Divisible homology classes

in the special linear group of a number field

Dominique Arlettaz and Piotr Zelewski

Introduction

Let F be a number field and SL(F) denote the infinite special linear group over F . The integral homology groups of SL(F) are in general not finitely generated but, it was shown by the first author in Section 2 of [A1] that, for all integers i ≥ 0 , H_{i}(SL(F); ℤ) is the direct sum of a free abelian group of finite rank and a torsion group. The next interesting problem consists in understanding the structure of this torsion subgroup.

Recently, G. Banaszak looked at the corresponding question for the algebraic Ktheory of number fields: more precisely, he investigated the subgroup D(i) of divisible elements in K_{i}F (see [B1], Chapter VIII, and [B2], Chapter II). The localization exact sequence in algebraic K-theory (see Section 5 of [Q1], Theorem 8 of [Q3] and Théorème 1 of [S])

⋯ ⟶ K_{i}O ⟶^{r_{∗}} K_{i}F ⟶ ⨁_{m} K_{i−1}(O/m) ⟶ ⋯,

where O is the ring of algebraic integers in F , r_{∗} the homomorphism induced by the inclusion r : O ↪ F , and where m runs over the set of maximal ideals of O, implies that D(i) is a subgroup of the image of r_{∗} , since K_{i−1}(O/m) is trivial if i is odd and finite cyclic if i is even; moreover, it follows from the finite generation of the groups K_{i}O for i ≥ 0 [Q2] that

D(i) = 0 if i is odd and D(i) is a finite group if i is even.

For any prime number ℓ, let D(i)_{ℓ} denote the ℓ-torsion subgroup of D(i) (in other words, the subgroup of ℓ-divisible ℓ-torsion elements in K_{i}F ). For i = 2n, n odd, Banaszak deduced that D(2n)_{ℓ} is in general non-trivial; subsequently, together with Kolster, he obtained the following description (see [B2], Theorem 3): if F is totally real, n an odd positive integer and ℓ an odd prime, the order of D(2n)_{ℓ} is exactly given by the ℓ-adic absolute value of

The second author wishes to thank the Swiss National Science Foundation and the Leverhulme Foundation for financial support while this research was being carried out.

\frac{w_{n+1}(F) ζ_{F} (−n)}{∏_{ν|ℓ} w_{n}(F_{ν})} ,

where ζ_{F} (−) is the Dedekind zeta function of F , w_{n}(k) the biggest integer s such that the exponent of the galois group Gal(k(μ_{s})/k) divides n for a field k (here μ_{s} is an s-th primitive root of unity), and F_{ν} the completion of F at ν . For instance, if F is the field of rationals ℚ, n an odd integer and ℓ an odd prime, the order of D(2n)_{ℓ} is equal to the ℓ-adic absolute value of the numerator of \frac{B_{n+1}}{n + 1} , where B_{n+1} is the (n + 1)-st Bernoulli number. Notice that the knowledge of D(2n) is of particular interest since it is related to the Lichtenbaum-Quillen conjecture (see [B2], Section II.2) and to étale K-theory (see [BZ]).

The purpose of the present paper is to study the divisible elements in homology of the infinite special linear group of a number field. Denote by \overline{D}(i) the subgroup of divisible elements in H_{i}(SL(F); ℤ) , and for a prime ℓ, by \overline{D}(i)_{ℓ} the ℓ-torsion subgroup of \overline{D}(i) (observe that \overline{D}(i) is a torsion group because of the result of [A1] mentioned above). In the first section (see Theorem 1.1), we prove that

\overline{D}(i) is a finite group for any i ≥ 0 .

In Section 2, we use the fact that the group SL(F) has the same homology as the simply connected infinite loop space BSL(F)^{+} obtained by performing the plus construction on the classifying space of SL(F) and consider the Hurewicz homomorphism

h_{i} : K_{i}F ≅ π_{i}BSL(F)^{+} ⟶ H_{i}(BSL(F)^{+}; ℤ) ≅ H_{i}(SL(F); ℤ)

for i ≥ 2 . We concentrate our attention to its restriction h_{i} : D(i) → \overline{D}(i) for i = 2n and show the following assertion (see Corollary 2.5):

For any n ≥ 1 , h_{2n} : D(2n)_{ℓ} ⟶ \overline{D}(2n)_{ℓ} is a split injection if ℓ > n.

