Universal Norms of p-Units

in Some Non-Commutative Galois Extensions

dedicated to Professor John Coates on the occasion of his 60th birthday

Kazuya Kato

Received: September 27, 2005 Revised: June 30, 2006

2000 Mathematics Subject Classification: Primary 14M25; Secondary 14F20

1 Introduction.

Fix a prime number p. Let F be a finite extension of ℚ and let F_{∞} be an algebraic extension of F. We will consider the ℤ_{p}-submodule U(F_{∞}/F) of O_{F} [1/p]^{×} ⊗ ℤ_{p} defined by

U(F_{∞}/F) = Image(\underleftarrow{lim}_{L}(O_{L}[1/p]^{×} ⊗ ℤ_{p}) → O_{F} [1/p]^{×} ⊗ ℤ_{p}),

where L ranges over all finite extensions of F contained in F_{∞} and where the inverse limit is taken with respect to the norm maps. In the case F_{∞} is the cyclotomic ℤ_{p}-extension of F, the understanding of U(F_{∞}/F) is related to profound aspects in Iwasawa theory studied by Coates and other people, as we will shortly recall in §3. Concerning bigger Galois extensions F_{∞}/F, the following result is (essentially) contained in Corollary 3.23 of Coates and Sujatha [4] (see §3 of this paper).

Assume F_{∞}/F is a Galois extension and Gal(F_{∞}/F) is a commutative p-adic Lie group. Assume also that there is only one place of F lying over p. Then U(F_{∞}/F) is of finite index in O_{F} [1/p]^{×} ⊗ ℤ_{p}.

We ask what happens in the case of non-commutative Lie extensions. The purpose of this paper is to prove the following theorem, which was conjectured by Coates.

Theorem 1.1. Let a_{1}, ⋯, a_{r} ∈ F, and let

F_{n} = F (ζ_{p^{n}}, a_{1}^{1/p^{n}}, ⋯, a_{r}^{1/p^{n}}), F_{∞} = ∪_{n≥1}F_{n},

where ζ_{p^{n}} denotes a primitive p^{n}-th root of 1. Let F^{cyc} be the cyclotomic ℤ_{p}- extension of F. Then:

(1) The quotient group U(F^{cyc}/F)/U(F_{∞}/F) is finite.

(2) If there is only one place of F lying over p, then U(F_{∞}/F) is of finite index in O_{F} [1/p]^{×} ⊗ ℤ_{p}.

An interesting point in the proof is that we use the finiteness of the higher K-groups K_{2n}(O_{F}) for n ≥ 1, for this result on the muliplicative group K_{1}. The author does not have any result on \underleftarrow{lim}_{L} O_{F} [1/S]^{×} without ⊗ℤ_{p}. The plan of this paper is as follows. In §2, we review basic facts. In §3, we review some known results in the case F_{∞}/F is an abelian extension. In §4 and §5, we prove Theorem 1.1 (we will prove a slightly stronger result Theorem 5.1). The author expresses his hearty thanks to John Coates for suggesting this subject and for advice, and to Ramdorai Sujatha for advice and the hospitality in Tata Institute where a part of this work was done.

2 Basic facts.

We prepare basic facts related to U(F_{∞}/F). Most materials appear in Coates and Sujatha [4]. We principally follow their notation.

2.1. Let p be a prime number, and let F be a finite extension of ℚ. In the case p = 2, we assume F is totally imaginary, for simplicity. Let F_{∞} be a Galois extension of F such that the Galois group G = Gal(F_{∞}/F) is a p-adic Lie group and such that only finitely many finite places of F ramify in F_{∞}. Let ℤ_{p}[[G]] be the completed group ring of G, that is, the inverse limit of the group rings ℤ_{p}[G/U] where U ranges over all open subgroups of G.

2.2. We define ℤ_{p}[[G]]-modules

Z^{i}(F_{∞}) and Z_{S}^{i}(F_{∞}) (i ≥ 0)

where S is a finite set of finite places of F contaning all places of F lying over p. Let

Z_{S}^{i}(F_{∞}) = \underleftarrow{lim}_{L} H^{i}(O_{L}[1/S], ℤ_{p}(1))

where L ranges over all finite extensions of F contained in F_{∞}, O_{L}[1/S] denotes the subring of L consisting of all elements which are integral at any finite place of L not lying over S, and H^{i} is the étale cohomology. In the case S is the set of all places of F lying over p, we denote Z_{S}^{i}(F_{∞}) simply by Z^{i}(F_{∞}).

Universal Norms of p-Units … 553

Since

(1) H^{1}(O_{L}[1/S], ℤ_{p}(1)) ≃ O_{L}[1/S]^{×} ⊗ ℤ_{p}

by Kummer theory,

(2) Z_{S}^{1}(F_{∞}) ≃ \underleftarrow{lim}_{L}(O_{L}[1/S]^{×} ⊗ ℤ_{p}).

Note that H^{i}(O_{L}[1/S], ℤ_{p}(1)) are finitely generated ℤ_{p}-modules and Z^{i}(F_{∞}) are finitely generated ℤ_{p}[[G]]-modules. These modules are zero if i ≥ 3 for the reason of cohomological dimension (here in the case p = 2, we use our assumption F is totally imaginary).

2.3. Let U_{S}(F_{∞}/F) be the image of \underleftarrow{lim}_{L}(O_{L}[1/S]^{×} ⊗ ℤ_{p}) in O_{F} [1/S]^{×} ⊗ ℤ_{p}. Here L ranges over all finite extensions of F contained in F_{∞}. The main points of the preparation in this section are the isomorphisms (1b) and (2b) below.

(1) Assume S contains all finite places of F which ramify in F_{∞}. Then there are canonical isomorphisms

(1a) H_{0}(G, Z_{S}^{2}(F_{∞})) ≃ H^{2}(O_{F} [1/S], ℤ_{p}(1)),
(1b) H_{1}(G, Z_{S}^{2}(F_{∞})) ≃ (O_{F} [1/S]^{×} ⊗ ℤ_{p})/U_{S}(F_{∞}/F).

