A NONCOMMUTATIVE WEIERSTRASS PREPARATION THEOREM AND APPLICATIONS TO IWASAWA THEORY

OTMAR VENJAKOB

Abstract. In this paper and a forthcoming joint one with Y. Hachimori [15] we study Iwasawa modules over an infinite Galois extension k_{∞} of a number field k whose Galois group G = G(k_{∞}/k) is isomorphic to the semidirect product of two copies of the p-adic numbers ℤ_{p} . After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group X_{nr} of the p-Hilbert class field over k_{∞}. As a main tool we prove a Weierstrass preparation theorem for certain skew power series rings. One striking result in our work is the discovery of the abundance of faithful torsion Λ(G)-modules, i.e. non-trivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra 𝔽_{p}[[G]] with coefficients in the finite field 𝔽_{p} is a unique factorization domain in the sense of Chatters [8].

1. Introduction

One of the highlights of noncommutative Iwasawa theory so far is the structure theory of Iwasawa modules due to Coates, Schneider and Sujatha. In [10] they proved that every torsion Λ(G)-module decomposes - up to pseudo-isomorphism - into the direct sum of cyclic modules, where Λ(G) = ℤ_{p}[[G]] denotes the Iwasawa algebra of a pro-p-group G belonging to a certain class of p-adic groups, viz the class of p-valued ones. In this paper we want to illustrate some features of their theory in the possibly easiest noncommutative example: we take as group the semi-direct product G = H ⋊ Γ of two copies of the p-adic integers ℤ_{p} ≅ H ≅ Γ with a non-trivial action of Γ on H.

Over a commutative ring, a finitely generated torsion module always has a nonzero annihilator in the ring (and special choices of such elements give rise to p-adic L-functions in classical Iwasawa theory), in module theory over a noncommutative ring Λ one distinguishes between faithful and bounded modules, i.e. (Λ-torsion) modules M whose annihilator ideal Ann(M) = {λ ϵ Λ|λm = 0 for all m ϵ M} is zero or non-trivial, respectively. We show that for the semidirect product G the Iwasawa algebra Λ = Λ(G) admits both types of modules. But what is most striking is that those modules which most naturally occur in arithmetic (see below as well as [15]), viz the Λ-modules which are finitely generated as Λ(H)-module, are all faithful (or pseudo-null). More precisely, for the images of such M in the quotient category Λ-mod/C of the category Λ-mod of finitely generated Λ-modules by the localizing subcategory C of pseudo-null modules even the following holds

Date: 06.03.02.

Key words and phrases. Iwasawa algebra, skew power series rings, Weierstrass preparation theorem, faithful modules, p-Hilbert class field.

During this research, the author has been supported by the EU Research Training Network “Arithmetical Algebraic Geometry” at the Department of Pure Mathematics and Mathematical Statistics, Cambridge.

Theorem (Theorem 6.3). The image in Λ-mod/C of a Λ-module M which is finitely generated as Λ(H)-module is a completely faithful, cyclic object.

Here an object M of the quotient category is called completely faithful if all its nonzero subquotient objects are faithful (for further details we refer the reader to section 6).

In section 8 we will consider p-adic Lie extensions k_{∞} over a number field k whose Galois group G = G(k_{∞}/k) is isomorphic to a semidirect product of the type studied above. Such examples arise by adjoining the p-power roots of an element α ϵ k which is not a root of unity to the cyclotomic ℤ_{p} -extension of k, assuming that the latter field contains the pth roots of unity μ_{p}, or μ_{4} if p = 2. If we take for example α = p and k = ℚ(μ_{p}), we shall show that the Galois group X_{nr} of the p-Hilbert class field gives rise to a completely faithful object.

At this opportunity we should mention a further result on the module X_{nr}. The remarkable duality between the inverse and direct limit of the p-primary ideal class groups in the tower k_{∞} of number fields which was observed by Iwasawa for the cyclotomic ℤ_{p} -extension - compare also with results of McCallum [21] or Nekovar [23] in the case of multiple ℤ_{p} -extensions - holds in the following (noncommutative) case

Theorem (Theorem 8.2). Let k = ℚ(μ_{p}) and k_{∞} = k(μ_{p^{∞}}, p^{p^{−∞}}). Then, there is an injective homomorphism of Λ(G)-modules

Cl(k_{∞})(p)^{∨} → E^{1}(X_{nr})

with pseudo-null cokernel, i.e. the Pontryagin dual of the direct limit of the p-ideal class groups is pseudo-isomorphic to the Iwasawa-adjoint of the inverse limit. In particular, if X_{nr} is pseudo-null, then the ideal class group Cl(k_{∞})(p) = 0 vanishes.

It is easy to see that X_{nr} is non-zero if and only if p is an irregular prime. We know no example at present where X_{nr} is not pseudo-null, and it may well be that is always the case. In a forthcoming paper [15] (with Y. Hachimori) we will prove that certain Selmer groups or rather their Pontryagin dual are indeed nonzero completely faithful objects in the quotient category. Thus faithful modules definitely occur in the world of arithmetic. Hence, in these cases, one cannot hope to obtain a link with L-functions via a global annihilator or a characteristic ideal, which simply does not exist for such completely faithful modules.

To obtain the algebraic results on modules over Λ(G), where now in general G is a uniform pro-p-group which is isomorphic to the semidirect product H ⋊ Γ of a uniform pro-p-group H with Γ ≅ ℤ_{p}, we proceed as follows: Firstly, we identify Λ(G) with the skew power series ring = R[[Y; σ, δ]] over the subalgebra R := Λ(H) of Λ(G). Secondly, we study in some generality skew power series rings A := R[[Y; σ, δ]] the definition of which is given in section 2 and prove a generalized Division Lemma and Weierstrass Preparation Theorem for such rings (see Theorem 3.1 and corollary 3.2). Here we give only one remarkable consequence of this preparation theorem. By Λ(G)-mod^{H} and C^{H} we denote the category of Λ(G)-modules which are finitely generated as Λ(H)-module, and its full subcategory of those pseudo-null Λ(G)-modules, which are finitely generated over Λ(H), respectively. The skew polynomial ring R[Y; σ; δ] ≅ R[Z, σ], Z = Y + 1, possesses as a ring of quotients Q[Z; σ] where Q = Q(H) denotes the skewfield of fractions of Λ(H). We prove the following

Theorem (Theorem 5.5). The quotient category Λ(G)-mod^{H}/C^{H} can be identified with a full subcategory of the category Q[Z; σ]-mod of finitely generated Q[Z; σ]- modules

Λ(G)-mod^{H}/C^{H} ⊆ Q(H)[Z; σ]-mod.

Under this identification, the equivalence classes of simple objects of the quotient category correspond uniquely to equivalence classes of maximal left ideals of Q[Z; σ] which contain a distinguished polynomial. Furthermore, bounded objects correspond to bounded Q[Z; σ]-modules.

An analogous statement holds if we replace Q[Z; σ] by the skew Laurent polynomial ring Q[Z, Z^{−1}; σ]. In the case where G ≅ ℤ_{p} ⋊ ℤ_{p} with non-trivial action, an easy calculation shows that the ring Q[Z, Z^{−1}; σ] is simple, hence does not admit any bounded module and the first theorem above follows.

For a commutative power series ring in several variables over a complete discrete valuation ring the Weierstrass preparation theorem is one key means of proving the unique factorization property. At least for the completed group algebra (of the semidirect product G) with coefficients in the finite field 𝔽_{p} a similar result holds which can hopefully be extended to the full Iwasawa algebra with ℤ_{p} -coefficients:

Theorem (Theorem 7.1). The ring 𝔽_{p}[[G]] = 𝔽_{p}[[X, Y; σ, δ]] is an UFD in the sense of Chatters [8].

For the precise definition of a (noncommutative) unique factorization domain (UFD) we refer the reader to section 7. To the author’s knowledge this is the first example of a complete noncommutative ring of global cohomological dimension greater than 1, which is a UFD.

Acknowledgements. I would like to thank John Coates most warmly not only for inviting me to DPMMS, Cambridge University, but also for his great interest and helpful comments on this project. Also I am very grateful to Yoshitaka Hachimori for valuable, stimulating discussions and reading parts of the manuscript. Susan Howson is thanked for helpful suggestions. Finally, I thank the EU Research Training Network “Arithmetic Algebraic Geometry” for its support and DPMMS for its hospitality.

Convention. In this paper, the ring R is always associative and with one. When we are talking about properties related to R like being “noetherian”, an “ideal”, a “unit”, of a certain “global dimension”, etc. we always mean the left and right property if not otherwise stated. But by an R-module we usually mean left R-module (not a bi-module). By a local ring R we mean a ring in which the non-units form a proper ideal, which is then automatically maximal as left, right and two-sided ideal. Equivalently, R has both a unique left and a unique right maximal ideal, which amounts to the same as the quotient R/J(R) of R by its Jacobson radical being a skewfield.

2. Skew power series rings

Let R be a ring, σ : R → R a ring endomorphism and δ : R → R a σ-derivation of R, i.e. a group homomorphism satisfying

δ(rs) = δ(r)s + σ(r)δ(s) for all r, s ϵ R.

Then the (formal) skew power series ring

R[[X; σ, δ]]

is defined to be the ring whose underlying set coincides with the set of usual (formal) power series R[[X]] but whose multiplication is induced by the relation

Xr = σ(r)X + δ(r). (2.1)

This implies

X^{n}r = ∑_{i=0}^{n}(X^{n}r)_{i}X^{i}

for certain elements (X^{n}r)_{i} ϵ R and in general the multiplication is given by

(∑ r_{i}X^{i})(∑ s_{j}X^{j}) = ∑_{i,j} r_{i}X^{i}s_{j}X^{j}
= ∑_{n} c_{n}X^{n}

with

c_{n} = ∑_{j≤n;i} r_{i}(X^{i}s_{j})_{n−j} (2.2)

(Note that (X^{i}s_{j})_{k} = 0 for i < k). To grant that the above series converges to an element c_{n} ϵ R we have to impose further conditions: Either δ = 0 (then X^{n}r = σ^{n}(r)X^{n}, i.e. the sum c_{n} = ∑_{i+j=n} r_{j}σ^{i}(s_{j}) is finite) or we assume that R is a complete ring with respect to the I-adic topology, I some σ-invariant ideal, and that it holds

δ(R) ⊆ I, δ(I) ⊆ I^{2}.

This implies by induction that δ(I^{k}) ⊆ (I^{k+1}) for all k ≥ 0 (I^{0} = R by convention) and the following lemma shows that the multiplication law in R[[X; σ, δ]] is well defined.

Lemma 2.1. (i) If r ϵ I^{k} then (X^{n}r)_{i} ϵ I^{k+n−i} for all i, n ≥ 0.

(ii) If σ is an isomorphism satisfying σ(I) = I, then for any n ϵ ℕ : r ϵ I^{k} \ I^{k+1} if and only if (X^{n}r)_{n} ϵ I^{k} \ I^{k+1}. In particular, r ≠ 0 if and only if (X^{n}r)_{n} ≠ 0.

Proof. All statements follow by induction, the case n = 1 following from the defining relation Xr = σ(r)X + δr, while the induction step uses

(X^{n+1}r)_{j} = σ((X^{n}r)_{j−1}) + δ((X^{n}r)_{j})

and (X^{n}r)_{n} = σ^{n}(r). □

Our interest in this kind of rings stems from applications in algebraic number theory where the following example naturally occurs, see section 8.

