ON the TATE MODULE of a NUMBER FIELD II 1. Introduction

ON THE TATE MODULE OF A NUMBER FIELD II

SOOGIL SEO

Abstract. We generalize a result of Kuz’min on a Tate module of a number field k. For a fixed prime p, Kuz’min described the inverse limit of the p part of the p-ideal class groups over the cyclotomic Zp-extension in terms of the global and local universal norm groups of p-units. This result plays a crucial rule in studying arithmetic of p-adic invariants especially the generalized Gross conjecture. We extend his result to the S-ideal class group for any finite set S of primes of k. We prove it in a completely different way and apply it to study the properties of various universal norm groups of the S-units.

1. Introduction ∪ For a number field k and an odd prime p, let k∞ = n kn be the cyclotomic n Zp-extension of k with kn the unique subfield of k∞ of degree p over k. For m ≥ n, let Nm,n = Nkm/kn denote the norm map from km to kn and let Nn = Nn,0 denote the norm map from kn to the ground field k0 = k. H × ≥ Huniv For a subgroup n of kn , n 0, let k be the universal norm subgroup of H k defined as follows ∩ Huniv H k = Nn n n≥0 univ and let (Hn ⊗Z Zp) be the universal norm subgroup of Hk ⊗Z Zp is defined as follows ∩ univ (Hk ⊗Z Zp) = Nn(Hn ⊗Z Zp). n≥0 The natural question whether the norm functor commutes with intersections of subgroups k× was already given in many articles and is related with algebraic and arithmetic problems including the generalized Gross conjecture and the Leopoldt conjecture. See for example, Bertrandias and Payan [1], Coleman [4], Gras [5], Gross [6], Jaulent [9], Kolster [10], Kuz’min [11] and [12]. Many applications are mainly for the global units and the global S-units for a set S of finite primes of k. In spite that it is very simple to describe the problem, it is very difficult to find solutions even in simpler cases. In [15], [16] and [17], we try to give certain informations on these problems up to torsion, that is, the ranks of various groups. The difficulties occur in the infinite class field theory with its complicate idele topology over k∞/k. We will search for a criterion over which the two functors of the universal and the intersection commutes exactly for the global units and the global S-units of k. Firstly we will generalize a result of Kuz’min with a new and clear proof. Secondly we will apply this result to the question mentioned. For a finite cyclic extension K/k and for a finite set S of primes of k, let Uk(S) be the global S-units of k. Let S′ be the set of primes of K lying over each prime 1 2 SOOGIL SEO v ∈ S, S′ = {w|v ; v ∈ S}. ′ ′ We will often write UK (S) for the S -units UK (S ) of K. Let ∏ ∏ × × JK,S = Uv kv w∈ /S′ w∈S′ be the S-idele group where we identify UK (S) with a subgroup of JK,S via the diagonal imbedding ϕK,S : UK (S) −→ JK,S. For K = kn, we write

Un(S) = Ukn (S), ϕn = ϕkn,S. We have the following exact sequence −→ −→ −→ϕn b 0 1 ker(ϕn) Uk(S) H (Gn,Jkn,S) where Gn = G(kn/k) denotes the Galois group of kn/k. Then Hasse’s local-global norm theorem for k× and the fact that every global norm is a local norm imply that ⊂ ∩ × NnUn(S) ker(ϕn) = Uk(S) Nnkn . By tensoring with Zp, the exact sequence induces

−→ ⊗ Z −→ ⊗ Z −→ϕn b 0 1 ker(ϕn) = ker(ϕn) p Uk(S) p H (Gn,Jkn,S) together with ⊗ Z ⊂ ⊗ Z ∩ × ⊗ Z Nn(Un(S) p) ker(ϕn) p = (Uk(S) Nnkn ) p. We let ∩ ∩ ⊗ Z ker(ϕ∞(S)) = ker(ϕn) = (ker(ϕn) p) n≥0 n≥0 for the intersection over all layers. Let Γ denote the procyclic group G(k∞/k) and for each n ≥ 0, let Γn = n G(k∞/kn) be the unique subgroup of Γ with index p . Let S = Sn be a ﬁnite set of primes of kn. Let Cln be the ideal class group of kn and let