(2) Assume F_{∞} contains the cyclotomic ℤ_{p}-extension F^{cyc}. Then we have canonical isomorphisms

(2a) H_{0}(G, Z^{2}(F_{∞}/F)) ⊗_{ℤ_{p}} ℚ_{p} ≃ H^{2}(O_{F} [1/p], ℤ_{p}(1)) ⊗_{ℤ_{p}} ℚ_{p},
(2b) H_{1}(G, Z^{2}(F_{∞})) ⊗_{ℤ_{p}} ℚ_{p} ≃ (O_{F} [1/p]^{×} ⊗ ℤ_{p})/U(F_{∞}/F) ⊗_{ℤ_{p}} ℚ_{p}.

Here H_{m}(G,?) = Tor_{m}^{ℤ_{p}[[G]]}(ℤ_{p},?) denotes the G-homology. Note that H_{m}(G, M) are finitely generated ℤ_{p}-modules for any finitely generated ℤ_{p}[[G]]- module M.

(1a) and (1b) follow from the spectral sequence

E_{2}^{i,j} = H_{−i}(G, Z_{S}^{j}(F_{∞})) ⇒ E_{∞}^{i} = H^{i}(O_{F} [1/S], ℤ_{p}(1)),

the isomorphisms 2.2 (1) (2), and the fact Z_{S}^{j}(F_{∞}) = 0 for j ≥ 3. The above spectral sequence is given in [9] Proposition 8.4.8.3 in the case G is commutative. In general, we have the above spectral sequence by [6] 1.6.5 (3). The proofs of (2a) and (2b) are given in 2.6 later.

2.4. By Kummer theory and by the well known structure theorem of the Brauer group of a global field, we have an exact sequence

(1) 0 → Pic(O_{F} [1/S]){p} → H^{2}(O_{F} [1/S], ℤ_{p}(1)) → ⊕_{v∈S}ℤ_{p} ⟶^{sum} ℤ_{p} → 0,

where {p} denotes the p-primary part. Let

Y_{S}(F_{∞}) = \underleftarrow{lim}_{L} Pic(O_{L}[1/S])\{p\},

where L ranges over all finite extensions of F contained in F_{∞}. In the case S is the set of all places of F lying over p, we denote Y_{S}(F_{∞}) simply by Y (F_{∞}). Then the exact sequences (1) with F replaced by L give an exact sequence of ℤ_{p}[[G]]-modules

(2) 0 → Y_{S}(F_{∞}) → Z_{S}^{2}(F_{∞}) → ⊕_{v∈S}ℤ_{p}[[G]] ⊗_{ℤ_{p}[[G_{v}]]} ℤ_{p} → ℤ_{p} → 0

where for each v ∈ S, G_{v} ⊂ G is the decomposition group of a place of F_{∞} lying over v. If S contains all finite place of F which ramify in F_{∞}, the composite homomorphism

(3) (O_{F} [1/S]^{×} ⊗ ℤ_{p})/U(F_{∞}/F) ≃ H_{1}(G, Z_{S}^{2}(F_{∞}))
→ ⊕_{v∈S}H_{1}(G, ℤ_{p}[[G]] ⊗_{ℤ_{p}[[G_{v}]]} ℤ_{p}) = ⊕_{v∈S}H_{1}(G_{v}, ℤ_{p})

induced by (1b) and (2) coincides with the homomorphism induced by the reciprocity maps

F_{v}^{×} → G_{v}^{ab}(p) ≃ H_{1}(G_{v}, ℤ_{p})

of local class field theory, where G_{v}^{ab} denotes the abelian quotient of G_{v} and (p) means the pro-p part.

2.5. Assume F_{∞} ⊃ F^{cyc}. Then we have isomorphisms

Z^{1}(F_{∞}) →^{≃} Z_{S}^{1}(F_{∞}), Y (F_{∞}) →^{≃} Y_{S}(F_{∞}).

The first isomorphism shows U(F_{∞}/F) = U_{S}(F_{∞}/F). In fact, for each finite extension L of F contained in F_{∞}, we have an exact sequence

0 → O_{L}[1/p]^{×} ⊗ ℤ_{p} → O_{L}[1/S]^{×} ⊗ ℤ_{p} →
→ ⊕_{w}ℤ_{p} → Pic(O_{L}[1/p]){p} → Pic(O_{L}[1/S]){p} → 0

where w ranges over all places of L lying over S but not lying over p. If L′ is a finite extension of F such that L ⊂ L′ ⊂ F_{∞}, and if w′ is a place of L′ lying over w, the transition map from ℤ_{p} at w′ to ℤ_{p} at w is the multiplication by the degree of the residue extension of w′/w. Since the residue extension of v in F^{cyc}/F for v not lying over p is a ℤ_{p}-extension, this shows that the inverse limit of ⊕_{w}ℤ_{p} for varying L is zero. Hence we have the above isomorphisms.

2.6. We prove (2a) (2b) of 2.3. Take S containing all finite places of F which ramify in F_{∞}. Let T be the set of all elements of S which do not lie over p.

By 2.4 (2) and by Y (F_{∞}) →^{≃} Y_{S}(F_{∞}) in 2.5, we have an exact sequence of ℤ_{p}[[G]]-modules

0 → Z^{2}(F_{∞}) → Z_{S}^{2}(F_{∞}) → ⊕_{v∈T} ℤ_{p}[[G]] ⊗_{ℤ_{p}[[G_{v}]]} ℤ_{p} → 0.

This gives a long exact sequence

⋯ → H_{m}(G, Z^{2}(F_{∞})) → H_{m}(G, Z_{S}^{2}(F_{∞})) →
→ ⊕_{v∈T} H_{m}(G_{v}, ℤ_{p}) → H_{m−1}(G, Z^{2}(F_{∞})) → · · · .