Example 2.2. Let G be the semi-direct product G = Γ_{1} ⋊ Γ_{2} of the pro-p-groups Γ_{1} and Γ_{2}, both isomorphic to the additive group of p-adic integers ℤ_{p}, where the action of Γ_{2} on Γ_{1} is given by a continuous group homomorphism ρ : Γ_{2} → Aut(Γ_{1}) ≅ ℤ_{p}^{∗}. Then the completed group algebras ℤ_{p}[[G]] and 𝔽_{p}[[G]] of G with coefficients in ℤ_{p} and 𝔽_{p} respectively are isomorphic to the skew power series rings

ℤ_{p}[[G]] ≅ ℤ_{p}[[X]][[Y; σ, δ]] =: ℤ_{p}[[X, Y; σ, δ]],
𝔽_{p}[[G]] ≅ 𝔽_{p}[[X]][[Y; σ, δ]] =: 𝔽_{p}[[X, Y; σ, δ]],

where

(i) X := γ_{1} − 1, Y := γ_{2} − 1 for some generators γ_{i} of Γ_{i}, i = 1, 2.

(ii) Setting R := ℤ_{p}[[X]] (𝔽_{p}[[X]]) and ϵ := ρ(γ_{2}) the ring automorphism σ

is given induced by X ↦ (X + 1)^{ϵ} − 1, while δ = σ − id.

Indeed, conferring Lazard ’s work [19] (see also [13]) the ring ℤ_{p}[[G]] (𝔽_{p}[[G]]) has X^{i}Y^{j}, 0 ≤ i, j < ∞, as a (complete) ℤ_{p}(𝔽_{p})-basis and it is straight forward to verify that the relation between X and Y is described via σ and δ. Furthermore, R is obviously complete with respect to the topology induced by its maximal ideal 𝔪 which is generated by X and p. Since σ is induced by choosing another generator of Γ_{1} under the isomorphism ℤ_{p}[[X]] ≅ ℤ_{p}[[Γ_{1}]] we see that σ(𝔪) = 𝔪. Thus we only have to verify that δ(𝔪^{k}) ⊆ 𝔪^{k+1} for k = 0, 1 : Since δ and σ are ℤ_{p} (𝔽_{p})- linear the claim follows immediately from the special case r = X observing also that δ(ℤ_{p}) = 0 (and analogously for 𝔽_{p}):

δX = σ(X) − X
= (X + 1)^{ϵ} − 1 − X
= ∑_{i≥1} ( \begin{matrix}ϵ \\ i \end{matrix} ) X^{i} − X
= (ϵ − 1)X^{k} + terms of higher degrees.

Since p|(ϵ − 1) we conclude δ(X) ⊆ 𝔪^{2}.

Example 2.3. The previous example generalizes immediately to the following situation: Let H be a uniform pro-p group on which the group Γ ≅ ℤ_{p} acts via a continuous group homomorphism ρ : Γ → Aut(H) and assume that the image of ρ is contained in Γ(H^{p}) := {γ ϵ Aut(H)|[γ, H] ⊆ H^{p}} where [γ, h] = h^{γ}h^{−1} for h ϵ H. Then it is easy to see that the semidirect product G = H ⋊ Γ is again an uniform pro-p group. Its Iwasawa algebras Λ(G) := ℤ_{p}[[G]] and Ω(G) := 𝔽_{p}[[G]] of G with coefficients in ℤ_{p} and 𝔽_{p} are isomorphic to the skew power series rings over the Iwasawa algebras Λ(H) and Ω(H), respectively,

Λ(G) ≅ Λ(H)[[Y; σ, δ]],
Ω(G) ≅ Ω(H)[[Y; σ, δ]],

where Y := γ − 1 for some generators γ of Γ, the ring automorphism σ is induced by h ↦ h^{γ} and again δ = σ − id. Indeed, since for any h ϵ H

δ(h − 1) = σ(h − 1) − (h − 1)
= h^{γ} − h
= (g^{p} − 1)h (for some g ϵ H by assumption)
ϵ 𝔪^{2}(see the claim in the proof of [28, lemma 3.24]),

δ(𝔪) ϵ 𝔪^{2} and the conditions on σ and δ are fulfilled as above.

We call a uniform pro-p-group G polycyclic if it contains a finite chain of subgroups

1 = G_{0} ⊆ G_{1} ⊆ ⋯ G_{i} ⊆ G_{i+1} ⊆ ⋯ G_{n} = G

such that G_{i} is normal in G_{i+1} with quotient G_{i+1}/G_{i} ≅ ℤ_{p} for 0 ≤ i < n. Then, as S. Howson suggests, one can identify Λ(G) with the skew power series ring in n-variables

Λ(G) ≅ ℤ_{p}[[X_{0}, …, X_{n−1}; σ_{1}, δ_{1}, …, σ_{n}, δ_{n}]]

which is defined recursively by

Λ(G_{1}) = ℤ_{p}[[X_{0}]],
Λ(G_{i+1}) = Λ(G_{i})[[X_{i}; σ_{i}, δ_{i}]],

where σ_{i} and δ_{i} are defined as above.

Example 2.4. A rather trivial special case of the previous example is R[[X]] where R is any (noncommutative) ring, σ = id and δ = 0. In the Iwasawa theory of elliptic curves there occur naturally open subgroups G ⊆ GL_{2}(ℤ_{p}) such that G = H × Γ where H = SL_{2}(ℤ_{p}) ∩ G and Γ ≅ ℤ_{p} is the center of G. Then the Iwasawa algebra Λ(G) can be identified with the power series ring Λ(H)[[T]] in one (commuting) variable with coefficients in the sub Iwasawa algebra Λ(H).

In case δ = 0 the following properties of A = R[[X; σ]] reflect the correspondent ones of R ([22, thm. 1.4.5, thm. 7.5.3], [20, Chap. II Cor. 3.2.12, Chap. III Thm. 3.4.6]):

Proposition 2.5. Let A = R[[X; σ]]. Then the following holds:

(i) If σ is injective and R is an integral domain, then A is an integral domain. (ii) If σ is injective and R a division ring, then A is a principal left ideal domain (Note that in this case δ = 0 because of the above conditions).

(iii) If σ is an automorphism and R is left (or right) Noetherian, then A is left (respectively right) Noetherian. In this case, the left (or right) global dimensions of A and R are related by

glA = glR + 1.

(iv) If σ is an automorphism and R is a maximal Noetherian order, then A

is a maximal Noetherian order.

(v) If σ is an automorphism and R an Auslander regular ring, then A is an Auslander regular ring.

In order to prove similar results in the general case we first have to introduce some filtrations on A = R[[X; σ, δ]]. We assume for the rest of this section that R is a local ring with maximal ideal 𝔪 which is complete and separated with respect to its 𝔪-adic topology and set

F_{0}A := A, F_{n}A := ∏_{i=0}^{∞} 𝔪^{n}X^{i}, n ≥ 1,

where A is identified with A = ∏_{i=0}^{∞} RX^{i}, and

G_{0} := A, G_{n} := ∑_{i=0}^{n}(F_{n−i}A)X^{i}, n ≥ 1,

where the sum is a sum of left R-modules. By definition, both filtrations G_{i} and F_{i} are exhaustive and F_{i}A ⊆ F_{j}A respectively G_{i}A ⊆ G_{j}A holds for all i ≥ j. The following lemma shows that they are indeed filtrations of rings:

Lemma 2.6. For all n, m ≥ 0, it holds: F_{n}A · F_{m}A ⊆ F_{n+m}A and G_{n}A · G_{m}A ⊆ G_{n+m}A.

Proof. The assertion for the filtration F_{i} follows immediately from the formula 2.2, lemma 2.1 and by the fact that 𝔪^{k} is closed with respect to the 𝔪-adic topology for all k (see [20, Ch. I 3.1(c)]). Concerning the second filtration it suffices by distributivity to prove that

(F_{n−i}A)X^{i}(F_{m−j}A)X^{j} ⊆ ∑_{k=0}^{n+m}(F_{n+m−k}A)X^{k}

for all 0 ≤ i ≤ n and 0 ≤ j ≤ m holds. Using the multiplicativity of F_{i} just proved this will follow immediately once we have shown the relation

X^{i}(F_{k}A) ⊆ ∑_{j=0}^{i}(F_{k+i−j}A)X^{j}.

So let f = ∑ f_{i}X^{i} in F_{k}A. The defining relation 2.1 implies that Xf decomposes into the sum

Xf = ∑_{i=0}^{∞} δ(f_{i})X^{i} + (∑_{i=1}^{∞} σ(f_{i−1})X^{i−1})X,

the first summand of which lies in F_{k+1}A while the second one belongs to (F_{k}A)X by lemma 2.1. The proof is accomplished by induction. □

Note that the two-sided ideals G_{k}A (apply the lemma with n or m equal to 0) can be described as follows:

G_{k}A = (∏_{i=k}^{∞} RX^{i}) × 𝔪X^{k−1} × ⋯ × 𝔪^{k}X^{0} = ∏_{i=0}^{∞} 𝔪^{k−i}X^{i} (2.3)

for all k ≥ 0, where 𝔪^{l} := R for negative integers l.

Lemma 2.7. The topology on A induced by the filtration G_{i}A coincides with the product topology of A ≅ ∏_{i=0}^{∞} R, where R is endowed with its 𝔪-adic topology.

Proof. With respect to the product topology a neighbourhood basis of 0 ϵ A consists of

A_{k} := ∏_{i=0}^{k−1} 𝔪^{k}X^{i} × ∏_{i=k}^{∞} RX^{i}, k ≥ 0.

It is easy to see that the B_{k} and G_{i} are cofinal to each other and hence induce the same topology on A. □

Since the product of complete Hausdorff topological spaces is again a complete Hausdorff topological space we obtain

Corollary 2.8. The ring A is a complete Hausdorff topological ring with respect to its product topology respectively the topology which is induced by the filtration G_{i}.

Proposition 2.9. There is a canonical isomorphism of graded rings

gr_{G}A ≅ (gr_{𝔪}R)[X̄; σ̄],

where the latter ring bears the mixed grading with respect to the grading of gr_{𝔪}R and the ”X̄-adic” grading. Here X̄ means the image of X and σ̄ is induced by σ.

Proof. The above description 2.3 implies immediately that

G_{k}/G_{k+1} ≅ ∏_{i=0}^{k} 𝔪^{i}/𝔪^{i+1}X̄^{k−i}

holds. Thus

⨁_{k=0}^{∞} G_{k}/G_{k+1} ≅ ⨁_{i,l≥0} 𝔪^{i}/𝔪^{i+1}X̄^{l}
≅ ⨁_{i≥0} gr_{𝔪}RX̄^{i}
≅ (gr_{𝔪}R)[X̄; σ̄].
□

As a corollary it turns out that A = R[[X; σ, δ]] reflects the properties of gr_{𝔪}R :

Corollary 2.10. If gr_{𝔪}R is

(i) integral and σ̄ is injective, then A is integral.

(ii) left (or right) Noetherian and σ̄ is an automorphism, then A is left (respectively right) Noetherian. If, in addition, the left (or right) global dimension of gr_{𝔪}R is finite, then so is the left (respectively right) global dimension of A and it holds

glA ≤ glgr_{𝔪}R + 1.

(iii) a Noetherian maximal order, R complete and σ̄ is an automorphism, then

A is a maximal Noetherian order. (iv) Noetherian, R complete and σ̄ is an automorphism, then A is a Zariski-

ring.

(v) Auslander regular, R complete and σ̄ is an automorphism, then A is Auslander regular.

Proof. Assuming conditions (i) or (ii), [22, thm. 1.2.9] tells that gr_{G}A ≅ (gr_{𝔪}R)[X̄; σ̄] is integral, respectively Noetherian. This implies the same properties for the ring A by [22, thm 1.6.7, thm. 1.6.9]. The statement concerning the global dimensions follow similarly by [22, thm. 7.5.3 (iii), cor. 7.6.18]. Now let us assume (iii). Then gr_{G}A ≅ (gr_{𝔪}R)[X̄; σ̄] is a maximal order according to [20, Chap. II Cor. 3.2.12], thus A is a maximal order by [10, lem 2.6]. Under the condition (iv) A is complete with respect to the filtration G_{i}. Since gr_{G}A is Noetherian, A is a Zariski ring by [20, Chap. II thm. 2.1.2]. To prove (v) it suffices to show that gr_{G}A is Auslander regular ([20, Chap. III thm. 2.2.5]), which is contained in [20, Chap. III thm. 3.4.6 (1)]. □

We close this section proving that under our current conditions on R the skew power series ring A is a local ring.

