that TFC (and hence also TFKS) fails for the integral group ring ℤD_{32}. Since ℤD_{2^{n}} is a homomorphic image of ℤD_{2^{n+1}}, it follows that TFC fails for ℤD_{2^{n}} for n ≥ 5, yielding a slight improvement of [10, Theorem 8.1] in the case of binary dihedral groups.

As in the previous section, the first step is to embed ℤD_{32} in a maximal order. Let us write

D_{32} = ⟨x, y | x^{16} = y^{2} = 1, yxy = x^{−1}⟩.

Besides the five homomorphisms listed in the previous section, there is a homomorphism from ℤD_{8} into M_{2×2}(ℤ[\sqrt{2}]) induced by

x ↦ [\begin{matrix}0 & −1 \\ 1 & \sqrt{2} \end{matrix}] and y ↦ [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] .

For convenience of notation, let us set α = \sqrt{2 + \sqrt{2}}; there is also a homomorphism from ℤD_{8} into M_{2×2}(ℤ[α]) induced by

x ↦ [\begin{matrix}0 & −1 \\ 1 & α \end{matrix}] and y ↦ [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] .

If we set Γ = ℤ × ℤ × ℤ × ℤ × M_{2×2}(ℤ) × M_{2×2}(ℤ[\sqrt{2}]) × M_{2×2}(ℤ[α]), then it is routine to check that the ring homomorphism induced by

\begin{matrix}x ↦ (1, 1, −1, −1, [\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}] , [\begin{matrix}0 & −1 \\ 1 & \sqrt{2} \end{matrix}] , [\begin{matrix}0 & −1 \\ 1 & α \end{matrix}]) and \\ y ↦ (1, −1, 1, −1, [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] , [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] , [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}]) \end{matrix} (3.0.1)

is an embedding of ℤD_{32} into Γ. As in the previous section, Γ is a maximal order by [9, Theorem 8.7], using the fact that ℤ[\sqrt{2}] and ℤ[\sqrt{2 + \sqrt{2}}] are principal ideal domains. (See [11, page 352]. Note that ℤ[\sqrt{2}] and ℤ[\sqrt{2 + \sqrt{2}}] are the rings of integers in the real subfields of the cyclotomic fields ℚ[ζ_{8}] and ℚ[ζ_{16}], respectively.) We identify ℤD_{32} with its image in Γ, so that we read the arrows in (3.0.1) as equality.

Following [10], let N = ∑_{g∈D_{32}} g in ℤD_{32}, and set Δ = ℤD_{32} + ℤ · \frac{N}{2} ⊆ Γ. In addition to the generators x and y given in (3.0.1), the order Δ requires the generator

\frac{N}{2} = (16, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}]). (3.0.2)

If we take the thirty-two group elements (generated by x and y as in (3.0.1)) together with \frac{N}{2} as in (3.0.2), we obtain a spanning set for Δ as an additive group. Reducing this set of thirty-three vectors (for example, using the commmand “ihermite” in the software package “Maple”), one easily shows that the conductor from Γ into Δ is the ideal

C = 16ℤ × 16ℤ × 16ℤ × 16ℤ × 16M_{2×2}(ℤ) × 4\sqrt{2}M_{2×2}(ℤ[\sqrt{2}]) × 2αM_{2×2}(ℤ[α])

Moreover, modulo the conductor, C, the ring Δ̄ = Δ/C requires only nineteen generators as an additive group, namely those in Table 1 below.

The next step is the construction of a Δ-lattice M, which we accomplish by means of a conductor square. Let X be the Γ-lattice

X = ℤ × ℤ × ℤ × ℤ × M_{2×1}(ℤ) × M_{2×1}(ℤ[\sqrt{2}]) × M_{2×1}(ℤ[α]).

(That is, the projection of X onto each of the seven components is a non-zero indecomposable lattice.) Let h : Δ → X be the Δ-homomorphism given by right multiplication by

h = (1, 1, 1, 1, [\begin{matrix}1 \\ 1 \end{matrix}] , [\begin{matrix}0 \\ 1 \end{matrix}] , [\begin{matrix}1 \\ 0 \end{matrix}]), (3.0.3)

and note that X = Γ · im(h). Set X̄ = X/CX, and let ν : X → X̄ be the natural map. Set M̄ = Δ/ ker(hν), a Δ̄-module since (C)h ⊆ CX implies that C ⊆ ker(hν). Identify M̄ with its image (Δ)hν ⊆ X̄, and let ι : M̄ → X̄ be the inclusion map. Now im(ι) = im(hν), so that X = Γ · im(h) implies that X̄ = Γ · im(ι).

