IC/97/48

United Nations Educational Scientific and Cultural Organization

and

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

RANKS OF K_{n} AND G_{n} OF ORDERS AND GROUP RINGS OF FINITE GROUPS OVER INTEGERS IN NUMBER FIELDS

Aderemi O. Kuku

International Centre for Theoretical Physics, Trieste, Italy.

Introduction

In this paper, we continue our study of higher K-theory of orders and group rings of finite groups over integers in number fields.

In earlier papers, [7], [10], [11], [12], the author had proved that if R is the ring of integers in a number field F, and Λ any R-order in a semi-simple F -algebra Σ, then K_{n}(Λ), G_{n}(Λ) are finitely generated Abelian groups for all n ≥ 1.

We now obtain information about the ranks of these groups and show precisely that if Γ is a maximal order containing Λ, then rank K_{n}(Λ) = rankG_{n}(Λ) = rankK_{n}(Γ) = rankK_{n}(Σ) for all n ≥ 2. It then follows that if Λ = ZZG, then rank K_{n}(ZZG) = rankG_{n}(ZZG) = rankK_{n}(Γ) = rank K_{n}(QG).

Notation For any ring A, we write K_{n}(A) for the Quillen K-groups π_{n+1}(BQP (A)) = π_{n}(BGL(A)^{+}) where P (A) is the category of finitely generated projective A-modules. If A is Noetherian, we write G_{n}(A) for π_{n+1}(BQ\underline{M}(A)) where \underline{M}(A) is the category of finitely generated A-modules. If R is the ring of integers in a number field F, and Λ is an R-order in a semi-simple F -algebra Σ, we write SK_{n}(Λ) = Ker(K_{n}(Λ) → K_{n}(Σ)), SK_{n}(Λ̂\underline{p}) = Ker(K_{n}(Λ̂\underline{p}) → K_{n}(Σ̂\underline{p})) where Λ̂\underline{p} = R̂\underline{p}⊗\underline{p}Λ Σ̂\underline{p} = F̂\underline{p}⊗_{F} Σ are the completions of Λ and Σ respectively at a prime ideal \underline{p} of R.

1.1 In [7], [10], [11], [12] it was proved that if R is the ring of integers in a number field F, Λ any R-order in a semi-simple F -algebra Σ, then for all n ≥ 1, K_{n}(Λ), G_{n}(Λ) are finitely generated Abelian groups. Our main result on the ranks of these groups is the following.

Theorem 1.2 Let R be the ring of integers in a number field F, Λ any R-order in a semi-simple F -algebra Σ, Γ a maximal R-order containing Λ. Then for all n ≥ 2 rank K_{n}(Λ) = rankK_{n}(Γ) = rankG_{n}(Λ) = rank K_{n}(Σ).

The proof of 1.2 will be in several steps 1.3-1.5 below.

Theorem 1.3 Let R be the ring of integers in a number field F, Γ a maximal R-order in a semi-simple F -algebra Σ. Then the canonical map K_{n}(Γ) → K_{n}(Σ) has finite kernel and torsion cokernel for all n ≥ 2. Hence, rank K_{n}(Γ) = rankK_{n}(Σ).

Proof Since Γ, Σ are regular, we have K_{n}(Γ) ≃ G_{n}(Γ) and K_{n}(Σ) ≃ G_{n}(Σ). So, we show that G_{n}(Γ) → G_{n}(Σ) has finite kernel and cokernel. Now, SG_{n}(Γ) = SK_{n}(Γ) is finite for all n ≥ 1 (being finitely generated and torsion see [6], [9],[10]). Also the localisation sequence of Quillen yields

… → G_{n+1}(Γ) → G_{n+1}(Σ) → ⊕_{\underline{p}} G_{n}(Γ/\underline{p}Γ) → …

where \underline{p} runs through the prime ideals of R. Now, for n ≥ 1, each G_{n}(Γ/\underline{p}Γ) is finite since Γ/\underline{p}Γ) is finite. So, ⊕_{\underline{p}} G_{n}(Γ/\underline{p}Γ) is torsion. Hence G_{n+1}(Σ)/Im(G_{n+1}(Γ) is torsion, as required.

