Iwasawa Theory

Chapter 2
Module theory

2.1. Pseudo-isomorphisms

Definition 2.1.1. §

For an integral domain $R$ , a pseudo-null $R$ -module is an $R$ -module $M$ with annihilator ${Ann}_{R} (M)$ of height at least $2$ .

Definition 2.1.2. §

Let $R$ be an integral domain. An $R$ -module homomorphism $f : A \to B$ is a pseudo-isomorphism if it has pseudo-null kernel and cokernel.

The existence of a pseudo-isomorphism from one object to another is not in general an equivalence relation on the category of finitely generated $R$ -modules, as it is not symmetric.

Example 2.1.3. §

The quotient of $𝔽_{p} [x, y]$ by the maximal ideal $(x, y)$ of height $2$ is pseudo-null. However, as $(x, y)$ is not principal, there is no pseudo-isomorphism $𝔽_{p} [x, y] \to (x, y)$ .

Nevertheless, we can make the following definition, which does provide an equivalence relation.

Definition 2.1.4. §

We say that two modules $A$ and $B$ over an integral domain $R$ are pseudo-isomorphic if $A_{𝔭} ≅ B_{𝔭}$ for all height one prime ideals $𝔭$ of $R$ .

Notation 2.1.5. §

We write $A ≃ B$ if $A$ and $B$ are pseudo-isomorphic modules over an integral domain $R$ .

If there exists a pseudo-isomorphism from one $R$ -module to another, then they are pseudo-isomorphic. Recall that a prime ideal $𝔭$ of $R$ lies in the support of a $R$ -module $A$ if and only if $A_{𝔭} \neq 0$ .

Lemma 2.1.6. §

Let $f : A \to B$ be a pseudo-isomorphism of $R$ -modules, where $R$ is an integral domain. Then $A$ and $B$ are pseudo-isomorphic.

Proof.

Since localization is an exact functor, for any height one prime ideal $𝔭$ of $R$ , we have an exact sequence

0 \to {(\ker f)}_{𝔭} \to A_{𝔭} \to B_{𝔭} \to {(coker f)}_{𝔭} \to 0 .

Since any prime ideal $𝔮$ in the support of ${Ann}_{R} (\ker f)$ or ${Ann}_{R} (coker f)$ has height at least $2$ , while $R_{𝔭}$ has Krull dimension one, we have $𝔮 R_{𝔭} = R_{𝔭}$ . Since ${(\ker f)}_{𝔭}$ and ${(coker f)}_{𝔭}$ then have no prime ideals in their support, they are both zero. □

Lemma 2.1.7. §

Let $A$ and $B$ be modules over a commutative ring $R$ with $A$ finitely presented, and let $S$ be a multiplicative subset of $R$ . Then we have a canonical isomorphism

{Hom}_{S^{- 1} R} (S^{- 1} A, S^{- 1} B) ≅ S^{- 1} {Hom}_{R} (A, B) .

Proof.

S^{- 1} 𝐴≅ S^{- 1} R \otimes_{R} A,

adjointness of $Hom$ and $\otimes$ yields

{Hom}_{S^{- 1} R} (S^{- 1} A, S^{- 1} B) ≅ {Hom}_{R} (A, S^{- 1} B) .

(2.1.1)

The result now follows from (2.1.1) and

{Hom}_{R} (A, S^{- 1} R \otimes_{R} B) ≅ S^{- 1} R \otimes_{R} {Hom}_{R} (A, B) .

(2.1.2)

To see that (2.1.2) holds, note first that it holds if $A = R$ . It then holds for every free $R$ -module $A$ of finite rank, as finite direct sums commute with $Hom$ in the first variable and direct sums commute with tensor products. In general, choose a resolution

P_{1} \to P_{0} \to A

with $P_{0}$ and $P_{1}$ finitely generated free $R$ -modules, and use the fact that the contravariant functors of $A$ in question are exact on right-exact sequences of $R$ -modules, in particular as $S^{- 1} R$ is $R$ -flat. □

We recall from the theory of primary decomposition that every ideal $I$ in a noetherian ring is a minimal finite intersection of primary ideals, and the minimal ideals among the finitely many associated primes of $I$ that are the radicals of these primary ideals are the isolated primes of $I$ .

Lemma 2.1.8. §

Any finitely generated torsion module over a noetherian ring $R$ has only finitely many height one prime ideals in its support.

Proof.

A prime ideal $𝔭$ is in the support of a finitely generated $R$ -module $M$ if and only if and only if it contains $I = {Ann}_{R} (M)$ . Any height one prime ideal containing $I$ is an isolated prime in its primary decomposition, so there can be only finitely many. □

From now on in this section, we use $Λ$ to denote an integrally closed noetherian domain. Note that the localization of $Λ$ at any height one prime is still a integrally closed noetherian domain, and it has a unique nonzero prime, so it is a DVR.

Lemma 2.1.9. §

Let $A$ and $B$ be torsion $Λ$ -modules. Let $X$ be the finite set of height one prime ideals in the support of $A$ or $B$ . Set

S = Λ - ⋃_{𝔭 \in X} 𝔭 .

Let $f : A \to B$ be a $Λ$ -module homomorphism. Then $f$ is a pseudo-isomorphism if and only if the localized map

S^{- 1} f : S^{- 1} A \to S^{- 1} B

is an isomorphism.

Proof.

First, note that $X$ is indeed finite by Lemma 2.1.8. Let $X = {𝔭_{1}, \dots, 𝔭_{r}}$ , where the $𝔭_{i}$ are distinct. As localization is an exact functor, it suffices to show that a finitely generated torsion $Λ$ -module $M$ with height one support in $X$ is pseudo-null if and only if $S^{- 1} M = 0$ .

If $M$ is pseudo-null, then its annihilator has height at least $2$ , so is not contained in any prime ideal of height one. Thus, for each $1 \leq i \leq r$ , there exists an element $y_{i} \in {Ann}_{Λ} (M)$ , with $y_{i} \notin 𝔭_{i}$ . Since there also exists an element $x_{i} \in 𝔭_{i}$ with $x_{i} \notin 𝔭_{j}$ for all $j \neq i$ , we have

x = y_{1} x_{2} x_{3} \dots x_{r} + x_{1} y_{2} x_{3} \dots x_{r} + \dots + x_{1} x_{2} \dots x_{r - 1} y_{r} \in S \cap {Ann}_{Λ} (M)

and so $S^{- 1} M = 0$ .

Conversely, suppose that $S^{- 1} M = 0$ . Then $M_{𝔭} = 0$ for all $𝔭 \in S$ , and hence for all height one prime ideals $𝔭$ , which is to say that for each $𝔭$ , there exists $s \in {Ann}_{Λ} (M)$ with $s \notin 𝔭$ , from which it follows that ${Ann}_{Λ} (M) ⊈ 𝔭$ for any height one prime ideal $𝔭$ . Therefore, ${Ann}_{Λ} (M)$ has height at least $2$ . □

Proposition 2.1.10. §

Let $A$ be a finitely generated, torsion $Λ$ -module. Then $A$ is pseudo-isomorphic to a direct sum $\oplus_{i = 1}^{s} Λ ∕ 𝔭_{i}^{k_{i}}$ with $𝔭_{i}$ a height one prime of $Λ$ and $k_{i} \geq 1$ for all $1 \leq i \leq s$ and for $s \geq 0$ . Moreover, this decomposition is unique up to ordering.

Proof.

Let $S$ be the complement of the union of the height one prime ideals in the support of $A$ . Then $S^{- 1} A$ is a torsion module over the principal ideal domain $S^{- 1} Λ$ , so we have an isomorphism

g : S^{- 1} A \to_{}^{\sim} ⨁_{i = 1}^{s} S^{- 1} (Λ ∕ 𝔭_{i}^{k_{i}})

with the $𝔭_{i}$ and $k_{i}$ as in the statement. By Lemma 2.1.7, there exists a $Λ$ -module homomorphism $f : A \to \oplus_{i = 1}^{s} Λ ∕ 𝔭_{i}^{k_{i}}$ with $S^{- 1} f = g$ . By Lemma 2.1.9, the map $f$ is a pseudo-isomorphism. The uniqueness is clear from the uniqueness in the structure theorem for finitely generated $S^{- 1} Λ$ -modules. □

The following is now clear.

Corollary 2.1.11. §

Two finitely generated, torsion $Λ$ -modules $A$ and $B$ are pseudo-isomorphic if and only if there exists a pseudo-isomorphism $f : A \to B$ .

Remark 2.1.12. §

A module $M$ over an integral domain $R$ is torsion if and only if $M_{(0)} = 0$ , which is to say that its localization at $0$ is trivial. In particular, the $R$ -torsion submodule of such a module $M$ is the kernel of the localization map to $M_{(0)}$ .

Lemma 2.1.13. §

Let $A$ be a finitely generated $Λ$ -module, let $T$ denote its $Λ$ -torsion submodule, and set $Z = A ∕ T$ . Then there is a pseudo-isomorphism

A \to T \oplus Z .

Proof.

Supposing without loss of generality that $T \neq 0$ , let $S$ be the complement of the union of the height one primes in the support of $T$ . Then $S^{- 1} Λ$ is a principal ideal domain, and by the structure theorem for finitely generated modules over principal ideal domains, we have a projection map

ρ^{'} : S^{- 1} A \to S^{- 1} T,

which realizes the $S^{- 1} Λ$ -torsion submodule $S^{- 1} T$ of $S^{- 1} A$ as a direct summand. (To see that $S^{- 1} T$ is the $S^{- 1} Λ$ -torsion submodule of $S^{- 1} A$ , note that it is torsion and the quotient $S^{- 1} Z = S^{- 1} A ∕ S^{- 1} T$ is $S^{- 1} Λ$ -torsion-free, as the fact that $Z \to Z_{(0)}$ is injective implies that $S^{- 1} Z \to S^{- 1} Z_{(0)} = Z_{(0)}$ is as well.) In other words, if we let $ν : A \to Z$ be the quotient map and $ν^{'}$ denote its localization, then $(ρ^{'}, ν^{'}) : S^{- 1} A \to S^{- 1} T \oplus S^{- 1} Z$ is an isomorphism.

By Lemma 2.1.7, there exist $ρ \in Hom (A, T)$ and $s \in S$ such that $ρ = s ρ^{'}$ . We consider the map

(ρ, ν) : A \to T \oplus Z .

Its localization is an isomorphism as multiplication by $s$ is an isomorphism on $S^{- 1} T$ . Since the kernel and cokernel are a subgroup and a quotient of $T$ , respectively, they are supported on $S$ , and the triviality of their localizations at $S$ implies their pseudo-nullity. Thus, $(ρ, ν)$ is a pseudo-isomorphism. □

Notation 2.1.14. §

For a $Λ$ -module $A$ , let us use

A^{*} = {Hom}_{Λ} (A, Λ)

to denote its $Λ$ -dual.

Note that Lemma 2.1.7 tells us that ${(A^{*})}_{𝔭} ≅ {(A_{𝔭})}^{*}$ for any prime ideal $𝔭$ of $Λ$ , the latter module being defined as

{(A_{𝔭})}^{*} = {Hom}_{Λ_{𝔭}} (A_{𝔭}, Λ_{𝔭}),

so we simply write $A_{𝔭}^{*}$ . Let $𝒬$ denote the quotient field of $Λ$ .

Lemma 2.1.15. §

Let $Z$ be a finitely generated, torsion-free $Λ$ -module. The map $Z \to Z^{∗∗}$ is an injective pseudo-isomorphism.

Proof.

For any height one prime ideal $𝔭$ , the modules $Z_{𝔭}$ and $Z_{𝔭}^{∗∗}$ are free, being finitely generated torsion-free modules over the principal ideal domain $Λ_{𝔭}$ . Moreover, the natural map $Z_{𝔭} \to Z_{𝔭}^{∗∗}$ is an isomorphism, being identified with the map a finite rank free $Λ_{𝔭}$ -module to its $Λ_{𝔭}$ -double dual. That is, $Z \to Z^{∗∗}$ is a pseudo-isomorphism, which is injective as $Z$ is torsion-free. □

Lemma 2.1.16. §

Let $A$ be a finitely generated $Λ$ -module. Inside $A_{(0)}^{*}$ , we have

A^{*} = ⋂_{𝔭 \in X_{1}} A_{𝔭}^{*},

where $X_{1}$ denotes the set of height $1$ primes of $Λ$

Proof.

Since $A^{*}$ is torsion-free, it sits inside each $A_{𝔭}^{*}$ , hence in the intersection. Let $f \in A_{(0)}^{*}$ lie in $A_{𝔭}^{*}$ for each $𝔭$ of height one. Then $f : A \to Λ_{𝔭}$ for all $𝔭$ , so $f$ has image in $Λ = ⋂_{𝔭 \in X_{1}} Λ_{𝔭}$ . It follows that $f \in A^{*}$ , hence the result. □

Definition 2.1.17. §

We say that a finitely generated $Λ$ -module $A$ is reflexive if the natural map $A \to A^{∗∗}$ is an isomorphism.

Note that a reflexive $Λ$ -module is necessarily torsion-free, since the dual of a finitely generated $Λ$ -module is torsion-free.

Lemma 2.1.18. §

A finitely generated, torsion-free $Λ$ -module $Z$ is reflexive if and only if $Z$ is the intersection of the $Z_{𝔭}$ over all height one prime ideals $𝔭$ of $Λ$ .

Proof.

We note that Lemma 2.1.16 implies that

Z^{∗∗} = ⋂_{𝔭 \in X_{1}} Z_{𝔭}^{∗∗},

and we recall that the natural map $Z_{𝔭} \to Z_{𝔭}^{∗∗}$ is an isomorphism. As the diagram

A module and its double dual through localizations. A full diagram description follows.

Diagram description: A module and its double dual through localizations

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Z; column 2: Z superscript (star star).
Row 2, from left to right: column 1: intersection subscript (fraktur p in X subscript (1)) Z subscript (fraktur p); column 2: intersection subscript (fraktur p in X subscript (1)) Z superscript (star star) subscript (fraktur p).

Arrows and lines:

An arrow from Z to Z superscript (star star), without a label.
An arrow from Z to intersection subscript (fraktur p in X subscript (1)) Z subscript (fraktur p), without a label.
An arrow from Z superscript (star star) to intersection subscript (fraktur p in X subscript (1)) Z superscript (star star) subscript (fraktur p), labelled isomorphism symbol.
An arrow from intersection subscript (fraktur p in X subscript (1)) Z subscript (fraktur p) to intersection subscript (fraktur p in X subscript (1)) Z superscript (star star) subscript (fraktur p), labelled isomorphism symbol.

commutes, we have the result. □

We have the following immediate corollary of Lemmas 2.1.16 and 2.1.18.

Corollary 2.1.19. §

Let $A$ be a finitely generated $Λ$ -module. Then $A^{*}$ is reflexive.