We let M be the pullback of ι and ν; that is, M = (M̄)ιν^{−1} ⊆ X. From [4, Lemma 3.2] we get the conductor square

\begin{matrix}M & ⟶ & X \\ ↓ & & ^{ν}↓ \\ M̄ & ⟶^{ι} & X̄ \end{matrix}

Let h̄ : Δ̄ → X̄ be the induced map; we have that M̄ = im(h̄), and so X̄ = Γ·im(h̄).

Proposition 3.1. For the Δ-lattice M constructed above, the restricted genus D(M) is non-trivial.

Proof. By Theorem 1.1, the size of the restricted genus D(M) equals the number of double cosets

E^{∗}(βM̄)\E^{∗}(X̄)/E^{∗}(X),

where E^{∗}(X̄) denotes the automorphism group of X̄, E^{∗}(βM̄) denotes the subgroup of all automorphisms τ ∈ E^{∗}(X̄) such that (M̄)τ = M̄, and E^{∗}(X) denotes the subgroup of automorphisms in E^{∗}(X̄) which lift (via ν) to automorphisms of X.

Note first that the automorphism group of the Γ-lattice X is given by

E^{∗}(X) = ℤ^{∗} × ℤ^{∗} × ℤ^{∗} × ℤ^{∗} × ℤ^{∗} × ℤ[\sqrt{2}]^{∗} × ℤ[α]^{∗},

while the automorphism group of the Γ-module X̄ is given by

E^{∗}(X̄) = ℤ̄_{16}^{∗} × ℤ̄_{16}^{∗} × ℤ̄_{16}^{∗} × ℤ̄_{16}^{∗} × ℤ̄_{16}^{∗} × (ℤ[\sqrt{2}]/(4\sqrt{2}))^{∗} × (ℤ[α]/(2α))^{∗}.

Thus, each of the first five coordinates of the quotient E^{∗}(X̄)/E^{∗}(X) has the form

ℤ̄_{16}^{∗}/{±1},

\begin{matrix}s_{1} = & (1, 1, 1, 1, [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] , [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] , [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}]) \\ s_{2} = & (0, 2, 0, 2, [\begin{matrix}1 & 1 \\ 1 & 1 \end{matrix}] , [\begin{matrix}1 + \sqrt{2} & 1 \\ 3 & 1 + \sqrt{2} \end{matrix}] , [\begin{matrix}1 + α & 1 \\ 1 + \sqrt{2} & 1 + α + α\sqrt{2} \end{matrix}]) \\ s_{3} = & (0, 0, 2, 2, [\begin{matrix}1 & 1 \\ 3 & 1 \end{matrix}] , [\begin{matrix}1 & 1 \\ 1 & 1 + \sqrt{2} \end{matrix}] , [\begin{matrix}1 & 1 \\ 1 + \sqrt{2} & 1 + α + α\sqrt{2} \end{matrix}]) \\ s_{4} = & (0, 0, 0, 4, [\begin{matrix}0 & 0 \\ 2 & 2 \end{matrix}] , [\begin{matrix}0 & 0 \\ \sqrt{2} & 2 + \sqrt{2} \end{matrix}] , [\begin{matrix}0 & 0 \\ \sqrt{2} + α & 2 + α + α\sqrt{2} \end{matrix}]) \\ s_{5} = & (0, 0, 0, 0, [\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix}] , [\begin{matrix}0 & \sqrt{2} \\ \sqrt{2} & 2 \end{matrix}] , [\begin{matrix}\sqrt{2} & α + α\sqrt{2} \\ α & 2 \end{matrix}]) \\ s_{6} = & (0, 0, 0, 0, [\begin{matrix}0 & 2 \\ 2 & 0 \end{matrix}] , [\begin{matrix}\sqrt{2} & 0 \\ 2 & \sqrt{2} \end{matrix}] , [\begin{matrix}α + α\sqrt{2} & \sqrt{2} \\ 0 & 2 + α \end{matrix}]) \\ s_{7} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 4 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 2 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ \sqrt{2} & 2 \end{matrix}]) \\ s_{8} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 4 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 2 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 2 + \sqrt{2} \end{matrix}]) \\ s_{9} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix}] , [\begin{matrix}\sqrt{2} & α\sqrt{2} \\ α\sqrt{2} & 2 + \sqrt{2} \end{matrix}]) \\ s_{10} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}2\sqrt{2} & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}α\sqrt{2} & 0 \\ 0 & 0 \end{matrix}]) \\ s_{11} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 2 \\ 2 & 0 \end{matrix}] , [\begin{matrix}α\sqrt{2} & \sqrt{2} \\ \sqrt{2} & 2 + α\sqrt{2} \end{matrix}]) \\ s_{12} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 2\sqrt{2} \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & α\sqrt{2} \\ 0 & 0 \end{matrix}]) \\ s_{13} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 4 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 2 \end{matrix}]) \\ s_{14} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 2\sqrt{2} & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ α\sqrt{2} & 0 \end{matrix}]) \\ s_{15} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 4 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 2 \end{matrix}]) \\ s_{16} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 2\sqrt{2} \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & α\sqrt{2} \end{matrix}]) \\ s_{17} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix}]) \\ s_{18} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 2 \\ 0 & 2 \end{matrix}]) \\ s_{19} = & (0, 0, 0, 0, [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 2 & 2 \end{matrix}]) \end{matrix}