Theorem 1.4 Let R be the ring of integers in a number field F, Λ any R-order in a semi-simple F -algebra, Γ a maximal order containing Λ. Then, for all n ≥ 1, the map G_{n}(Γ) → G_{n}(Λ) induced by the functor \underline{M}(Γ) → \underline{M}(Λ) given by restriction of scalars has finite kernel and cokernel. Hence, rank G_{n}(Γ) = rankK_{n}(Γ) = rankG_{n}(Λ).

Proof There exists a non-zero element s ∈ R such that Λ ⊂ Γ ⊂ Λ[\frac{1}{s}). Let S = \{s^{i}\}, i ≥ 0. Then Λ_{S} = Λ ⊗_{R} R_{S} ≃ Γ ⊗_{R} R_{S} = Γ_{S}. We first show that for all n ≥ 1, α_{2n−1} : G_{2n−1}(Γ) → G_{2n−1}(Λ) has finite kernel and cokernel.

Consider the following commutative diagram of exact sequences:

… ⟶ G_{2n−1}(Γ/sΓ) ⟶^{β_{2n−1}} G_{2n−1}(Γ) ⟶^{ρ_{2n−1}} G_{2n−1}(Γ_{S}) ⟶ G_{2n−2}(Γ/sΓ) ⟶ …
↓  ↓α_{2n−1}  ↓δ  ↓
⟶ G_{2n−1}(Λ/sΛ) ⟶^{β′_{2n−1}} G_{2n−1}(Λ) ⟶^{ρ′_{2n−1}} G_{2n−1}(Λ_{S}) ⟶ G_{2n−2}(Λ/sΛ) ⟶ …
(I)

where δ is an isomorphism.

Since Γ/sΓ and Λ/sΛ are finite rings, we have that G_{2n−2}(Γ/sΓ) = G_{2n−2}(Λ/sΛ) = 0. From the middle commutative square of (I), we have an exact sequence 0 → ker α_{2n−1} → ρ_{2n−1} = Imβ_{2n−1}. Now, Imβ_{2n−1} is finite. Hence kerα_{2n−1} is finite. Also, kerρ′_{2n−1} = Imβ′_{2n−1} is finite. Hence coker α_{2n−1} is finite, as a homomorphic image of kerρ′_{2n−1}.

We next show that for all n ≥ 1, α_{2n} : G_{2n}(Γ) → G_{2n}(Λ) is injective with finite cokernel. Now consider the commutative diagram of exact sequences:

0 ⟶ G_{2n}(Γ) ⟶^{ρ_{2n}} G_{2n}(Γ_{S}) ⟶^{η_{2n}} G_{2n−1}(Γ/sΓ) ⟶ …
↓α_{2n}  ↓δ  ↓
0 ⟶ G_{2n}(Λ) ⟶^{ρ′_{2n}} G_{2n}(Λ_{S}) ⟶^{η′_{2n}} G_{2n−1}(Λ/sΛ) ⟶ …

(II)

where δ is an isomorphism.

¿From the exact sequence associated with the left commutative square, i.e. 0 → ker α_{2n} → ker ρ_{2n} → ker ρ′_{2n} → coker α_{2n} → … we have ker ρ_{2n} = 0 and hence that ker α_{2n} = 0.

Also Coker ρ_{2n} = Im η_{2n} is finite (since G_{2n−1}(Γ/sΓ) is finite). hence the result

Theorem 1.5 Let R be the ring of integers in a number field F, Λ any R-order in a semisimple F -algebra, Γ a maximal order containing Λ. Then for all n ≥ 1 the map K_{n}(Λ) → K_{n}(Γ) has finite kernel and cokernel. Hence rank K_{n}(Λ) = rank K_{n}(Γ).