Let G^{cyc} = Gal(F^{cyc}/F) and for v ∈ T, let G_{v}^{cyc} be the image of G_{v} in G^{cyc}. Then v is unramified in F^{cyc}/F, and we have a canonical isomorphism G_{v}^{cyc} ≃ ℤ_{p} which sends the Frobenius of v in G_{v}^{cyc} to 1 ∈ ℤ_{p}. Let H_{v} (v ∈ T) be the kernel of G_{v} → G_{v}^{cyc}. Since G is a p-adic Lie group and since the characteristic of the residue field of v is different from p, H_{v} is of dimension ≤ 1 as a p-adic Lie group. Furthermore, if H_{v} is infinite, for an element σ_{v} of G_{v} whose image in G_{v}^{cyc} is the Frobenius of v, the inner automorphism on H_{v} by σ_{v} is of infinite order as is seen from the usual description of the tame quotient of the absolute Galois group of F_{v}. These prove

(1) For v ∈ T, the kernel and the cokernel of the canonical map H_{m}(G_{v}, ℤ_{p}) → H_{m}(G_{v}^{cyc}, ℤ_{p}) are finite for any m.

Since the composition O_{F} [1/S]^{×} → H_{1}(G, Z_{S}^{2}(F_{∞})) → H_{1}(G_{v}^{cyc}, ℤ_{p}) = G_{v}^{cyc} ≃ ℤ_{p} for v ∈ T coincides with the v-adic valuation O_{F} [1/S]^{×} → ℤ, (1) shows that the cokernel of H_{1}(G, Z_{S}^{2}(F_{∞})) → ⊕_{v∈T} H_{1}(G_{v}, ℤ_{p}) is finite. Hence by the above long exact sequence, we have the following commutative diagram with exact rows in which the kernel of the first arrow of each row is finite.

H_{0}(G, Z^{2}(F_{∞})) → H_{0}(G, Z_{S}^{2}(F_{∞})) → ⊕_{v∈T} ℤ_{p} → 0
↓ ↓ ↓
H^{2}(O_{F} [1/p], ℤ_{p}(1)) → H^{2}(O_{F} [1/S], ℤ_{p}(1)) → ⊕_{v∈T} ℤ_{p} → 0

By this diagram and by 2.3 (1a), we have 2.3 (2a). We next prove 2.3 (2b). By the above (1), H_{2}(G_{v}, ℤ_{p}) is finite for v ∈ T. By this and by the case m = 1 of the above (1), we see that the complex 0 → H_{1}(G, Z^{2}(F_{∞})) → H_{1}(G, Z_{S}^{2}(F_{∞})) → ⊕_{v∈T} H_{1}(G_{v}^{cyc}, ℤ_{p}) has finite homology groups. By 2.3 (1b) and by U(F_{∞}/F) = U_{S}(F_{∞}/F) (2.5), the kernel of the last arrow of this complex is isomorphic to (O_{F} [1/p]^{×} ⊗ ℤ_{p})/U(F_{∞}/F). This proves 2.3 (2b).

3 Abelian extensions (Review).

In this section, we review the proof of the following result of Coates and Sujatha ([4] Cor. 3.23), and then recall some known facts on U(F^{cyc}/F).

Proposition 3.1. Assume F_{∞}/F is Galois and Gal(F_{∞}/F) is a commutative p-adic Lie group. Assume further that there is only one place of F lying over p. Then:

(1) U(F_{∞}/F) is of finite index in O_{F} [1/p]^{×} ⊗ ℤ_{p}.

(2) H_{m}(G, Y (F_{∞})) and H_{m}(G, Z^{2}(F_{∞})) are finite for any m.

In fact, this result was written in [4] in the situation Gal(F_{∞}/F) ≃ ℤ_{p}^{2}. This was because this result appeared in [4] in the study of the arithmetic of a ℤ_{p}^{2}- extension generated by p-power division points of an elliptic curve with complex multiplication. We just check here that the method of their proof works in this generality.

Proof. We may (and do) assume F_{∞} ⊃ F^{cyc}. In the case p = 2, to apply our preparation in §2, we assume F is totally imaginary without a loss of generality (we may replace F by a finite extension of F having only one place lying over p for the proof of 3.1).

(1) follows from the finiteness of H_{1}(G, Z^{2}(F_{∞})) in (2) by 2.3 (2b). We prove (2). We have H_{0}(G, Z^{2}(F_{∞})) ⊗_{ℤ_{p}} ℚ_{p} ≃ H^{2}(O_{F} [1/p], ℤ_{p}(1)) ⊗_{ℤ_{p}} ℚ_{p} by 2.3 (2a), and H^{2}(O_{F} [1/p], ℤ_{p}(1)) is finite by the exact sequence 2.4 (1) and by the assumption that there is only one place of F lying oer p. Hence H_{0}(G, Z^{2}(F_{∞})) ⊗_{ℤ_{p}} ℚ_{p} = 0. This shows that H_{m}(G, Z^{2}(F_{∞})) ⊗_{ℤ_{p}} ℚ_{p} = 0 for any m (Serre [11]). (Here the assumption G is commutative is essential. See 5.6.) This proves H_{m}(G, Z^{2}(F_{∞})) is finite for any m. Let v be the unique place of F lying over p. Then by class field theory, the decomposition group G_{v} of v in G is of finite index in G. By the exact sequence

H_{2}(G_{v}, ℤ_{p}) → H_{2}(G, ℤ_{p}) → H_{1}(G, Z^{2}(F_{∞})/Y (F_{∞})) → H_{1}(G_{v}, ℤ_{p}) → H_{1}(G, ℤ_{p})

obtained from 2.4 (2), this shows that H_{1}(G, Z^{2}(F_{∞})/Y (F_{∞})) and hence the kernel of H_{0}(G, Y (F_{∞})) → H_{0}(G, Z^{2}(F_{∞})) are finite. Hence H_{0}(G, Y (F_{∞})) is finite, and by Serre [11], H_{m}(G, Y (F_{∞})) is finite for any m.