We also observe that, in general, there are elements in \overline{D}(i)_{ℓ} which do not belong to the image of h_{i} : K_{i}F → H_{i}(SL(F); ℤ) ; for example, \overline{D}(i)_{ℓ} may be non-trivial even if i is odd or if i = 2n with n even. The last section is devoted to the following vanishing result (see Theorem 3.1):

If N is a positive integer and ℓ a prime number > N such that D(2n)_{ℓ} = 0 for 1 ≤ n ≤ N , then \overline{D}(i)_{ℓ} = 0 for 1 ≤ i ≤ 2N .

Let us finally mention that the structure of the integral homology groups of the infinite general linear group GL(F) may be deduced from the knowledge of the integral homology of SL(F) by the Künneth formula, because of the homotopy equivalence BGL(F)^{+} ≃ BSL(F)^{+}×BF^{×} which follows from the fact that BSL(F)^{+} is the universal cover of BGL(F)^{+} (see [A1], proof of Corollary 9).

1. A finiteness theorem

The first result on the structure of the integral homology groups of SL(F) is given by Theorem 7 of [A1]: for any i ≥ 0 , H_{i}(SL(F); ℤ) is the direct sum of a free abelian group of finite rank and a torsion group. It has the following consequence on the integral cohomology of SL(F) : for any i ≥ 0 , H^{i}(SL(F); ℤ) contains no divisible elements except 0 (see [A1], Corollary 8). The purpose of this section is to investigate the subgroup \overline{D}(i) of divisible elements in H_{i}(SL(F); ℤ) .

Theorem 1.1. For any i ≥ 0 , \overline{D}(i) is a finite group.

Proof. According to [Q3], Theorem 4 or [Q1], Section 7, Proposition 3.2, there is a fibration

∏_{m} BQP (O/m) ⟶ BQP (O) ⟶ BQP (F) ,

where BQP (A) denotes the classifying space of the Q-construction over the category of finitely generated projective A-modules for a ring A, ∏ the weak product (i.e., the direct limit of cartesian products with finitely many factors), and where m runs over the set of maximal ideals of O. By looping its base space, we obtain the fibration (see [Q3], Theorem 1)

ΩBQP (F) ≃ BGL(F)^{+} × K_{0}F ⟶^{g} ∏_{m} BQP (O/m) ⟶ BQP (O) .

Since K_{i}O ≅ π_{i+1}BQP (O) is finitely generated for all i ≥ 0 [Q2], the homotopy exact sequence of this fibration shows that, for all i ≥ 1 , the map g induces a C-isomorphism g_{∗} : K_{i}F → π_{i}(∏_{m} BQP (O/m)) , where C is the Serre class of all finitely generated abelian groups. But, g may be lifted to a map on the universal covers:

f : BSL(F)^{+} ≃ B̃GL(F)^{+} ⟶ ∏_{m} B̃QP (O/m) ,

which induces again a C-isomorphism on π_{i} for all i ≥ 2 . Consequently, the induced homomorphism f_{♯} : H_{i}(BSL(F)^{+}; ℤ) → H_{i}(∏_{m} B̃QP (O/m); ℤ) is also a C-isomorphism for i ≥ 2 . But by the Künneth formula, H_{i}(∏_{m} B̃QP (O/m); ℤ) is a direct sum of finitely generated abelian groups , since the integral homology groups of B̃QP (O/m) are finitely generated for all m, and has therefore no divisible elements except 0 . This implies that \overline{D}(i) is contained in the kernel of f_{♯} , hence, it is finitely generated. Finally, \overline{D}(i) is finite because one deduces clearly from the structure of H_{i}(SL(F); ℤ) that \overline{D}(i) is a torsion group.

We shall check that \overline{D}(i) is in general non-trivial (see Corollary 2.6).