Proposition 2.11. An element f = ∑ f_{i}X^{i} ϵ A is a unit (in A) if and only if the constant term f_{0} is a unit in R. In particular, A is a local ring.

Proof. The “only if” part follows by considering the canonical surjective map A → A/G_{1}A ≅ R/𝔪. Now assume without loss of generality that f_{0} = 1 (otherwise replace f by f_{0}^{−1}f ) and set h := f_{0} − f ϵ G_{1}A. Then, due to completeness of A with respect to G_{i}A, the “geometric series” ∑_{i≥0} h^{i} converges to an element g ϵ A, which is a left and right inverse of f = 1−h. That means that the non-units consist precisely of the elements of 𝔐 = G_{1}A. □

3. The Weierstrass preparation theorem

Let R be a (not necessarily commutative) local ring with maximal (left) ideal 𝔪 and suppose that R is separated and complete with respect to its 𝔪-adic topology. As usual we denote the residue class skewfield R/𝔪 by k. As before we assume that σ : R → R is a ring-automorphism satisfying σ(𝔪) = 𝔪, i.e. σ induces also a ring automorphism σ : k → k, while δ : R → R is a σ-derivation such that δ(R) ⊆ 𝔪, δ(𝔪) ⊆ 𝔪^{2}. In particular, δ operates trivially on k. Therefore there is a canonical surjective reduction map:

¯ : R[[X; σ, δ]] → k[[X; σ]].

In example 2.2 σ will turn out to be the identity on k. For any f = ∑ a_{i}X^{i} ϵ A := R[[X; σ, δ]] the reduced order ord^{red}(f) of f is defined to be the order of the reduced power series \overline{f}, i.e.

ord^{red}(f) = min{i|a_{i} ϵ R^{∗}}.

Theorem 3.1. Let f ϵ A be a power series with finite reduced order s := ord^{red}(f) < ∞. Then A is the direct sum

A = Af ⊕ ⨁_{i=0}^{s−1} RX^{i}

of the R-modules N := Af and M := ⨁_{i=0}^{s−1} RX^{i}.

Proof. (compare to Bourbaki’s proof in the commutative case [3, §3 no.8]) We first prove that N ∩ M = 0 :

For any element of this intersection one has an equality

r_{0} + r_{1}X + … + r_{s−1}X^{s−1} = (∑ b_{j}X^{j})f
= ∑ c_{n}X^{n}.

Comparing the coefficients we conclude that for k ≥ 0

0 =^{!} c_{s+k} = ∑ ; \begin{matrix}0 ≤ j ≤ s + k \\ i \end{matrix} b_{i}(X^{i}a_{j})_{s+k−j}

= b_{k}(X^{k}a_{s})_{k} + ∑ ; \begin{matrix}0 ≤ j ≤ s + k \\ i \\ (i, j) ≠ (k, s) \end{matrix} b_{i}(X^{i}a_{j})_{s+k−j}.

To prove that b_{j} = 0, j ≥ 0, it suffices to see that b_{k} ϵ 𝔪^{n} for all k and n. By double induction, let us assume that b_{i} ϵ 𝔪^{n−1} for all i and b_{i} ϵ 𝔪^{n} for i < k. Since a_{s} ϵ R^{∗}, so is (X^{k}a_{s})_{k} by lemma 2.1 and the following claim implies that b_{k} ϵ 𝔪^{n}.

Claim: b_{i}(X^{i}a_{j})_{s+k−j} ϵ 𝔪^{n} for all 0 ≤ j ≤ s + k, (i, j) ≠ (k, s).

For i < k the claim is obvious because of b_{i} ϵ 𝔪^{n} then. So let us assume i ≥ k, (i, j) ≠ (k, s). We have to deal with three cases separately:

(i) If i + j ≤ s + k then (X^{i}a_{j})_{s+k−j} = 0.

(ii) If i + j = s + k then j = s + k − i < s (note that i = k would imply that j = s), i.e. a_{j} ϵ 𝔪 by the definition of s and the claim follows from lemma 2.1.

(iii) If i + j > s + k then i − (s + k − j) ≥ 1, i.e. the claim follows from the same lemma.

We still have to prove that N + M = A. Since a_{s} ϵ R^{∗} and A is local we can write

f = ∑_{i=0}^{s−1} a_{i}X^{i} + gX^{s}

with g ϵ A^{∗}. Setting G := g^{−1} and

h := −G(∑_{i=0}^{s−1} a_{i}X^{i}) = −G(f − gX^{s}) = X^{s} − Gf

we claim that all coefficients of h are in 𝔪. Indeed,

h_{n} = ∑_{j≤min{n,s−1}} G_{i}(X^{i}a_{j})_{n−j} ϵ 𝔪

using lemma 2.1 because a_{j} ϵ 𝔪 for j < s.

Now let r be an arbitrary element in A. Recursively we can find a series q^{(n)}, n ϵ ℕ, of elements in A such that

q^{(0)}X^{s} ≡ r mod M

and

q^{(n+1)}X^{s} ≡ q^{(n)}h mod M for n ≥ 0

(Just cut off the terms of degrees less than s with respect to X and then factor out X^{s} on the right). Since q_{l}^{(n+1)} = (q^{(n)}h)_{s+l} it is easily seen that q_{l}^{(n+1)} ϵ 𝔪^{n+1} for all l using the fact that the coefficients of h lie all in 𝔪. Hence, due to the completeness of R, ∑ q^{(n)} converges to an element q ϵ A. The relations

(q^{(0)} + ⋯ + q^{(n)})X^{s} ≡ r + (q^{(0)} + ⋯ + q^{(n−1)})h mod M

imply that

r ≡ q(X^{s} − h) mod M
= (qG)f ϵ Af.
□

In analogy with the commutative situation we call a monic polynomial F = X^{s} + a_{s−1} + ⋯ + a_{1}X + a_{0} ϵ R[X; σ, δ] ⊆ R[[X; σ, δ]] distinguished or Weierstrass polynomial (of degree s) if a_{i} ϵ 𝔪 for all 0 ≤ i ≤ s − i. Here, R[X; σ, δ] denotes the skew polynomial ring in one variable over R with respect to σ and δ.

Corollary 3.2. Under the assumptions of the theorem f can be expressed uniquely as the product of an unit ϵ of A and a distinguished polynomial F ϵ A :

f = ϵF.

Proof. According to the theorem the element X^{s} ϵ A can be uniquely written as

X^{s} = vf − H

where H = ∑_{i=0}^{s−1} g_{i}X^{i} and v ϵ A. Reducing the coefficients mod 𝔪 we obtain

X^{s} = v̄f̄− ∑_{i=0}^{s−1} ḡ_{i}X^{i}.

Since ord(f̄) = ord^{red}(f) = s, i.e. ord(v̄f̄) = ord(v̄) + ord(f̄) ≥ s, the coefficients ḡ_{i} must all vanish. Therefore

F := X^{s} + H

is a distinguished polynomial and

X^{s} = v̄f̄,

i.e. ord(v̄) = ord(X^{s}) − ord(f̄) = 0. Hence, v is a unit in A and f = ϵF with ϵ := v^{−1}. □

Independently, D. Burns and C. Greither [12] have proved another generalized version of the Weierstrass preparation theorem for power series rings in one commuting variable over noncommutative (not necessarily local) rings.

Remark 3.3. Let J be a left ideal of A = R[[X; σ, δ]]. Then A/J is finitely generated as R-module if and only if J contains a Weierstrass polynomial. Indeed, if all elements of J reduce to zero in κ := k[[X; σ]], then M/𝔐M ≅ κ where 𝔐 := ker(¯ : A → κ). But since there is some surjection R^{n} → M and 𝔪 ⊆ 𝔐 the module M/𝔐M is a finitely generated k-module, a contradiction. The other implication is a direct consequence of theorem 3.1.

If, moreover, J = Af is principal, then A/J is a finitely generated R-module if and only if ord^{red}(F) is finite, i.e. if and only if Af = AF can also be generated by some Weierstrass polynomial F.

Of course, there exist right versions for all statements of this section. Indeed, since σ is assumed to be an automorphism, one gets a natural ring isomorphism

R[[X; σ, δ] ≅ [[X; σ′, δ′]]R,

where σ′ = σ^{−1}, δ′ = −δ ∘σ^{−1} and in the latter ring the coefficients are written on the right side of the variable X. Moreover, the reduced order is invariant under this isomorphism, i.e. the notion of an distinguished polynomial and its degree is independent of the representation as left or right power series.

4. Faithful modules

In this section we calculate the global annihilator ideals of certain modules over the Iwasawa algebra of some p-adic Lie group. First we concentrate on the example 2.2, i.e. A = R[[Y; σ, δ]] with R = ℤ_{p}[[X]] or R = 𝔽_{p}[[X]]. Recall that in this case δ is just given as σ − id. To exclude the case where G respectively A are commutative, we assume that ϵ ≠ 1.

Remark 4.1. (i) There is a canonical isomorphism

R[Y; σ, δ] ≅ R[Z_{;}σ], r ↦ r, Y ↦ Z − 1

of skew polynomial rings, because

Zr = (Y + 1)r
= Y r + r
= σ(r)Y + δr + r
= σ(r)Y + σ(r) − r + r
= σ(r)(Y + 1) = σ(r)Z.

(ii) Any ideal I of A which contains a polynomial 0 ≠ b = ∑_{i=0}^{s} b_{i}Z^{i} also contains a non-zero element r ϵ R. First note that Z^{i}γ = σ^{i}(γ)Z^{i} for γ := X + 1 ϵ R^{∗}, i.e. σ^{j}(γ)Z^{i} − Z^{i}γ = (σ^{j}(γ) − σ^{i}(γ))Z^{i} the latter being zero if and only if i = j because σ operates without fixpoint on Γ_{1}. Since I is two-sided it also contains the element

σ^{s}(γ)b − bγ = ∑_{i=0}^{s} b_{i}(σ^{s}(γ)Z^{i} − Z^{i}γ)

= ∑_{i=0}^{s−1} b_{i}(σ^{s}(γ) − σ^{i}(γ))Z^{i}

which is nonzero whenever one of the b_{i}, 0 ≤ i ≤ s−1, is nonzero because R is integral. Proceeding recursively one concludes that I contains an element of the form rZ^{i} for some 0 ≠ r ϵ R and i ϵ ℕ. But then it contains also r itself because Z = Y + 1 is a unit in A.

(iii) The same argument proves that the skew Laurent polynomial ring Q[Z, Z^{−1}; σ] over the maximal ring of quotients Q of R is simple (note that the ring automorphism σ extends uniquely to Q.)

Proposition 4.2. (i) Let R = 𝔽_{p}[[X]]. If M is an A-module which is not torsion as R-module, then its annihilator ideal over A vanishes:

Ann_{A}(M) = 0.

(ii) Now let R = ℤ_{p}[[X]]. If M is an A-module without p-torsion and such that the module M/p is not torsion as R/p = 𝔽_{p}[[X]]-module (for example if M is a free R-module), then Ann_{A}(M) = 0.

Recall that an A-module M is called faithful if the annihilator ideal Ann_{A}(M) is zero and otherwise M is called bounded. Before proving the proposition we want to draw an immediate conclusion from it in the case R = ℤ_{p}[[X]]. If M is an A-module which is finitely generated as R-module and of strictly positive R-rank rk_{R}M > 0, then the conditions of (ii) in the proposition are satisfied for N := M/tor_{ℤ_{p}}M. Indeed, by [17, cor. 1.11] it holds rk_{R/p}N/pN = rk_{R/p}(_{p}N) + rk_{R}N = rk_{R}N ≠ 0 in this case, where _{p}N denotes the kernel of multiplication by p. Thus Corollary 4.3. Every A-module M which is finitely generated as R-module and of strictly positive R-rank rk_{R}M > 0 is faithful.