3.2. In the rest of this section, we recall some known facts about U(F^{cyc}/F). Let G^{cyc} = Gal(F^{cyc}/F). For a place v of F lying over p, let G_{v}^{cyc} ⊂ G^{cyc} be the decomposition group of v (so G_{v}^{cyc} ≃ ℤ_{p}). Let (⊕_{v|p}G_{v}^{cyc})^{0} be the kernel of the canoncial map ⊕_{v|p}G_{v}^{cyc} → G^{cyc}. Let

α_{F} : (O_{F} [1/p]^{×} ⊗ ℤ_{p})/U(F^{cyc}/F) → (⊕_{v|p}G_{v}^{cyc})^{0}

be the homomorphism induced by the reciprocity maps of local fields F_{v}, which appeared in 2.4 (3). It is known that the following conditions (1) - (3) are equivalent.

(1) Ker (α_{F}) is finite. (That is, U(F^{cyc}/F) is of finite index in the kernel of O_{F} [1/p]^{×} ⊗ ℤ_{p} → (⊕_{v|p}G_{v}^{cyc})^{0}.)

(2) Coker (α_{F}) is finite.

(3) H_{0}(G^{cyc}, Y (F^{cyc})) is finite.

The equivalence of (1)-(3) is proved as follows. Though this is not at all an essential point, in the case p = 2, to apply our preparation in §2, we assume F is totally imaginary without a loss of generality (we can replace F by a finite extension of F for the proof of the equivalence). Let σ be a topological generator of G^{cyc}. Then H_{0}(G^{cyc}, Z^{2}(F^{cyc})) is isomorphic to the cokernel of σ − 1 : Z^{2}(F^{cyc}) → Z^{2}(F^{cyc}) and H_{1}(G^{cyc}, Z^{2}(F^{cyc})) is isomorphic to the kernel of it. Since Z^{2}(F^{cyc}) is a torsion ℤ_{p}[[G^{cyc}]]-module, this shows that the ℤ_{p}-rank of H_{1}(G^{cyc}, Z^{2}(F^{cyc})) ≃ (O_{F} [1/p] ⊗ ℤ_{p})/U(F^{cyc}/F) is equal to the ℤ_{p}-rank of H_{0}(G^{cyc}, Z^{2}(F^{cyc})) ≃ H^{2}(O_{F} [1/p], ℤ_{p}(1)) which is equal to the ℤ_{p}-rank of (⊕_{v|p}G_{v}^{cyc})^{0} by 2.4 (1). Hence (1) and (2) are equivalent. The exact sequence 2.4 (2) (take F_{∞} = F^{cyc} and S to be the set of all places of F lying over p) shows that Coker (α_{F}) is isomorphic to the kernel of H_{0}(G^{cyc}, Y (F^{cyc})) → H_{0}(G^{cyc}, Z^{2}(F^{cyc})) = H^{2}(O_{F} [1/p], ℤ_{p}(1)). The image of the last map is Pic(O_{F} [1/p]){p} by 2.4 (1) (2), and hence is finite. Hence Coker (α_{F}) is finite if and only if H_{0}(G^{cyc}, Y (F^{cyc})) is finite.

3.3. Greenberg [7] proved that H_{0}(G^{cyc}, Y (F^{cyc})) is finite if F is an abelian extension of ℚ (hence all (1) - (3) in 3.2 are satisfied in this case).

3.4. In the case F is totally real, by Coates [2] Theorem 1.13, H_{0}(G^{cyc}, Y (F^{cyc})) is finite if Leopoldt conjecture for F is true.

3.5. Let F be a CM field. Let F^{+} be the real part of F, and let H_{0}(G^{cyc}, Y (F^{cyc}))^{±} ⊂ H_{0}(G^{cyc}, Y (F^{cyc})) be the ±-part with respect to the action of the complex conjugation in Gal(F/F^{+}). Then by the above result of Coates, H_{0}(G^{cyc}, Y (F^{cyc}))^{+} is finite if Leopoldt conjecture for F^{+} is true. On the other hand, Conjecture 2.2 in Coates and Lichtenbaum [3] says that H_{0}(G^{cyc}, Y (F^{cyc}))^{−} is finite. In [8], Gross conjectured that the kernel and the cokernel of the (-)-part α_{F} of α_{F} is finite (this finiteness is also a consequence of Conjecture 2.2 of [3]), and formulated a conjecture which relates α_{F}^{−} to the leading terms of the Taylor expansions at s = 0 of p-adic Artin L-functions. Thus known conjectures support that the equivalent conditions (1) - (3) in 3.2 are satisfied by any CM field F. A natural question arises: Are (1) - (3) in 3.2 true for any number field F?

4 A result on Tor modules.

The purpose of this section is to prove Proposition 4.2 below.

4.1. For a compact p-adic Lie group G, for a ℤ_{p}[[G]]-module T, and for a continuous homomorphism G → ℤ_{p}^{×}, let T (χ) be the ℤ_{p}[[G]]-module whose underlying abelian group is that of T and on which ℤ_{p}[[G]] acts by ℤ_{p}[[G]] → ℤ_{p}[[G]] → End(T), where the first arrow is the automorphism σ ↦ χ(σ)σ (σ ∈ G) of the topological ring ℤ_{p}[[G]] and the second arrow is the original action of ℤ_{p}[[G]] on T. We call T (χ) the twist of T by χ.