We recall that a noetherian local ring $Ω$ is regular if its maximal ideal $𝔪$ is generated by the terms of a regular sequence ${(x_{i})}_{i = 1}^{d}$ , which is to say that $x_{i}$ is not a zero divisor in $Ω ∕ (x_{1}, \dots, x_{i - 1})$ for $1 \leq i \leq d$ . In this case, $Ω$ has Krull dimension $d$ . Equivalently, a noetherian local ring $Ω$ is regular of Krull dimension $d$ if its maximal ideal can be generated by $d$ elements, or if $\dim_{Ω ∕ 𝔪} 𝔪 ∕ 𝔪^{2} = d$ .

Proposition 2.1.20. §

Let $Ω$ be a regular local ring of Krull dimension $2$ . Then every finitely generated, reflexive $Ω$ -module is free.

Proof.

Let $A$ be a finitely generated and reflexive $Ω$ -module. As $Ω$ is regular, it has a regular sequence so a principal prime ideal $𝔭 = (f)$ . We first claim that $A ∕ 𝔭𝐴$ is a free $Ω ∕ 𝔭$ -module. As $Ω ∕ 𝔭$ is regular of Krull dimension $1$ , its maximal ideal is principal (and it is a domain), so it is a DVR. Thus, it suffices to show that $A ∕ 𝔭𝐴$ is torsion-free over $Ω ∕ 𝔭$ . For this, note that the exact sequence

0 \to {Hom}_{Ω} (A^{*}, Ω) \to_{}^{f} {Hom}_{Ω} (A^{*}, Ω) \to {Hom}_{Ω} (A^{*}, Ω ∕ 𝔭)

implies that the map

A^{∗∗} ∕ 𝔭 A^{∗∗} \to {Hom}_{Ω} (A^{*}, Ω ∕ 𝔭)

is injective. As

{Hom}_{Ω} (A^{*}, Ω ∕ 𝔭) ≅ {Hom}_{Ω ∕ 𝔭} (A^{*} ∕ 𝔭 A^{*}, Ω ∕ 𝔭)

is $Ω ∕ 𝔭$ -torsion free, the module $A^{∗∗} ∕ 𝔭 A^{∗∗}$ is $Ω ∕ 𝔭$ -torsion free. But $A$ is reflexive, so $A^{∗∗} ∕ 𝔭 A^{∗∗} ≅𝐴 ∕ 𝔭𝐴$ , proving the claim.

Next, let $s = \dim_{Ω ∕ 𝔪} A ∕ 𝔪𝐴$ , where $𝔪$ is the maximal ideal of $Ω$ . By Nakayama’s Lemma, there exists a minimal $Ω$ -generating set of $A$ with $s$ elements, which is to say a surjective map $π : Ω^{s} \to A$ . Since $A ∕ 𝔭𝐴$ is $Ω ∕ 𝔭$ -free, and by what we have just said of rank $s$ , the induced surjection ${(Ω ∕ 𝔭)}^{s} \to A ∕ 𝔭𝐴$ is necessarily an isomorphism. Therefore, multiplication by $f$ is surjective on $\ker π$ , and Nakayama’s lemma then tells us that $\ker π = 0$ . □

Remark 2.1.21. §

Any regular local ring is a UFD by a theorem of Auslander and Buchsbaum. In particular, regular local rings are integrally closed domains.

Theorem 2.1.22. §

Let $Λ$ be a regular local ring of Krull dimension at most $2$ . Let $A$ be a finitely generated $Λ$ -module. Then there exists a pseudo-isomorphism

A \to Λ^{r} \oplus ⨁_{i = 1}^{s} Λ ∕ 𝔭_{i}^{k_{i}}

for some $r, s \geq 0$ and height one primes $𝔭_{i}$ and integers $k_{i} \geq 1$ for $1 \leq i \leq s$ . Moreover, $r$ and $s$ are unique, and the prime powers are unique up to ordering.

Proof.

Suppose first that $A$ is torsion-free. By Lemma 2.1.15, the map $A \to A^{∗∗}$ is an injective pseudo-isomorphism, and by Proposition 2.1.20, we have that $A^{∗∗} ≅ Λ^{r}$ for some $r$ . This is then the unique $r$ for which there exists a pseudo-isomorphism $A \to Λ^{r}$ , being that it is then the dimension of $A_{(0)}$ over the quotient field of $Λ$ .

The result for torsion modules is Proposition 2.1.10. We can combine the torsion-free and torsion cases by applying Lemma 2.1.13 and the decompositions in each case. The uniqueness follows from the uniqueness in the two cases. □

2.2. Power series rings

Let $𝒪$ be a complete commutative local noetherian ring with maximal ideal $𝔪$ and finite residue field of characteristic $p$ . We study the ring $Λ = 𝒪 ⟦ T ⟧$ , beginning with the following analogue of the division algorithm.

Proposition 2.2.1 (Division algorithm). §

Let $f, g \in Λ$ , and suppose that $f \notin 𝔪 Λ$ . Let $n$ be the largest integer such that $f \in 𝔪 Λ + (T^{n})$ . Then we may write

g = 𝑞𝑓 + r

for a unique $q \in Λ$ and $r \in 𝒪 [T]$ with $\deg r < n$ .

Proof.

Suppose without loss of generality that $n > 0$ . Let $u \in 𝒪^{\times}$ be the coefficient of $T^{n}$ in $f$ . Let $a \in Λ$ and $b \in 𝒪 [T]$ be such that

f = a T^{n} + b,

where $\deg b < n$ . Note that $b \in 𝔪 Λ$ by choice of $n$ , and since $a - u$ lies in the maximal ideal of $Λ$ , we have $a \in Λ^{\times}$ . Let $q_{0}^{'} \in Λ$ and $r_{0} \in 𝒪 [T]$ be such that

g = q_{0}^{'} T^{n} + r_{0},

where $\deg r_{0} < n$ . Setting $q_{0} = a^{- 1} q_{0}^{'}$ , we have

g = q_{0} a T^{n} + r_{0} \equiv q_{0} f + r_{0} mod 𝔪 Λ .

Let $g_{1} = g - q_{0} f - r_{0} \in 𝔪 Λ$ , and repeat the process to obtain $q_{1} \in 𝔪 Λ$ and $r_{1} \in 𝔪 𝒪 [T]$ with $\deg r_{1} < n$ and

g_{1} \equiv q_{1} f + r_{1} mod 𝔪^{2} Λ .

Note then that

g \equiv (q_{0} + q_{1}) f + (r_{0} + r_{1}) mod 𝔪^{2} Λ .

Recursively, we may then construct

\begin{array}{rcl} q = q_{0} + q_{1} + q_{2} + \dots \in Λ & and & r = r_{0} + r_{1} + r_{2} + \dots \in 𝒪 [T] \end{array}

such that $g = 𝑞𝑓 + r$ and $\deg r < n$ .

As for uniqueness, if $g = q^{'} f + r^{'}$ with $r^{'} \in 𝒪 [T]$ and $\deg r^{'} < n$ , then

(q - q^{'}) f + (r - r^{'}) = 0 .

We then need only show that if $c \in Λ$ and $d \in 𝒪 [T]$ with $\deg d < n$ satisfy $𝑐𝑓 + d = 0$ , then $c = d = 0$ . Suppose that this is not the case, and let $k \geq 0$ be such that $c, d \in 𝔪^{k} Λ$ but not both $c$ and $d$ are contained in $𝔪^{k + 1} Λ$ . We see that $𝑐𝑓$ is congruent to a multiple of $T^{n}$ modulo $𝔪^{k + 1} Λ$ , which forces $d \in 𝔪^{k + 1} Λ$ , as $\deg d < n$ . But then $𝑐𝑓 \in 𝔪^{k + 1} Λ$ , and since $f \notin 𝔪 Λ$ , this forces $c \in 𝔪^{k + 1} Λ$ , a contradiction. □

Definition 2.2.2. §

A distinguished (or Weierstrass) polynomial $f \in Λ$ is a polynomial with leading coefficient $1$ that satisfies

f (T) \equiv T^{\deg f} mod 𝔪 Λ .

Theorem 2.2.3 (Weierstrass preparation). §

Let $g \in Λ$ with $g \notin 𝔪 Λ$ . Then there exist a unique distinguished polynomial $f$ and unit $u \in Λ^{\times}$ such that

g = 𝑢𝑓 .

Proof.

We begin with existence. Let $n$ be the maximal such that $g \in 𝔪 Λ + (T^{n})$ , and let $u_{0} \in 𝒪^{\times}$ be the coefficient of $T^{n}$ in $g$ . Using the division algorithm, write

T^{n} = 𝑞𝑔 + r

for some unique $q \in Λ$ and $r \in 𝒪 [T]$ with $\deg r < n$ . Since all of the terms of $g$ of degree less than $n$ lie in $𝔪$ , we have $r \in 𝔪 Λ$ . If we set $f = T^{n} - r$ , then $f$ is a distinguished polynomial. Moreover, the constant coefficient $q_{0}$ of $q$ satisfies $q_{0} u_{0} \equiv 1 mod T$ , so is a unit in $𝒪$ , and therefore $q \in Λ^{\times}$ . Letting $u = q^{- 1}$ , we have $g = 𝑢𝑓$ , as desired. The uniqueness of $f$ and $u$ is forced by the uniqueness of $q$ and $r$ . □

We then have the following corollaries of the Weierstrass preparation theorem.

Corollary 2.2.4. §

Suppose that $𝒪$ is a PID. Then the ring $Λ$ is a unique factorization domain.

Proof.

Let $π \in 𝔪$ be a generator. For $g \in π^{n} Λ - π^{n + 1} Λ$ , we may apply the Weierstrass preparation theorem to factor $π^{- n} g$ into a polynomial $f$ times a unit, and then use the fact that $𝒪 [T]$ is a UFD to factor $f$ into a product of irreducible polynomials, each of which is a Weierstrass polynomial times a unit. This gives the desired factorization of $g$ as a product of a power of $π$ , finitely many irreducible Weierstrass polynomials, and a unit. Clearly any other factorization is equivalent (up to unit and ordering) to such a factorization. □

Remark 2.2.5. §

In fact, it is more generally true that if $𝒪$ is a regular local ring, then so is $Λ = 𝒪 ⟦ T ⟧$ . For such $𝒪$ , the ring $𝒪 ⟦ T_{1}, T_{2}, \dots, T_{r} ⟧$ is then of course a UFD as well.

Suppose from now on that $𝒪$ is the valuation ring of a finite extension of $ℚ_{p}$ . Let $π \in 𝒪$ be a uniformizer. We may view $𝒪$ as sitting inside $ℂ_{p}$ . Given $f \in Λ$ and $a \in ℂ_{p}$ with $| a |_{p} < 1$ , the evaluation $f (a)$ converges to an element of $ℂ_{p}$ .

Corollary 2.2.6. §

Let $g \in Λ$ be nonzero. There exist only finitely many $a \in ℂ_{p}$ with $| a |_{p} < 1$ such that $g (a) = 0$ .

Proof.

By Weierstrass preparation, we have $g = π^{μ} 𝑢𝑓$ with $μ \geq 0$ , $u \in Λ^{\times}$ , and $f$ a Weiserstrass polynomial. As $u$ is a unit, one cannot have $a \in ℂ_{p}$ with $| a |_{p} < 1$ such that $u (a) = 0$ . Therefore, $g (a) = 0$ if and only if $f (a) = 0$ , and so the result follows from the fact that $f$ is a polynomial. □

Corollary 2.2.7. §

Let $g \in 𝒪 [T]$ , and let $f$ be a distinguished polynomial in $𝒪 [T]$ with $f$ dividing $g$ in $Λ$ . Then $g ∕ f \in 𝒪 [T]$ .

Proof.

Let $n = \deg f$ . Suppose $α \in ℂ_{p}$ is a root of $f$ . If $| α |_{p} > 1$ , then $0 = | f (α) |_{p} = | α |_{p}^{n} > 1$ , which is impossible. If $| α |_{p} = 1$ , then

0 = f (α) \equiv α^{n} mod 𝔪_{ℂ_{p}},

where $𝔪_{ℂ_{p}}$ denotes the maximal ideal of $ℂ_{p}$ , which is again impossible. If $| α |_{p} < 1$ , then set $q = g ∕ f \in Λ$ . Since $q (α)$ converges, we have $g (α) = 0$ . Let $𝒪^{'}$ denote the valuation ring of the splitting field of $f$ . We divide $g$ and $f$ by $T - α$ inside $𝒪^{'} [T]$ and repeat the process with the resulting polynomials, which we denote $f_{1}$ and $g_{1}$ . After $n$ iterations, we have obtain $f_{n} = 1$ , and since $f$ is monic, $g_{n}$ is the polynomial $q \in 𝒪 [T]$ . □

Next, let us consider ideals in $Λ$ .

Definition 2.2.8. §

Two elements $f, g \in Λ$ are said to be relatively prime if the only elements in $Λ$ that divide both $f$ and $g$ are units.

Lemma 2.2.9. §

Suppose that $f, g \in Λ$ are relatively prime. Then $(f, g)$ has finite index in $Λ$ .

Proof.

Suppose that $h \in (f, g)$ is a polynomial of minimal degree (which exists by Weierstrass preparation), and suppose it is exactly divisible by a power $π^{n}$ of $π$ . Assume first that $h$ has positive degree. Let $h^{'} \in Λ$ be defined by $h = π^{n} h^{'}$ . Without loss of generality, suppose that $h^{'}$ does not divide $f$ . The division algorithm produces $q \in Λ$ and $r \in 𝒪 [T]$ with $\deg r < \deg h^{'}$ such that $f = q h^{'} + r$ . Then $π^{n} r \in (f, g)$ , which forces $r = 0$ by the minimality of the degree of $h$ . But then $h^{'}$ divides $f$ , which is a contradiction, so $h$ must be of degree $0$ .

So now, suppose that $n$ is minimal such that $π^{n} \in (f, g)$ . At least one of $f$ and $g$ is not divisible by $π$ : suppose it is $f$ , and assume without loss of generality that $f$ is a distinguished polynomial. We have $(π^{n}, f) \subseteq (f, g)$ , but

Λ ∕ (π^{n}, f) ≅ (𝒪 ∕ π^{n} 𝒪) [T] ∕ (\bar{f}),

where $\bar{f}$ is the image of $f$ in $𝒪 ∕ π^{n} 𝒪 [T]$ , and the quotient ring is a finite ring by the division algorithm, as $𝒪$ has finite residue field. □

Proposition 2.2.10. §

Every prime ideal of $Λ$ is one of $0$ , $(π, T)$ , $(π)$ , or $(f)$ , where $f$ is an irreducible distinguished polynomial.

Proof.