Proof (of prop. 4.2). By the following lemma the second statement is a consequence of the first one, which can be proven as follows. First note that the R-torsion submodule M_{R−tor} of M is an A-module, because X is a normal element of A (see below). Thus M has a quotient N without R-torsion and since Ann_{A}(M) ⊆ Ann_{A}(N) we may assume without loss of generality that M itself is a torsionfree R-module. Now, if Ann_{A}(M) contains a non-trivial element b ϵ A this can be written as X^{n}b̃ with ord^{red}(b̃) < ∞ because R is a principal ideal domain all ideals of which are generated by some power of X. Due to the Weierstrass preparation theorem b̃ is a Weierstrass polynomial in the variable Y up to a unit, i.e. we may assume that b is already a Weierstrass polynomial because neither the unit nor X^{n} can annihilate M by assumption. By the above remark then the annihilator ideal contains a nonzero element r ϵ R which contradicts the assumption that M is a torsionfree R-module. □

Remark 4.4. By similar arguments one can show that every nonzero two-sided ideal J of A = 𝔽_{p}[[X, Y; σ, δ]] contains some power of X. It follows immediately that for every two-sided ideal J of A = ℤ_{p}[[X, Y; σ, δ]] the cyclic module A/J is finitely generated over ℤ_{p}[[Y]] ≅ Λ(Γ) (for any lift Γ ⊆ G). In particular, by an easy induction argument on the number of generators, each finitely generated bounded A-module is finitely generated over Λ(Γ).

Lemma 4.5. Let Λ := ℤ_{p}[[G]] and Ω := 𝔽_{p}[[G]] ≅ Λ/p the completed group algebras of some pro-finite group G with coefficients in ℤ_{p} and 𝔽_{p} respectively and assume that both rings are Noetherian integral domains. If M is an A-module without p-torsion then the following holds

Ann_{Λ}(M) ≠ 0 ⇒ Ann_{Ω}(M/p) ≠ 0.

Proof. If f is any nonzero element in Ann_{Λ}(M) and f = p^{n}f′ is the unique factorization such that f′ is not divisible by p then f′ is also an element in the annihilator of M because the latter module does not have any p-torsion by assumption. Thus the image of f′ in Ω is obviously a non-zero element of Ann_{Ω}(M/p). □

In the following we study the behaviour of the annihilator ideal under induction. So let H be a closed subgroup of a p-adic Lie group G. If M denotes a finitely generated Λ(H)-module, then we write Ind_{G}^{H}M := Λ(G) ⊗_{Λ(H)} M for the Λ(G)- module which is induced from M.

Let us assume for a moment that H = (h_{1}, …, h_{r}) is a uniform pro-p-group with minimal system of generators h_{1}, …, h_{r}. Then, using the isomorphism Λ(H) ≅ ∏_{(l_{1},...,l_{r}) ϵ ℕ^{r}} ℤ_{p}(h_{1} − 1)^{l_{1}} ⋯ (h_{r} − 1)^{l_{r}}, it is easily seen that the ℤ_{p}-linear span ∑ ℤ_{p}h_{n} is dense in Λ(H) where h_{n} := h_{1}^{ν_{1}^{n}} ⋯ h_{r}^{ν_{r}^{n}} for some bijection ℕ ≅ ℕ^{r}, n → ν^{n} = (ν_{1}^{n}, …, ν_{r}^{n}). Since every p-adic analytic group possesses an uniform subgroup of finite index it follows immediately that there exists (for arbitrary H as above) a countable set H = {h_{n} ϵ H|n ϵ ℕ} such that its ℤ_{p}-linear span is dense in Λ(H). For a left ideal I and an element r of a ring Λ we will denote by (I : r) := {λ ϵ Λ|λr ϵ I} the annihilator of r + I ϵ Λ/I.

Lemma 4.6. Let I be a left ideal of Λ := Λ(H) and set M := Λ/I. Then, for any choice of H, the annihilator ideal equals the countable intersection

Ann_{Λ}(M) = ⋂_{n=0}^{∞} I_{n}

of the left ideals I_{n} := Ih_{n}^{−1} = (I : h_{n}).

Proof. In general, it holds

Ann_{Λ}(M) = ⋂_{m ϵ M} Ann_{Λ}(m)
= ⋂_{r ϵ Λ}(I : r).

Thus Ann_{Λ}(M) is contained in ⋂_{n=0}^{∞} I_{n}. On the other hand any element λ of the latter intersection annihilates r + I ϵ M for any r in the ℤ_{p}-linear span of H. Since this span is dense in Λ the statement follows by the continuity of the action of Λ on M. □

In order to compare Ann_{Λ(G)}(Ind_{G}^{H}M) with Λ(G)Ann_{Λ(H)}(M) the following lemma will be crucial. Note that Λ(G) is a flat Λ(H)-module ([25, proof of lem 5.5]).

Lemma 4.7. Let I_{i}, i ϵ ℕ, be a set of left ideals in Λ(H). Then the following holds:

Λ(G) ⋂_{i} I_{i} = ⋂_{i} Λ(G)I_{i}.

A similar statement holds for an arbitrary filtered index set T if the ideals form a descending chain I_{s} ⫆ I_{t} for t ≥ s.

Proof. Due to the flatness of Λ(G) the functor Λ(G) ⊗_{Λ(H)} − interchanges with finite intersections. Hence we can replace the ideals I_{i} by the ideals J_{i} := ⋂_{n=0}^{i} I_{n} to come into the situation of the second statement, in which the intersection can be expressed as a projective limit: ⋂_{T} I_{t} = lim ; ⟵_{T}I_{t}. Since inverse limits interchange

with the completed tensor product Λ(G)ˆ⊗_{Λ(H)}− and the latter coincides with the usual tensor product if applied to finitely generated Λ(H)-modules (see [5]) we obtain

Λ(G) ⋂_{T} I_{t} = Λ(G) ⊗_{Λ(H)} ⋂_{T} I_{t}
= Λ(G)ˆ⊗_{Λ(H)} ⋂_{T} I_{t}
= Λ(G)ˆ⊗_{Λ(H)} lim ; ⟵_{T}I_{t}
= lim ; ⟵_{T}Λ(G)ˆ⊗_{Λ(H)}I_{t}
= lim ; ⟵_{T}Λ(G) ⊗_{Λ(H)} I_{t}
= lim ; ⟵_{T}Λ(G)I_{t}.
□

Proposition 4.8. Let M be a finitely generated Λ(H)-module. Then the global annihilator ideal of Ind_{G}^{H}M is contained in ideal generated by the global annihilator of M :

Ann_{Λ(G)}(Ind_{G}^{H}M) ⊆ Λ(G)Ann_{Λ(H)}(M).

In particular, if M is a faithful Λ(H)-module, then Ind_{G}^{H}M is a faithful Λ(G)- module.

Proof. Since a finitely generated Λ(H)-module M is the sum of finitely many cyclic submodules M_{i} and Ann_{Λ(H)}(M) = ⋂_{i} Ann_{Λ(H)}(M_{i}) this statement is easily reduced to the case of an cyclic module M = Λ(H)/I. Then Ind_{G}^{H}M is isomorphic to Λ(G)/Λ(G)I and it holds

Ann_{Λ(G)}(Λ(G)/Λ(G)I) ⊆ ⋂_{H} Λ(G)Ih^{−1} = Λ(G)Ann_{Λ(H)}(M)

by the previous lemmata □

This result should be compared to a theorem of Harris [16] which tells that the module Ind_{G}^{H}ℤ_{p} is bounded whenever 2 dim H > dim G. Using this proposition we can produce a series of faithful Λ(G)-modules where G is either an appropriate pro-p-subgroup of SL_{2}(ℤ_{p}) or GL_{2}(ℤ_{p}) (and thus SL_{n}(ℤ_{p}) or GL_{n}(ℤ_{p})). Indeed,

the closed subgroup H of GL_{2}(ℤ_{p}) generated by the matrices t = ( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} )

and c = ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) , where e.g. a = 1 + p and b = 1, i.e. more or less “the”

(pro-p-) Borel subgroup of SL_{2}(ℤ_{p}), is isomorphic to the semidirect product

H ≅ C ⋊ T of C = c^{ℤ_{p}} = ( \begin{matrix}1 & ℤ_{p} \\ 0 & 1 \end{matrix} ) and T = t^{ℤ_{p}}. Hence all the induced

modules Ind_{G}^{H}(Λ(H)/Λ(H)F) where F denotes a distinguished polynomial in Λ(H) ≅ ℤ_{p}[[X, Y; σ, δ]] are faithful. We should mention that the faithfulness of Ind_{G}^{H}(Λ(H)/Λ(H)(t − 1)) ≅ Ind_{G}^{T}ℤ_{p} was proved by Greenberg (private communication) using “p-adic harmonic analysis.”

5. Modules up to pseudo-isomorphism

In this section let R denote a noetherian local ring with maximal ideal 𝔪_{R} which is compact with respect to its 𝔪_{R}-adic topology. Furthermore, let σ be a ring automorphism and δ a σ-derivation as in section 2. We write A for the (compact) ring R[[Y; σ, δ]] and B for the (non-compact) ring R[Y; σ, δ]. If 𝔪_{A} denotes the unique maximal ideal of A, then 𝔪_{B} := B ∩ 𝔪_{A} is a maximal ideal of B and they are the kernels of the canonical R-algebra homomorphisms

𝔪_{A} = ker(A = R[[Y; σ, δ]] → R/𝔪_{R}),
𝔪_{B} = ker(B = R[Y; σ, δ] → R/𝔪_{R}),

which are both induced by Y ↦ 0.

Lemma 5.1. With the above notation it holds that

(i) 𝔪_{A} = 𝔪_{R}A + Y A = A𝔪_{R} + AY and 𝔪_{B} = 𝔪_{R}B + Y B = B𝔪_{R} + BY. (ii) For any A-module M, the submodules 𝔪_{A}^{i}M = 𝔪_{B}^{i}M coincide as B-

modules. In particular,

A = lim ; ⟵_{i}A/𝔪_{A}^{i} = lim ; ⟵_{i}B/𝔪_{B}^{i} and
M = lim ; ⟵_{i}M/𝔪_{A}^{i}M = lim ; ⟵_{i}M/𝔪_{B}^{i}M.

Proof. Due to compactness and the fact that 𝔪_{R} is finitely generated over R, it is easy to see that 𝔪_{R}A = F_{1}A and thus 𝔪_{R}A+T A = 𝔪_{R}A+AT = G_{1}A = 𝔪_{A}. The proof for B is similar, but easier, while the “left-versions” follow by symmetry. The second item follows immediately from the first one. □

By A-mod and B-mod we denote the category of finitely generated A- and Bmodules, respectively. For the full subcategories consisting of modules which are even finitely generated over R we write A-mod^{R} and B-mod^{R}, respectively. Similarly, we write mod-A, ^{R}mod-A, etc. for the correspondent categories of right modules.

Lemma 5.2. Let M be in A-mod^{R}. Then any B-submodule N ⊆ M is an Asubmodule and any B-quotient of M is an A-quotient.

Proof. The first statement obviously implies the second one. Since all rings under consideration are noetherian we may assume that N = Bm ⊆ M is a cyclic B-submodule. Then the A-submodule Am ⊆ M is finitely generated over R and thus the left ideal Ann_{A}(m) contains a distinguished polynomial. By the Weierstrass preparation theorem, A/Ann_{A}(m) ≅ B/Ann_{B}(m) and the statement for the submodules follows. □

Recall that the grade j_{A}(M) is the minimal integer i such that

E_{A}^{i}M := Ext_{A}^{i}(M, A)

does not vanish.