Remark 1.2. The same argument proves that the subgroup of divisible elements in H_{i}(Ω^{s}BSL(F)^{+}; ℤ) is also finite for all i ≥ 0 and s ≥ 0 .

2. The Hurewicz homomorphism

Denote by X_{F} the 1-connected Ω-spectrum whose 0-th space is the infinite loop space BSL(F)^{+} : the homotopy groups of X_{F} are the K-groups of F in dimensions ≥ 2 . This spectrum is of interest for algebraic K-theory because of the following result.

Theorem 2.1. For i ≥ 2 , the Hurewicz homomorphism h̃_{i} : K_{i}(F; ℤ_{(ℓ)}) → H_{i}(X_{F}; ℤ_{(ℓ)}) is an isomorphism if ℓ is a prime number > \frac{i+1}{2} .

Proof. Since the spectrum X_{F} is 1-connected, its Postnikov k-invariants k^{i+1}(X_{F}) are cohomology classes of finite order ρ_{i} for i ≥ 3 , and ρ_{i} is only divisible by primes p ≤ \frac{i+1}{2} (see [A3], Theorem 1.5). Now, let us write X_{F} [i] for the i-th Postnikov section of X_{F} (i.e., X_{F} [i] is a spectrum with π_{j}X_{F} [i] = 0 for j > i, π_{j}X_{F} ≅ π_{j}X_{F} [i] for j ≤ i), and for any prime number ℓ, (X_{F} [i])_{(ℓ)} for its localization at ℓ, which has the property that π_{j}(X_{F} [i])_{(ℓ)} ≅ (K_{j}F)_{(ℓ)} ≅ K_{j}(F; ℤ_{(ℓ)}) for j ≤ i. If ℓ > \frac{i+1}{2} , all k-invariants of (X_{F} [i])_{(ℓ)} are trivial and (X_{F} [i])_{(ℓ)} is a wedge of Eilenberg-MacLane spectra:

(X_{F} [i])_{(ℓ)} ≃ ⋁_{j=2}^{i} Σ^{j}H(K_{j}(F; ℤ_{(ℓ)}))

(for any abelian group G, H(G) denotes the Eilenberg-MacLane spectrum having all homotopy groups trivial except for G in dimension 0 ). Then, it is easy to compute

H_{i}(X_{F}; ℤ_{(ℓ)}) ≅ H_{i}(X_{F} [i]_{(ℓ)}; ℤ) ≅ ⨁_{j=2}^{i} H_{i}(Σ^{j}H(K_{j}(F; ℤ_{(ℓ)})); ℤ) .

But it follows from [C], Théorème 2 or [A3], Proposition 1.3 that H_{i}(Σ^{j}H(K_{j}(F; ℤ_{(ℓ)})); ℤ) is trivial if j < i < j +2ℓ−2 . Consequently, the condition i < 2ℓ−1 produces the desired assertion since H_{i}(X_{F}; ℤ_{(ℓ)}) ≅ H_{i}(Σ^{i}H(K_{i}(F; ℤ_{(ℓ)}), i); ℤ) ≅ K_{i}(F; ℤ_{(ℓ)}) .

Remark 2.2. From the theorem, it is true that K_{i}F and H_{i}(X_{F}; ℤ) have isomorphic subgroups of ℓ-torsion divisible elements if ℓ ≥ \frac{i+1}{2} . We shall prove in another paper that for any bounded below spectrum, the cokernel of the Hurewicz homomorphism is a group of finite exponent. Consequently, all divisible elements in H_{i}(X_{F}; ℤ) belong to the image of the Hurewicz homomorphism h̃_{i} : K_{i}F → H_{i}(X_{F}; ℤ) (but, may be, they are images of elements which are not divisible in K_{i}F ).

Remark 2.3. If we look at integers i ≥ 1 , we may also consider the Hurewicz homomorphism K_{i}(F; ℤ_{(p)}) → H_{i}(Y_{F}; ℤ_{(p)}) , where Y_{F} denotes the 0-connected Ω-spectrum whose 0-th space is BGL(F)^{+} : then, the conclusion of Theorem 2.1 holds for primes ℓ > \frac{i}{2} + 1 .