Suppose that $𝔭$ is a nonzero prime ideal in $Λ$ with $𝔭 \neq (π)$ . By the primality of $𝔭$ , there then exists a distinguished polynomial $f$ in $𝔭$ that is irreducible and not divisible by $π$ . So choose such an $f$ : if $𝔭 = (f)$ , we are done. Otherwise, there exists $g \in 𝔭$ with $g \notin (f)$ , and therefore by Lemma 2.2.9, there exists $π^{n} \in (f, g)$ for some $n \geq 1$ . Since $𝔭$ is prime, we then have $π \in 𝔭$ , and since $f \equiv T^{\deg f} mod π$ , we have $T^{\deg f} \in 𝔭$ . Again, primality of $𝔭$ then forces $T \in 𝔭$ , and finally, $𝔭 = (π, T)$ by the maximality of $(π, T)$ . □

Remark 2.2.11. §

We have that $Λ ∕ (π) ≅ (𝒪 ∕ 𝔪) ⟦ T ⟧$ , while $Λ ∕ (f)$ for a distinguished polynomial $f$ is free of rank $\deg f$ over $𝒪$ .

Lemma 2.2.12. §

A finitely generated $Λ$ -module is pseudo-null if and only if it is finite.

Proof.

Suppose that $M$ is a finitely generated, pseudo-null $Λ$ -module. To say that ${Ann}_{Λ} (M)$ has height at least $2$ is to say that it contains two relatively prime elements, hence has finite index in $Λ$ . On the other hand, if $M$ is a finite $Λ$ -module, then

{Ann}_{Λ} (M) = ⋂_{m \in M} {Ann}_{Λ} (m),

and ${Ann}_{Λ} (m)$ must be of finite index in $Λ$ , since $m$ generates a finite $Λ$ -module isomorphic to $Λ ∕ {Ann}_{Λ} (m)$ . It follows that ${Ann}_{Λ} (M)$ has finite index in $Λ$ , and therefore has height $2$ . □

It follows that a pseudo-isomorphism of $Λ$ -modules, for $𝒪$ a valuation ring in a finite extension of $ℚ_{p}$ , is a $Λ$ -module homomorphism with finite kernel and cokernel.

Theorem 2.2.13 (Structure theorem for finitely generated $Λ$ -modules). §

Let $M$ be a finitely generated $Λ$ -module. Then there exists a pseudo-isomorphism

M \to Λ^{r} \oplus ⨁_{i = 1}^{s} Λ ∕ (f_{i}^{k_{i}}) \oplus ⨁_{j = 1}^{t} Λ ∕ (π^{l_{j}})

for some $r, s, t \geq 0$ , $k_{i} \geq 1$ and $f_{i}$ a distinguished irreducible $Λ$ -polynomial for $1 \leq i \leq s$ , and $l_{j} \geq 1$ for $1 \leq j \leq t$ . Moreover, these quantities are unique up to reordering.

Proof.

This follows directly from Theorem 2.1.22 and the fact that the height one prime ideals in $Λ$ are $(π)$ and the ideals $(f)$ for $f$ an irreducible distinguished polynomial. □

2.3. Completed group rings

For a profinite group $G$ , we use $U ⊴^{o} G$ to denote that $U$ is an open normal subgroup of $G$ .

Definition 2.3.1. §

Let $G$ be a profinite group, and let $𝒪$ be a commutative ring. We define the completed $𝒪$ -group ring of $G$ to be the inverse limit

𝒪 ⟦ G ⟧ = \underset{U ⊴^{o} G}{lim_{\leftarrow}} 𝒪 [G ∕ U]

with respect to the quotient maps $𝒪 [G ∕ V] \to 𝒪 [G ∕ U]$ for $V \leq U$ .

Remark 2.3.2. §

In the case that $G$ is finite, we have $𝒪 ⟦ G ⟧ = 𝒪 [G]$ , the usual group ring.

We shall study completed group rings only for certain very special classes of rings $𝒪$ and profinite groups $G$ . In particular, let us assume that $𝒪$ is local and complete with respect to a maximal ideal $𝔪$ , which is to say that

𝒪 ≅ \underset{n}{lim_{\leftarrow}} 𝒪 ∕ 𝔪^{n} 𝒪 .

Remark 2.3.3. §

Since $𝒪$ is complete with respect to the maximal ideal $𝔪$ , we have

𝒪 ⟦ G ⟧ ≅ \underset{\begin{array}{c} U ⊴^{o} G \\ n \geq 0 \end{array}}{lim_{\leftarrow}} (𝒪 ∕ 𝔪^{n} 𝒪) [G ∕ U]

Definition 2.3.4. §

The augmentation ideal $I_{G}$ of $𝒪 ⟦ G ⟧$ is equal to

\ker (𝒪 ⟦ G ⟧ \to_{}^{𝜖} 𝒪),

where $𝜖$ is the augmentation map, the inverse limit of the $𝒪$ -linear maps $𝒪 [G ∕ U] \to 𝒪$ that take every group element to $1$ .

Remark 2.3.5. §

The map $𝜖$ is surjective, and therefore it induces an isomorphism

𝒪 ⟦ G ⟧ ∕ I_{G} ≅ 𝒪

We require the following lemma.

Lemma 2.3.6. §

Let $k$ be a field of characteristic $p$ , and let $G$ be a finite abelian $p$ -group. Then $k [G]$ is a local ring with maximal ideal the augmentation ideal in $k [G]$ .

Proof.

Suppose that $𝐺≅ \oplus_{i = 1}^{r} ℤ ∕ p^{n_{i}} ℤ$ for some $n_{i} \geq 1$ and $r \geq 0$ , and let $g_{i}$ be the inverse image of a generator of the $i$ th component under this isomorphism. It is easy to see that

k [G] ≅𝑘 [X_{1}, X_{2}, \dots, X_{r}] ∕ (X_{1}^{p^{n_{1}}} - 1, X_{2}^{p^{n_{2}}} - 1, \dots, X_{r}^{p^{n_{r}}} - 1)

under the map that takes $g_{i}$ to $X_{i}$ . Moreover, $X_{i}^{p^{n_{i}}} - 1 = {(X_{i} - 1)}^{p^{n_{i}}}$ for each $i$ since $k$ has characteristic $p$ . Setting $T_{i} = X_{i} - 1$ the resulting ring

k [T_{1}, T_{2}, \dots, T_{r}] ∕ (T_{1}^{p^{n_{1}}}, T_{2}^{p^{n_{2}}}, \dots, T_{r}^{p^{n_{r}}})

is local with maximal ideal $(T_{1}, T_{2}, \dots, T_{r})$ . (This is well-known, but note that if $f \notin (T_{1}, T_{2}, \dots, T_{r})$ , then $f$ has nontrivial constant coefficient, and we may construct an inverse by successive approximation, working modulo higher and higher total degrees.) The inverse image of this ideal under our isomorphism is the augmentation ideal of $k [G]$ . □

We now let $𝒪$ be a commutative noetherian local ring that is complete with the topology defined by its maximal ideal $𝔪$ .

Proposition 2.3.7. §

Let $𝒪$ be a complete commutative noetherian local ring with finite residue field characteristic $p$ , and let $G$ be a topologically finitely generated abelian pro- $p$ group. Then the algebra $𝒪 ⟦ G ⟧$ is a local ring with maximal ideal $𝔪 𝒪 ⟦ G ⟧ + I_{G}$ .

Proof.

We note that

𝒪 ⟦ G ⟧ ∕ (𝔪 + I_{G}) ≅ 𝒪 ∕ 𝔪,

so $𝔪 + I_{G}$ is maximal. If $𝔐$ is any maximal ideal of $𝒪 ⟦ G ⟧$ , then we have an injection

𝒪 ∕ (𝔐 \cap 𝒪) \to 𝒪 ⟦ G ⟧ ∕ 𝔐,

which forces $𝒪 ∕ (𝔐 \cap 𝒪)$ to be a field, hence $𝔐 \cap 𝒪$ to be maximal in $𝒪$ , and therefore $𝔐 \cap 𝒪$ to be equal to $𝔪$ .

Moreover, we have

𝒪 ⟦ G ⟧ ∕ 𝔪 𝒪 ⟦ G ⟧ ≅𝑘 ⟦ G ⟧,

where $k = 𝒪 ∕ 𝔪$ . This follows from the fact that $𝔪 𝒪 ⟦ G ⟧$ is an inverse limit of a countable inverse system of modules $𝔪 \cdot (𝒪 ∕ 𝔪^{n}) [G ∕ U]$ with surjective maps, as this implies that ${lim_{\leftarrow}}^{1}$ of the system vanishes. (Here, the countability of the system is guaranteed by the assumption of finite generation on $G$ .)

The problem is reduced to showing that the augmentation ideal of $k ⟦ G ⟧$ is its only maximal ideal. As the quotient of $k ⟦ G ⟧$ by a maximal ideal surjects onto the quotient of $k [G ∕ U]$ by the image of that maximal ideal for every open normal subgroup $U$ of $G$ , it suffices to demonstrate our claim in the case of a finite abelian $p$ -group $G$ . However, that result is just Lemma 2.3.6. □

For any $r \geq 0$ , recall that

𝒪 ⟦ T_{1}, T_{2}, \dots, T_{r} ⟧ ≅ \underset{n}{lim_{\leftarrow}} 𝒪 [T_{1}, T_{2}, \dots, T_{r}] ∕ (T_{1}^{n}, T_{2}^{n}, \dots, T_{r}^{n}) .

The latter $𝒪$ -modules in the inverse limit are free of finite rank over $𝒪$ , and so can be given the $𝔪$ -adic topology, and the inverse limit then defines a topology on the power series ring itself.

The following lemma will be of use to us.

Lemma 2.3.8. §

Suppose that $𝒪$ is a complete commutative local noetherian ring with finite residue field, and let $𝔪$ denote its maximal ideal. Let $r \geq 1$ . The following sets of ideals provide bases of open neighborhoods of $0$ that all define the same topology on the ring $R = 𝒪 [T_{1}, T_{2}, \dots, T_{r}]$ :

i.

${I_{s, t} ∣ s, t \geq 1}$ , where $I_{s, t} = 𝔪^{s} R + (T_{1}^{t}, T_{2}^{t}, \dots, T_{r}^{t})$ ,

ii.

${𝔐^{n} ∣ n \geq 1}$ , where $𝔐 = 𝔪𝑅 + (T_{1}, T_{2}, \dots, T_{r})$ ,

iii.

${J_{s, t} ∣ s, t \geq 1}$ , where

J_{s, t} = 𝔪^{s} R + (ω_{t} (T_{1}), ω_{t} (T_{2}), \dots, ω_{t} (T_{r}))

and we define

ω_{n} (T) = {(T + 1)}^{p^{n}} - 1

for any $T$ and any $n \geq 0$ .

In particular, $R$ is isomorphic to the inverse limit of the quotients modulo the ideals in any of these sets.

Proof.

To show that two of the sets of ideals define the same topology is exactly to show that every ideal in each of the two sets contains an ideal in the other set. Note that

{(T_{1}, T_{2}, \dots, T_{r})}^{(t - 1) r + 1} \subseteq (T_{1}^{t}, T_{2}^{t}, \dots, T_{r}^{t}) .

We then see that

\begin{array}{rcl} I_{1, t} \supseteq 𝔐^{(t - 1) r + 1} & and & J_{1, t} \supseteq 𝔐^{(p^{t} - 1) r + 1}, \end{array}

and from this we obtain that

\begin{array}{rcl} I_{s, t} \supseteq I_{1, t}^{s} \supseteq 𝔐^{s ((t - 1) r + 1)} & and & J_{s, t} \supseteq J_{1, t}^{s} \supseteq 𝔐^{s ((p^{t} - 1) r + 1)} . \end{array}

On the other hand, we have

\begin{array}{rcl} 𝔐^{n} \supseteq I_{n, n} & and & 𝔐^{n} \supset J_{n, n}, \end{array}

where the latter containment uses that

ω_{n} (T_{i}) = \sum_{j = 1}^{p^{n}} (\binom{p^{n}}{j}) T_{i}^{j} \in (p^{n} T_{i}, p^{n - 1} T_{i}^{p}, p^{n - 2} T_{i}^{p^{2}}, \dots, T_{i}^{p^{n}}) \subset (𝔪^{n} + 𝔪^{n - 1} T_{i} + \dots + T_{i}^{n}) R \subset 𝔐^{n} .

Therefore, the topology defined by the powers of $𝔐$ agrees both with the topologies defined by the ideals $I_{s, t}$ and by the ideals $J_{s, t}$ . The final remark follows from the first part, as the set of $I_{s, t}$ defines the natural topology on the power series ring. □

Theorem 2.3.9. §

Let $𝒪$ be a complete commutative local noetherian ring with finite residue field of characteristic $p$ . Suppose that $𝐺≅ ℤ_{p}^{r}$ for some $r$ , and let ${γ_{i} ∣ 1 \leq i \leq r}$ be a generating set of $G$ . Then there is a unique topological isomorphism

𝒪 ⟦ G ⟧ \to_{}^{\sim} 𝒪 ⟦ T_{1}, T_{2}, \dots, T_{r} ⟧

that takes $γ_{i} - 1$ to $T_{i}$ .

Proof.

Let $U_{n}$ be the open subgroup of $G$ generated by ${γ_{i}^{p^{n}} ∣ 1 \leq i \leq r}$ for some $n \geq 0$ . We note that

𝒪 [G ∕ U_{n}] \to 𝒪 [T_{1}, T_{2}, \dots, T_{r}] ∕ (ω_{n} (T_{1}), ω_{n} (T_{2}), \dots, ω_{n} (T_{r}))

via the map that takes $γ_{i}$ to $T_{i} + 1$ . Moreover, note that $ω_{m} (T_{i})$ divides $ω_{n} (T_{i})$ for $m \leq n$ , and these isomorphisms between group and polynomial rings are compatible with the canonical quotient maps on both sides. Since the groups $U_{n}$ form a basis of open neighborhoods of $0$ in $G$ , we have

𝒪 ⟦ G ⟧ ≅ \underset{n}{lim_{\leftarrow}} 𝒪 [T_{1}, T_{2}, \dots, T_{r}] ∕ (ω_{n} (T_{1}), ω_{n} (T_{2}), \dots, ω_{n} (T_{r}))

On the other hand, we have

𝒪 ⟦ T_{1}, T_{2}, \dots, T_{r} ⟧ ≅ \underset{n}{lim_{\leftarrow}} 𝒪 [T_{1}, T_{2}, \dots, T_{r}] ∕ (T_{1}^{n}, T_{2}^{n} \dots, T_{r}^{n}) .

Since $𝒪 ≅ lim_{\leftarrow} 𝒪 ∕ 𝔪^{s}$ as well, that the two inverse limits are isomorphic follows from the equality of the topologies defined by the sets of ideals in (i) and (iii) of Lemma 2.3.8. □

Remark 2.3.10. §

We remark that the theorem implies that $𝒪 ⟦ ℤ_{p}^{k} ⟧$ is noetherian, as a power series ring in finitely many variables over a noetherian ring is noetherian, and Lemma 2.3.8 implies that it is complete with respect to its unique maximal ideal.

2.4. Invariants of $Λ$ -modules

Let $𝒪$ be the valuation ring of a $p$ -adic field, and let $π$ be a uniformizer of the maximal ideal $𝔪$ of $𝒪$ . Set $Λ = 𝒪 ⟦ T ⟧$ . We can use the structure theorem to construct invariants attached to a finitely generated $Λ$ -module.