Cln(S) = Cln/⟨cl(v)⟩v∈S be the S-ideal class group of kn where cl(v) ∈ Cln is the ideal class containing v. We deﬁne the S-Tate module Tk(S) of k following Kuz’min as the inverse limit of Cln(S) with respect to the norm maps

Tk(S) = lim←− An(S) n where An(S) = Cln(S) ⊗ Zp. For the group In(S) of fractional S-ideals of kn, let ⊗ Z I∞(S) = lim←− im(In(S) p) be the inverse limit of the image im(In(S) ⊗ Zp) of (In(S) ⊗ Zp) with respect to the norm maps inside An(S) induced from the exact sequence

1 −→ Pn(S) ⊗ Zp −→ In(S) ⊗ Zp −→ An(S) −→ 1 where Pn(S) denotes the S-principal ideals of kn. The following theorem is a generalized version of Kuz’min’s result for any set S of primes of k. Hence when S is empty, then the Tate module can be described in terms of the global units. We will prove the main result in a totally different way ON THE TATE MODULE OF A NUMBER FIELD II 3 using genus theory for p-ideal class group via cohomology theory rather than the infinite class field theory as was used by Kuz’min. The advantage of our proof is that it describes clearly the isomorphism and it is easy to generalize for an arbitrary set S compared to Kuz’min’s proof. Let On(S) be the ring of S-integers of kn. For each generating class (cl(an) ⊗ ∈ Γ ∈ × − O 1)n Tk(S) and αn kn such that (γ 1)an = αn n(S), let × ⊗ Z univ ∩ ⊗ Z Γ ker(ϕ∞(S)) (k p) (Uk(S) p) ∞ → ψ (S): Tk(S) univ = univ (Uk(S) ⊗ Zp) (Uk(S) ⊗ Zp) be defined as ψ∞(S)((cl(an) ⊗ 1)) = Nn(αn) ⊗ 1 which is well defined modulo Nn(Un(S) ⊗ Zp) for all n. Theorem 1.1. Let k be a number field and let S be any finite set of primes of k. Then ψ∞(S) defines an isomorphism

Γ ∼ × univ Tk(S) = (k ⊗ Zp) ∩ (Uk(S) ⊗ Zp) ∞ → ψ (S): Γ univ . I∞(S) (Uk(S) ⊗ Zp)

Let Sk(p) be the set of primes of k dividing p. Since the universal norm group and the norm comparable group are identical over the compact subgroups, Theorem 1.1 can be written as the following corollary. Corollary 1.2. For a ﬁnite set S of primes of k, Γ ∼ × ⊗ Z univ ∩ ∩ ⊗ Z Tk(S) →= (k p) (Uk(S Sp) p) Γ univ I∞(S) (Uk(S ∩ Sp) ⊗ Zp) Γ Tk(S ∩ Sp) = Γ . I∞(S ∩ Sp) Γ We remark that the statement that for the set S of the primes dividing p, Tk(S) is ﬁnite is an equivalent version of the generalized Gross conjecture (see [6], [9] and [10]). Note that for a set S with S ⊃ Sk(p), it recovers the result of Kuz’min ∼ × ⊗ Z univ Γ →= (k p) Tk(Sp) univ . (Uk(Sp) ⊗ Zp)

When S ∩ Sp is the empty set ϕ, Corollary 1.2 contains information on the global units Uk since Uk(S ∩ Sp) is equal to Uk. By comparing these cases that S ∩ Sp = ϕ and that S ⊃ Sp, we have the following proposition. Proposition 1.3. Let S be a ﬁnite set of primes containing all the primes dividing p. If the primes in S are principal over all intermediate layers, then the identity map induces the following isomorphism × univ × univ (k ⊗ Zp) ∩ (Uk ⊗ Zp) ∼ (k ⊗ Zp) univ = univ . (Uk ⊗ Zp) (Uk(S) ⊗ Zp) In particular, it follows that univ univ (Uk(S) ⊗ Zp) ∩ (Uk ⊗ Zp) = (Uk ⊗ Zp) and univ ∩ univ Uk(S) Uk = Uk . 4 SOOGIL SEO