To be able to prove 1.5, we first prove the following

Theorem 1.6 Let R be the ring of integers in a number field F, Λ any R-order in a semisimple F -algebra Σ, Γ a maximal order containing Λ. Then for all n ≥ 1 the map K_{n}(Λ) → K_{n}(Γ) (induced by the inclusion map Λ ↦ Γ), is an isomorphism mod torsion.

Proof First note that since every R-order is a ZZ-order, there exists a non-zero integer s such that Λ ⊂ Γ ⊂ Λ(\frac{1}{s}). Put \underline{q} = sΓ. Then we have a Cartesian square

\begin{matrix}Λ & ⟶ & Γ \\ ↓ & & ↓ \\ Λ/q & ⟶ & Γ/q \end{matrix}  (I)

Now, by tensoring (I) with ZZ(\frac{1}{s}), if we write A(\frac{1}{s}) for A ⊗ ZZ(\frac{1}{s}) for any Abelian group A, we have long exact Mayer-Vietoris sequence (see [2] or [18])

… K_{n+1}(Γ/\underline{q})(\frac{1}{s}) →^{ρ} K_{n}(Λ)(\frac{1}{s}) →^{η} K_{n}(Γ)(\frac{1}{s}) ⊕ K_{n}(Λ/\underline{q})(\frac{1}{s})

→^{α} K_{n}(Γ/\underline{q})(\frac{1}{s}) → K_{n−1}(Λ)(\frac{1}{s}) → …  (II)

Now Λ/\underline{q} and Γ/\underline{q} are finite rings and so K_{n}(Λ/\underline{q}) and K_{n}(Γ/\underline{q}) are finite groups (see [11] 1.1).

Now from the commutative diagram

\begin{matrix}K_{n}(Λ)(\frac{1}{s}) ⟶^{η} K_{n}(Γ)(\frac{1}{s}) ⊕ K_{n}(Λ/q)(\frac{1}{s}) \\ ↘ δ  ↙ α \\ K_{n}(Γ)(\frac{1}{s}) \end{matrix}  (III)

where α is the projection onto the first factor, we have an exact sequence

0 → kerη → kerδ → kerα → cokerη → cokerδ

→ cokerα → 0 .

We want to see that kerδ and coker δ are torsion.

Now ker η is torsion since kerη = Imρ (from II) and Imρ is torsion since K_{n+1}(Γ/\underline{q})(\frac{1}{s}) is torsion.

Also, kerα is tosion since K_{n}(Λ/\underline{q})(\frac{1}{s}) is torsion. Hence, kerδ is torsion.

Now cokerα = 0 and coker η is torsion since K_{n}(Λ/\underline{q})(\frac{1}{s}) is torsion. Hence from (III) cokerδ is torsion.

Since kerδ and cokerδ are torsion, it follows that K_{n}(Λ) → K_{n}(Γ) is an isomorphism mod torsion for all n ≥ 1.

Proof of 1.5 Let α_{n} : K_{n}(Λ) → K_{n}(Γ) denote the map. By 1.6, ker α_{n} is torsion. Also for all n ≥ 1, K_{n}(Λ) is finitely generated see [11] 2.1. Hence Kerα_{n} is finitely generated as a subgroup of K_{n}(Λ). Now, by 1.6, coker α_{n} is torsion. Also K_{n}(Γ) is finitely generated and K_{n}(λ) is finitely generated. Hence coker α_{r} is finitely generated, and so, coker α_{n} is finite. So rank K_{n}(Γ) = rankK_{n}(Λ).

As a fall out from the above, we now have the following result which also proves that SK_{n}(Λ) is finite for any R-order Λ.

Theorem 1.6 Let R be the ring of integers in a number field F, Λ any R-order in a semi-simple F -algebra Σ. Then the canonical map K_{n}(Λ) → K_{n}(Σ) has finite kernel and torsion cokernel.