It is also useful to consider the Hurewicz homomorphism on the space level

h_{i} : K_{i}(F; ℤ_{(ℓ)}) ⟶ H_{i}(BSL(F)^{+}; ℤ_{(ℓ)})

for i ≥ 2 , and the commutative diagram

K_{i}(F; ℤ_{(ℓ)}) ≅ π_{i}(BSL(F)^{+}; ℤ_{(ℓ)}) ⟶^{h_{i}} H_{i}(BSL(F)^{+}; ℤ_{(ℓ)}) ≅ H_{i}(SL(F); ℤ_{(ℓ)})
↓ ≅  ↓ σ
K_{i}(F; ℤ_{(ℓ)}) ≅ π_{i}(X_{F}; ℤ_{(ℓ)})  \underrightarrow{_{h_{i}}}  H_{i}(X_{F}; ℤ_{(ℓ)}) ,

where σ denotes the iterated homology suspension. Thus, Theorem 2.1 has the following immediate consequence (see also [A2], Section 2).

Corollary 2.4. If i is a positive integer and ℓ a prime number > \frac{i+1}{2} , then the Hurewicz homomorphism h_{i} : K_{i}(F; ℤ_{(ℓ)}) → H_{i}(SL(F); ℤ_{(ℓ)}) is a split injection.

Since we know that D(i) = 0 for odd i’s, let us consider i = 2n and obtain the following splitting result.

Corollary 2.5. If n is a positive integer and ℓ a prime number > n, then the Hurewicz homomorphism h_{2n} : D(2n)_{ℓ} → \overline{D}(2n)_{ℓ} is a split injection.

Of course, if F is totally real, i = 2n an even integer with n odd and ℓ a prime > n, then Banaszak’s formula for the order of D(2n)_{ℓ} asserts that \overline{D}(2n)_{ℓ} is non-trivial for suitable n and ℓ. If F = ℚ for instance, D(2n)_{ℓ} is non-trivial if ℓ is an irregular prime and n an odd integer such that ℓ divides the numerator of \frac{B_{n+1}}{n + 1} . Actually, it

turns out that, in general, \overline{D}(i)_{ℓ} is bigger than D(i)_{ℓ} ( ℓ > \frac{i+1}{2} ).

Theorem 2.6. Let F be a totally real number field, i a positive integer and ℓ a prime number ≥ \frac{i+1}{2} . There are non-trivial ℓ-torsion divisible elements in H_{i}(SL(F); ℤ) which do not belong to the image of the Hurewicz homomorphism h_{i} : K_{i}F → H_{i}(SL(F); ℤ) . In particular, H_{i}(SL(F); ℤ) may contain non-trivial divisible elements even if i is odd or if i = 2n with n even.

Proof. Since ℓ is a prime > \frac{i+1}{2} , all k-invariants of the localized i-th Postnikov section (BSL(F)^{+}[i])_{(ℓ)} of BSL(F)^{+} are trivial since this is the case for the spectrum (X_{F} [i])_{(ℓ)} . Therefore, (BSL(F)^{+}[i])_{(ℓ)} is a product of Eilenberg-MacLane spaces:

(BSL(F)^{+}[i])_{(ℓ)} ≃ ∏_{j=2}^{i} K(K_{j}(F; ℤ_{(ℓ)}), j)

This homotopy equivalence and the Künneth formula provide a calculation of

H_{i}(SL(F); ℤ_{(ℓ)}) ≅ H_{i}((BSL(F)^{+}[i])_{(ℓ)}; ℤ) ≅ H_{i}(∏_{j=2}^{i} K(K_{j}(F; ℤ_{(ℓ)}), j); ℤ) :

this homology group has not only H_{i}(K(K_{i}(F; ℤ_{(ℓ)}), i); ℤ) ≅ K_{i}(F; ℤ_{(ℓ)}) as direct summand, but also mixed terms, for instance of the form