Lemma 5.3. Let M be in A-mod^{R}. Then the following holds:

(i) There is a natural identification of A-modules

M = A ⊗_{B} M,

where M is considered as B-module via restriction. In particular, A ⊗_{B} − is faithfully flat on the full subcategory of B-mod^{R} whose objects are restrictions of A-mod^{R}.

(ii) For all i there is a natural isomorphism of right A-modules

E_{B}^{i}(M) ⊗_{B} A ≅ E_{A}^{i}(M).

Moreover, E_{A}^{i}(M) and E_{B}^{i}(M) are in ^{R}mod-A and ^{R}mod-B, respectively, and E_{B}^{i}(M) bears a natural right A-module structure which extends the right B-module structure.

(iii) The grades of M as A- and B-module coincide:

j_{A}(M) = j_{B}(M).

In particular, M is a pseudo-null A-module if and only if it is a pseudonull B-module.

Proof. By [20, Ch. II, prop. 1.8 (3)] there is a natural isomorphism A ⊗_{B} M ≅ lim ; ⟵_{i}M/𝔪_{B}^{i}. Thus the first statement follows from lemma 5.1. The isomorphism

in (ii) is a standard fact of homological algebra using the flatness of A over B. If M is finitely generated over R we can find distinguished polynomials F_{1}, …, F_{n} and a surjection

⨁_{i=1}^{r} A/AF_{i} ↠ M.

Denoting the kernel of it by N the long exact E^{•}-sequence (for both A and B) gives

0 = E^{0}(N) → E^{1}(M) → ⨁_{i=1}^{r} E^{1}(A/AF_{i})

E^{1}(N) → E^{2}(M) → 0
E^{i}(N) ≅ E^{i+1}(M) for all i ≥ 3.

An easy calculation using the “right” version of the Weierstrass preparation theorem shows that

E_{A}^{1}(A/AF_{i}) ≅ A/F_{i}A ≅ B/F_{i}B ≅ E_{B}^{1}(B/BF_{i})

as right modules. Thus E^{1}(M) is finitely generated over R and E_{B}^{1}(M) bears a natural right A-module structure by lemma 5.2 (indeed its right version). Applying this also to N instead of M we obtain the statement (ii) by induction. Now (iii) follows from (i) and (ii). To prove (iv) let M be a pseudo-null A-module which is finitely generated as R-module. Then

E_{C}^{i}(C ⊗_{B} M) ≅ E_{B}^{i}(M) ⊗_{B} C = 0

for all i because C has global dimension less or equal to 1. It follows that Q⊗_{R}M ≅ C⊗_{B}M = 0, i.e. M is R-torsion. Since gr_{𝔪_{A}}A is a commutative noetherian regular equidimensional integral domain which is an algebra over some field, the other direction follows as in [25, lem. 6.4]. □

Assume now that

• R is still a compact noetherian local ring with maximal ideal 𝔪_{R} as before, but now such that

• gr_{𝔪_{R}}R is a commutative noetherian regular equidimensional integral domain which is an algebra over some field and

• σ induces the identity on the residue class field k, i.e. gr_{𝔪_{A}}A = gr_{𝔪_{B}}B is also commutative.

Then, by 2.10 and its proof, both A and B are Auslander regular and without zero divisor. Thus there is a dimension theory for A- and B-modules, for a thorough treatment of which we refer the reader to [10] or [28]. The previous lemma shows that the canonical (co-)dimension filtrations of M ϵ A-mod^{R} considered as Aor B-module coincide. Recall that M is called pseudo-null if j_{A}(M) ≥ 2, i.e. if E_{A}^{0}M = E_{A}^{1}M = 0 (similarly for B). Since R is without zero divisors the set S := R \ {0} is an Ore set of R and thus the total ring of quotients Q := R_{S} exists. By [11, prop. 2.12] α and δ extend uniquely to Q, thus S is also an Ore Set for B and there is a natural isomorphism of rings

C := B_{S} = Q[X; σ, δ]

(see below for further explanation in the case where δ = σ − id). Then it is well-known that C is a principal ideal domain (division algorithm). Now we are able to prove the following characterization of pseudo-null A-modules which are finitely generated over R :

Proposition 5.4. Let M be in A-mod^{R}. Then the following holds: M is pseudonull if and only if rk_{R}M = 0, i.e. if and only if M is a torsion R-module.

Proof. Let M be a pseudo-null A-module which is finitely generated as R-module. Then

E_{C}^{i}(C ⊗_{B} M) ≅ E_{B}^{i}(M) ⊗_{B} C = 0

for all i because C has global dimension less or equal to 1. It follows that Q⊗_{R}M ≅ C⊗_{B}M = 0, i.e. M is R-torsion. Since gr_{𝔪_{A}}A is a commutative noetherian regular equidimensional integral domain which is an algebra over some field, the other direction follows as in [25, lem. 6.4] using Hilbert-Samuel polynomials. □

We write C for the full subcategory of A-mod consisting of pseudo-null modules. Since C is a Serre subcategory the (abelian) quotient category A-mod/C exists together with a natural functor

q : A-mod → A-mod/C.

By a sort of homomorphism theorem for abelian categories one may identify the analogous quotient category of A-mod^{R} with a full subcategory of the prior one

A-mod^{R}/C ∩ A-mod^{R} ⊆ A-mod/C.

Henceforth we will denote C ∩ A-mod^{R} just by C^{R}. On the other hand restriction of scalars and tensoring with C ⊗_{B} − gives rise to an exact functor of abelian categories

Φ : A-mod^{R} → B-mod^{R} → C-mod,

the kernel of which is just C^{R} by the above proposition. By the universal property of quotient categories, A-mod^{R}/C^{R} can be identified with a full subcategory of C-mod, which we call by abuse of notation C-mod^{R}. In terms of left ideals,i.e. of cyclic modules this can be rephrased as follows: Let I be a left ideal of A which contains a distinguished polynomial, F say, or equivalently M := A/I is in A-mod^{R}. Then I_{B} := I ∩ B is a nonzero proper left ideal of B and its legalization I_{C} := (I_{B})_{S} is at least a nonzero left ideal of C. But if we assume that M is not pseudo-null then I_{C} ⫋ C by the above inclusion of categories. Thus we get a map

{I ◁_{l} A|A/I ϵ A-mod^{R}, q(A/I) ≠ 0} → {J ◁_{l} C|J contains a WP in B}
I ↦ I_{C},

where ◁_{l} indicates left ideal and “WP” stands for Weierstrass polynomial. There is also a map in the reverse direction:

J ↦ J_{B} := J ∩ B ↦ J_{A} := AJ_{B}.

It is easy to see that always (J_{A})_{C} = J and I ⊆ (I_{C})_{A} with pseudo-null quotient. Hence, if we restrict the left set to those left ideals I such that A/I does not contain any pseudo-null submodule, then we obtain a bijection

{I ◁_{l} A|0 ≠ A/I ϵ A-mod^{R}, (A/I)_{ps} = 0} ↔ {J ◁_{l} C|J contains a WP in B},

where, for an A-module M we denote by M_{ps} its maximal pseudo-null submodule. Since simple objects in A-mod^{R}/C^{R} can be represented by some cyclic module A/I, one immediately obtains

Theorem 5.5. There exists an equivalence of categories

A-mod^{R}/C^{R} ≅ C-mod^{R},

under which the equivalence classes of simple objects of the quotient category correspond uniquely to equivalence classes of maximal left ideals of C which contain a distinguished polynomial. Furthermore, C-mod^{R} is the smallest abelian subcategory of C-mod which is closed under extensions and contains the simple modules corresponding to the latter class of maximal left ideals.

Here the equivalence classes in the quotient category are just the classes of to each other isomorphic objects, while two left ideals J_{1}, J_{2} of C are equivalent if and only if there exists an isomorphism C/J_{1} ≅ C/J_{2} of C-modules.

We end this section by applying the above result to prime ideals. For simplicity we assume now that δ = σ − id. For any subring B̃ ⊆ A containing the ring B := R[Y; σ, δ] ≅ R[Z; σ] (Y = Z + 1) we write Spec_{R}(B̃) for the set of those prime ideals 𝔭 of B̃ such that B̃/𝔭 is a finitely generated R-module. Note that the zero ideal is not contained in Spec_{R}(B̃).

Lemma 5.6. Let 𝔭 be in Spec_{R}(A). The intersection with B̃, 𝔭 ↦ 𝔭 ∩ B̃ induces an injective map

Spec_{R}(A) → Spec_{R}(B̃).

Proof. For 𝔭 ϵ Spec_{R}(A), B̃/𝔭∩B̃ = A/𝔭 by the Weierstrass preparation theorem. Since B̃ is dense in A there is only one A-module structure on B̃/𝔭∩B̃ extending the B-module structure which implies injectivity. □

Now we specify B̃ to be the skew Laurent polynomial ring

B̃ := R[Z, Z^{−1}; σ].

Since the ring-automorphism extends uniquely to the skewfield of fractions Q of R, the ring B̃ embeds into

C̃ := Q[Z, Z^{−1}; σ]

which is - as legalization of Q[Z, σ] - a principal ideal domain. Moreover, Q[Z, Z^{−1}; σ] is also the quotient ring B̃_{S} of B̃ with respect to the set S = R \ {0} of regular elements of R. Indeed, since σ is an automorphism, the elements of R[Z, Z^{−1}; σ] and Q[Z, Z^{−1}; σ] can be written either with coefficients on the left or right site of the powers of Z. Let 𝔭 be in Spec_{R}(B̃) and assume that B̃/𝔭 has no R-torsion. Then, 𝔭 ∩ R = 0 and thus 𝔭_{S} ⫋ B̃_{S} = C is a proper prime ideal, see [22, prop. 2.1.16]. Denoting by Spec_{R,tf}(A) the subset of Spec_{R}(A) consisting of those prime ideals 𝔭 such that A/𝔭 is without R-torsion we obtain

Proposition 5.7. Under the above conditions there is a natural embedding

Spec_{R,tf}(A) ⊆ Spec_{R,tf}(B̃) ⊆ Spec(C̃) \ {0}(⊆ Spec(Q[Z; σ]) \ {0}).

By G(A) and G_{R}(A) we denote the group of (fractional) c-ideals and the subgroup which is generated by c-prime ideals in Spec_{R,tf}(A), respectively. For the definition and a review of its basic properties we refer the reader to [10, §4]. Note that under our current assumptions A is a maximal order (loc. cit. lem. 2.6). Since for any maximal order O, G(O) is the free abelian group on the set of all prime c-ideals, the proposition implies

Corollary 5.8. There is an injective group homomorphism

G_{R}(A) ↪ G(Q[Z, Z^{−1}; σ]).

6. Completely faithful objects

In this section we are going to apply the results of the previous ones to certain Iwasawa algebras. In particular we are interested in the existence of bounded and faithful objects in the quotient category.

First we recall some general definitions. For this purpose let A be any maximal order satisfying that every finitely generated torsion A-module induces an object of finite length in the quotient category A-mod/C. The annihilator ideal of an object M of A-mod/C is defined by Ann(M) := ∑_{q(M)≅M} Ann_{A}(M), where q denotes again the natural functor

q : A-mod → A-mod/C.

Note that by [26, lem 2.5] in conjunction with [7, cor of thm. 2.5] Ann(q(M)) = Ann_{A}(M/M_{ps}) where M_{ps} denotes the maximal pseudo-null submodule of M. Recall that the object M is called completely faithful if Ann(N) = 0 for any nonzero subquotient N of M. It is called bounded if Ann(N) ≠ 0 for any subobject N ⊆ M (In particular, the image q(M) of any pseudo-null module M is both bounded and completely faithful, by definition).

We should also mention that for a A-torsion module M, its image M := q(M) decomposes uniquely into a direct sum M ≅ M_{cf} ⊕M_{b} where M_{cf} is completely faithful and M_{b} bounded, cf. [10, prop. 4.1]. Moreover, M_{cf} or more general all completely faithful objects of finite length M of q(A-mod) are cyclic, i.e. M ≅ q(Λ/L) for some non-zero left ideal L ⊆ Λ (loc. cit. lemma 2.7 and prop. 4.1).