2. Proofs of the main results Let L/k be a finite Galois extension. For a finite set S of finite primes of k, let Ik(S),Pk(S) and Uk(S) be the group of S-ideals of k, the subgroup of principal S-ideals in Ik(S) and the S-units of k respectively. Similarly we define IL(S),PL(S) ′ ′ and UL(S) as the the group of S -ideals of L, the subgroup of principal S -ideals ′ ′ in IL(S) and the S -units of L respectively where S denotes the prime ideals of ′ L dividing the prime ideals of S. Let ClL(S) = IL(S)/PL(S) be the S -ideal class group of L and A (S) = Cl (S) ⊗ Z . L L ∪ p Z Over the cyclotomic p-extension k∞ = n kn of k, let

Tk(S) = lim←− An(S) the inverse limit of An(S) with respect to the norm maps which will be called the S-Tate module. For the group In(S) of fractional S-ideals of kn, let ⊗ Z I∞(S) = lim←− im(In(S) p) be the inverse limit of the image im(In(S) ⊗ Zp) of (In(S) ⊗ Zp) with respect to the norm maps inside An(S) induced from the exact sequence

1 −→ Pn(S) ⊗ Zp −→ In(S) ⊗ Zp −→ An(S) −→ 1 where Pn(S) denotes the S-principal ideals of kn. In his papers (see Proposition 1.1 of [15] and Proposition 7.5 of [11]), Kuz’min described the S-Tate module Tk(S) in terms of the local and global universal norm groups using infinite class field theory when S consists of the primes dividing p. As we mentioned in the introduction, we generalize this result of Kuz’min for any set S and prove it in a different and clear way. Theorem 2.1. Let k be a number field and S be a finite set of finite primes of k. Then ψ∞(S) defines an isomorphism

Γ ∼ × univ Tk(S) = (k ⊗ Zp) ∩ (Uk(S) ⊗ Zp) ∞ → ψ (S): Γ univ . I∞(S) (Uk(S) ⊗ Zp) Proof. We start with the following lemma.

Lemma 2.2. Let M be a Z-module. Then for Z-modules N1 and N2 of M, we have

(N1 ∩ N2) ⊗Z Zp = (N1 ⊗Z Zp) ∩ (N2 ⊗Z Zp).

Proof. Since Z is a commutative Noetherian ring, the Z-module Zp is ﬂat by (iii) of Theorem 3 of §3.4 of [3]. The lemma follows from Proposition 6 and Remark 1 of §2.7 of [3].

We will often identify two modules appearing as in Lemma 2.2. For a Z[Gn] sub- H × module n of kn , the following identiﬁcation will also be used

(NnHn) ⊗Z Zp = Nn(Hn ⊗Z Zp) which follows immediately from the deﬁnitions of the norm map and the tensor product. ON THE TATE MODULE OF A NUMBER FIELD II 5

× For a ﬁnite Galois extension L/k, it follows from PL(S) = L /Uk(S) and the ′ S -ideal class group ClL(S) = IL(S)/PL(S) that there exist exact sequences of G-modules G G G 1 (1) 1 −→ PL(S) −→ IL(S) −→ ClL(S) −→ H (G, PL(S)) −→ 1,

× G 1 (2) 1 −→ Uk(S) −→ k −→ PL(S) −→ H (G, UL(S)) −→ 1,

1 2 2 × (3) 1 −→ H (G, PL(S)) −→ H (G, UL(S)) −→ H (G, L ).