Table 1. Generators of Δ̄ a cyclic group of order four. We shall need only the first five coordinates of E^{∗}(X̄)/E^{∗}(X). (In fact, the quotient (ℤ[\sqrt{2}]/(4\sqrt{2}))^{∗}/ℤ[\sqrt{2}]^{∗} has order two, while (ℤ[α]/(2α))^{∗}/ℤ[α]^{∗} is trivial.)

The problem is to compute the cosets E^{∗}(βM̄)\E^{∗}(X̄). Suppose that τ ∈ E^{∗}(βM̄), so that τ is an automorphism of X̄ which restricts to an automorphism of M̄. Now the ring Δ̄ = Δ/C is local, so the map h̄ : Δ̄ → M̄ is a projective cover, and hence the automorphism τ lifts to an automorphism σ of Δ̄. This yields a commutative diagram

\begin{matrix}Δ̄ & ⟶^{σ} & Δ̄ \\ h̄↓ & & h̄↓ \\ X̄ & ⟶^{τ} & X̄ \end{matrix} (3.1.0)

in which σ and τ are automorphisms, and h̄ is the surjection given in (3.0.3), reduced modulo the conductor C. That is,

σh̄ = h̄τ. (3.1.1)

Since τ is an automorphism of X̄, we can write

τ = (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}, t_{7}) (3.1.2)

where t_{1}, ..., t_{5} ∈ ℤ̄_{16}^{∗}, t_{6} ∈ (ℤ[\sqrt{2}]/(4\sqrt{2}))^{∗}, and t_{7} ∈ (ℤ[α]/(2α))^{∗}. Combining (3.0.3) and (3.1.2) we obtain, for the right-hand side of (3.1.1),

h̄τ = (t_{1}, t_{2}, t_{3}, t_{4}, [\begin{matrix}t_{5} \\ t_{5} \end{matrix}] , [\begin{matrix}0 \\ t_{6} \end{matrix}] , [\begin{matrix}t_{7} \\ 0 \end{matrix}]). (3.1.3)

For the left-hand side of (3.1.1), note that σ is an automorphism of Δ̄, so, using Table 1, we can view the map σ as right multiplication by the element

σ = ∑_{i=1}^{19} n_{i}s_{i}

of Δ̄, for some integers n_{1}, ..., n_{19}, so that

σh̄ = ∑_{i=1}^{19} n_{i}s_{i}h̄. (3.1.4)

Comparing the first entries of (3.1.3) and (3.1.4), we get from the equation (3.1.1) that

t_{1} ≡ n_{1} (mod 16), (3.1.5)

(since, among s_{1}, ..., s_{19}, only s_{1} has a non-zero first entry). Similarly, comparing the second entries of (3.1.3) and (3.1.4), we get from equation (3.1.1) that

t_{2} ≡ n_{1} + 2n_{2} (mod 16), (3.1.6)

the third entries yield

t_{3} ≡ n_{1} + 2n_{3} (mod 16), (3.1.7)

and the fourth entries yield

t_{4} ≡ n_{1} + 2n_{2} + 2n_{3} + 4n_{4} (mod 16). (3.1.8)

The fifth entries of (3.1.3) and (3.1.4) are 2 × 1 matrices over ℤ (mod 16), so the equation (3.1.1) yields two congruences

t_{5} ≡ n_{1} + 2n_{2} + 2n_{3} + 2n_{5} + 2n_{6} (mod 16) and

t_{5} ≡ n_{1} + 2n_{2} + 4n_{3} + 4n_{4} + 2n_{5} + 2n_{6} + 4n_{7} + 4n_{8} (mod 16). (3.1.9)

Subtracting the first congruence from the second in (3.1.9) yields the congruence

0 ≡ 2n_{3} + 4n_{4} + 4n_{7} + 4n_{8} (mod 16). (3.1.10)

The sixth entries of (3.1.3) and (3.1.4) are 2 × 1 matrices over ℤ[\sqrt{2}] (mod 4\sqrt{2}), so the equation (3.1.1) yields two congruences, the first of which is

0 ≡ n_{2} + n_{3} + \sqrt{2}n_{5} + 2n_{11} + 2\sqrt{2}n_{12} (mod 4\sqrt{2}). (3.1.11)