Proposition 4.2. Let G be a compact p-adic Lie group, let H be a closed normal subgroup of G, and assume that we are given a finite family of closed normal subgroups H_{i} (0 ≤ i ≤ r) of G such that {1} = H_{0} ⊂ H_{1} ⊂ ⋯ ⊂ H_{r} = H, H_{i}/H_{i−1} ≃ ℤ_{p} for 1 ≤ i ≤ r and such that the the action of G on H_{i}/H_{i−1} by inner automorphisms is given by a homomorphism χ_{i} : G/H → ℤ_{p}^{×}. Let M be a finitely generated ℤ_{p}[[G]]-module, and let M′ be a subquotient of the ℤ_{p}[[G]]-module M. Let m ≥ 0. Then there is a finite family (S_{i})_{1≤i≤k} of ℤ_{p}[[G/H]]-submodules of H_{m}(H, M′) satisfying the following (i) and (ii).

(i) 0 = S_{0} ⊂ S_{1} ⊂ ⋯ ⊂ S_{k} = H_{m}(H, M′).

(ii) For each i (1 ≤ i ≤ k), there are a subquotient T of the ℤ_{p}[[G/H]]- module H_{0}(H, M) and a family (s(j))_{1≤j≤r} of non-negative integers s(j) such that ♯{j|s(j) > 0} ≥ m and such that S_{i}/S_{i−1} is isomorphic to the twist T (∏_{1≤j≤k} χ_{j}^{s(j)}) of T.

Note

H_{m}(H, M) = Tor_{m}^{ℤ_{p}[[H]]}(ℤ_{p}, M) = Tor_{m}^{ℤ_{p}[[G]]}(ℤ_{p}[[G/H]], M)

for ℤ_{p}[[G]]-modules M. A key point in the proof of Proposition 3.1 was that for commutative rings, Tor_{m} vanishes if Tor_{0} vanishes. This is not true for non-commutative rings. In the next section, we will use the above relation of Tor_{0} and Tor_{m} in a non-commutative situation for the proof of Theorem 1.1.

4.3. We denote this proposition with fixed r by (A_{r}). Let (B_{r}) be the case M = M′ of (A_{r}).

Since (B_{r}) is a special case of (A_{r}), (B_{r}) follows from (A_{r}). In 4.4, we show that conversely, (A_{r}) follows from (B_{r}). In 4.5, we prove (B_{1}). In 4.6, for r ≥ 1, we prove (B_{r}) assuming (A_{r−1}) and (B_{1}). These give a proof of Prop.4.2.

4.4. We can deduce (A_{r}) from (B_{r}) as follows. Let M′′ be the quotient of the ℤ_{p}[[G]]-module M such that M′ is a ℤ_{p}[[G]]-submodule of M′′. We have an exact sequence of ℤ_{p}[[G/H]]-modules

H_{m+1}(H, M′′/M′) → H_{m}(H, M′) → H_{m}(H, M′′).

Then (A_{r}) for the pair (M, M′) is obtained from (B_{r}) applied to M′′/M′ and to M′′ since H_{0}(H, M′′/M′) and H_{0}(H, M′′) are quotients of the ℤ_{p}[[G/H]]- module H_{0}(H, M).

4.5. We prove (B_{1}). Assume r = 1. Let χ = χ_{1}. Note that H ≃ ℤ_{p}. Let α be a topological generator of H, and let N = α − 1 ∈ ℤ_{p}[[G]]. Let I = Ker (ℤ_{p}[[G]] → ℤ_{p}[[G/H]]) = ℤ_{p}[[G]]N = Nℤ_{p}[[G]]. We have (1) For σ ∈ G, σNσ^{−1} is expressed as a power series in N with coefficients in ℤ_{p} which is congruent to χ(σ)N mod N^{2}. In particular, σNσ^{−1} ≡ χ(σ)N mod I^{2}.

In fact, σNσ^{−1} = α^{χ(σ)} − 1 = (1 + N)^{χ(σ)} − 1 = χ(σ)N + ∑_{n≥2} c_{i}N^{i} for some c_{i} ∈ ℤ_{p}. Concerning H_{m}(H, M) (m ≥ 0), we have:

(2) N(M) is a ℤ_{p}[[G]]-submodule of M, I kills M/N(M), and there is an isomorphism of ℤ_{p}[[G/H]]-modules

H_{0}(H, M) ≃ M/N(M).

(3) Ker (N : M → M) is a ℤ_{p}[[G]]-submodule of M, I kills Ker (N : M → M), and there is an isomorphism of ℤ_{p}[[G/H]]-modules

H_{1}(H, M) ≃ Ker (N : M → M)(χ).

(4) H_{m}(H, M) = 0 fo m ≥ 2.

We prove (2)–(4). We have a projective resolution

0 → I → ℤ_{p}[[G]] → ℤ_{p}[[G/H]] → 0

of the right ℤ_{p}[[G]]-module Z_{p}[[G/H]]. Since H_{m}(H,?) = Tor_{m}^{ℤ_{p}[[G]]}(ℤ_{p}[[G/H]],?), H_{0}(H, M) (resp. H_{1}(H, M)) is isomorphic to the cokernel (resp. kernel) of I ⊗_{ℤ_{p}[[G]]} M → M, and H_{m}(H, M) = 0 for all m ≥ 2. This proves (2) and (4). Furthermore,

H_{1}(H, M) ≃ Ker (I ⊗_{ℤ_{p}[[G]]} M → M) ≃ I ⊗_{ℤ_{p}[[G]]} Ker (N : M → M)
≃ I/I^{2} ⊗_{ℤ_{p}[[G/H]]} Ker (N : M → M).

Consider the bijection

Ker (N : M → M) → I/I^{2} ⊗_{ℤ_{p}[[G/H]]} Ker (N : M → M) ; x ↦ N ⊗ x.

By the above (1), for σ ∈ G, we have σN ⊗ x = χ(σ)Nσ ⊗ x = χ(σ)N ⊗ σx in I/I^{2} ⊗_{ℤ_{p}[[G/H]]} Ker (N : M → M). Hence

I/I^{2} ⊗_{ℤ_{p}[[G/H]]} Ker (N : M → M) ≃ Ker (N : M → M)(χ)

as ℤ_{p}[[G/H]]-modules. This proves (3). Let

V_{n} = Ker (N^{n} : M → M) (n ≥ 0), V = ∪_{n}V_{n}.