From the equation (1) and Clk(S) = Ik(S)/Pk(S), there exists the commutative diagram 1 −−−−→ P (S) −−−−→ I (S) −−−−→ Cl (S) −−−−→ 1 k k k    y y yιk,L

G G G 1 1 −−−−→ PL(S) −−−−→ IL(S) −−−−→ ClL(S) −−−−→ H (G, PL(S)) −−−−→ 1. Applying the snake lemma to the above diagram, we obtain 1 G (4) 1 −→ ker(ιk,L) −→ H (G, UL(S)) −→ IL(S) /Ik(S) 1 −→ coker(ιk,L) −→ H (G, PL(S)) −→ 1 where we used G ∼ 1 PL(S) /Pk(S) = H (G, UL(S)) from the equation (2). Moreover if G(L/k) = ⟨γ⟩ is cyclic, then from the equation (3), it follows that ∩ × 1 ∼ −1 ∼ Uk(S) NL/kL H (G, PL(S)) = H (G, PL(S)) = NL/kUL(S) where the isomorphism is the connecting homomorphism deﬁned by (α) = αOL(S) 7→ NL/k(α) for the ring OL(S) of S-integers of L. G G G By letting im(IL(S) ) the image of IL(S) inside ClL(S) , we have the following isomorphism G ∩ × ClL(S) ∼ Uk(S) NL/kL ψL : G = im(IL(S) ) NL/kUL(S) G where the isomorphism is deﬁned to be for each class cl(a) ∈ ClL(S) of a ∈ IL(S) × and α ∈ L such that (γ − 1)a = αOL(S)

ψL(cl(a)) = NL/k(α) which is well deﬁned modulo NL/kUL(S). Since all groups above are p-group, we have the following isomorphism of compact groups G ∩ × ⊗ Z ClL(S) ⊗Z Zp ∼ (Uk(S) NL/kL ) Z p G = . im(IL(S) ⊗Z Zp) NL/k(UL(S)) ⊗Z Zp It follows from the trivial isomorphisms G ∼ G ∼ ClL(S) ⊗Z Zp = (ClL(S) ⊗Z Zp) ,NL/k(UL(S)) ⊗Z Zp = NL/k(UL(S)) ⊗Z Zp) that G ∩ × ⊗ Z (ClL(S) ⊗Z Zp) ∼ (Uk(S) NL/kL ) Z p ψL : G = . im(IL(S) ⊗Z Zp) NL/k(UL(S) ⊗Z Zp) 6 SOOGIL SEO

G For a subﬁeld F of L over k and for each class cl(a) ∈ ClL(S) of a ∈ IL(S) such that (γ − 1)a = αOL(S), by applying the norm map NL/F , we have

(γ − 1)NL/F (a) = NL/F (γ − 1)a = NL/F (αOL(S)) = NL/F (α)OF (S) since the extensions k ⊂ F ⊂ L are abelian. Hence we have

ψF (NL/F cl(a)) = NF/k(NL/F (α)) = NL/k(α) = ψL(cl(a)) which induces the following commutative diagram

ψ Ω −−−−→L Θ L L   (5) yNL/F yid

ψF ΩF −−−−→ ΘF where Ω• and Θ• represent

G(F/k) × (Cl (S) ⊗Z Z ) (Uk(S) ∩ N F ) ⊗Z Zp Ω = F p , Θ = F/k F G(F/k) F im(IF (S) ⊗Z Zp) NF/k(UF (S) ⊗Z Zp)

Now let L vary all subﬁelds kn of the cyclotomic Zp-extension of k. For an inverse system of Galois modules {Mn}n≥0 of Gn = G(kn/k), it is obvious that 0 0 0 ←−lim H (Gn,Mn) = lim←− H (G∞,Mn) = H (G∞, ←−lim Mn). By taking the inverse limits in (5) with respect to the norm maps on the left and the inclusion maps on the right, ψkn induces the isomorphism ψ∞(S) = lim ψkn × ⊗ Z Gn ∩ ⊗ Z (Cln(S) Z p) ∼ (Uk(S) Nkn/kkn ) Z p (6) ψ∞(S) : lim → lim . ←− Gn ⊗ Z ←− ⊗ Z im(In(S) Z p) Nkn/k(Un(S) Z p) Since all the groups appearing above are compact, by taking the inverse limits which are exact over the following exact sequence

⊗ Z Gn G G (Cln(S) Z p) −→ ⊗Z Z n −→ ⊗Z Z n −→ −→ 1 im(In(S)) p) (Cln(S) p) G 1 (im(In(S)) ⊗Z Zp) n it follows that