Proof from the commutative diagram

K_{n}(Λ) ⟶^{β} K_{n}(Σ)

α ↘ ↙ γ

K_{n}(Γ)

we have an exact sequence

0 → kerα → SK_{n}(Λ) → SK_{n}(Γ) → \frac{K_{n}(Γ)}{ImK_{n}(Λ)} → \frac{K_{n}(Σ)}{ImK_{n}(Λ)} → \frac{K_{n}(Σ)}{Im(K_{n}(Γ)} → 0 .

Now, by 1.5 kerα is finite and by [8], SK_{n}(Γ) = SG_{n}(Γ) is finite for all n ≥ 1. Hence from the exact sequence (I) above, SK_{n}(Λ) is finite.

Also, by 1.5, \frac{K_{n}(Γ)}{ImK_{n}(Λ)} is finite, and by 1.3, \frac{K_{n}(Σ)}{Im(K_{n}Γ)} is torsion. Hence the result.

1.7 Remarks (i) The above results hold for Λ = RG where G is any finite group

(ii) The ranks of K_{n}(R) and K_{n}(F) are well known and are due to A. Borel (see [1] or [4]).

More precisely, let r_{1} be the number of embeddings of F in IR and r_{2} the number of distinct conjugate pairs of embeddings of F in ℂ with image not contained in IR. Then

rank K_{n}(F) = \{ \begin{matrix}1 & if & n = 0 \\ ∞ & if & n = 1 \\ 0 & if & n = 2k k > 0 \\ r_{1} + r_{2} & if & n = 4k + 1 \\ r_{2} & if & n = 4k + 3 \end{matrix}

rank K_{n}(R) = \{ \begin{matrix}1 & if & n = 0 \\ r_{1} + r_{2} − 1 & if & n = 1 \\ r_{1} + r_{2} & if & n = 4k + 1 \\ r_{2} & if & n = 4k + 3 \\ 0 & if & n = n = 2k k > 0 \end{matrix}

It then means that if Σ is a direct product of matrix algebras over fields and Γ is a maximal order in Σ, then rank K_{n}(Γ) = rank K_{n}(Σ) is completely determined since Σ = ΠM_{n_{i}}(F_{i}) and Γ = ∏ M_{n_{i}}(R_{i}) where R_{i} is the ring of integers in F_{i}. Also, by 1.2, this is equal to rank G_{n}(Λ) as well as rank K_{n}(Λ) if Λ is any R-order contained in Γ.

However, if Σ does not split, there exists a Galois extension E of F which splits Σ, in which case we can reduce the problem to that of computation of ranks of K_{n} of fields.

(iii) In [6], B. Jahren proves that if G is a finite group with r irreducible real representations, c of them of complex type, then for all n > 1, we have

rank K_{n}(ZZG) = \{ \begin{matrix}r & if & n ≡ 1(4) \\ c & if & n ≡ 3(4) \\ 0 & if & n is even . \end{matrix}

(iv) Even though we do not have finite generation results for K_{n}(Λ), G_{n}(Λ), of padic orders Λ, the following results (1.8) concerning finite kernel and cokernel are quite interesting.

Theorem 1.8 Let R be the ring of integers in a p-adic field F, Λ any R-order in a semisimple F -algebra Σ which is unramified over its centre, Γ a maximal R-order containing Λ. Then, for all n ≥ 2, (i) the canonical map K_{n}(Γ) → K_{n}(Σ) has finite kernel and cokernel; (ii) the canonical map G_{n}(Λ) → G_{n}(Σ) has finite kernel and cokernel (iii) α_{n} : G_{n}(Γ) → G_{n}(Λ) has finite kernel and cokernel where α_{n} is the map induced

by the functor \underline{M}(Γ) → \underline{M}(Λ) given by restriction of scalars.

Proof From the commutative diagram G_{n}(Γ) ⟶ G_{n}(Λ)

↘ β ↙ γ
K_{n}(Σ)

We have an exact sequence

0 → kerα → SK_{n}(Γ) → SG_{n}(Λ) ⟶^{δ} \frac{G_{n}(Λ)}{ImG_{n}(Γ)}
⟶^{η} \frac{K_{n}(Σ)}{Im(G_{n}(Γ)} → \frac{K_{n}(Σ)}{ImG_{n}(Λ)} → 0  (II)

Note that SK_{n}(Γ) is finite or zero ∀n ≥ 1 since Σ is unramified over its centre see [8]. Also SG_{n}(Λ) is finite or zero (see [8]). Also from (II) above, kerα is finite or zero ∀n ≥ 1 as a subgroup of SK_{n}(Γ).