Definition 2.4.1. §

Let $M$ be a finitely generated $Λ$ -module, pseudo-isomorphic to

Λ^{r} \oplus ⨁_{i = 1}^{s} Λ ∕ (f_{i}^{k_{i}}) \oplus ⨁_{j = 1}^{t} Λ ∕ (π^{l_{j}})

for some $r, s, t \geq 0$ , $k_{i} \geq 1$ and $f_{i}$ a distinguished irreducible $Λ$ -polynomial for $1 \leq i \leq s$ , and $l_{j} \geq 1$ for $1 \leq j \leq t$ .

i.

The $λ$ and $μ$ -invariants of $M$ are

\begin{array}{rcl} λ (M) = \sum_{i = 1}^{s} k_{i} \deg f_{i} & and & μ (M) = \sum_{j = 1}^{t} l_{j}, \end{array}

respectively.

ii.

The characteristic polynomial of $M$ is

char (M) = π^{μ (M)} \prod_{i = 1}^{s} f_{i}^{k_{i}},

and the characteristic ideal of $M$ is the ideal ${char}_{Λ} (M)$ of $Λ$ generated by $char (M)$ .

We remark that the characteristic polynomial is multiplicative in exact sequences, as follows from the following lemma.

Lemma 2.4.2. §

Let

0 \to A \to_{}^{ι} B \to_{}^{π} C \to 0

be a short exact sequence of finitely generated, torsion $Λ$ -modules. Then $char (B) = char (A) char (C)$ .

Proof.

Let $X$ be the set of height one prime ideals in the support of $B$ , and let $S = Λ - ⋃_{𝔭 \in X} 𝔭$ . Identifying $S^{- 1} A$ , $S^{- 1} B$ , and $S^{- 1} C$ with direct sums of quotients of $S^{- 1} Λ$ by height one prime ideals, that the characteristic ideals of these modules are multiplicative in $S^{- 1} Λ$ is a standard result in the theory of modules over a principal ideal domain. The lemma follows easily from this. □

We next consider the quotients of finitely generated, torsion $Λ$ -modules. Recall that

ω_{n} (T) = {(T + 1)}^{p^{n}} - 1

for any $n \geq 0$ .

Remark 2.4.3. §

Suppose $Γ$ is a procyclic group isomorphic to $ℤ_{p}$ , and let $γ \in Γ$ be a topological generator. Let $Γ_{n}$ denote the quotient of $Γ$ of order $p^{n}$ . Recall that we have an isomorphism $𝒪 ⟦ Γ ⟧ \to_{}^{\sim} Λ$ that takes $γ - 1$ to $T$ . Then $γ^{p^{n}} - 1$ is taken to $ω_{n}$ , so we have that

𝒪 [Γ_{n}] ≅ Λ ∕ (ω_{n}) .

We then have that

Λ ≅ \underset{n}{lim_{\leftarrow}} Λ ∕ (ω_{n}) .

Moreover, the quotient $M ∕ ω_{n} M$ of a $Λ$ -module $M$ is identified with the $Γ^{p^{n}}$ -coinvariant group $M_{Γ^{p^{n}}}$ of $M$ .

Lemma 2.4.4. §

If $M$ is a finitely generated $Λ$ -module, then the canonical maps

M \to_{}^{\sim} \underset{n}{lim_{\leftarrow}} M ∕ ω_{n} M \to_{}^{\sim} \underset{m, n}{lim_{\leftarrow}} M ∕ (π^{m}, ω_{n}) M

are isomorphisms.

Proof.

Since $Λ$ is noetherian and $M$ is finitely generated, there exists a presentation of $M$ as a $Λ$ -module:

Λ^{r} \to Λ^{s} \to M \to 0

for some $r, s \geq 0$ . Since tensor product is right exact and

Λ ∕ (π^{m}, ω_{n}) \otimes_{Λ} 𝑀≅𝑀 ∕ (π^{m}, ω_{n}) M,

we have that

{(Λ ∕ (π^{m}, ω_{n}))}^{r} \to {(Λ ∕ (π^{m}, ω_{n}))}^{s} \to M ∕ (π^{m}, ω_{n}) M \to 0 .

is exact as well. As the inverse limit is exact on finite groups, the resulting inverse limit

Λ^{r} \to Λ^{s} \to \underset{m, n}{lim_{\leftarrow}} M ∕ (π^{m}, ω_{n}) M \to 0

is exact, so there is a canonical isomorphism

M \to_{}^{\sim} \underset{m, n}{lim_{\leftarrow}} M ∕ (π^{m}, ω_{n}) M .

Since the latter map factors as

M \to \underset{n}{lim_{\leftarrow}} M ∕ ω_{n} M \to \underset{m, n}{lim_{\leftarrow}} M ∕ (π^{m}, ω_{n}) M,

we are done if we can show the second of these maps is injective. By left exactness of the inverse limit, this will follow from the injectivity of the maps

M_{n} \to \underset{m}{lim_{\leftarrow}} M_{n} ∕ π^{m} M_{n},

where we have set $M_{n} = M ∕ ω_{n} M$ . For this, note that Nakayama’s Lemma tells us that $A = ⋂_{m} π^{m} M_{n} = 0$ , since $𝜋𝐴 = A$ . □

Remark 2.4.5. §

The proof of Lemma 2.4.4 goes through with $ω_{n}$ replaced by any sequence $f_{n}$ of distinguished polynomials with $f_{m} ∣ f_{n}$ for $m \leq n$ and $f_{m} \neq f_{n}$ if $m < n$ .

For $n \geq m$ , we set $ω_{n, m} = ω_{n} ∕ ω_{m}$ . Let us also set $ω_{n, - 1} = ω_{n}$ .

Lemma 2.4.6. §

Let $M$ be a finitely generated torsion $Λ$ -module containing no elements of finite order. Then there exists an integer $n_{0} \geq - 1$ such that $ω_{n, n_{0}} M = p^{n - n_{0}} M$ for all $n \geq n_{0}$ .

Proof.

Since $M$ has no $p$ -torsion, we have $μ (M) = 0$ . The structure theorem implies the existence of a pseudo-isomorphism

ϕ : M \to ⨁_{i = 1}^{s} Λ ∕ (f_{i})

with $f_{i}$ distinguished, and which must be injective as, again, $M$ has no $p$ -torsion. As $\prod_{i = 1}^{s} f_{i}$ annihilates $M$ , we have that $T^{λ (M)}$ annihilates $M ∕ 𝜋𝑀$ . It follows that ${(T + 1)}^{p^{m}}$ acts as the identity on $M ∕ 𝜋𝑀$ for any $m$ with $p^{m} \geq λ (M)$ . Fix such an $m$ , and let $n_{0}$ be an integer such that $p^{n_{0}} \geq p^{m} (e + 1)$ , where $e + 1$ is the ramification index of $π$ in $𝒪$ .

For $𝜃 \in {End}_{Λ} (M)$ given by the action of $T + 1$ , the exact sequence

0 \to {End}_{Λ} (M) \to_{}^{π} {End}_{Λ} (M) \to {End}_{Λ} (M ∕ 𝜋𝑀)

implies that

𝜃^{p^{m}} - 1 \in π {End}_{Λ} (M) .

For any $n \geq n_{0}$ , we then have

𝜃^{p^{n}} - 1 = {((𝜃^{p^{m}} - 1) + 1)}^{p^{n - m}} - 1 \in (π^{p^{n - m}}, 𝑝𝜋) {End}_{Λ} (M) = 𝑝𝜋 {End}_{Λ} (M) .

Let $ψ \in {End}_{Λ} (M)$ with $𝜃^{p^{n}} = 1 + 𝑝𝜋𝜓$ . Since

ω_{n + 1, n} = \sum_{c = 0}^{p - 1} {(T + 1)}^{c p^{n}},

we have that $ω_{n + 1, n}$ acts on $M$ as

\sum_{c = 0}^{p - 1} {(1 + 𝑝𝜋𝜓)}^{c} \in p + \sum_{c = 0}^{p - 1} 𝑐𝑝𝜋𝜓 + p^{2} {End}_{Λ} (M) \subseteq p + 𝑝𝜋 {End}_{Λ} (M) .

For $\bar{M} = M ∕ 𝑝𝜋𝑀$ , we therefore have that $ω_{n + 1, n} \cdot \bar{M} = p \cdot \bar{M}$ . This forces

ω_{n + 1, n} \cdot M = p \cdot M

by Nakayama’s lemma, which implies the result. □

We now have the following result on the orders of quotients of finitely generated, torsion $Λ$ -modules.

Theorem 2.4.7. §

Let $M$ be a finitely generated, torsion $Λ$ -module, and let $n_{0} \geq - 1$ be such that $char (M)$ and $ω_{n, n_{0}}$ are relatively prime for all nonnegative $n \geq n_{0}$ . Set $λ = λ (M)$ and $μ = μ (M)$ . Let $q$ denote the order of the residue field $k$ of $𝒪$ , and let $e$ denote the ramification index of $𝒪$ over $ℤ_{p}$ . Then there exists an integer $ν \in ℤ$ such that

| M ∕ ω_{n, n_{0}} M | = q^{p^{n} μ + 𝑛𝑒𝜆 + ν}

for all sufficiently large $n \geq 0$ .

Proof.

Our proof consists of four steps. In the first, we treat the case of finite $M$ . In the second, we reduce to the case of direct sums of quotients of $Λ$ . In the third, we treat the quotients of $Λ$ by powers of $π$ , and in the fourth, we treat the quotients of $Λ$ by distinguished polynomials. For simplicity of notation, let us set $ω_{n}^{'} = ω_{n, n_{0}}$ .

Note first that if $M$ is finite, then $M ∕ ω_{n}^{'} 𝑀≅𝑀$ for $n$ sufficiently large, as follows from Lemma 2.4.4, noting Remark 2.4.5. In this case, $q^{ν}$ is then just the order of $M$ . To see that $ν$ is an integer and not just a rational number, note that $M$ has a filtration ${π^{i} M ∣ i \geq 0}$ and the graded quotients $π^{i} M ∕ π^{i + 1} M$ are finite-dimensional $k$ -vector spaces, so of order a power of $q$ . It follows that
$| M | = \prod_{i = 0}^{\infty} | π^{i} M ∕ π^{i + 1} M |$
is a power of $q$ as well.
In the general case, consider the map
ϕ : M → N = ⨁ i=1sΛ∕(f iki)⊕⨁ j=1tΛ∕(πlj)
constructed in Theorem 2.2.13. It has finite kernel and cokernel, and the induced maps
ϕn: M∕ωn′M → N∕ω n′N
fit into a commutative diagram
Diagram description: Kernel and cokernel comparison for an Iwasawa module map
The structural squares and triangles displayed here commute.
Objects, listed by row and column:
- Row 1, from left to right: column 2: 0; column 3: 0; column 4: 0.
- Row 2, from left to right: column 2: kernel phi; column 3: kernel phi; column 4: kernel phi subscript (n).
- Row 3, from left to right: column 2: M; column 3: M; column 4: M / omega prime subscript (n)M; column 5: 0.
- Row 4, from left to right: column 1: 0; column 2: N; column 3: N; column 4: N / omega prime subscript (n)N; column 5: 0.
- Row 5, from left to right: column 2: cokernel phi; column 3: cokernel phi; column 4: cokernel phi subscript (n); column 5: 0.
- Row 6, from left to right: column 2: 0; column 3: 0; column 4: 0.
Arrows and lines:
1. An arrow from 0 (row 1, column 2) to kernel phi (row 2, column 2), without a label.
2. An arrow from 0 (row 1, column 3) to kernel phi (row 2, column 3), without a label.
3. An arrow from 0 (row 1, column 4) to kernel phi subscript (n), without a label.
4. An arrow from kernel phi (row 2, column 2) to kernel phi (row 2, column 3), labelled omega prime subscript (n).
5. An arrow from kernel phi (row 2, column 2) to M (row 3, column 2), without a label.
6. An arrow from kernel phi (row 2, column 3) to kernel phi subscript (n), without a label.
7. An arrow from kernel phi (row 2, column 3) to M (row 3, column 3), without a label.
8. An arrow from kernel phi subscript (n) to M / omega prime subscript (n)M, without a label.
9. An arrow from M (row 3, column 2) to M (row 3, column 3), labelled omega prime subscript (n).
10. An arrow from M (row 3, column 2) to N (row 4, column 2), labelled phi.
11. An arrow from M (row 3, column 3) to M / omega prime subscript (n)M, without a label.
12. An arrow from M (row 3, column 3) to N (row 4, column 3), labelled phi.
13. An arrow from M / omega prime subscript (n)M to 0 (row 3, column 5), without a label.
14. An arrow from M / omega prime subscript (n)M to N / omega prime subscript (n)N, labelled phi subscript (n).
15. An arrow from 0 (row 4, column 1) to N (row 4, column 2), without a label.
16. An arrow from N (row 4, column 2) to N (row 4, column 3), labelled omega prime subscript (n).
17. An arrow from N (row 4, column 2) to cokernel phi (row 5, column 2), without a label.
18. An arrow from N (row 4, column 3) to N / omega prime subscript (n)N, without a label.
19. An arrow from N (row 4, column 3) to cokernel phi (row 5, column 3), without a label.
20. An arrow from N / omega prime subscript (n)N to 0 (row 4, column 5), without a label.
21. An arrow from N / omega prime subscript (n)N to cokernel phi subscript (n), without a label.
22. An arrow from cokernel phi (row 5, column 2) to cokernel phi (row 5, column 3), labelled omega prime subscript (n).
23. An arrow from cokernel phi (row 5, column 2) to 0 (row 6, column 2), without a label.
24. An arrow from cokernel phi (row 5, column 3) to cokernel phi subscript (n), without a label.
25. An arrow from cokernel phi (row 5, column 3) to 0 (row 6, column 3), without a label.
26. An arrow from cokernel phi subscript (n) to 0 (row 5, column 5), without a label.
27. An arrow from cokernel phi subscript (n) to 0 (row 6, column 4), without a label.
where the map $ω_{n}^{'} : N \to N$ is injective since one cannot have $ω_{n}^{'} g \in (f_{i}^{k_{i}})$ (or in $(π^{l_{j}})$ ) for some $i$ (resp., $j$ ) unless $g \in (f_{i}^{k_{i}})$ (resp., $(π^{l_{j}})$ ) as $ω_{n}^{'}$ is relatively prime to each $f_{i}$ by assumption (and to $π$ by definition). Now, for sufficiently large $n$ , we have that multiplication by $ω_{n}^{'}$ is the zero map on $\ker ϕ$ and $coker ϕ$ , as $\ker ϕ$ and $coker ϕ$ are finite. Therefore, the snake lemma tells us that, for such $n$ , we have $coker ϕ ≅ coker ϕ_{n}$ and an exact sequence
0 →ker⁡ϕ →ker⁡ϕn → coker ⁡ ϕ → 0.
Defining $η \geq 0$ by
qη = |ker⁡ϕ| = |ker⁡ϕn| |coker ⁡ ϕn|,
we have that
|M∕ωn′M| = qη∏ i=1s|Λ∕(ω n′,f iki)|⋅∏ j=1t|Λ∕(ω n′,πlj)|
for the same sufficiently large $n$ . This reduces the theorem to modules of the form $M = Λ ∕ (π^{l})$ for some $l \geq 1$ or $M = Λ ∕ (f)$ with $f$ a (a power of an irreducible) distinguished polynomial relatively prime to every $ω_{n}^{'}$ .
Suppose now that $M = Λ ∕ (π^{l})$ for some $l \geq 1$ . We then have
$M ∕ ω_{n}^{'} M = Λ ∕ (ω_{n}^{'}, π^{l}) ≅ (𝒪 ∕ π^{l} 𝒪) ⟦ T ⟧ ∕ (ω_{n}^{'}),$
Since $ω_{n}^{'}$ is a distinguished polynomial of degree $p^{n} - p^{n_{0}}$ , the latter ring is isomorphic to ${(𝒪 ∕ π^{l} 𝒪)}^{p^{n}}$ as an $𝒪$ -module. We therefore have that
$| M ∕ ω_{n}^{'} M | = q^{p^{n} l - p^{n_{0}} l} .$
Note that $μ (M) = l$ , and we can take $ν = - p^{n_{0}} l$ for this $M$ .
Finally, suppose that $M = Λ ∕ (f)$ for some distinguished polynomial $f$ relatively prime to every $ω_{n}^{'}$ . By Lemma 2.4.6, we have that there exists $n_{1} \geq n_{0}$ such that
$ω_{n, n_{1}} M = p^{n - n_{1}} M$
for all $n \geq n_{1}$ . We also have an exact sequence
$0 \to M ∕ ω_{n_{1}}^{'} M \to_{}^{ω_{n, n_{1}}} M ∕ ω_{n}^{'} M \to M ∕ ω_{n, n_{1}} M \to 0,$
and therefore we have
$M ∕ ω_{n, n_{1}} 𝑀≅𝑀 ∕ p^{n - n_{1}} 𝑀≅ {(𝒪 ∕ π^{e (n - n_{1})} 𝒪)}^{λ},$
the latter isomorphism being of $𝒪$ -modules. Defining $ν \in ℤ$ by
$q^{ν} = | M ∕ ω_{n_{1}}^{'} M | \cdot q^{- n_{1} 𝑒𝜆},$
we then have
$| M ∕ ω_{n}^{'} M | = q^{𝑛𝑒𝜆 + ν},$
as desired.