Gn ⊗ Z Γ (Cln(S) ⊗Z Zp) ∼ (lim←−(Cln(S) Z p)) (7) lim = ←− ⊗ Z Gn ⊗ Z Γ (im(In(S)) Z p) (lim←− (im(In(S)) Z p) and similarly that × × ∩ ⊗Z Z lim(U (S) ∩ N k ) ⊗Z Z (Uk(S) Nkn/kkn ) p ∼ ←− k kn/k n p (8) lim = . ←− N (U (S) ⊗Z Z ) lim N (U (S) ⊗Z Z ) kn/k n p ←− kn/k n p It follows from (7) and (8) that

Γ × univ Tk(S) ∼ ker(ϕ∞(S)) (k ⊗ Zp) ∩ (Uk(S) ⊗ Zp) Γ = univ = univ . I∞(S) (Uk(S) ⊗ Zp) (Uk(S) ⊗ Zp) This completes the proof of Theorem 2.1.

For an ideal a of Ikn (S), let cl(a) denote the ideal class containing a inside

Clkn (S). Let Sk(p) be the set of primes of k dividing p. ON THE TATE MODULE OF A NUMBER FIELD II 7

Lemma 2.3. Let S be a finite set of finite primes of k. Then ∏ Γ ⟨ ⟩ I∞(S) = ←−lim cl( pn) c pn∈Sk(p)∩S where the limit is taken over all the primes dividing p which are not in S. In special, if either S contains all the primes dividing p or the primes dividing p are principal, then Γ I∞(S) = 0. Proof. Note that each prime which is prime to p is unramified over the cyclotomic Zp-extension k∞/k. Suppose that primes lying over p are contained in S. Then if a ∈ I (S)Gn then for each prime ideal P lying over a prime p dividing a, we have n ∏ P | a P|p since the Galois group acts transitively on In(S) and hence ∏ ∑ e(P)f(P) pn | Nkn/k( P) = p = p Nkn/ka P|p where e(P)(= 1) and f(P) represent the ramification index and respectively the residue class degree of P over p. It follows from the definition of the norm map over the ideal class group that

Gn ←−lim im(In(S) = 0 where the inverse limit is taken with respect to the norm maps. From Lemma 2.3, we may use the following notational convention ∏ Γ ∩ c ⟨ ⟩ I∞(S) = (Sk(p) S )∞ = ←−lim cl( pn) . c pn∈Sk(p)∩S For subgroups H of k× which are finite type over Z, let lim (H ⊗Z ) be the n n ←−n≥0 n p inverse limit of Hn ⊗ Zp with respect to the norm maps. Let π denote the natural projection H ⊗ Z → H ⊗ Z π :←− lim( n p) k p. n≥0 comp If we define the norm compatible subgroup (Hk ⊗ Zp) of Hk ⊗ Zp to be H ⊗ Z comp H ⊗ Z ( k p) = π(lim←−( n p)). n≥0 The following lemma is a direct consequence of the basic properties of the inverse limit over the compact groups (cf. [2], [14] and [18]). H × Z Lemma 2.4. For subgroups n of kn which are finite type over , univ comp (Hk ⊗ Zp) = (Hk ⊗ Zp) .

It is well known that over the cyclotomic Zp-extension k∞ of k, all primes which are prime to p are unramified. Moreover all primes of k are finitely decomposed in k∞. Hence the residue class degree fv at each prime v of k which is prime to p is infinite over k∞/k. This shows that if an element α of ∩ ∩ × ⊗ Z univ × ⊗ Z × ⊗ Z (k p) = (Nn(kn p)) = (Nn(kn ) p) n n 8 SOOGIL SEO is divisible by a prime q of k, then α is infinitely p-divisible at q unless q | p. Hence α is divisible only by primes dividing p. The same argument will show that k×univ is contained in the p-units Uk(Sp) of k. This leads to the following lemma.