Since by hypothesis Σ = ∏ M_{n_{i}}(F_{i}) for some p-adic fields F_{i}, then Γ = ∏ M_{n_{i}}(R_{i}) where R_{i} are the ring of integers of F_{i}. So, showing that \frac{K_{n}(Σ)}{ImG_{n}(Γ)} is finite reduces to showing that if R is the ring of integers of a p-adic field F, then \frac{K_{n}(F)}{ImK_{n}(R)} is finite for all n ≥ 2.

But this is clear from the localisation sequence

… K_{n+1}(F) → K_{n}(R) → K_{n}(F) → K_{n−1}(R/\underline{m}) → SK_{n−1}(R)

.

Hence \frac{K_{n}(F)}{ImK_{n}(R)} is finite or zero for all n ≥ 2. ¿From (II) above, \frac{K_{n}(Σ)}{ImG_{n}(Λ)} is finite as a homomorphic image of \frac{K_{n}(Σ)}{Im(G_{n}Γ)}. Also from (II), \frac{G_{n}(Λ)}{ImG_{n}(Γ)}/Kerη ⊆ Imη ⊆ \frac{K_{n}(Σ)}{Im(G_{n}(Γ))}. Now, kerη = Imδ is finite. Hence \frac{G_{n}(Λ)}{ImG_{n}(Γ)} is finite.

References

[1] A. Borel, “Stable real cohomology of arithmetic groups”, Ann. Scient. Ec. Norm. Sup. 4, Serie 7, 1974, 235-272.

[2] R. Charney, A Note on Excusion in K-Theory, Springer Lecture Notes 1046.

[3] A. Dress and A.O. Kuku, “The Cartan map for equivariant higher K-groups”, Communications in Algebra 9 (7) 727-746 (1981).

[4] D. Grayson, “A survey of K-theory of fields”, Contemporary Mathematics 83, 31-55, 1989.

[5] B. Jahren, “On the rational K-theory of group-rings of finite groups (preprint).

[6] A.O. Kuku, “Some finiteness theorems in the K-theory of orders in p-adic algebras”, J. London Math. Society (2) (13) 1976, 122-128.

[7] A.O. Kuku, “SK_{n} of orders and G_{n} of finite rings”, Springer Lecture Notes 551, 1976, 60-68.

[8] A.O. Kuku, “SG_{n} of orders and group-rings”, Math. Zeit. 165, 291-295 (1979).

[9] A.O. Kuku, “Higher algebraic K-theory of group-rings and orders in algebras over number fields”, Communications in Algebra 10 (8), 805-816 (1982).

[10] A.O. Kuku, “K-theory of group-rings of finite groups over maximal orders in division algebras”, J. Algebra 91, 1, 18-31 (1984).

[11] A.O. Kuku, “K_{n}, SK_{n} of integral group-rings and orders”, Contemporary Mathematics (55) 333-338 (1986).

[12] A.O. Kuku, “Some finiteness results in the higher K-theory of orders and grouprings”, Topology and its Applications (25) , 1987, 185-191.

[13] R. Oliver, Whitehead Groups of Finite Groups, Cambridge University Press (1988).

[14] D.G. Quillen, “Higher algebraic theory I”, Springer Lecture Notes 341.

[15] I. Renier and K. Roggenkamp, Integral Representations, Springer Lecture Notes 744, 1979.

[16] E. Spanier, Algebraic Topology, McGraw Hill.

[17] A. Suslin, Stability in Algebraic K-Theory, Springer Lecture Notes 966 (1982) 304-333.

[18] C. Weibel, “Mayer-Vieloris sequences and module structures on NK_{n}.