K_{2m}(F; ℤ_{(ℓ)}) ⊗ (K_{j_{1}}(F; ℤ_{(ℓ)}) ⊗ K_{j_{2}}(F; ℤ_{(ℓ)}) ⊗ ⋯ ⊗ K_{j_{s}}(F; ℤ_{(ℓ)})) ,

where 2m + j_{1} + j_{2} + ⋯ + j_{s} = i; however, the right hand side of this tensor product may include a free ℤ_{(ℓ)}-module if j_{1}, j_{2}, …, j_{s} are ≡ 1 (mod 4 ) and ≥ 5 (see [Bo]). If this occurs for m odd, then all elements of D(2m)_{ℓ} are divisible in the above mixed term. Consequently, \overline{D}(i)_{ℓ} contains not only D(i)_{ℓ} , but also D(2m)_{ℓ} for suitable choices of m ≤ \frac{i−5}{2} . This may happen even if i is odd or if i = 2n with n even.

Example 2.7. Take F = ℚ and ℓ = 691 . It is known that D(22)_{691} is non-trivial (see [B1], Section VIII.3) and that K_{j}ℚ/torsion is infinite cyclic if j ≡ 1 (mod 5 ) and ≥ 5 . The argument introduced in the previous proof exhibits for instance non-trivial elements in \overline{D}(27)_{691} , in \overline{D}(36)_{691}/D(36)_{691} , and in \overline{D}(66)_{691}/D(66)_{691} .

It is easy to deduce from Theorem 2.1 that the divisible elements detected by Theorem 2.6 vanish under σ.

Corollary 2.8. If i is a positive integer and ℓ a prime number > \frac{i+1}{2} , then the iterated homology suspension σ : H_{i}(SL(F); ℤ_{(ℓ)}) → H_{i}(X_{F}; ℤ_{(ℓ)}) satisfies σ(\overline{D}(i)_{ℓ}/D(i)_{ℓ}) = 0 .

Remark 2.9. As we mentioned in the introduction, all divisible elements in K_{i}F belong to the image of the homomorphism r_{∗} : K_{i}O → K_{i}F induced by the inclusion r : O ↪ F . If i is a positive integer and ℓ a prime > \frac{i+1}{2} , it follows obviously from Theorem 2.1 that the ℓ-torsion divisible elements in H_{i}(X_{F}; ℤ) are also elements of the image of the induced homomorphism r_{∗} : H_{i}(X_{O}; ℤ) → H_{i}(X_{F}; ℤ) . We do not know the answer of the following question: is \overline{D}(i)_{ℓ} contained in the image of r_{∗} : H_{i}(SL(O); ℤ) → H_{i}(SL(F); ℤ) ?

3. A vanishing theorem

The study of the Serre spectral sequence of the fibration

∏_{m} BQP (O/m) ⟶ BQP (O) ⟶ BQP (F)

(introduced in Section 1) shows that H_{i}(SL(F); ℤ) contains in general a lot of ℓ-torsion

elements for all primes ℓ. The goal of this section is to prove that for certain choices of the integer i and the prime ℓ, the group H_{i}(SL(F); ℤ) has no non-trivial ℓ-torsion divisible elements.

Theorem 3.1. If N is a positive integer and ℓ a prime number > N with the property that D(2n)_{ℓ} = 0 for all positive n ≤ N , then \overline{D}(i)_{ℓ} = 0 for all positive i ≤ 2N .

Proof. As in the proof of Theorem 2.6, the assumption ℓ > N provides a homotopy equivalence

(BSL(F)^{+}[2N])_{(ℓ)} ≃ ∏_{j=2}^{2N} K(K_{j}(F; ℤ_{(ℓ)}), j) .

According to [B2], Section II.1, Corollary 1, the vanishing of D(2n)_{ℓ} implies the splitting

K_{2n}(F; ℤ_{(ℓ)}) ≅ K_{2n}(O; ℤ_{(ℓ)}) ⊕ (⨁_{m} K_{2n−1}(O/m; ℤ_{(ℓ)})) .