×univ × univ Lemma 2.5. k ⊂ Uk(Sp), (k ⊗ Zp) ⊂ Uk(Sp) ⊗ Zp. By Lemmas 2.4 and 2.5, we have the following corollary. Corollary 2.6. For a ﬁnite set S of primes of k, univ univ (Uk(S) ⊗ Zp) = (Uk(S ∩ Sp) ⊗ Zp) . Proof. It follows from Lemmas 2.4 and 2.5 that univ comp comp comp (Uk(S) ⊗ Zp) = (Uk(S) ⊗ Zp) = (Uk(S) ⊗ Zp) ∩ (Uk(Sp) ⊗ Zp) comp = (Uk(S ∩ Sp) ⊗ Zp) univ = (Uk(S ∩ Sp) ⊗ Zp) .

Theorem 2.1, Lemma 2.5 and Corollary 2.6 lead to the following corollary. Corollary 2.7. For a ﬁnite set S of primes of k, Γ ∼ × ⊗ Z univ ∩ ∩ ⊗ Z Tk(S) →= (k p) (Uk(S Sp) p) Γ univ I∞(S) (Uk(S ∩ Sp) ⊗ Zp) Γ Tk(S ∩ Sp) = Γ . I∞(S ∩ Sp)

Note that for a set S with S ⊃ Sp, it recovers the result of Kuz’min ∼ × ⊗ Z univ Γ →= (k p) Tk(S) univ . (Uk(Sp) ⊗ Zp)

Note also that for a set S with S ∩ Sp = ϕ, Uk(S ∩ Sp) is equal to the global units Uk of k and hence Γ ∼ × ⊗ Z univ ∩ ⊗ Z Tk(S) →= (k p) (Uk p) Γ univ . I∞(S) (Uk ⊗ Zp) Proposition 2.8. Let S be a ﬁnite set of primes containing all the primes dividing p. If the primes in S are principal over all intermediate layers, then the identity map induces the following isomorphism × univ × univ (k ⊗ Zp) ∩ (Uk ⊗ Zp) ∼ (k ⊗ Zp) univ = univ . (Uk ⊗ Zp) (Uk(S) ⊗ Zp) In particular, it follows that univ univ (Uk(S) ⊗ Zp) ∩ (Uk ⊗ Zp) = (Uk ⊗ Zp) and univ ∩ univ Uk(S) Uk = Uk . Proof. From Corollary 2.7 when S is the empty set ϕ, the assumption implies that ∼ × ⊗ Z univ ∩ ⊗ Z Γ = (k p) (Uk p) ∞ → ψ : Tk univ (Uk ⊗ Zp) ON THE TATE MODULE OF A NUMBER FIELD II 9 where ψ∞ = ψ∞(ϕ),Tk = Tk(ϕ),Uk = Uk(ϕ) and for a ﬁnite set S of primes containing all the primes dividing p, ∼ × ⊗ Z univ ∩ ⊗ Z Γ = (k p) (Uk(S) p) ∞ → ψ (S): Tk(S) univ . (Uk(S) ⊗ Zp) We have the following commutative diagram

ψ∞ T Γ −−−−→ W k    yπ yid

Γ ψ∞(S) Tk(S) −−−−→ WS where × univ × univ (k ⊗ Zp) ∩ (Uk ⊗ Zp) (k ⊗ Zp) ∩ (Uk(S) ⊗ Zp) W = univ , WS = univ (Uk ⊗ Zp) (Uk(S) ⊗ Zp) and π is the natural projection map which reduces to the identity map by the assumption. Note that all maps appearing in the diagram are isomorphic. From the commutative diagram, we have the ﬁrst claim of the proposition × univ × univ (k ⊗ Zp) ∩ (Uk ⊗ Zp) ∼ (k ⊗ Zp) univ = univ . (Uk ⊗ Zp) (Uk(S) ⊗ Zp) In particular, it follows that univ univ (Uk(S) ⊗ Zp) ∩ (Uk ⊗ Zp) = (Uk ⊗ Zp) .