Then, since ℤ_{p}[[G]]N^{n} = N^{n}ℤ_{p}[[G]], V_{n} is a ℤ_{p}[[G]]-submodule of M. Since ℤ_{p}[[G]] is Noetherian and M is a finitely generated ℤ_{p}[[G]]-module, V = V_{n} for some n. That is, N is nilpotent on V. Since Ker (N : M/V → M/V) = 0, we have H_{1}(H, M/V) = 0 by (3). Hence

(5) H_{1}(H, V) = H_{1}(H, M),

(6) H_{0}(H, V) → H_{0}(H, M) is injective.

Consider the monodromy filtration (W_{i})_{i} on the abelian group V given by the nilpotent endomorphism N in the sense of Deligne [5] 1.6. It is an increasing filtration characterized by the properties N(W_{i}) ⊂ W_{i−2} for all i, and N^{i} : gr_{i}^{W} →^{≃} gr_{−i}^{W} for all i ≥ 0.

(7) W_{i} are ℤ_{p}[[G]]-submodules of V.

In fact, for σ ∈ G, the filtration (σW_{i})_{i} also has the characterizing property of (W_{i})_{i} by (1). Now we define an increasing filtration (W′_{i})_{i} of the ℤ_{p}[[G/H]]-module H_{0}(H, V) and an increasing filtration (W′′_{i})_{i} on the ℤ_{p}[[G/H]]-module H_{1}(H, V) = H_{1}(H, M) as follows. By identifying H_{0}(H, V) with Coker (N : V → V), let W′_{i} = W_{i}(Coker (N : V → V)) (i.e. the image of W_{i} in Coker (N : V → V)). By identifying H_{1}(H, V) with Ker (N : V → V)(χ), let W′′_{i} = W_{i}(Ker (N : V → V))(χ) (i.e. (W_{i} ∩ Ker (N : V → V))(χ)). Then W′′_{0} = H_{1}(H, M), and W′′_{i} = 0 if i is sufficiently small. We prove:

(8) For any i ≥ 0,

gr_{−i}^{W′′} ≃ gr_{i}^{W′}(χ^{i+1})

as ℤ_{p}[[G/H]]-modules.

By the injectivity of H_{0}(H, V) → H_{0}(H, M) (6), this proves (B_{1}).

We prove (8). By (1), we have

(9) The map N : gr_{i}^{W} → gr_{i−2}^{W} satisfies σNσ^{−1} = χ(σ)N for σ ∈ G.

Let P_{i} ⊂ gr_{i}^{W} (i ≤ 0) be the primitive part Ker (N : gr_{i}^{W} → gr_{i−2}^{W}) ([5] 1.6.3). Then for i ≥ 0, the canonical map gr_{−i}^{W}(Ker (N : V → V)) → P_{−i} is an isomorphism of ℤ_{p}[[G/H]]-modules ([5] 1.6.6). Furthermore, we have a bijection P_{−i} →^{≃} gr_{i}^{W} (Coker (N : V → V)) as the composition

P_{−i} → gr_{−i}^{W} ←^{N^{i}} gr_{i}^{W} → gr_{i}^{W} (Coker (N : V → V))

([5] 1.6.4, 1.6.6, and the dual statement of 1.6.6 for Coker (N)). By (9), this gives an isomorphism of ℤ_{p}[[G/H]]-modules P_{−i} ≃ gr_{i}^{W} (Coker (N : V → V))(χ^{i}). Hence we have (8).

4.6. Let r ≥ 1. We prove (B_{r}) assuming (A_{r−1}) and (B_{1}). Let J = H_{1}. By the spectral sequence

E_{2}^{−i,−j} = H_{i}(H/J, H_{j}(J, M)) ⇒ E_{∞}^{−m} = H_{m}(H, M)

in which H_{j}(J, M) = 0 for j ≥ 2, we have an exact sequence of ℤ_{p}[[G/H]]- modules

(1) H_{m−1}(H/J, H_{1}(J, M)) → H_{m}(H, M) → H_{m}(H/J, H_{0}(J, M)).

We consider H_{m−1}(H/J, H_{1}(J, M)) first. By (B_{1}) applied to the triple (G, J, M), H_{1}(J, M) is a successive extension of twists of subquotients of H_{0}(J, M) by χ_{1}^{i} (i ≥ 1). By (A_{r−1}) applied the triple (G/J, H/J, H_{0}(J, M)), H_{m−1}(H/J,?) of these subquotients of H_{0}(J, M) are successive extensions of twists of subquotients of H_{0}(H/J, H_{0}(J, M)) = H_{0}(H, M) by ∏_{2≤j≤r} χ_{j}^{s(j)} such that s(j) ≥ 0 for all j and such that ♯({j | s(j) > 0} ≥ m − 1. Hence H_{m−1}(H/J, H_{1}(J, M)) is a successive extension of twists of subquotients of H_{0}(H, M) by ∏_{1≤j≤r} χ_{j}^{s(j)} such that s(j) ≥ 0 for all j and such that ♯({j | s(j) > 0} ≥ m. We consider H_{m}(H/J, H_{0}(J, M)) next. By (B_{r−1}) (which is assumed since we assume (A_{r−1})) applied to the triple (G/J, H/J, H_{0}(J, M)), H_{m}(H/J, H_{0}(J, M)) is a successive extension of twists of subquotients of H_{0}(H/J, H_{0}(J, M)) = H_{0}(H, M) by ∏_{2≤j≤r} χ_{i}^{s(j)} such that s(j) ≥ 0 for all j and such that ♯({j | s(j) > 0} ≥ m. By these properties of H_{m−1}(H/J, H_{1}(J, M)) and H_{m}(H/J, H_{0}(J, M)), the exact sequence (1) proves (B_{r}) (assuming (A_{r−1}) and (B_{1})).