Now let G = H ⋊ Γ be an uniform pro-p-group as in example 2.3 and set R := Λ(H) as well as

A := Λ(G) ≅ Λ(H)[[Y; σ, δ]] = R[[Y; σ, δ]].

For some left ideal I of A, consider the cyclic A-module M = A/I and assume that it is finitely generated as R-module and has no non-zero pseudo-null submodule. Then, by proposition 5.4, M is torsionfree as R-module. We shall denote the annihilator ideal of M by J = Ann_{A}(M).

Lemma 6.1. In the above situation, assume that M is bounded, i.e. J ≠ 0. Then the module Λ/J is finitely generated over R, too, and has no non-zero pseudo-null submodule, either.

Proof. We use the same notations as in lemma 4.6 and 4.7 (Λ being replaced by A). Since I_{n} = Ih_{n}^{−1} and h_{n} is a unit in A, obviously A/I_{n} ≅ A/I and we obtain an injection

A/J_{i} ⊆ ∏_{n=0}^{i} A/I_{n} ≅ ∏ M.

Thus, for all i ≥ 0, the module A/J_{i} has no non-zero pseudo-null submodule. Now, assume that infinitely many subquotients J_{i}/J_{i+1} of the chain of left ideals

⋯ ⊆ J_{i+1} ⊆ J_{i} ⊆ ⋯ ⊆ J_{1} ⊆ J_{0} = I

are non trivial (here we assume without loss of generality that h_{0} = 1, i.e. J_{0} = I_{0} = I). Then, since 0 ≠ J_{i}/J_{i+1} ⊆ A/J_{i+1} cannot be pseudo-null, the image q(J_{i}/J_{i+1}) would be non-zero for infinitely many i contradicting the fact that all torsion A-modules have finite length in the quotient category. Thus the above chain becomes stationary, say J = J_{k}, and hence

A/J = A/J_{k} ⊆ ∏_{n=0}^{k} M

is finitely generated over R and without non-zero pseudo-null submodule. □

This lemma tells us that the identification

A-mod^{R}/C^{R} ≅ C-mod^{R}

in theorem 5.5 respects “boundedness”, i.e. an object M = q(M) is bounded if and only if C ⊗_{B} M is a bounded C-module. In particular, q(A/I) is bounded if and only if I_{C} is a bounded ideal of C, i.e. contains a nonzero ideal. Also, using [10, prop. 4.4], it implies that G_{R}(A) ≅ G((A-mod^{R}/C^{R})^{b}) where the latter denotes the Grothendieck group of the full subcategory of bounded objects (A-mod^{R}/C^{R})^{b} of A-mod^{R}/C^{R}. Hence, by the results of the previous section we obtain the following

Proposition 6.2. There is an injective group homomorphism

G((A-mod^{R}/C^{R})^{b}) ↪ G(Q[Z, Z^{−1}; σ]).

There are two special cases of particular interest. First, if G is the semidirect product of two copies of ℤ_{p}, we have seen that Q[Z, Z^{−1}; σ] is simple (cf. remark 4.1), thus G(Q[Z, Z^{−1}; σ]) = 0 and we obtain the following

Theorem 6.3. Let G be a non-abelian group isomorphic to ℤ_{p} ⋊ ℤ_{p}. Then the image q(M) of any M ϵ A-mod^{R}, is completely faithful, i.e. the subcategory q(A-mod^{R}) of A-mod/C consists only of completely faithful objects. In particular, all objects M of q(A-mod^{R}) are cyclic, i.e. M ≅ q(Λ/L) for some non-zero left ideal L ⊆ Λ.

Of course, the theorem is also an immediate consequence of corollary 4.3 and proposition 6.5.

Second, if G is an open subgroup of GL_{2}(ℤ_{p}) such that G = H × Γ where H = G ∩ SL_{2}(ℤ_{p}) and Γ = center(G), then σ is the identity, i.e. the variable Z or Y commute with Q. It is well known and easy to check that then every twosided ideal of Q[Y] can be generated by an element in Z(Q)[Y] ⊆ Q[Y] where Z(Q) denotes the center of the skewfield Q. In [18] S. Howson has proved that Z(R) = ℤ_{p} in this case. In order to determine the bounded A-modules which are finitely generated over R and not pseudo-null it might be crucial to answer the following

Question 6.4. Let H be any open pro-p-subgroup of SL_{2}(ℤ_{p}). Is the center Z(Q) of the skewfield of fractions Q of Λ(H) (i) equal to ℚ_{p}, (ii) algebraic over ℚ_{p} or (iii) has it transcendent elements over ℚ_{p}? Are the elements of Q\Z(Q) algebraic or transcendent over Z(Q)?

In any case, we know that the simple objects which can be represented by Amodules which are finitely generated over R correspond to those maximal left ideals of Q[Y] which contain a distinguished polynomial. If item (i) of the question holds it would probably tell us that the only bounded objects are those which are induced from the center of G.

Since Λ(H) ⊆ Λ(G) for the canonical normal subgroup H of G, we may consider any Λ(G)-module M also as Λ(H)-module. If such M is even finitely generated over Λ(H), then it is known that dim_{Λ(H)} M ≥ dim_{Λ(G)} ([25, lem. 6.4]). We now will prove that in the case where G ≅ ℤ_{p} ⋊ ℤ_{p}. equality holds.

Proposition 6.5. Let G ≅ ℤ_{p} ⋊ ℤ_{p}. With the above notations it holds that dim_{Λ(H)} M = dim_{Λ(G)} . In particular, M is a pseudo-null Λ(G)-module if and only if it is a torsion Λ(H)-module.

Thus a pseudo-null Λ(G)-module which is finitely generated over Λ(H) is - as Λ(H)-module up to finite modules - the direct sum of a finitely generated ℤ_{p} -module and summands of the form ℤ/p^{n}[[H]].

Proof. Since M is zero-dimensional (as Λ(G)- or Λ(H)-module) if and only if it is finite, we only have to prove

Claim: If M is torsion-free as Λ(H)-module than it is not a pseudo-null module.

This is just the statement of proposition 5.4, but we want to give a different argument which might be interesting on its own: Since X is a normal element of Λ(G), M has a descending chain of submodules {X^{n}M}_{n ϵ ℕ}. If M is torsionfree as Λ(H) ≅ ℤ_{p}[[X]]-module , then it es easy to see that the subquotients X^{n+1}M/X^{n}M are isomorphic to M/XM as ℤ_{p} -module. Since the latter module cannot be finite by the strong Nakayama lemma, it follows that these subquotients have dimension 1 both over Λ(G) and Λ(H) (see [28, prop. 3.5 (ii),cor. 4.8]). Choosing a good filtration on M (and taking the induced filtrations on X^{n}M) we get an infinite descending chain of associated graded modules

grM ⫌ gr(XM) ⫌ … ⫌ gr(X^{n}M) ⫌ …

with subquotients gr(X^{n}M)/gr(X^{n+1}M) ≅ gr(X^{n}M/X^{n+1}M) of Krull dimension 1 over grΛ(G) ≅ 𝔽_{p}[X_{0}, X_{1}, X_{2}] by [28, thm. 3.22]. This implies that grM has Krull dimension 2, i.e. dim_{Λ(G)} M = 2, by the next lemma. □

Lemma 6.6. Let R be a commutative Noetherian ring with finite Krull dimension, M a finitely generated R-module which has an infinite chain of submodules

M = M_{0} ⫌ M_{1} ⫌ … ⫌ M_{i} ⫌ …

with non-zero subquotients M_{i}/M_{i+1} of Krull-dimension α. Then, the Krull dimension of M is (strictly) greater than α.

Proof. Assume the contrary. Then deg(M) := ∑_{dim R/𝔭=α} length_{A_{𝔭}}M_{𝔭} is finite (compare [4, VIII § 1.5]). Since the function deg is additive (loc.cit.), all subquotients contribute a nonzero positive number to deg(M) by assumption, a contradiction.

□

Remark 6.7. The lemma corresponds to the more or less well known fact that for a commutative Noetherian ring the classical Krull-dimension coincides with the Gabriel-dimension or, depending on the definition, with the Gabriel-dimension minus one, see the notes “Krull dimension” in [14, p. 239].

Question 6.8. For which p-adic Lie groups without p-torsion, does it hold that the Gabriel-dimension of every finitely generated Λ(G)-module coincides with its Bjoerk-dimension, respectively the Krull-dimension of the associated graded module?

Of course the question is answered positively for all abelian p-adic Lie groups without p-torsion.

7. Unique Factorisation Rings

Chatters and Jordan studied in [9] a class of Noetherian (= left and right Noetherian) rings which in the commutative case consists of the Noetherian unique factorisation domains:

A prime Noetherian ring A is called unique factorisation ring (UFR) if every non-zero prime ideal of A contains a non-zero principal prime ideal.

If A satisfies the descending chain condition on prime ideals (e.g. if the classical Krull dimension dim(Spec(A)) of A is finite, where Spec(A) is the poset of prime ideals, see [14, Appendix 3.]), then the above condition is equivalent to

Every height-1 prime ideal of A is principal.

The meaning of principal ideal I is that I = aA = Aa for some element a ϵ A. Such elements are called normal and if I is a prime ideal in addition than a is called prime element. The class of UFRs contains the following much more restricted one (Remark (5) in [9]):

A Noetherian integral domain A which contains at least one height-1 prime ideal is called unique factorisation domain (UFD) if every height-1 prime P of A is principal and A/P is an integral domain.

We also should mention that any UFR A is a unique factorisation ring in the sense of [1], i.e. A is a maximal order and any prime c-ideal - a prime ideal which is (left or right) reflexive - is principal (cf. the remark after prop. 1.5 in [1]).

Now we come back to our example 2.2, where A = 𝔽_{p}[[G]] is the completed group algebra of the semidirect product G = ℤ_{p}⋊ℤ_{p}. It seems that A is the first example of a complete respectively power series ring which admits unique factorisation.

Theorem 7.1. The ring A = 𝔽_{p}[[G]] = 𝔽_{p}[[X, Y; σ, δ]] is a UFD. Furthermore, (X) := AXA is the only prime ideal of height one in A and X is - up to multiplication by units - the only prime element of A.

Proof. Since (X) is the kernel of

¯ : 𝔽_{p}[[X, Y; σ, δ]] → 𝔽_{p}[[Y]]

it is a prime ideal which coincides with XA and by symmetry also with AX (if the coefficients are written on the right of the powers of Y). By the principal ideal theorem [22, thm. 4.1.11] (X) has height 1. Now assume that there is some height-1 prime ideal P of A different from (X). Then P contains an element p with p̄ ≠ 0 in 𝔽_{p}[[Y]]. By the Weierstrass preparation theorem p is in R[Y; σ, δ] up to a unit where R = 𝔽_{p}[[X]]. The remark 4.1 (ii) implies that P ∩ R ≠ 0, i.e. X^{n} ⊆ P ∩ R ⊆ P for some n because all ideals of R are of the form (X^{n}). Since P is prime it follows that (X) ⊆ P, a contradiction. Thus the unique height-1 prime (X) is principal with A/(X) being integral, i.e. A is a UFD. □

Similarly, it is seen that X^{n}, n ϵ ℕ, are - up to multiplication by units - the only normal elements of A.

Let C(X) be the set of elements of A which are regular modulo (X) and S = {X^{n}|n ϵ ℕ}. Then it is known (and in this example easy to verify) that the classical (left and right) localisations A_{(X)} := A_{C(X)} and A_{S} of A with respect to C(X) and S exist. They have the following properties

(i) A_{(X)} is a bounded local principal ideal domain with a single prime ideal XA_{(X)} and whose only (left or right or two-sided) ideals are of the form X^{n}A_{(X)} (see [8] prop. 2.8 and its proof in conjunction with th. 2.7, compare also with [6, prop. 2.5 ] ).