Therefore, K_{2n}(F; ℤ_{(ℓ)}) is a direct sum of finitely generated ℤ_{(ℓ)}-modules and the same is true for H_{k}(K(K_{2n}(F; ℤ_{(ℓ)}), 2n); ℤ) , for all k ≥ 1 ( 2 ≤ 2n ≤ 2N ). On the other hand, K_{j}(F; ℤ_{(ℓ)}) is finitely generated if j is odd because of the localization exact sequence. We may finally conclude by the Künneth formula that, for i ≤ 2N ,

H_{i}(SL(F); ℤ_{(ℓ)}) ≅ H_{i}((BSL(F)^{+}[2N])_{(ℓ)}; ℤ) ≅ H_{i}(∏_{j=2}^{2N} K(K_{j}(F; ℤ_{(ℓ)}), j); ℤ)

is again a direct sum of finitely generated ℤ_{(ℓ)}-modules, and hence has no non-trivial ℓ-torsion divisible elements, since the ℓ-torsion subgroup of any finitely generated ℤ_{(ℓ)}- module is finite. In other words, we get \overline{D}(i)_{ℓ} = 0 for i ≤ 2N .

Remark 3.2. It is shown in [BG] that, for F = ℚ, the Kummer-Vandiver conjecture [W, p.157] holds if and only if D(2n)_{ℓ} = 0 for n even, ℓ odd. It is known (loc.cit.) that this conjecture holds for ℓ < 125′000 . Thus, the formula in the introduction for the order of D(2n)_{ℓ} , for n odd, makes it easy to check the hypothesis of Theorem 3.1 for ℓ < 125′000 and F = ℚ.

References

[A1] D. Arlettaz: On the homology of the special linear group over a number field, Comment. Math. Helv. 61 (1986), 556–564.

[A2] D. Arlettaz: The Hurewicz homomorphism in algebraic K-theory, J. Pure Appl. Algebra 71 (1991), 1–12.

[A3] D. Arlettaz: The order of the differentials in the Atiyah-Hirzebruch spectral sequence, K-Theory 6 (1992), 347–361.

[B1] G. Banaszak: Algebraic K-theory of number fields and rings of integers and the Stickelberger ideal, Ann. of Math. 135 (1992), 325–360.

[B2] G. Banaszak: Generalization of the Moore exact sequence and the wild kernel for higher K-groups, to appear in Compositio Math..

[BG] G. Banaszak and W. Gajda: Euler systems for higher K-groups, in preparation.

[BZ] G. Banaszak and P. Zelewski: Continuous K-theory, submitted to the Proceedings of the K-theory conference, Strasbourg 1992.

[Bo] A. Borel: Cohomologie réelle stable de groupes S-arithmétiques classiques, C. R. Acad. Sci. Paris Sér. A 274 (1972), 1700–1702.

[C] H. Cartan: Algèbres d’Eilenberg-MacLane et homotopie, exposé 11, Séminaire H. Cartan Ecole Norm. Sup. (1954/1955).

[Q1] D. Quillen: Higher algebraic K-theory I, in Higher K-theories, Lecture Notes in Math. 341 (Springer 1973), 85–147.

[Q2] D. Quillen: Finite generation of the groups K_{i} of rings of algebraic integers, in Higher K-theories, Lecture Notes in Math. 341 (Springer 1973), 179–198.

[Q3] D. Quillen: Higher K-theory for categories with exact sequences, in New Developments in Topology, London Math. Soc. Lecture Note Ser. 11 (Cambridge University Press 1974), 95–103.

[S] C. Soulé: Groupes de Chow et K-théorie des variétés sur un corps fini, Math. Ann. 268 (1984), 317–345.

[W] L. Washington: Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83 (Springer, 1982).

Dominique Arlettaz Institut de mathématiques Université de Lausanne CH–1015 Lausanne, Switzerland e-mail: dominique.arlettaz@ima.unil.ch Piotr Zelewski Department of Mathematics and Statistics University of Newcastle Newcastle upon Tyne NE1 7RU, U.K. e-mail: p.m.zelewski@newcastle.ac.uk