□

We next wish to consider results which give us conditions that allow us to compute invariants of $Λ$ -modules from their quotients. For this, the following lemma is useful.

Lemma 2.4.8. §

Let $ϕ : M \to N$ be a pseudo-isomorphism of $Λ$ -modules, and let $f \in Λ$ be a distinguished polynomial. Then the induced map $ϕ_{f} : M ∕ 𝑓𝑀 \to N ∕ 𝑓𝑁$ is also a pseudo-isomorphism, and moreover, we have

\begin{array}{rcl} | \ker ϕ_{f} | \leq | \ker ϕ | | coker ϕ | & and & | coker ϕ_{f} | \leq | coker ϕ | . \end{array}

Similarly, using $A [f]$ to denote the kernel of $f : A \to A$ for any $Λ$ -module $A$ , the induced map $_{f} ϕ : M [f] \to N [f]$ is also a pseudo-isomoprhism, and we have

\begin{array}{rcl} | \ker_{f} ϕ | \leq | \ker ϕ | & and & | {coker}_{f} ϕ | \leq | \ker ϕ | | coker ϕ | . \end{array}

Proof.

Consider first the diagram

Multiplication by f on a quotient-kernel sequence. A full diagram description follows.

Diagram description: Multiplication by f on a quotient-kernel sequence

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: M / kernel phi; column 3: N; column 4: cokernel phi; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: M / kernel phi; column 3: N; column 4: cokernel phi; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to M / kernel phi (row 1, column 2), without a label.
An arrow from M / kernel phi (row 1, column 2) to N (row 1, column 3), labelled phi.
An arrow from M / kernel phi (row 1, column 2) to M / kernel phi (row 2, column 2), labelled f.
An arrow from N (row 1, column 3) to cokernel phi (row 1, column 4), without a label.
An arrow from N (row 1, column 3) to N (row 2, column 3), labelled f.
An arrow from cokernel phi (row 1, column 4) to 0 (row 1, column 5), without a label.
An arrow from cokernel phi (row 1, column 4) to cokernel phi (row 2, column 4), labelled f.
An arrow from 0 (row 2, column 1) to M / kernel phi (row 2, column 2), without a label.
An arrow from M / kernel phi (row 2, column 2) to N (row 2, column 3), labelled phi.
An arrow from N (row 2, column 3) to cokernel phi (row 2, column 4), without a label.
An arrow from cokernel phi (row 2, column 4) to 0 (row 2, column 5), without a label.

The snake lemma then yields an exact sequence

0 \to (M ∕ \ker ϕ) [f] \to N [f] \to (coker ϕ) [f] \to M ∕ (𝑓𝑀 + \ker ϕ) \to N ∕ 𝑓𝑁 \to coker ϕ_{f} \to 0 .

(2.4.1)

The kernel of $ϕ_{f}$ has order at most the products of the orders of the kernels of the maps $M ∕ 𝑓𝑀 \to M ∕ (𝑓𝑀 + \ker ϕ)$ and $M ∕ (𝑓𝑀 + \ker ϕ) \to N ∕ 𝑓𝑁$ . The first clearly has order at most $| \ker ϕ |$ , and by (2.4.1), the second has order at most $| coker ϕ |$ . The statement on $coker ϕ_{f}$ is also clear from the exact sequence.

As for $_{f} ϕ$ , the snake lemma applied to

Multiplication by f on a kernel sequence. A full diagram description follows.

Diagram description: Multiplication by f on a kernel sequence

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: kernel phi; column 3: M; column 4: M / kernel phi; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: kernel phi; column 3: M; column 4: M / kernel phi; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to kernel phi (row 1, column 2), without a label.
An arrow from kernel phi (row 1, column 2) to M (row 1, column 3), without a label.
An arrow from kernel phi (row 1, column 2) to kernel phi (row 2, column 2), labelled f.
An arrow from M (row 1, column 3) to M / kernel phi (row 1, column 4), labelled phi.
An arrow from M (row 1, column 3) to M (row 2, column 3), labelled f.
An arrow from M / kernel phi (row 1, column 4) to 0 (row 1, column 5), without a label.
An arrow from M / kernel phi (row 1, column 4) to M / kernel phi (row 2, column 4), labelled f.
An arrow from 0 (row 2, column 1) to kernel phi (row 2, column 2), without a label.
An arrow from kernel phi (row 2, column 2) to M (row 2, column 3), without a label.
An arrow from M (row 2, column 3) to M / kernel phi (row 2, column 4), labelled phi.
An arrow from M / kernel phi (row 2, column 4) to 0 (row 2, column 5), without a label.

yields exactness of

0 \to (\ker ϕ) [f] \to M [f] \to (M ∕ \ker ϕ) [f] \to \ker ϕ ∕ f \ker ϕ,

Together with (2.4.1), this implies that $_{f} ϕ$ has finite kernel contained in $\ker ϕ$ and finite cokernel of order at most $| \ker ϕ | \cdot | coker ϕ |$ . □

Definition 2.4.9. §

Let $A$ be a finitely generated $𝒪$ -module. The $π$ -rank $r_{π} (A)$ of $A$ is the dimension of $A ∕ 𝜋𝐴$ as a vector space over $k = 𝒪 ∕ π 𝒪$ .

Proposition 2.4.10. §

Let $M$ be a finitely generated, torsion $Λ$ -module. Then $μ (M) = 0$ if and only if the quantities $r_{π} (M ∕ ω_{n} M)$ are bounded as $n$ varies.

Proof.

Consider a pseudo-isomorphism

ϕ : M \to N = ⨁_{i = 1}^{s} Λ ∕ (f_{i}^{k_{i}}) \oplus ⨁_{j = 1}^{t} Λ ∕ (π^{l_{j}}) .

Then $ϕ_{ω_{n}} : M ∕ ω_{n} M \to N ∕ ω_{n} N$ has finite kernel and cokernel of bounded order by Lemma 2.4.8, so it suffices to check the result for $N$ . Note that the $π$ -rank of $Λ ∕ (f, ω_{n})$ for $f$ a distinguished polynomial is bounded by $\deg f$ , since $Λ ∕ (f) ≅ 𝒪^{\deg f}$ as an $𝒪$ -module. On the other hand, $Λ ∕ (π^{l}, ω_{n})$ is isomorphic to ${(𝒪 ∕ π^{l})}^{p^{n}}$ as an $𝒪$ -module, so has unbounded $π$ -rank. □

Similarly, we have the following proposition for the $λ$ -invariant.

Proposition 2.4.11. §

Let $M$ be a finitely generated, torsion $Λ$ -module. Then $λ (M)$ is equal to the following quantities:

i.: ${rank}_{𝒪} M$ and
ii.: the maximal integer $λ$ such that $M$ has a quotient isomorphic to ${(𝒪 ∕ π^{n} 𝒪)}^{λ}$ as an $𝒪$ -module for every $n$ .

Proof.

Consider a pseudo-isomorphism

ϕ : M \to N = ⨁_{i = 1}^{s} Λ ∕ (f_{i}^{k_{i}}) \oplus ⨁_{j = 1}^{t} Λ ∕ (π^{l_{j}}) .

Let $n > μ (M)$ . Then $Λ ∕ (π^{l_{j}})$ has trivial $𝒪$ -rank and no quotient of the form $𝒪 ∕ π^{n} 𝒪$ , since $n > l_{j}$ . On the other hand, $Λ ∕ (f_{i}^{k_{i}})$ is isomorphic to $𝒪^{k_{i} \deg f_{i}}$ as an $𝒪$ -module by Remark 2.2.11, so has a quotient of the form ${(𝒪 ∕ π^{n} 𝒪)}^{m}$ for exactly those $m \leq k_{i} \deg f_{i}$ . Therefore, the result holds for $N$ .

By definition, $𝒪$ -rank is not affected by pseudo-isomorphism, so $λ (M) = {rank}_{𝒪} M$ . Moreover, if $λ = {rank}_{𝒪} M$ , then the quotient of $M$ modulo its $π$ -power torsion subgroup is a finitely generated torsion-free $𝒪$ -module of rank $λ$ , hence is isomorphic to $𝒪^{λ}$ and has a quotient isomorphic to ${(𝒪 ∕ π^{n} 𝒪)}^{m}$ for exactly those $m \leq λ$ . □

Finally, for finitely generated $Λ$ -modules which are not necessarily $Λ$ -torsion, we have the following result on $Λ$ -ranks.

Proposition 2.4.12. §

Let $M$ be a finitely generated $Λ$ -module. Then we have

{rank}_{Λ} (M) = {rank}_{𝒪} (M ∕ T M) - {rank}_{𝒪} (M [T]) .

Moreover, we have

{rank}_{𝒪} (M ∕ ω_{n} M) = p^{n} {rank}_{Λ} (M) + c

for some $c \geq 0$ for all sufficiently large $n$ .

Proof.

Again consider a pseudo-isomorphism

ϕ : M \to N = Λ^{r} \oplus ⨁_{i = 1}^{s} Λ ∕ (f_{i}^{k_{i}}) \oplus ⨁_{j = 1}^{t} Λ ∕ (π^{l_{j}}) .

Then ${rank}_{Λ} (M) = {rank}_{Λ} (N)$ , and by Lemma 2.4.8, we have

\begin{array}{rcl} {rank}_{𝒪} (M ∕ T M) = {rank}_{𝒪} (N ∕ T N) & and & {rank}_{𝒪} (M [T]) = {rank}_{𝒪} (N [T]), \end{array}

or more strongly, that $M [T] \to N [T]$ is a pseudo-isomorphism.

Given this, the proof of the first part is reduced to case that $M = N$ . Since $Λ ∕ T Λ$ has $𝒪$ -rank $1$ and $Λ ∕ (f)$ for a distinguished polynomial $f$ has

Λ ∕ (f, T) ≅ 𝒪 ∕ f (0) 𝒪,

we have that the $𝒪$ -rank of the latter module is nonzero, and then equal to $1$ , if and only if $T$ divides $f$ . Finally, $Λ ∕ (T, π^{l}) ≅ 𝒪 ∕ π^{l} 𝒪$ for $l \geq 1$ and so has trivial $𝒪$ -rank. It follows that ${rank}_{𝒪} (N ∕ T N) = r + s$ , where $s$ is the number of $f_{i}$ equal to $T$ . As for $N [T]$ , note that $Λ [T] = 0$ and $Λ ∕ (π^{l}) [T] = 0$ , while $Λ ∕ (f) [T]$ is nonzero, and then of $𝒪$ -rank $1$ , if and only if $T$ divides $f$ . Therefore, we have ${rank}_{𝒪} N [T] = s$ , and part a follows.

We note that ${rank}_{𝒪 ⟦ ω_{n} ⟧} (M) = p^{n} {rank}_{Λ} (M)$ , since $Λ$ has rank $p^{n}$ over $𝒪 ⟦ ω_{n} ⟧$ . The first part applied with $T$ replaced by $ω_{n}$ then implies

{rank}_{𝒪} (M ∕ ω_{n} M) = p^{n} {rank}_{Λ} (M) + {rank}_{𝒪} (M [ω_{n}]) .

It suffices then to show that ${rank}_{𝒪} (M [ω_{n}])$ is bounded in $n$ . But this follows as $ω_{n, m}$ is relatively prime to ${char}_{Λ} (M)$ for $n$ sufficiently large for all $m$ . □

2.5. Pontryagin duality

Let $A$ be a locally compact, Hausdorff topological abelian group.

Definition 2.5.1. §

The Pontryagin dual of $A$ is defined to be the topological group

A^{\lor} = {Hom}_{cts} (A, ℝ ∕ ℤ)

with the compact-open topology, which is to say, with basis of open sets of the form

B (K, U) = {f \in A^{\lor} ∣ f (K) \subseteq U},

where $K \subset A$ is compact and $U \subset ℝ ∕ ℤ$ is open.

Of course, if $f : A \to B$ is a continuous map of locally compact, Hausdorff abelian groups, then there is a natural map $f^{\lor} : B^{\lor} \to A^{\lor}$ given by $f^{\lor} (φ) = φ \circ f$ .

The following is the key theorem regarding the Pontryagin dual, which we state without proof.