Let T be a ﬁnite set of primes of k. Let Nn = Nkn/k be the norm map for kn/k. Write Nn(T ) = NnUn(T ). Then the inverse limit of Nn(T ) with respect to the inclusion maps is equal to N univ ←−lim n(T ) = Uk(T ) . n≥0

Similarly since Nn(T ) ⊗ Zp = (Nn Un(T )) ⊗ Zp = Nn (Un(T ) ⊗ Zp), the inverse limit of Nn(T ) ⊗ Zp with respect to the inclusion maps is equal to N ⊗ Z ⊗ Z univ ←−lim( n(T ) p) = (Uk(T ) p) . n≥0 ◦ × ∩ By letting Nnkn(T ) = Nnkn Uk(T ), it follows from the inclusions ◦ pn Uk(T ) ⊃ Nnkn(T ) ⊃ Nn(T ) ⊃ Uk(T ) ◦ that Nnkn(T ) /Nn(T ) is a p-group. It follows from Lemma 2.5 that ◦ univ ∩ ×univ ←−lim Nnkn(T ) = k Uk(T ) = k if T = S. n≥0 By taking the inverse limits to the following exact sequence ◦ ◦ Nnkn(T ) 1 −→ Nn(T ) −→ Nnkn(T ) −→ −→ 1 Nn(T ) we have the left exact sequence ◦ −→ univ −→ univ ∩ −→ Nnkn(T ) (9) 1 Uk(T ) k Uk(T ) ←−lim N . n≥0 n(T ) 10 SOOGIL SEO

◦ ⊗ Z × ⊗ Z ∩ ⊗ Z It follows from (Nnkn(T ) ) p = Nn(kn p) (Uk(T ) p) that ◦ ⊗ Z × ⊗ Z univ ∩ ⊗ Z ←−lim((Nnkn(T ) ) p) = (k p) (Uk(T ) p) × univ = (k ⊗ Zp) if T = S. By taking the inverse limits to the following exact sequence ◦ ◦ Nnkn(T ) 1 −→ (NnUn(T )) ⊗ Zp −→ (Nnkn(T ) ) ⊗ Zp −→ ( ) ⊗ Zp −→ 1 Nn(T ) we have the exact sequence ◦ −→ ⊗Z univ −→ ×⊗Z univ∩ ⊗Z −→ Nnkn(T ) ⊗Z −→ 1 (Uk(T ) p) (k p) (Uk(T ) p) ←−lim( p) 1 Nn(T ) ◦ ◦ where the inverse limit is exact since (Nnkn(T ) )⊗Zp is compact. Since Nnkn(T ) /Nn(T ) is a p-group, it follows from (9) and the isomorphism ◦ ◦ Nnkn(T ) ∼ Nnkn(T ) ⊗ Z ←−lim( N ) = ←−lim( N p) n≥0 n(T ) n≥0 n(T ) that kuniv ∩ U (T ) N k (T )◦ k ⊂ lim( n n ⊗ Z ). univ ←− N p Uk(T ) n≥0 n(T ) Hence there is an injective map univ ∩ × ⊗ Z univ ∩ ⊗ Z −→ k Uk(T ) −→ (k p) (Uk(T ) p) 1 univ univ . Uk(T ) (Uk(T ) ⊗ Zp) When T = ϕ, we can read oﬀ kuniv ∩ U (k× ⊗ Z )univ ∩ (U ⊗ Z ) 1 −→ k −→ p k p univ ⊗ Z univ Uk (Uk p) and when T = S ⊃ Sp, univ × ⊗ Z univ −→ k −→ (k p) 1 univ univ . Uk(S) (Uk(S) ⊗ Zp) This leads to the following commutative diagram 1 −−−−→ N −−−−→ W ϕ ϕ  ∼ y y=

1 −−−−→ NS −−−−→ WS where for a finite set T of finite primes of k or T = ϕ, univ × univ k ∩ Uk(T ) (k ⊗ Zp) ∩ (Uk(T ) ⊗ Zp) NT = univ , WT = univ . Uk(T ) (Uk(T ) ⊗ Zp) It follows from the first claim that the second vertical map is an isomorphism when the primes in S are principal. Hence the first vertical map is also injective. This leads to the second claim of the proposition univ ∩ univ Uk(S) Uk = Uk . This completes the proof of the proposition. ON THE TATE MODULE OF A NUMBER FIELD II 11

References

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Department of Mathematics, Yonsei University, 134 Sinchon-Dong, Seodaemun-Gu, Seoul 120-749, South Korea e-mail: [email protected]