(ii) A_{S} is simple ([9, lem. 2.1]).

(iii) A = A_{(X)} ∩ A_{S} ([9, thm. 2.3]). This should be compared also with Chamarie’s result 3.3-3.5 in [7]: While A_{S} is ”responsible” for the faithful modules the bounded ring A_{(X)} covers the bounded modules (in the quotient category of A-mod modulo the Serre subcategory of pseudo-null A-modules, see [10] or [28]).

Remark 7.2. Let A = 𝔽_{p}[[G]] and J be a left ideal of A such that M := A/J does not contain any pseudo-null submodule. Then M is faithful in the quotient A-mod/C if and only if no power X^{n}, n ϵ ℕ, is contained in J. Indeed, under the assumptions the annihilator of the image of M in the quotient category coincides with the usual annihilator ideal Ann_{A}(M). If M is bounded, the latter ideal is a c-ideal in A, i.e. a power of the unique prime ideal (X) ([10, lem 4.3]). In particular, X^{n} is contained in J for some n. On the other hand, assume that X^{n} ϵ J. Then J contains obviously the two-sided ideal AX^{n}.

We should also mention that the fact that 𝔽_{p}[[G]] is a UFR in the sense of [1] can be derived more easily from the following general theorem:

Theorem 7.3. Let A be a local noetherian integral ring which is a maximal order (in its ring of quotients). If the global homological dimension of A is less or equal to 2 then A is a UFR in the sense of [1].

Proof. Conferring prop. 1.8. (2) in [1] it is sufficient to prove that every reflexive ideal I is principal: Denote by I^{+} := Hom_{A}(I, A) the dual of I and choose a projective resolution of I^{+}

0 → P_{1} → P_{0} → I^{+} → 0.

Applying the functor −^{+} we obtain an exact sequence

0 → I → P_{0}^{+} → P_{1}^{+} → Ext_{A}^{1}(I^{+}, A) → 0

because I ≅ I^{++} is reflexive. Now it follows by homological algebra that I has projective dimension equal to zero (since the projective dimension of Ext_{A}^{1}(I^{+}, A) is less or equal to 2). But as A is local it follows that I is a free left A-module. By symmetry it is also a right principal ideal and the theorem follows. □

Recall that a principal ideal domain (PID) is an integral ring all left and right ideals of which are principal.

Corollary 7.4. Let R be a local PID and π a generator of the unique maximal ideal 𝔪 = (π) of R and assume that R is complete in its π-adic topology. If σ is a ring endomorphism of R which induces an automorphism σ̄ of κ := R/(π), then, for any σ-derivation δ as in section 2, the skew power series ring R[[Y; σ, δ]] is an UFD in the sense of [1].

Proof. Since κ is a skew field its global dimension is zero. Hence, by corollary 2.10 and [22, thm. 7.5.3 (iii)], glA ≤ glgr_{π}R + 1 = glκ[x] + 1 = 2 and the result follows from proposition 2.11, corollary 2.10 (iii) and the theorem. □

In general, there is a canonical homomorphism

G(R[Y; σ, δ]) → G(R[[Y; σ, δ),

induced by I → Î:= A⊗_{B} I ≅ I ⊗_{B} A. In those cases where this map is surjective (e.g. for 𝔽_{p}[[G]] as above) the UFR property for B in the sense of [1] implies the UFR property for A (supposed the latter ring is a maximal order). Anyway, one obtains the following implication

Proposition 7.5. Let R be a compact noetherian local ring with maximal ideal 𝔪_{R}. If R[Y; σ, δ] is a UFR in the sense of [1], then all ideals in G_{R}(A) are principal.

We should mention that theorem 2.4 and proposition 1.8 (2) of [1] imply the following: If a noetherian ring R is a UFR in their sense, then so is R[Z, σ]. E.g., if Λ(H) is a UFR, then all ideals in G_{Λ(H)}(Λ(G)) are principal, where H ⊆ G are the groups of example 2.3 or in particular of example 2.4 where H ⊆ G ⊆ GL_{2}(ℤ_{p}).

Proof. Let I be a prime c-ideal of A such that A/I is finitely generated over R, set I_{B} := B ∩ I as before and consider the following exact and commutative diagram

[figure: 0; (I_{B})^{++}; B^{++}; E^{1}E^{1}(B/I_{B}); 0; I_{B}; φ_{I_{B}}; B; φ_{B}; B/I_{B}; 0]

Here, the first line is easily obtained by forming twice the long exact E^{•}-sequence of the bottom line and the vertical homomorphisms are (induced by) the canonical maps from a module to its bi-dual. By the snake lemma, lemma 5.2 and 4.3 (i), the functor A ⊗_{B} − is faithfully flat on the cokernel W of φ_{I_{B}}. But A ⊗_{B} W ≅ cokern(φ_{I}) (as A-module) is zero by assumption. Thus, by left-right symmetry, I_{B} is also a reflexive ideal and maps onto I, i.e. the following restriction of the above homomorphism is surjective

G_{R}(B) ↠ G_{R}(A).

The result follows. □

In section 4 we have seen that Λ(G) possesses an abundance of faithful modules. Now we will describe an infinite series of bounded modules which induces an infinite series of pairwise non-isomorphic simple objects in the quotient category by pseudo-null modules. Indeed, by [10, prop. 4.4] the prime c-ideals, i.e. those prime ideals which are reflexive (both as left and right Λ(G-)module), correspond to the annihilator ideals of bounded simple objects. Thus it is sufficient to find an infinite series of prime ideals which are principal and thus reflexive.

To this end we set ω_{−1} := 1, ω_{n} := (X +1)^{p^{n}} −1, n ≥ 0, and denote the cyclotomic polynomial by

ξ_{n} := \frac{ω_{n}}{ω_{n−1}} , n ≥ 0.

Then, ξ_{0} = ω_{0} = X, while

ξ_{n} = ∑_{i=0}^{p−1}(1 + X)^{ip^{k−1}}, k ≥ 1.

Proposition 7.6. The polynomials ξ_{n} are prime elements of Λ = Λ(G), i.e. (ξ_{n}) := Λξ_{n} = ξ_{n}Λ is a prime ideal of Λ for every n ≥ 0. Furthermore, the quotient Λ/(ξ_{n}) is isomorphic to ℤ_{p}[ζ_{p^{n}}][[Y; \overline{σ}, \overline{δ}]], where ζ_{p^{n}} denotes a primitive p^{n}-root of unit and \overline{σ} and \overline{δ} are induced by σ and δ, respectively.

Proof. First, we show that ω_{m} and ξ_{n} are normal elements in Λ : Since ω_{n}Λ = ∏ ℤ_{p}[[X]]ω_{n}Y^{i} is the kernel of the canonical map

ℤ_{p}[[X, Y; σ, δ]] ≅ Λ(G) → Λ((H/H_{n}) ⋊ Γ) ≅ (ℤ_{p}[[X]]/ℤ_{p}[[X]]ω_{n})[[Y; \overline{σ}, \overline{δ}]],

this is a two-sided ideal, which by symmetry coincides with Λω_{n}, thus ω_{n} is normal. Hence, we obtain

ω_{n−1}ξ_{n}Λ = ω_{n}Λ = Λω_{n} = Λω_{n−1}ξ_{n} = ω_{n−1}Λξ_{n},

i.e. ξ_{Λ} = Λξ_{n} because Λ does not have zero divisors.

Since (ℤ_{p}[[X]]/ℤ_{p}[[X]]ξ_{n})[[Y; σ, δ]] ≅ ℤ_{p}[ζ_{p^{n}}][[Y; σ, δ]] is integral by cor. 2.10 (note that grℤ_{p}[ζ_{p^{n}}] ≅ 𝔽_{p}[π], where π is the image of X respectively ζ_{p^{n}} − 1). □

8. The “False Tate Curve”

In this section we will study a Galois extension of a number field k which arises from a p-adic Galois representation which is analogous to the local presentation induced by a Tate elliptic curve. Thus, following J. Coates, we call it “false Tate curve.”

So let α ϵ k^{×} be a unit of k which is not a root of unity. Consider the discrete G_{k}- module A := k̄^{×}/Δ_{α}, where k̄ denotes a fixed algebraic closure of k, G_{k} = G(k̄/k) the absolute Galois group of k and Δ_{α} = α^{ℤ} the subgroup of k̄^{×} which is generated by α. Then the p^{n}-division points A_{p^{n}} of A contain the p^{n}-roots of unity μ_{p^{n}} ⊆ k̄^{×} and we have natural exact sequences of G_{k}-modules for every n

0 → μ_{p^{n}} → A_{p^{n}} → ℤ/p^{n}ℤ → 0,

where the latter module has trivial G_{k} action and where the surjection A_{p^{n}} → ℤ/p^{n}ℤ is given as follows: If β ϵ k̄^{×} represents an element of A_{p^{n}}, i.e. β^{p^{n}} = α^{c} for some c ϵ ℤ, then its class βΔ_{α} is send to c mod p^{n}.

Writing T_{p}(B) := lim ; ⟵_{n}B_{p^{n}} the Tate-module and V_{p}(B) := T_{p}(B) ⊗_{ℤ_{p}} ℚ_{p} for the p-

adic representation associated with an arbitrary discrete G_{k} module B, we obtain the exact sequences

0 → ℤ_{p}(1) → T_{p}(A) → ℤ_{p} → 0,

where ℤ_{p}(1) = T_{p}(μ) is the first Tate-twist of ℤ_{p}, and

0 → ℚ_{p}(1) → V_{p}(A) → ℚ_{p} → 0.

If G denotes the kernel of the action ρ : G_{k} → GL(V) of G_{k} on V = V_{p}(A) :

G := ker(ρ : G_{k} → GL(V));

then the field k_{∞} := k̄^{G} which is determined by G via Galois theory is of course k(A_{p^{∞}}) = k(μ_{p^{∞}}, α^{p^{−∞}}), the field, which arises by adjoining the p-power roots of unity as well as p-power roots of α to k.

For the rest of the paper, we assume that k contains the p-th roots of unity μ_{p} ⊆ k. By k_{cycl} we denote the cyclotomic ℤ_{p}-extension k_{cycl} = k(μ_{p^{∞}}) of k, by H := G(k_{∞}/k_{cycl}) and Γ := G(k_{cycl}/k) the Galois groups of the extensions k_{∞}/k_{cycl} and k_{cycl}/k respectively. Then Kummer theory tells us that

Hom(H, μ_{p^{∞}}) ≅ im(Δ_{α} ⊗_{ℤ} ℚ_{p}/ℤ_{p} → k_{cycl}^{×} ⊗_{ℤ} ℚ_{p}/ℤ_{p}) ≅ ℚ_{p}/ℤ_{p}

as G_{k}-modules. In particular, H ≅ ℤ_{p}(1) as Γ-module, i.e. G is isomorphic to the (non-abelian) semi-direct product

G ≅ H ⋊ Γ = ℤ_{p}(1) ⋊ ℤ_{p}.

Of course, the arithmetic in k_{∞}/k depends decisively on the choice of α. If we take α = p for example we have the following properties.

Lemma 8.1. Let k = ℚ(μ_{p}) and k_{∞} = k(μ_{p^{∞}}, p^{p^{−∞}}). Then the extension k_{∞}/k is (i) totally ramified at the unique place over p, in particular there is just one

prime of k_{∞} above p. (ii) unramified outside p.

Proof. (The proof arose from a discussion with Y. Hachimori) It suffices to prove these statements for the finite extensions k(μ_{p^{n}}, p^{p^{−n}}.) Then item (ii) follows from [2, Lemma 5].