Theorem 2.5.2 (Pontryagin duality). §

Let $L$ denote the category of locally compact, Hausdorff topological abelian groups, let $𝒞$ denote the category of compact, Hausdorff topological abelian groups, and let $𝒟$ denote the category of discrete topological abelian groups. Then the Pontryagin dual provides a self-inverse contravariant functor from $L$ to its itself. Moreover, it induces contravariant equivalences of categories between $𝒞$ and $𝒟$ in both directions.

Remark 2.5.3. §

If $A$ is a profinite or discrete torsion, then in fact

A^{\lor} = {Hom}_{cts} (A, ℚ ∕ ℤ),

while if $A$ is pro- $p$ or discrete $p$ -torsion, then we have

A^{\lor} = {Hom}_{cts} (A, ℚ_{p} ∕ ℤ_{p}) .

Moreover, we note that if $A$ is discrete, then every homomorphism from it is continuous. On the other hand, if $A$ is a finitely generated $ℤ_{p}$ -module, then every $ℤ_{p}$ -linear homomorphism is continuous, so

A^{\lor} = {Hom}_{ℤ_{p}} (A, ℚ_{p} ∕ ℤ_{p}) .

Remark 2.5.4. §

If $A$ has the additional structure of a topological $G$ -module for a profinite group $G$ , then $A^{\lor}$ has the continuous $G$ -action given by

(g \cdot f) (a) = f (g^{- 1} a)

for $g \in G$ , $f \in A^{\lor}$ and $a \in A$ .

Remark 2.5.5. §

Pontryagin duality induces a nondegenerate continuous pairing

A \times A^{\lor} \to ℚ_{p} ∕ ℤ_{p}, (a, f) \mapsto f (a) .

If $A$ is also a topological $G$ -module, then the latter pairing is $G$ -equivariant.

Here is another interesting result.

Proposition 2.5.6. §

a.: If $A$ is a compact, Hausdorff topological $ℤ_{p}$ -module, then $A$ is profinite.
b.: If $A$ is a discrete topological $ℤ_{p}$ -module, then $A$ is $ℤ_{p}$ -torsion.

Proof.

Let us start with part b. Since $A$ is discrete, every element $a \in A$ has $p^{n} a = 0$ for some $n \geq 0$ by continuity of the action. As for part a, we note that the dual of a compact $ℤ_{p}$ -module is $A$ a discrete $ℤ_{p}$ -module, hence $ℤ_{p}$ -torsion. Then $A^{\lor}$ is the direct limit of the finite submodules generated by any finite set of its elements, so $A$ is the topologically the inverse limit of the Pontryagin duals of those submodules, and therefore $A$ is profinite. □

Corollary 2.5.7. §

Every finite topological $ℤ_{p}$ -module has the discrete topology.

Example 2.5.8. §

Since $ℤ_{p}$ is procyclic, a continuous homomorphism from it is determined by where $1$ is sent. Since $ℤ_{p}$ is a free pro- $p$ group, we can send $1$ to any element. Therefore, we have $ℤ_{p}^{\lor} = ℚ_{p} ∕ ℤ_{p}$ .

Definition 2.5.9. §

We say an locally compact module over a profinite ring $R$ is cofinitely generated if its Pontryagin dual is a finitely generated right $R$ -module.

2.6. Iwasawa adjoints

We continue to suppose that $Λ = 𝒪 ⟦ T ⟧$ for a valuation ring $𝒪$ of a $p$ -adic field with uniformizer $π$ . Let $F$ denote the quotient field of $𝒪$ . We will be most interested in Pontryagin duals of $Λ$ -modules.

Definition 2.6.1. §

Let $ι : Λ \to Λ$ be the unique continuous $𝒪$ -linear ring homomorphism satisfying $ι (T) = {(T + 1)}^{- 1} - 1$ .

We can convert the canonical right action on the Pontryagin dual of a $Λ$ -module to a left action using an involution, as follows.

Proposition 2.6.2. §

If $M$ is a locally compact, Hausdorff topological $Λ$ -module, then $M^{\lor}$ is as well, with respect to the action

(λ \cdot φ) (m) = φ (ι (λ) m)

(2.6.1)

for $λ \in Λ$ , $m \in M$ , and $φ \in M^{\lor}$ .

Let $s \geq 0$ be such that $π^{s}$ generates the different of $𝒪 ∕ ℤ_{p}$ . Then the $𝒪$ -balanced pairing

𝒪 \times 𝒪 \to ℤ_{p}, (x, y) \mapsto {Tr}_{F ∕ ℚ_{p}} (π^{- s} 𝑥𝑦)

(2.6.2)

is perfect. For a locally compact, Hausdorff topological $Λ$ -module $M$ , we have a left $Λ$ -module structure on ${Hom}_{𝒪} (M, F ∕ 𝒪)$ as in (2.6.1), with $φ$ now in ${Hom}_{𝒪} (M, F ∕ 𝒪)$ .

Proposition 2.6.3. §

For every finitely or cofinitely generated $𝒪$ -module $A$ , there exists an isomorphism

A^{\lor} ≅ {Hom}_{𝒪} (A, F ∕ 𝒪) .

These can be chosen to be natural in $A$ in a manner that is canonical up to the choice of uniformizer $π$ of $𝒪$ . Moreover, if $A$ is a $Λ$ -module, then the isomorphism is of $Λ$ -modules.

Proof.

The perfect pairing of (2.6.2) yields an isomorphism $𝒪 ≅ Hom (𝒪, ℤ_{p})$ and therefore the composite $𝒪$ -module isomorphism

F ∕ 𝒪 ≅ 𝒪 \otimes_{ℤ_{p}} ℚ_{p} ∕ ℤ_{p} ≅ {Hom}_{ℤ_{p}} (𝒪, ℤ_{p}) \otimes_{ℤ_{p}} ℚ_{p} ∕ ℤ_{p} ≅ {Hom}_{ℤ_{p}} (𝒪, ℚ_{p} ∕ ℤ_{p}) .

Since $A$ is (co)finitely generated over $ℤ_{p}$ , we have the following $Λ$ -module isomorphisms

A^{\lor} ≅ {Hom}_{ℤ_{p}} (A, ℚ_{p} ∕ ℤ_{p}) ≅ {Hom}_{𝒪} (A, {Hom}_{ℤ_{p}} (𝒪, ℚ_{p} ∕ ℤ_{p})) ≅ {Hom}_{𝒪} (A, F ∕ 𝒪),

and naturality is easily checked. □

We have the following analogue of Proposition 2.5.6.

Proposition 2.6.4. §

a.: Every compact $Λ$ -module is an inverse limit of finite $Λ$ -modules.
b.: Every discrete $Λ$ -module is a direct limit of finite $Λ$ -modules.

Proof.

By Pontryagin duality, it suffices to prove part b. For this, we again note that the continuity of the $Λ$ -action on a discrete module $M$ ensures that, for any $m \in M$ , the annihilator ${Ann}_{Λ} (m)$ is an open ideal of $Λ$ . But then $M$ is the union of its finite $Λ$ -submodules $Λ \cdot m$ for $m \in M$ . □

Note that if $M$ is a finitely generated $Λ$ -module, we endow it with the topology under which $(π^{m}, ω_{n}) M$ forms a basis of open submodules of $M$ .

Definition 2.6.5. §

Let $M$ be a finitely generated, torsion $Λ$ -module, and set $M_{n} = M ∕ ω_{n, m} M$ for $n \geq m$ and some fixed $m \geq - 1$ with $ω_{n, m}$ relatively prime to $char (M)$ for all $n$ . Set

α (M) = \underset{n}{lim_{\leftarrow}} M_{n}^{\lor} ≅ {(\underset{n}{lim_{\to}} M_{n})}^{\lor},

where $M_{n} \to M_{n + 1}$ is induced by the map $m \mapsto ω_{n + 1, n} m$ on $M$ . Then the $Λ$ -module $α (M)$ is called the Iwasawa adjoint to $M$ .

Remarks 2.6.6. §

a.: We leave it to the reader to check that the definition of $α (M)$ does not depend on $m$ .
b.: If $ϕ : M \to N$ is a $Λ$ -module homomorphism, where $M$ and $N$ are finitely generated and $Λ$ -torsion, then we obtain a natural $Λ$ module homomorphism $α (ϕ) : α (N) \to α (M)$ .

Lemma 2.6.7. §

The contravariant functor $α$ is left exact.

Proof.

To see the exactness, note that

M_{n} ≅𝑀 \otimes_{Λ} Λ ∕ (ω_{n, m}),

the tensor product is right exact, the Pontryagin dual is an exact contravariant functor, and the inverse limit is exact on finite abelian groups. □

Lemma 2.6.8. §

If $M$ is a finite $Λ$ -module, then $α (M) = 0$ .

Proof.

Since $M$ is finite, the map $ω_{n, m} : M \to M$ is zero for $n$ sufficiently large relative to a fixed $m$ . The result follows. □

Lemma 2.6.9. §

If $M$ is a finitely generated, torsion $Λ$ -module with $μ (M) = 0$ , then there are natural isomorphisms

α (M) ≅ {Hom}_{ℤ_{p}} (M, ℤ_{p}) ≅ {Hom}_{𝒪} (M, 𝒪)

as $Λ$ -modules. Here, $Λ$ acts on both ${Hom}_{ℤ_{p}} (M, ℤ_{p})$ and ${Hom}_{𝒪} (M, 𝒪)$ by

(λ \cdot ϕ) (m) = ϕ (ι (λ) m) .

Proof.

Let $N$ be the $p$ -power torsion submodule of $M$ . By Lemma 2.6.7 and Lemma 2.6.8, the map $α (M ∕ N) \to α (M)$ is an isomorphism, so we can and do suppose that $M$ is $p$ -torsion-free.

Since $M$ is finitely generated over $ℤ_{p}$ , we have that for sufficiently large $m$ and $n \geq m$ that $ω_{n, m}$ acts on $M$ by multiplication by $p^{n - m}$ by Lemma 2.4.6. Therefore, we see that

α (M) ≅ \underset{n}{lim_{\leftarrow}} {(M ∕ p^{n} M)}^{\lor} ≅ \underset{n}{lim_{\leftarrow}} {Hom}_{ℤ_{p}} (M ∕ p^{n} M, ℤ ∕ p^{n} ℤ) ≅ {Hom}_{ℤ_{p}} (M, ℤ_{p}) .

For the other isomorphism, we note that

{Hom}_{ℤ_{p}} (M, ℤ_{p}) ≅ {Hom}_{ℤ_{p}} (M \otimes_{𝒪} 𝒪, ℤ_{p}) ≅ {Hom}_{𝒪} (M, Hom (𝒪, ℤ_{p})) ≅ {Hom}_{𝒪} (M, 𝒪),

the latter isomorphism using the pairing of (2.6.2), and all of these isomorphisms are of $Λ$ -modules. □

Proposition 2.6.10. §

Let $ϕ : M \to N$ be a pseudo-isomorphism of finitely generated, torsion $Λ$ -modules. Then the induced map $α (ϕ) : α (N) \to α (M)$ an injective pseudo-isomorphism.

Proof.

As the inverse limit is exact on finite modules, in order to show that $α (ϕ)$ is a pseudo-isomorphism it suffices to show that the maps $N_{n}^{\lor} \to M_{n}^{\lor}$ have kernel and cokernel of bounded order. By exactness of the Pontryagin dual, this reduces to proving that $M_{n} \to N_{n}$ has kernel and cokernel of bounded order, which follows from Lemma 2.4.8.

Finally, by Lemma 2.6.7, we have that the sequence

0 \to α (coker ϕ) \to α (N) \to_{}^{α (ϕ)} α (M)

is exact. The injectivity of $α (ϕ)$ then follows from Corollary 2.6.8. □

Definition 2.6.11. §

For a $Λ$ -module $M$ , we let $M^{ι}$ denote the $Λ$ -module that is $M$ as a set but on which the $Λ$ -action $\cdot_{ι}$ is

λ \cdot_{ι} m = ι (λ) m

for $λ \in Λ$ and $m \in M$ .

Lemma 2.6.12. §

a.: For any positive integer $ℓ$ , we have $α (Λ ∕ (π^{l})) ≅ Λ ∕ (π^{l})$ .
b.: For any distinguished polynomial $f$ , we have $α (Λ ∕ (f)) ≅ Λ ∕ (ι (f))$ .

Proof.

For part a, set $γ = T + 1$ and let $M = Λ ∕ (π^{l})$ . Then any element in $M_{n} = M ∕ ω_{n} M$ (taking $m = - 1$ ) may be uniquely written as

f = \sum_{i = 0}^{p^{n} - 1} a_{i} γ^{i}

modulo $ω_{n} = γ^{p^{n}} - 1$ , for some $a_{i} \in 𝒪 ∕ π^{l} 𝒪$ for $0 \leq i \leq p^{n} - 1$ . Let us identify $M_{n}^{\lor}$ with ${Hom}_{𝒪} (M_{n}, F ∕ 𝒪)$ as in Proposition 2.6.3. We define a map

ψ_{n} : M_{n} \to M_{n}^{\lor}

by setting

ψ_{n} (f) (γ^{i}) = \frac{a_{i}}{π^{l}},

and extending $𝒪$ -linearly. Then $ψ_{n}$ is clearly an injective homomorphism, and it is also easily seen that the $ψ_{n} (γ^{i})$ form a $𝒪$ -basis of $M_{n}^{\lor}$ , so $ψ_{n}$ is surjective as well. Moreover, $ψ_{n}$ is a map of $Λ$ -modules as

ψ_{n} (𝛾𝑓) (γ^{i}) = \frac{a_{i - 1}}{π^{l}} = ψ_{n} (f) (γ^{i - 1}) = (γ \cdot ψ_{n} (f)) (γ^{i}) .

The diagram

Compatibility with dual transition maps. A full diagram description follows.

Diagram description: Compatibility with dual transition maps

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: M subscript (n plus 1); column 2: M subscript (n plus 1) superscript (vee).
Row 2, from left to right: column 1: M subscript (n); column 2: M subscript (n) superscript (vee).

Arrows and lines:

An arrow from M subscript (n plus 1) to M subscript (n plus 1) superscript (vee), labelled psi subscript (n plus 1).
An arrow from M subscript (n plus 1) to M subscript (n), without a label.
An arrow from M subscript (n plus 1) superscript (vee) to M subscript (n) superscript (vee), labelled omega subscript (n plus 1,n) superscript (vee).
An arrow from M subscript (n) to M subscript (n) superscript (vee), labelled psi subscript (n).

commutes since

ω_{n + 1, n}^{\lor} (ψ_{n + 1} (f)) (γ^{i}) = ψ_{n + 1} (f) (ι (ω_{n + 1, n}) γ^{i}) = \sum_{j = 0}^{p - 1} ψ_{n + 1} (f) (γ^{i + p^{n} j}) = ψ_{n} (f) (γ^{i}) .

In the inverse limit, we obtain $α (Λ ∕ (π^{l})) ≅ Λ ∕ (π^{l})$ .