The seventh entries of (3.1.3) and (3.1.4) are 2 × 1 matrices over ℤ[α] (mod 2α), so the equation (3.1.1) yields two congruences, the second of which is

\begin{matrix}0 ≡ (1 + \sqrt{2})n_{2} + (1 + \sqrt{2})n_{3} + (\sqrt{2} + α)n_{4} + αn_{5} + \sqrt{2}n_{7} + \\ α\sqrt{2}n_{9} + \sqrt{2}n_{11} + α\sqrt{2}n_{14} + 2n_{19} (mod 2α). \end{matrix} (3.1.12)

Then (3.1.10) implies that

n_{3} ≡ 0 (mod 2), (3.1.13)

so that (3.1.11) implies that

n_{2} ≡ n_{5} ≡ 0 (mod 2) (3.1.14)

(keeping in mind that n_{1}, ..., n_{19} are integers). In the ring ℤ[α], the ideal (2) = (\sqrt{2})^{2} = (α)^{4}, so combining, (3.1.13) and (3.1.14) with (3.1.12), we conclude that

n_{4} ≡ 0 (mod 2) (3.1.15)

as well.

Suppose that t_{1} ≡ 3 (mod 16) while t_{2} ≡ ±1 (mod 16) and t_{3} ≡ ±1 (mod 16). Then (3.1.5) says that n_{1} ≡ 3 (mod 16) also. So, since n_{2} is even (by (3.1.14)), we get from (3.1.6) that 2n_{2} ≡ 12 (mod 16) (and t_{2} ≡ −1 (mod 16)). Similarly, since n_{3} is even (by (3.1.13), we get from (3.1.7) that 2n_{3} ≡ 12 (mod 16) (and t_{3} ≡ −1 (mod 16)). Finally, since n_{4} is even (by (3.1.15)), we get from (3.1.8) that t_{4} ≡ 3 + 12 + 12 + 0 ≡ 11 (mod 16) or t_{4} ≡ 3 + 12 + 12 + 8 ≡ 3 (mod 16); in particular, t_{4} ≢ ±1 (mod 16).

By a similar argument, if t_{1} ≡ −3 (mod 16) while t_{2} ≡ ±1 (mod 16) and t_{3} ≡ ±1 (mod 16), then t_{4} ≢ ±1 (mod 16). But we showed that each of the first five coordinates of the quotient E^{∗}(X̄)/E^{∗}(X) has the form ℤ̄_{16}^{∗}/{±1}. Therefore, it follows that the automorphism (3, 1, 1, 1, 1, 1, 1) ∈ E^{∗}(X̄) cannot be in the same double coset as the identity automorphism (1, 1, 1, 1, 1, 1, 1) ∈ E^{∗}(X̄).

As an immediate consequence of Proposition 3.1 and Theorem 1.3, we obtain the following.

Theorem 3.2. Torsion free cancellation fails for ℤD_{32}.

Corollary 3.3. The torsion free Krull-Schmidt-Azumaya Theorem fails for ℤD_{32}.

4. Remarks on ℤD_{16}

From Theorem 2.4 we get that TFKS (and hence TFC) holds for ℤD_{8} and of course also for ℤD_{4} (since ℤD_{4} is a homomorphic image of ℤD_{8}). From Theorem 3.2 we get that TFC (and hence TFKS) fails for ℤD_{32} and so also for ℤD_{2^{n}} for n > 5 (since then ℤD_{32} is a homomorphic image of ℤD_{2^{n}}). This leaves open the question as to whether or not TFC or TFKS holds for the integral group ring ℤD_{16}. Two remarks are in order.

First, the construction of the maximal overorder of ℤD_{32} in §3 shows that a maximal overorder of ℤD_{16} consists of the order Γ = ℤ × ℤ × ℤ × ℤ × M_{2×2}(ℤ) × M_{2×2}(ℤ[\sqrt{2}]). It is easy to show that TFKS holds for Γ, so that, by Theorems 1.3 and 1.5, TFC holds for ℤD_{16} if and only if TFKS holds for ℤD_{16} if and only if every restricted genus of ℤD_{16}-lattices consists of a single isomorphism class.

Second, we have generated numerous “small” ℤD_{16}-lattices by computer and, using Theorem 1.1, computed the restricted genus of each. In each case the restricted genus was trivial. This suggests that, if there is a ℤD_{16}-lattice with non-trivial restricted genus, it will be “large” (requiring at least three generators).

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Department of Mathematics, University of Georgia, Athens, Georgia 30602

E-mail address: phindman@alpha.math.uga.edu

Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431–0991

E-mail address: klingler@acc.fau.edu

Department of Mathematics, University of Toledo, Toledo, Ohio 43606-3390

E-mail address: codentha@math.utoledo.edu