5 Some non-commutative Galois extensions.

Theorem 1.1 in Introduction is contained in Corollary 5.2 of the following Theorem 5.1, for the extension F_{∞}/F in Theorem 1.1 satisfies the assumption of Theorem 5.1 with n(i) = 1 for all i.

Theorem 5.1. Assume that F_{∞} is a Galois extension of F, F_{∞} ⊃ ∪_{n}F (ζ_{p^{n}}), and that there is a finite family of closed normal subgroups H_{i} (1 ≤ i ≤ r) of G = Gal(F_{∞}/F) satisfying the following condition. Let F^{cyc} be the cyclotomic ℤ_{p}-extension of F and let H be the kernel of G → G^{cyc} = Gal(F^{cyc}/F). Then {1} = H_{0} ⊂ H_{1} ⊂ ⋯ ⊂ H_{r}, H_{r} is an open subgroup of H, and for 1 ≤ i ≤ r, H_{i}/H_{i−1} ≃ ℤ_{p} and the action of G on it by inner automorphism is the n(i)-th power of the cyclotomic character G → ℤ_{p}^{×} for some positive integer n(i) > 0. Let S be any finite set of finite places of F containing all places lying over p. Then the kernel and the cokernel of the canonical maps

H_{m}(G, Z_{S}^{2}(F_{∞})) → H_{m}(G^{cyc}, Z_{S}^{2}(F^{cyc})),
H_{m}(G, Y (F_{∞})) → H_{m}(G^{cyc}, Y (F^{cyc}))

are finite for any m. In particular (since H_{m}(G^{cyc},?) = 0 for m ≥ 2), H_{m}(G, Z_{S}^{2}(F_{∞})) and H_{m}(G, Y (F_{∞})) are finite for any m ≥ 2.

Corollary 5.2. Let the assumption be as in Theorem 5.1. Then:

(1) The quotient group U(F^{cyc}/F)/U(F_{∞}/F) is finite.

(2) If there is only one place of F lying over p, then U(F_{∞}/F) is of finite index in O_{F} [1/p]^{×} ⊗ ℤ_{p}, and H_{m}(G, Y (F_{∞})) and H_{m}(G, Z^{2}(F_{∞})) are finite for any m.

(3) If F is an abelian extension over ℚ, then H_{m}(G, Y (F_{∞})) is finite for any m.

In fact, by 2.3 (2b), (1) of Corollary 5.2 follows from the finiteness of the kernel and the cokernel of H_{1}(G, Z^{2}(F_{∞})) → H_{1}(G^{cyc}, Z^{2}(F^{cyc})) which is a special case of Theorem 5.1. (2) follows from (1) and the case F_{∞} = F^{cyc} of Proposition 3.1. (3) follows from (1) and the result of Greenberg introduced in 3.3.

Corollary 5.3. Let the assumption be as in Theorem 5.1. Then H_{m}(G, Z^{1}(F_{∞})) for m ≥ 1 and the kernel of the canonical map H_{0}(G, Z^{1}(F_{∞})) → O_{F} [1/p]^{×} ⊗ ℤ_{p} are finite.

In fact, for S containing all finite places which ramify in F_{∞}, since Z^{1}(F_{∞}) →^{≃} Z_{S}^{1}(F_{∞}) (2.5), the spectral sequence in 2.3 shows that H_{m}(G, Z^{1}(F_{∞})) for m ≥ 1 is isomorphic to H_{m+2}(G, Z_{S}^{2}(F_{∞})), and the kernel of H_{0}(G, Z^{1}(F_{∞})) → O_{F} [1/p]^{×} ⊗ ℤ_{p} is isomorphic to H_{2}(G, Z_{S}^{2}(F_{∞})). Hence this corollary follows from the finiteness of H_{m}(G, Z_{S}^{2}(F_{∞})) for m ≥ 2 in Theorem 5.1.

5.4. We prove Theorem 5.1. First in this 5.4, we show that the kernel and the cokernel of H_{m}(G, Z_{S}^{2}(F_{∞})) → H_{m}(G^{cyc}, Z_{S}^{2}(F^{cyc})) are finite for any m assuming that S contains all finite places of F which ramify in F_{∞},. We may replace F by a finite extension of F. Hence we may assume that H_{r} = H, ∪_{n≥1}F (ζ_{p^{n}}) = F^{cyc}, and that in the case p = 2, F is totally imaginary. Let 𝔭 be the augmentation ideal of ℤ_{p}[[G^{cyc}]]. It is a prime ideal of ℤ_{p}[[G^{cyc}]]. By the spectral sequence E_{2}^{−i,−j} = H_{i}(G^{cyc}, H_{j}(H,?)) ⇒ E_{∞}^{−m} = H_{m}(G,?), it is sufficient to prove that H_{i}(G^{cyc}, H_{m}(H, Z_{S}^{2}(F_{∞}))) is finite for any i and for any m ≥ 1. For a finitely generated ℤ_{p}[[G^{cyc}]]-module M, H_{i}(G^{cyc}, M) is isomorphic to M/𝔭M if i = 0, to the part of M annihilated by 𝔭 if i = 1, and is zero if i ≥ 2. Applying this taking M = H_{m}(H, Z_{S}^{2}(F_{∞})), we see that it is sufficient to prove