For part b, suppose that $M = Λ ∕ (f)$ with $f$ a distinguished polynomial of degree $d$ . Let us define $𝜖 : Λ \to 𝒪$ by setting $𝜖 (g)$ equal to the coefficient of $T^{d - 1}$ in $r$ , where $r \in 𝒪 [T]$ is the unique polynomial of degree less than $d$ with $g = 𝑞𝑓 + r$ for some $q \in Λ$ . We then define

𝜃 : Λ ∕ (f) \to {Hom}_{𝒪} {(Λ ∕ (f), 𝒪)}^{ι}

𝜃 (\bar{g}) (\bar{h}) = 𝜖 (𝑔h),

where $\bar{g}, \bar{h} \in Λ ∕ (f)$ and $g, h \in Λ$ are lifts of $\bar{g}$ and $\bar{h}$ respectively. This is clearly well-defined, and moreover it is a $Λ$ -module homomorphism, since

𝜃 (λ \bar{g}) (\bar{h}) = 𝜖 (𝜆𝑔h) = 𝜃 (\bar{g}) (λ \bar{h}) = (λ \cdot 𝜃 (\bar{g})) (\bar{h}) .

If $r \in 𝒪 [T]$ is nonzero of degree less $k$ than $d$ , then letting $\bar{r}$ denote the image of $r$ in $Λ ∕ (f)$ , we have

𝜃 (\bar{r}) (T^{d - 1 - k}) = 𝜖 (T^{d - 1 - k} r) \neq 0,

which means that $\bar{r} \notin \ker 𝜃$ , so $𝜃$ is injective. A count of $𝒪$ -ranks now tells us that $α$ has finite cokernel.

In fact, $𝜃$ is surjective, as for any $0 \leq k \leq d - 1$ and $g = \sum_{i = 0}^{d - 1} a_{i} T^{i}$ , we have that

T^{k} g - \sum_{j = 1}^{k} a_{d - j} T^{k - j} f \equiv \sum_{i = k}^{d - 1} a_{i - k} T^{i} mod π,

since $f$ is distinguished, and hence

𝜃 (T^{k}) (\bar{g}) \equiv a_{d - 1 - k} mod π .

Since the functions $ϕ_{k} \in {Hom}_{𝒪} {(Λ ∕ (f), 𝒪)}^{ι}$ with $ϕ_{k} (g) = a_{d - 1 - k}$ generate ${Hom}_{𝒪} {(Λ ∕ (f), 𝒪)}^{ι}$ and agree with the $𝜃 (T^{k})$ modulo $π$ , the $𝜃 (T^{k})$ do as well by Nakayama’s lemma. In other words, $𝜃$ is an isomorphism $Λ ∕ (f) \to α {(Λ ∕ (f))}^{ι}$ , and part b follows as ${(Λ ∕ (f))}^{ι} ≅ Λ ∕ (ι (f))$ . □

Theorem 2.6.13. §

Let $M$ be a finitely generated, torsion $Λ$ -module. Then $α (M)$ is a finitely generated, torsion $Λ$ -module that is pseudo-isomorphic to $M^{ι}$ . Moreover, $α (M)$ contains no nontrivial finite $Λ$ -submodules.

Proof.

Consider a pseudo-isomorphism

𝜃 : N = ⨁_{i = 1}^{s} Λ ∕ (f_{i}^{k_{i}}) \oplus ⨁_{j = 1}^{t} Λ ∕ (π^{l_{j}}) \to M,

which exists by the structure theorem and Proposition 2.1.11. Note that $α (𝜃) : α (M) \to α (N)$ is an injective pseudo-isomorphism. If we can show that $α (N)$ is pseudo-isomorphic to $N^{ι}$ , then clearly $α (M)$ will be pseudo-isomorphic to $M^{ι}$ , as pseudo-isomorphism is an equivalence relation on finitely generated, torsion $Λ$ -modules. Moreover, if $α (N)$ has no nonzero finite $Λ$ -submodules, then neither does $α (M)$ , being isomorphic to a submodule of $α (N)$ . By the additivity of the adjoint functor, it then suffices to assume that $M$ is a quotient of $Λ$ by a height one prime ideal, but this is covered by Lemma 2.6.12. □

2.7. The group ring of a cyclic $p$ -group

Let us suppose that $G$ is a cyclic group of order $p$ . In this section, we wish to study the structure theory of modules over $ℤ_{p} [G]$ that are finitely generated, free $ℤ_{p}$ -modules. From our study of modules over $Λ = ℤ_{p} ⟦ T ⟧$ (or representation theory over $ℚ_{p}$ ), we are easily able to classify such modules up to pseudo-isomorphism.

Let $N_{G} \in ℤ_{p} [G]$ denote the norm element, and let $X = ℤ_{p} [G] ∕ N_{G}$ , which is noncanonically isomorphic to the augmentation ideal $I_{G}$ via the map $x \mapsto (g - 1) x$ , for $g \in G$ a generator.

Lemma 2.7.1. §

Let $A$ be a finitely generated $ℤ_{p} [G]$ -module, where $G$ is cyclic of order $p$ . Then there are $s, t \geq 0$ and a homomorphism

ϕ : A \to X^{s} \oplus ℤ_{p}^{t}

with finite kernel and cokernel, and $\ker ϕ = 0$ if and only if $A$ is $p$ -torsion free.

Proof.

We remark that for a given generator $g$ of $G$ , we have an isomorphism

ψ : Λ ∕ (ω_{1}) \to_{}^{\sim} ℤ_{p} [G]

determined by $ψ (T) = g - 1$ . Any element of $A$ generates a cyclic $ℤ_{p} [G]$ -module, which may then be viewed as a quotient of $Λ ∕ (ω_{1})$ . Since $ω_{1} = T \cdot ω_{1, 0}$ and $ω_{1, 0}$ is irreducible, we have $Λ ∕ (ω_{1}, f)$ is finite for a power series $f \in Λ$ if $f$ is not a unit times a product of a power of $T$ and a power of $ω_{1, 0}$ . This leaves three possibilities for nontrivial $p$ -torsion free quotients of $Λ ∕ (f)$ , which are $Λ ∕ (f) ≅ ℤ_{p} [G]$ , $Λ ∕ (T) ≅ ℤ_{p}$ , and $Λ ∕ (ω_{1, 0}) ≅𝑋$ , since $ψ (ω_{1, 0}) = N_{G}$ . Therefore, the structure theorem for finitely generated $Λ$ -modules tells us that $A$ is pseudo-isomorphic to a direct sum of copies of the latter two $ℤ_{p} [G]$ -modules, $ℤ_{p}$ and $X$ . □

Remark 2.7.2. §

The $ℤ_{p} [G]$ -module $ℤ_{p} [G]$ is pseudo-isomorphic to $X \oplus ℤ_{p}$ . Explicitly, letting $𝜖$ denote the augmentation map, we have

\begin{array}{rcl} ℤ_{p} [G] \to X \oplus ℤ_{p}, & a \mapsto ((g - 1) a, 𝜖 (a)) \\ X \oplus ℤ_{p} \to ℤ_{p} [G], & (x, b) \mapsto x + b N_{G}, \end{array}

and both of these maps are injective with cokernel isomorphic to $ℤ ∕ 𝑝ℤ$ .

We now state the main result of the section.

Theorem 2.7.3. §

Let $A$ be a finitely generated $ℤ_{p} [G]$ -module that is $p$ -torsion free. Then there is an isomorphism

ϕ : A \to ℤ_{p} {[G]}^{r} \oplus X^{s} \oplus ℤ_{p}^{t}

of $ℤ_{p} [G]$ -modules for some $r, s, t \geq 0$ .

Proof.

Since $ℤ_{p} [G]$ is $ℤ_{p} [G]$ -projective, we also have that if $𝐵≅ ℤ_{p} {[G]}^{r}$ is the maximal $ℤ_{p} [G]$ -free quotient of $A$ , then setting $A^{'} = \ker (A \to B)$ , we have an isomorphism

𝐴≅ A^{'} \oplus ℤ_{p} {[G]}^{r},

where $A^{'}$ has no free $ℤ_{p} [G]$ -quotient. We may therefore assume that $A$ itself has no free $ℤ_{p} [G]$ -quotient.

Consider the sequence

0 \to A^{G} \to A \to I_{G} A \to 0 .

Since $A$ is $p$ -torsion free, we must have ${(I_{G} A)}^{G} = 0$ , since there is an injective pseudo-isomorphism

ψ : A \to X^{s} \oplus ℤ_{p}^{t}

for some $s, t \geq 0$ , and ${(I_{G} X)}^{G} = 0$ , while $I_{G} ℤ_{p} = 0$ . In particular, we have that $A^{G} ≅ ℤ_{p}^{t}$ , and there is an injective pseudo-isomorphism from $I_{G} A$ to $X^{u}$ for the above $u$ , since $I_{G} X ≃ X$ .

Let ${x_{1}, \dots, x_{m}}$ be a minimal generating set of $I_{G} A$ as a $ℤ_{p} [G]$ -module. We note that $ℤ_{p} [G] x_{i}$ is isomorphic to a finite index submodule of $X$ , and it is therefore a power $I_{G}^{n} X$ for some $n$ . (Here, note that $𝑝𝑋 \in I_{G} X$ .) The map $X \to I_{G}^{n}$ given by $x \mapsto {(g - 1)}^{n} x$ for a generator $g \in G$ , is an isomorphism, so in fact we have $ℤ_{p} [G] x_{i} ≅𝑋$ .

If $y \in ℤ_{p} [G] x_{i} \cap ℤ_{p} [G] x_{j}$ , then by minimality we clearly must have

y \in I_{G} x_{i} \cap I_{G} x_{j}

since $ℤ_{p} [G] x_{i} ≅𝑋$ has $I_{G} x_{i}$ as its unique maximal improper submodule. We then have $x_{i}^{'} \in ℤ_{p} [G] x_{i}$ and $x_{j}^{'} \in ℤ_{p} [G] x_{j}$ with

y = (g - 1) x_{i}^{'} = (g - 1) x_{j}^{'},

which forces $x_{i}^{'} - x_{j}^{'} \in {(I_{G} A)}^{G}$ . In other words, we have $x_{i}^{'} = x_{j}^{'}$ , contradicting minimality. We therefore have $m = s$ and $I_{G} 𝐴≅ I_{G}^{s}$ .

We now know that $A$ fits in an exact sequence

0 \to ℤ_{p}^{t} \to A \to_{}^{π} X^{s} \to 0,

which we claim splits. To see this, write $X^{s} = ⟨ x_{1}, \dots, x_{s} ⟩$ . Then $z_{i} = N_{G} {\tilde{x}}_{i}$ is an element of $ℤ_{p}^{t}$ , and the sequence splits if and only if $z_{i} \in p ℤ_{p}^{t}$ for all $i$ , since this means exactly that there exist $y_{i} \in ℤ_{p}^{t}$ with $z_{i} = p y_{i}$ and therefore $N_{G} ({\tilde{x}}_{i} - y_{i}) = 0$ , which tells us that $⟨ {\tilde{x}}_{i} - y_{i} ⟩ ≅𝑋$ . The $ℤ_{p} [G]$ -linear map taking $x_{i}$ to $\tilde{x_{i}} - y_{i}$ then determines the splitting. If not, we have that some ${\tilde{x}}_{i}$ generates a direct summand of $A$ isomorphic to $ℤ_{p} [G]$ , since $z_{i}$ (for some $i$ ) may be taken as part of a basis ${z_{i}, w_{2}, \dots, w_{t}}$ of $ℤ_{p}^{t}$ , and

A = ⟨ {\tilde{x}}_{1}, \dots, {\tilde{x}}_{s}, w_{2}, \dots, w_{t} ⟩ ≅ ℤ_{p} [G] \oplus ⟨ {\tilde{x}}_{1}, \dots, {\tilde{x}}_{i - 1}, {\tilde{x}}_{i + 1}, \dots, {\tilde{x}}_{s}, w_{2}, \dots, w_{t} ⟩ .

Since we have assumed that $A$ has no $ℤ_{p} [G]$ -quotient, the latter cannot happen, so the sequence splits, as desired. □

2.8. Eigenspaces

In this section, we suppose that $Δ$ is a finite abelian group. For a fixed prime $p$ , we consider the group

Δ^{*} = Hom (Δ, {\bar{ℚ_{p}}}^{\times})

of $p$ -adic characters of $Δ$ . Let $𝒪$ denote the $ℤ_{p}$ -algebra generated by the roots of unity of order dividing the exponent of $Δ$ , and let $E$ denote the quotient field of $𝒪$ . For $χ \in Δ^{*}$ , we let $𝒪_{χ}$ the $ℤ_{p}$ -algebra generated by the values of $χ$ , and let $E_{χ}$ denote its fraction field. Cearly, the ring $𝒪$ contains $𝒪_{χ}$ .

What we shall call eigenspaces of a $ℤ_{p} [Δ]$ -module shall in general, in fact, be quotients. Note that $χ \in Δ^{*}$ induces a map $\tilde{χ} : 𝒪 [Δ] \to 𝒪$ , which restricts to a map $ℤ_{p} [Δ] \to 𝒪_{χ}$ .

Definition 2.8.1. §

Let $A$ be an $𝒪 [Δ]$ -module, and let $ψ \in Δ^{*}$ . We define the $ψ$ -eigenspace of $A$ as

A^{ψ} = A \otimes_{𝒪 [Δ]} 𝒪,

where the map $𝒪 [Δ] \to 𝒪$ in the tensor product is $\tilde{χ}$ .

Remark 2.8.2. §

If $p ∤ | Δ |$ , then the canonical map $A \to A^{ψ}$ induces an isomorphism

{a \in A ∣ 𝛿𝑎 = ψ (δ) a for all δ \in Δ} \to_{}^{\sim} A^{ψ} .

It is the former module that might more typically be called an eigenspace. It can be interpreted as the $Δ$ -invariant group of the twist $A (ψ)$ of $A$ that is $A$ as an $𝒪$ -module but on which $δ \in Δ$ acts as $ψ (δ) δ$ does on $A$ . Our eigenspace $A^{ψ}$ is instead the $Δ$ -coinvariant group of $A (ψ)$ .

Notation 2.8.3. §

For $ψ \in Δ^{*}$ , set

e_{ψ} = \frac{1}{| Δ |} \sum_{δ \in Δ} ψ (δ) δ^{- 1} \in E [Δ] .

Note that

σ e_{ψ} = ψ (σ) e_{ψ}

for every $σ \in Δ$ , and in particular

𝒪 [Δ] e_{ψ} = 𝒪 e_{ψ}

as an $𝒪 [Δ]$ -submodule of $E [Δ]$ .

Proposition 2.8.4. §

We have a canonical decomposition of rings and $E [Δ]$ -modules

E [Δ] ≅ \prod_{ψ \in Δ^{*}} E e_{ψ} .

If $p ∤ | Δ |$ , we similarly have a decomposition

𝒪 [Δ] ≅ \prod_{ψ \in Δ^{*}} 𝒪 e_{ψ} .

Proof.