Now consider the local extensions K = ℚ_{p}(μ_{p^{n}}) and L = K(p^{p^{−n}}) of ℚ_{p}. Since the extension ℚ_{p}(p^{p^{−1}})/ℚ_{p} is not Galois, no pth root of p can be contained in the cyclic extension K/ℚ_{p}. Hence, it follows from Kummer theory that the degree of L over K is [L : K] = p^{n}, i.e. [L : ℚ_{p}] = [ℚ(μ_{p^{n}}, p^{p^{−n}}) : ℚ](= (p − 1)p^{2n−1}) and in particular p does not split in k(μ_{p^{n}}, p^{p^{−n}}). Since the maximal abelian quotient G^{ab} of G = G(L/ℚ_{p}) ≅ G(L/K) ⋊ G(K/ℚ_{p}) is isomorphic to

G^{ab} ≅ G(L/K)_{G(K/ℚ_{p})} ⊕ G(K/ℚ_{p}) = G(K/ℚ_{p})

(note that G(L/K) ≅ ℤ/p^{n}(1) has no non-zero G(K/ℚ_{p})-invariant quotient because G(K/ℚ_{p}) acts via the cyclotomic character on G(L/K)), the only cyclic extensions of ℚ_{p} in L are contained in K and cannot be unramified. Hence p is totally ramified in k(μ_{p^{n}}, p^{p^{−n}}) for all n. □

Let L be the maximal unramified abelian p-extension of k_{∞}. Then its Galois group X_{nr} = G(L/k_{∞}) is a compact Λ(G)-module where Λ(G) denotes the Iwasawa algebra

Λ(G) = ℤ_{p}[[G]] ≅ ℤ_{p}[[X, Y; σ, δ]].

If we define H_{n} and Γ_{n} to be the unique subgroup of H and Γ of index p^{n}, respectively, then G_{n} := H_{n} ⋊ Γ_{n} is an open normal subgroup of G of index p^{2n}. Let us write k_{n,m} := k_{∞}^{H_{n}⋊Γ_{n}} = k(μ_{p^{m+1}}, α^{p^{−n}}), 0 ≤ n, m ≤ ∞, for the intermediate fields of k_{∞}/k. Analogous to the module X_{nr} we have the Λ(Γ)- modules X_{n}r(k_{n,∞} for every field k_{n,∞} = k_{cycl}(α^{p^{−n}}). By class field theory there are canonical isomorphisms of Λ(Γ)- and Λ(G)-modules respectively:

X_{nr}(k_{n,∞}) ≅ lim ; ⟵_{m}Cl(k_{n,m})(p)
X_{nr} ≅ lim ; ⟵_{n,m}Cl(k_{n,m})(p)
≅ lim ; ⟵_{n}X_{nr}(k_{n,∞}).

Here Cl(K)(p) denotes the p-part of the ideal class group of a (possibly infinite) extension K of ℚ. Y. Ochi has proven that X_{nr} is a finitely generated torsion Λ(G)-module.

Theorem 8.2. Let μ_{p} ⊆ k and k_{∞} = k(A_{p^{∞}}) as above and suppose that there is only one prime 𝔭 of k above p and that 𝔭 is totally ramified in k_{∞}/k. If k_{∞}/k is unramified outside 𝔭, then there are canonical isomorphisms

(X_{nr})_{H_{n}} ≅ X_{nr}(k_{n,∞})

and

(X_{nr})_{G_{n}} ≅ Cl(k_{n,n})(p).

If, in addition, the μ-invariant μ(X_{nr}(k_{cycl})) = 0 is zero, then the following holds:

(i) The μ-invariant is zero for any intermediate field μ(X_{nr}(k_{n,∞})) = 0 and X_{nr} is a finitely generated Λ(H)-module. In particular, the image of X_{nr} in the quotient category is a (possibly zero) completely faithful, hence cyclic object.

(ii) There is an injective homomorphism of Λ(G)-modules

Cl(k_{∞})(p)^{∨} → E^{1}(X_{nr})

with pseudo-null cokernel, i.e. the Pontryagin dual of the direct limit of the p-ideal class groups is pseudo-isomorphic to the Iwasawa-adjoint of the inverse limit. In particular, if X_{nr} is pseudo-null, then the ideal class group Cl(k_{∞})(p) = 0 vanishes.

(iii) There exists a constant c ≥ 0 independent of n such that

λ(X_{nr}(k_{n,∞})) = rk_{Λ(H)}(X_{nr})p^{n} + c

holds for all sufficiently large n.

We should mention that until now no single example of X_{nr} not being pseudonull is known. Before we can prove the theorem we need some preparation. For a discrete G_{k}-module N we set N^{∗} := Hom_{ℤ_{p}}(N, ℚ_{p}/ℤ_{p}) where this module is endowed with the discrete topology.

Lemma 8.3. For every Λ-module M there is a natural exact sequence

0 → (lim ; ⟶_{n}M_{G_{n}}(p))^{∨} → E^{1}(M) → (lim ; ⟶_{n}H^{1}(G_{n}, M^{∨})^{∗} ⊗ ℚ_{p}/ℤ_{p})^{∨} → 0.

In particular, if M is pseudo-null, then the direct limit lim ; ⟶_{n}M_{G_{n}}(p) = 0 vanishes.

Proof. This is just the Pontryagin dual of [24, (5.4.13)(i)] for r = 1. □

Proposition 8.4. Let M be a Λ(G)-module which is finitely generated over Λ(H).

(i) Assume that M_{G_{n}} is finite. Then (M_{H_{n}})^{Γ_{n}} is finite, too. (ii) Assume that M has no non-zero pseudo-null submodule. Then it holds

that

H^{1}(G_{n}, M^{∨}) ≅ ((M_{H_{n}})^{Γ_{n}})^{∨}.

If, in addition, M_{G_{n}} is finite for all n, then we obtain an isomorphism

(lim ; ⟶_{n}M_{G_{n}})^{∨} ≅ E^{1}(M).

(iii) Assume again that M_{G_{n}} is finite for all n. Then there is a canonical

injection

(lim ; ⟶_{n}M_{G_{n}})^{∨} → E^{1}(M)

with pseudo-null cokernel.

Proof. The first item follows from the exact sequence

0 → (M_{H_{n}})^{Γ_{n}} → M_{H_{n}} → M_{H_{n}} → M_{G_{n}} → 0

noting that M_{H_{n}} is a finitely generated ℤ_{p} -module. Now assume that M has no non-zero pseudo-null submodule, i.e. M is a torsion-free Λ(H)-module (cf. [25, lem 6.4 ]). Thus (M^{H_{n}})_{Γ_{n}} = 0 and the first part of (ii) follows from the Hochschild-Serre long exact sequence

0 → ((M_{H_{n}})^{Γ_{n}})^{∨} → H^{1}(G_{n}, M^{∨}) → ((M^{H_{n}})_{Γ_{n}})^{∨} = 0

while the second part is a consequence of the first part, (i) and lemma 8.3. To prove (iii) we denote by M_{0} the maximal pseudo-null submodule of M and we set N := M/M_{0}. The exact sequence

0 → M_{0} → M → N → 0

induces via the long exact sequence of E^{i} the exact sequence

E^{0}(M_{0}) = 0 → E^{1}(N) → E^{1}(M) → 0 = E^{1}(M_{0}) (8.4)

using the fact that E^{0}(M_{0}) = E^{1}(M_{0}) = 0 vanishes for pseudo-null modules. On the other hand we obtain taking co-invariants

0 = N^{H_{n}} → (M_{0})_{H_{n}} → M_{H_{n}} → N_{H_{n}} → 0

and

(N_{H_{n}})^{Γ_{n}} → (M_{0})_{G_{n}} → (M)_{G_{n}} → (N)_{G_{n}} → 0.

First note that (N)_{G_{n}} and hence (N_{H_{n}})^{Γ_{n}} are finite for all n by (i). Thus also (M_{0})_{G_{n}} is finite for all n. So, by lemma 8.3, lim ; ⟶_{n}M_{G_{n}} ≅ lim ; ⟶_{n}N_{G_{n}} and the result

follows from the above isomorphism 8.4 and (ii). □

Proof. (of the theorem) The first statement is proved by the same arguments as for ℤ_{p}-extensions, see e.g. [24, lem. 11.1.5]. Now assume that the μ-invariant vanishes. Nakayama’s lemma tells that X_{nr} is a finitely generated Λ(H)-module, i.e. X(k_{n,∞}) = (X_{nr})_{H_{n}} a finitely generated ℤ_{p}-module and thus μ(X(k_{n,∞})) = 0. Item (ii) follows from the above proposition while (iv) is a consequence of the next lemma. □

Lemma 8.5. Let M be a Λ(G)-module which is finitely generated as Λ(H)-module of rank rk_{Λ(H)}M = d. Then there is a constant c ≥ 0 such that

rk_{ℤ_{p}}M_{H_{n}} = dp^{n} + c

for all sufficiently large n.

Proof. Since M is finitely generated as Λ := Λ(H) =≅ ℤ_{p}[[X]]-module, by the (commutative) structure theory we obtain an exact sequence

0 → M/M_{0} → Λ(H)^{d} ⊕ ⨁_{i=0}^{s} Λ/ΛF_{i} → E → 0,

where M_{0} denotes the maximal finite submodule of M, E is finite and the F_{i} ϵ Λ, 0 ≤ i ≤ s, are Weierstrass polynomials. Taking H_{n}-coinvariants and tensoring with ℚ_{p} one sees immediately that

rk_{ℤ_{p}}M_{H_{n}} = rk_{ℤ_{p}}ℤ_{p}[H/H_{n}]^{d} + ∑_{i=0}^{s} rk_{ℤ_{p}}(Λ/ΛF_{i})_{H_{n}}

holds. Since the second summand is bounded by ∑_{i=0}^{s} rk_{ℤ_{p}}Λ/ΛF_{i} < ∞, it becomes stationary for large n and the result follows. □

The asymptotic behaviour of the λ-invariant suggest the following asymptotic growth of the order of the p-ideal class group. If e_{n} denotes the precise exponent of #Cl(k_{n,n})(p) = p^{e_{n}} with respect to p, then I guess that

e_{n} = (rk_{Λ(H)}X_{nr} · n + O(1))p^{n}

holds.

We finish this paper with a corollary one certain units which are linked to the module X_{nr} via Kummer theory. To this end, we denote by O_{K} the ring of integers of a number field K and its units by E(K) := O_{K}^{×}. We set E(k_{∞}) = \underrightarrow{lim}E(k′)

where the direct limit runs through all finite subextensions k|k′|k_{∞}. Then

E(k_{∞}) := (E(k_{∞}) ⊗_{ℤ} ℚ_{p}/ℤ_{p})^{∨}

is a compact Λ(G)-module which fits into the following exact and commutative diagram (see [27, prop. 4.15])

[figure: 0; Cl(k_{∞})(p)^{∨}; X_{Σ}(−1); E(k_{∞}); 0,; 0; Cl(k_{∞})(p)^{∨}; tor_{Λ}X_{Σ}(−1); tor_{Λ}E(k_{∞}); 0.]

Here, Σ := S_{p} ∪ S_{∞} and X_{Σ}(−1) = H^{1}(G_{Σ}(k_{∞}), μ_{p^{∞}})^{∨} is the first negative Tate twist of the Galois group G_{Σ}(k_{∞})^{ab}(p) of the maximal outside Σ unramified abelian p-extension k_{Σ}^{ab}(p) over k_{∞}. With other words, G_{Σ}(k_{∞}) is the Galois group of the maximal outside Σ unramified extension k_{Σ} of k over k_{∞} and G_{Σ}(k_{∞})^{ab}(p) its maximal abelian pro-p quotient. It follows from [27, thm 3.1.5, cor. 3.1.6], that tor_{Λ}X_{Σ}(−1) is pseudo-isomorphic (in the quotient category) to E^{1}(X_{nr}). Finally, as a consequence of the above theorem we obtain

Corollary 8.6. Under the assumptions of the theorem the torsion submodule tor_{Λ}E(k_{∞}) of E(k_{∞}) is pseudo-null.

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