(1) H_{m}(H, Z_{S}^{2}(F_{∞}))_{𝔭} = 0 for any m ≥ 1,

where (?)_{𝔭} denotes the localization at the prime ideal 𝔭. We apply Proposition 4.2 to the case M = M′ = Z_{S}^{2}(F_{∞}). By this proposition, to prove (1), it is sufficient to show that for any subquotient T of the ℤ_{p}[[G^{cyc}]]- module H_{0}(H, M) = Z_{S}^{2}(F^{cyc}) and for any integer k ≥ 1, we have T (k)_{𝔭} = 0. Here T (k) is the k-th Tate twist. It is sufficient to prove that H_{0}(G^{cyc}, T (k)) is finite. Since Z_{S}^{2}(F^{cyc}) is a finitely generated torsion ℤ_{p}[[G^{cyc}]]-module, the ℤ_{p}[[G^{cyc}]]-module T is a successive extension of ℤ_{p}[[G^{cyc}]]-modules which are either finite or isomorphic to ℤ_{p}[[G^{cyc}]]/𝔮 for some prime ideal 𝔮 of ℤ_{p}[[G^{cyc}]] of height one. We may assume T ≃ ℤ_{p}[[G^{cyc}]]/𝔮. Then there is a ℤ_{p}[[G^{cyc}]]- homomorphism Z_{S}^{2}(F^{cyc}) → T with finite cokernel. Hence it is sufficient to prove that H_{0}(G^{cyc}, Z^{2}(F^{cyc})(k))) is finite for any k ≥ 1. But

H_{0}(G^{cyc}, Z^{2}(F^{cyc})(k))) ≃ H^{2}(O_{F} [1/S], ℤ_{p}(k + 1)).

The last group is finite by Soulé [12]. In fact, by Quillen [10] and Borel [1], K_{2k}(O_{F} [1/S]) is finite, and by Soulé [12], we have a surjective Chern class map from K_{2k}(O_{F} [1/S]) to H^{2}(O_{F} [1/S], ℤ_{p}(k + 1)).

5.5. We complete the proof of Theorem 5.1. Let S be a finite set of finite places of F which contains all places of F lying over p. Take a finite set S′ of finite places of F such that S ⊂ S′ and such that S′ contains all finite places of F which ramify in F_{∞}. By comparing the exact sequence 2.4 (2) for F_{∞}/F and that for F^{cyc}/F, we see that the finiteness of the kernel and the cokernel of H_{m}(G, Z_{S}^{2}(F_{∞})) → H_{m}(G^{cyc}, Z_{S}^{2}(F^{cyc})) for all m and that of H_{m}(G, Y (F_{∞})) → H_{m}(G^{cyc}, Y (F^{cyc})) for all m are consequences of the following (1) - (3).

(1) The kernel and the cokernel of H_{m}(G, Z_{S′}^{2}(F_{∞})) → H_{m}(G^{cyc}, Z_{S′}^{2}(F^{cyc})) are finite for all m.

(2) The kernel and the cokernel of H_{m}(G, ℤ_{p}) → H_{m}(G^{cyc}, ℤ_{p}) are finite for all m.

(3) The kernel and the cokernel of H_{m}(G_{v}, ℤ_{p}) → H_{m}(G_{v}^{cyc}, ℤ_{p}) are finite for all m and for all finite places v of F. Here G_{v} ⊂ G denotes a decomposition group of a place of F_{∞} lying over v, and G_{v}^{cyc} denotes the image of G_{v} in G^{cyc}.

We proved (1) already in 5.4. (2) and (3) follow from the case M = M′ = ℤ_{p} of Proposition 4.2.

Remark 5.6. There is an example of a p-adic Lie extension F_{∞}/F for which there is only one place of F lying over p but U(F_{∞}/F) is not of finite index in O_{F} [1/p]^{×} ⊗ ℤ_{p}. For example, let F = ℚ, let E be an elliptic curve over F with good ordinary reduction at p, and let F_{∞} be the field generated over F by p^{n}-division points of E for all n. Then U(F_{∞}/F) = {1} and is not of finite index in O_{F} [1/p]^{×} ⊗ ℤ_{p} = ℤ[1/p]^{×} ⊗ ℤ_{p} ≃ ℤ_{p}. In fact U(F_{∞}/F) must be killed by the reciprocity map of local class field theory of ℚ_{p} into G_{p}^{ab}(p) ≃ ℤ_{p}^{2}, where G_{p} ⊂ G = Gal(F_{∞}/F) denotes the decomposition group at p, and G_{p}^{ab}(p) denotes the pro-p part of the abelian quotient of G_{p}. The image of p ∈ ℤ[1/p]^{×} in G_{p}^{ab}(p) is of infinite order. This proves U(F_{∞}/F) = {1}. In this case, H_{0}(G, Z^{2}(F_{∞})) is finite, but H_{1}(G, Z^{2}(F_{∞})) is not finite.

Remark 5.7. There is an example of a p-adic Lie extension F_{∞}/F for which G = Gal(F_{∞}/F) ≃ ℤ_{p}^{2} and H_{0}(G, Y (F_{∞}/F)) is not finite. Let K be an imaginary quadratic field in which p splits, let K_{∞} be the unique Galois extension of K such that Gal(K_{∞}/K) ≃ ℤ_{p}^{2}, let F be a finite extension of K in which p splits completely, and let F_{∞} = F K_{∞}. Then the ℤ_{p}-rank of H_{1}(G, Y (F_{∞})) is ≥ [F : K] − 1 as is shown below. Hence it is not zero if F ≠ K. In fact, from the exact sequence 2.4 (2) with S the set of all places of F lying over p, we can obtain

rank_{ℤ_{p}}H_{1}(G, Y (F_{∞})) ≥

≥ (∑_{v∈S} rank_{ℤ_{p}}H_{1}(G_{v}, ℤ_{p})) − rank_{ℤ_{p}}H_{1}(G, ℤ_{p}) − rank_{ℤ} O_{F} [1/p]^{×}.

But rank_{ℤ_{p}}H_{1}(G_{v}, ℤ_{p}) = 2 for any v ∈ S, rank_{ℤ_{p}}H_{1}(G, ℤ_{p}) = 2, rank_{ℤ}O_{F} [1/p]^{×} = 3[F : K]−1 by Dirichlet’s unit theorem, and hence the right hand side of the above inequality is 2[F : ℚ] − 2 − (3[F : K] − 1) = [F : K] − 1.

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Kazuya Kato

kazuya@kusm.kyoto-u.ac.jp 566

Documenta Mathematica · Extra Volume Coates (2006)