One need only remark that the $e_{ψ}$ are mutually orthogonal idempotents that sum to $1$ , as is a basic fact of character theory (in this case for a finitely generated abelian group). □

The following lemma is useful to note.

Lemma 2.8.5. §

Let $ψ \in Δ^{*}$ . For any $E [Δ]$ -module $A$ (or $𝒪 [Δ]$ -module $A$ if $p ∤ | Δ |$ ), we have $A^{ψ} = e_{ψ} A$ .

Proof.

If $a \in e_{ψ} A$ , then $e_{ψ} a = a$ , as $e_{ψ}$ is an idempotent. Conversely, if $a \in A^{(ψ)}$ , then

e_{ψ} a = \frac{1}{| Δ |} \sum_{δ \in Δ} ψ {(δ)}^{- 1} 𝛿𝑎 = a,

as $𝛿𝑎 = ψ (δ) a$ . □

The following is a consequence of Proposition 2.8.4.

Proposition 2.8.6. §

For every $E [Δ]$ -module $A$ , there is an internal direct sum decomposition

𝐴≅ ⨁_{ψ \in Δ^{*}} A^{ψ} .

If $p ∤ | Δ |$ , then this decomposition holds for $𝒪 [Δ]$ -modules as well.

Proof.

We have

𝐴≅𝐴 \otimes_{𝒪 [Δ]} 𝒪 [Δ] ≅𝐴 \otimes_{𝒪 [Δ]} ⨁_{ψ \in Δ^{*}} 𝒪 e_{ψ} ≅ ⨁_{ψ \in Δ^{*}} A \otimes_{𝒪 [Δ]} 𝒪 e_{ψ} ≅ ⨁_{ψ \in Δ^{*}} e_{ψ} A \otimes_{𝒪 [Δ]} 𝒪 [Δ] ≅ ⨁_{ψ \in Δ^{*}} A^{ψ},

with the second step being Proposition 2.8.4 and the last step following from Lemma 2.8.5. □

Eigenspaces of an $𝒪 [Δ]$ -module behave well under tensor products and homomorphism groups, as seen in the following result.

Lemma 2.8.7. §

Let $A$ and $B$ be $𝒪 [Δ]$ -modules with $A = A^{χ}$ and $B = B^{ψ}$ for some $χ, ψ \in Δ^{*}$ . We then have

A \otimes_{𝒪} B = {(A \otimes_{𝒪} B)}^{𝜒𝜓}

and

{Hom}_{𝒪} (A, B) = {Hom}_{𝒪} {(A, B)}^{χ^{- 1} ψ} .

Proof.

For $a \in A$ and $b \in B$ , we have

δ (a \otimes b) = δ (a) \otimes δ (b) = χ (δ) a \otimes ψ (δ) b = 𝜒𝜓 (δ) \cdot a \otimes b .

For $ϕ \in {Hom}_{𝒪} (A, B)$ , we have

(δ \cdot ϕ) (a) = 𝛿𝜙 (δ^{- 1} a) = ψ (δ) ϕ (χ {(δ)}^{- 1} a) = ψ χ^{- 1} (δ) ϕ (a) .

□

We next consider a slightly different notion of eigenspaces, in this case for $ℤ_{p} [Δ]$ -modules.

Definition 2.8.8. §

Let $A$ be a $ℤ_{p} [Δ]$ -module, and let $χ \in Δ^{*}$ . The $χ$ -eigenspace $A^{(χ)}$ of $A$ is defined as

A^{(χ)} = A \otimes_{ℤ_{p} [Δ]} 𝒪_{χ},

where the map $ℤ_{p} [Δ] \to 𝒪_{χ}$ is given by $\tilde{χ}$ .

Notation 2.8.9. §

For $χ \in Δ^{*}$ , set

ẽ_{χ} = \frac{1}{| Δ |} \sum_{δ \in Δ} {Tr}_{E_{χ} ∕ ℚ_{p}} (χ (δ)) δ^{- 1} \in ℤ_{p} [Δ],

where ${Tr}_{E_{χ} ∕ ℚ_{p}} : E_{χ} \to ℚ_{p}$ denotes the trace map.

Notation 2.8.10. §

For a field $E$ , let $G_{E}$ denote its absolute Galois group, which is to say the Galois group of the extension of $E$ given by a fixed separable closure.

Definition 2.8.11. §

We say that two $p$ -adic characters $χ, ψ : Δ \to {\bar{ℚ_{p}}}^{\times}$ are conjugate if there exists $σ \in G_{ℚ_{p}}$ such that $χ = σ \circ ψ$ .

Remark 2.8.12. §

If $χ$ and $ψ$ are conjugate, then $𝒪_{χ} = 𝒪_{ψ}$ .

Remark 2.8.13. §

If $A$ is also a $ℚ_{p}$ -vector space or $p ∤ | Δ |$ , then the canonical map $ẽ_{χ} A \to A^{(χ)}$ is an isomorphism. Note that while $A^{(χ)}$ has an $𝒪_{χ}$ -module structure, the $ℤ_{p} [Δ]$ -module $ẽ_{χ} A$ is only endowed with such a structure when a choice of character $ψ$ in the conjugacy class of $χ$ is made.

Let $Σ$ denote the set of conjugacy classes in $Δ^{*}$ . We let $[χ]$ denote the conjugacy class of $χ \in Δ^{*}$ . We then have the following.

Lemma 2.8.14. §

Let $A$ be a $ℤ_{p} [Δ]$ -module, and let $χ \in Δ^{*}$ . We have

A^{(χ)} \otimes_{𝒪_{χ}} 𝒪 ≅ {(A \otimes_{ℤ_{p}} 𝒪)}^{χ} .

If $A$ is also a $ℚ_{p}$ -vector space or $p ∤ | Δ |$ , then we also have

A^{(χ)} \otimes_{ℤ_{p}} 𝒪 ≅ ⨁_{ψ \in [χ]} {(A \otimes_{ℤ_{p}} 𝒪)}^{ψ}

Proof.

For the first isomorphism, we merely note that

A^{(χ)} \otimes_{𝒪_{χ}} 𝒪 ≅𝐴 \otimes_{ℤ_{p} [Δ]} e_{χ} 𝒪_{χ} \otimes_{𝒪_{χ}} 𝒪 ≅𝐴 \otimes_{ℤ_{p} [Δ]} e_{χ} 𝒪 ≅ {(A \otimes_{ℤ_{p}} 𝒪)}^{χ} .

Let $Δ_{χ} = Δ ∕ \ker χ$ , which is a cyclic group, generated by an element we call $δ_{χ}$ . Note that $ψ \in Δ^{*}$ is conjugate to $χ$ if and only if $ψ$ factors through $Δ_{χ}$ and there exists $σ \in G_{ℚ_{p}}$ such that $ψ (δ_{χ}) = σ (χ (δ_{χ}))$ . Hence, the characters in $[χ]$ are in one-to-one correspondence with the $G_{ℚ_{p}}$ -conjugates of $χ (δ_{χ})$ . Let $ξ = χ (δ_{χ})$ , and suppose that $Φ \in ℤ_{p} [X]$ is its minimal polynomial. We then have

𝒪 \otimes_{ℤ_{p}} ℤ_{p} [ξ] ≅ 𝒪 \otimes_{ℤ_{p}} ℤ_{p} [X] ∕ (Φ (X)) ≅ 𝒪 [X] ∕ (Φ (X)) ≅ \prod_{ξ^{'}} 𝒪 [X] ∕ (X - ξ^{'}) ≅ \prod_{ξ^{'}} 𝒪,

where $ξ^{'}$ runs over the $G_{ℚ_{p}}$ -conjugates of $ξ$ , and the composite map takes $1 \otimes ξ$ to $ξ^{'}$ in the $ξ^{'}$ -coordinate. Reinterpreting this, we have

𝒪 \otimes_{ℤ_{p}} e_{χ} 𝒪_{χ} ≅ ⨁_{ψ \in [χ]} e_{ψ} 𝒪

as $𝒪 [Δ]$ -modules, where the map takes $1 \otimes e_{χ}$ to $e_{ψ}$ in the $ψ$ -coordinate. (Note that $e_{ψ} = σ e_{χ}$ if $ψ = 𝜎𝜒$ , if we let $σ$ act on the coefficients of $e_{χ}$ .) Therefore, we may conclude that

A^{(χ)} \otimes_{ℤ_{p}} 𝒪 ≅𝐴 \otimes_{ℤ_{p} [Δ]} e_{χ} 𝒪_{χ} \otimes_{ℤ_{p}} 𝒪 ≅ ⨁_{ψ \in [χ]} A \otimes_{ℤ_{p} [Δ]} e_{ψ} 𝒪 ≅ ⨁_{ψ \in [χ]} {(A \otimes_{ℤ_{p}} 𝒪)}^{ψ} .

□

Proposition 2.8.15. §

For every $ℚ_{p} [Δ]$ -module $A$ , and every $ℤ_{p} [Δ]$ -module $A$ if $p ∤ | Δ |$ , there is a direct sum decomposition

𝐴≅ ⨁_{[χ] \in Σ} A^{(χ)}

of $ℤ_{p} [Δ]$ -modules, where the sum is over the conjugacy classes in $Σ$ .

Proof.

We define

Φ : A \to ⨁_{[χ] \in Σ} A^{(χ)}

as the product of the surjective maps $A \to A \otimes_{ℤ_{p} [Δ]} e_{χ} 𝒪_{χ}$ that take $a$ to $a \otimes e_{χ}$ . We first show that $Φ$ is an isomorphism after tensoring with $𝒪$ . That is,

Φ \otimes {id}_{𝒪} : A \otimes_{ℤ_{p}} 𝒪 \to ⨁_{[χ] \in Σ} A^{(χ)} \otimes_{ℤ_{p}} 𝒪 .

By Lemma 2.8.14, the right-hand side is isomorphic to

⨁_{[χ] \in Σ} ⨁_{ψ \in [χ]} {(A \otimes_{ℤ_{p}} 𝒪)}^{ψ} ≅ ⨁_{ψ \in Δ^{*}} {(A \otimes_{ℤ_{p}} 𝒪)}^{ψ}

under the map that takes $(a \otimes e_{χ}) \otimes 1$ to $(a \otimes 1) \otimes e_{ψ}$ . The composite map is then the map that takes $a \otimes 1$ to $(a \otimes 1) \otimes e_{ψ}$ , and this is an isomorphism by Proposition 2.8.6. Thus, we have that $Φ \otimes {id}_{𝒪}$ is an isomorphism, and as $𝒪$ is a free $ℤ_{p}$ -module, we have that $Φ$ is an isomorphism. □

Even if $p ∣ | Δ |$ , we have a weaker direct sum decomposition of $ℤ_{p} [Δ]$ -modules.

Notation 2.8.16. §

Let $Υ$ denote the set of maximal ideals of $ℤ_{p} [Δ]$ .

Remark 2.8.17. §

Every $𝔪 \in Υ$ is the kernel of a composite map $\tilde{χ} : ℤ_{p} [Δ] \to_{}^{\tilde{χ}} 𝒪 \to \bar{𝔽_{p}}$ . Thus, $Υ$ may be identified with the set of equivalence classes of characters in $Δ^{*}$ under which two characters are considered equivalent if the above compositions are $G_{𝔽_{p}}$ -conjugate. We write $ψ \in 𝔪$ if $ψ \in Δ^{*}$ lies in the equivalence class corresponding to $𝔪$ .

The proof of the following is left to the reader. Perhaps the easiest way to think of it is that each $A_{𝔪}$ is just $A^{(ρ)}$ for $ρ$ a $p$ -adic character of the prime-to- $p$ part of the group $Δ$ .

Proposition 2.8.18. §

For any $ℤ_{p} [Δ]$ -module $A$ , there is a canonical direct sum decomposition

𝐴≅ ⨁_{𝔪 \in Υ} A_{𝔪}

We have $A_{𝔪}^{(χ)} ≅ A^{(χ)}$ for $χ \in 𝔪$ , and if $p ∤ | Δ |$ , then $𝔪 = [χ]$ and $A_{𝔪} ≅ A^{(χ)}$ for any $χ \in 𝔪$ . If $A$ is a $ℚ_{p}$ -vector space, then we have that

A_{𝔪} ≅ ⨁_{[χ] \subset 𝔪} A^{(χ)} .

Chapter 2 Module theory

2.1. Pseudo-isomorphisms

Definition 2.1.1. §

Definition 2.1.2. §

Example 2.1.3. §

Definition 2.1.4. §

Notation 2.1.5. §

Lemma 2.1.6. §

Lemma 2.1.7. §

Lemma 2.1.8. §

Lemma 2.1.9. §

Proposition 2.1.10. §

Corollary 2.1.11. §

Remark 2.1.12. §

Lemma 2.1.13. §

Notation 2.1.14. §

Lemma 2.1.15. §

Lemma 2.1.16. §

Definition 2.1.17. §

Lemma 2.1.18. §

Corollary 2.1.19. §

Proposition 2.1.20. §

Remark 2.1.21. §

Theorem 2.1.22. §

2.2. Power series rings

Proposition 2.2.1 (Division algorithm). §

Definition 2.2.2. §

Theorem 2.2.3 (Weierstrass preparation). §

Corollary 2.2.4. §

Remark 2.2.5. §

Corollary 2.2.6. §

Corollary 2.2.7. §

Definition 2.2.8. §

Lemma 2.2.9. §

Proposition 2.2.10. §

Remark 2.2.11. §

Lemma 2.2.12. §

Theorem 2.2.13 (Structure theorem for finitely generated Λ-modules). §

2.3. Completed group rings

Definition 2.3.1. §

Remark 2.3.2. §

Remark 2.3.3. §

Definition 2.3.4. §

Remark 2.3.5. §

Lemma 2.3.6. §

Proposition 2.3.7. §

Lemma 2.3.8. §

Theorem 2.3.9. §

Remark 2.3.10. §

2.4. Invariants of Λ-modules

Definition 2.4.1. §

Lemma 2.4.2. §

Remark 2.4.3. §

Lemma 2.4.4. §

Remark 2.4.5. §

Lemma 2.4.6. §

Theorem 2.4.7. §

Lemma 2.4.8. §

Definition 2.4.9. §

Proposition 2.4.10. §

Proposition 2.4.11. §

Proposition 2.4.12. §

2.5. Pontryagin duality

Definition 2.5.1. §

Theorem 2.5.2 (Pontryagin duality). §

Remark 2.5.3. §

Remark 2.5.4. §

Remark 2.5.5. §

Proposition 2.5.6. §

Corollary 2.5.7. §

Example 2.5.8. §

Definition 2.5.9. §

2.6. Iwasawa adjoints

Definition 2.6.1. §

Proposition 2.6.2. §

Proposition 2.6.3. §

Proposition 2.6.4. §

Definition 2.6.5. §

Remarks 2.6.6. §

Chapter 2
Module theory

Theorem 2.2.13 (Structure theorem for finitely generated $Λ$ -modules). §

2.4. Invariants of $Λ$ -modules

2.7. The group ring of a cyclic $p$ -group