IWASAWA λ INVARIANTS, MASSEY PRODUCTS, AND PSEUDO-NULL MODULES

By

Peikai Qi

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Mathematics—Doctor of Philosophy

2025

ABSTRACT

The first chapter of this thesis serves as an introduction. The remaining two chapters each focus on

different topics, all related to the theme of Iwasawa theory. These chapters are adapted from two

separate papers, yet they form a coherent whole in the following sense.

The second chapter studies the relation between Iwasawa λ-invariants and Massey products. For

imaginary quadratic fields, Gold’s criterion provides a connection between the Iwasawa λ-invariant

of the cyclotomic Zp-extension and cup products. Sands [San93] used this criterion to construct

infinitely many imaginary quadratic fields with λ > 1. Massey products can be viewed as a higher-

order generalization of cup products. In Chapter 2, we generalize Gold’s criterion by establishing

a connection between the Iwasawa λ-invariant and Massey products. This generalization also

recovers results of McCallum and Sharifi.

In the third chapter, we study the quotients of the pseudo-null module. Let Λ be an Iwasawa

algebra and M be a pseudo-null Λ-module. Take a regular element T in Λ. We prove a criterion

for determining when the quotient module M/T M is again pseudo-null over Λ/T . This result

provides a useful tool for reducing questions over Zd

p-extensions to those over Zd−1

p

-extensions.

Though the topics of two chapters seems unrelated, they all are motivated by studying the

Gold’s criterion. There are two main directions in which Gold’s criterion can be generalized. The

first is by replacing cup products with Massey products—this is the focus of Chapter 2. The second

is by considering S-ramified Zp-extensions instead of the classical cyclotomic Zp-extensions. By

the joint papers of Matt Stokes and the author [QS24a][QS24b], we find that the S-ramified Zp-

extensions of CM field behaves similarly to the cyclotomic Zp-extension of totally real field. For

cyclotomic Zp-extension of totally real fields , Greenberg conjectured that the Iwasawa module

formed by the inverse limit of class group is a pseudo-null module. Hence, we need to consider

the structure of pseudo-null module to better understand S-ramified Zp-extensions. We summarize

the properties of S-ramified Zp-extensions before Chapter 3 to motivate the work done in Chapter

3. The summary is a reproduction of the joint research work [QS24a][QS24b] by Matt Stokes and

the author.

Copyright by
PEIKAI QI
2025

ACKNOWLEDGEMENTS

There are many people I would like to thank for their support throughout my graduate journey.

First and foremost, I am deeply grateful to my parents for their unwavering love, encouragement,

and support. Without them, I would not have been able to pursue a Ph.D. in mathematics.

I would like to express my sincere thanks to my advisor, Professor Preston Wake, for his

invaluable guidance and mentorship. He first introduced me to the world of modular forms and

then led me into the rich landscape of Iwasawa theory. During the pandemic, when I struggled with

depression and lost over 20 pounds, his encouragement helped me persevere through a very difficult

time. I am deeply thankful for his care—both academically and personally—and for his patience

in answering my countless (and sometimes naive) questions. His mentoring style and approach to

research will continue to influence me throughout my career and life.

I also thank the faculty of the Department of Mathematics, especially Professors Igor Rapinchuk,

Georgios Pappas, Rajesh Kulkarni, Aaron Levin, and Teena Gerhardt. Their excellent courses

introduced me to the beauty of algebraic geometry and number theory, and I learned a great deal

from discussions with them.

I am grateful to the department staff, in particular Taylor Alvarado, for patiently answering my

many questions about departmental policies. Their support made my graduate life much smoother.

I would also like to thank my collaborator, Matt Stokes, who encouraged me to explore a

generalization of Gold’s criterion by considering other types of Zp-extensions. The preface of

Chapter 3 is adapted from our joint work and is the motivation for Chapter 3. I am grateful for his

insights and collaboration.

To my friends at Michigan State University—thank you. Thank you for the mathematical

discussions, for keeping me company during difficult times, for taking care of me when I broke my

collarbone, and for your encouragement when things felt overwhelming. Your presence meant so

much to me.

I also want to thank myself. Despite dealing with neck pain, back pain, stomachaches, knee

pain, seasonal allergies, and various other ailments, I managed to complete this degree.

I am

iv

thankful for my persistence.

Finally, thanks to the sunshine, thanks to flowers, thanks to the rivers, thanks to the squirrels,

thanks to the cold snow, thanks to the warm wind, thanks to the beauty of the life. Thanks for the

world having me!

v

TABLE OF CONTENTS

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

CHAPTER 2

IWASAWA λ INVARIANT AND MASSEY PRODUCT . . . . . . . . .

1

5

CHAPTER 3

QUOTIENT OF PSEUDO-NULL MODULE . . . . . . . . . . . . . . . 57

BIBLIOGRAPHY .

. .

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

vi

CHAPTER 1

INTRODUCTION

Let K be a number field. We say that K∞/K is a Zp-extension if there exists a tower of field

extensions

K = K0 ⊂ K1 ⊂ K2 ⊂ · · · ⊂ Kl ⊂ · · · ⊂ K∞

such that Gal(Kl/K) ∼= Z/plZ and K∞ = (cid:83)

l Kl. Iwasawa proved that there exist constants µ, λ, ν

such that, for sufficiently large l,

#Cl(Kl)[p∞] = pµpl+λl+ν,

where Cl(Kl)[p∞] denotes the p-primary part of the class group.

Let S be the set of primes of K lying above p, and let KS be the maximal extension of K

unramified outside S. Define GK,S = Gal(KS/K).

Theorem 1.0.1 (Gold’s criterion [Gol74]). Let K be an imaginary quadratic field, and let hK =

#Cl(K). Let K∞/K be the cyclotomic Zp-extension. Suppose that p ∤ hK and that p splits in K,

i.e., pOK = p0˜p0. Then

λ ≥ 2 ⇐⇒ αp−1 ≡ 1

(mod ˜p2

0) ⇐⇒ χ ∪ α = 0,

where α is a generator of phK

0 , which can be viewed as an element in cohomology group
H 1(Gal(KS/K), µp) by Kummer theory and χ ∈ H 1(Gal(KS/K), Zp) is the character corre-

sponding to the cyclotomic Zp-extension.

The first equivalence is the original formulation of Gold’s criterion. The second equivalence

follows from Poitou–Tate duality. Thus, Gold’s criterion establishes a bridge between the Iwasawa

λ-invariant and cup products.

Let µn be the group of n-th roots of unity.

Theorem 1.0.2 (McCallum-Sharifi[MS03]). Let K = Q(µp) ⊂ Q(µp2) ⊂ · · · ⊂ Q(µpl) ⊂ · · · ⊂
Q(µp∞) be the cyclotomic Zp-extension. Decompose Cl(Q(µp))[p∞] = (cid:76)
i εiCl(Q(µp))[p∞] as

1

direct sum of eigenspaces with respect to the action of Gal(Q(µp)/Q). Under some conditions, we

have

λi ≥ 2 ⇐⇒ χ ∪ αi = 0

where λi is the Iwasawa invariant that corresponds to the i-th piece and αi is an element in K ∗

constructed from the i-th piece and χ is an element in H 1(Gal(KS/K), Zp) corresponding to the

cyclotomic Zp-extension.

Both theorems have the same form: “λ ≥ 2 ⇐⇒ χ ∪ α = 0”, which motivates us to find a

common generalization. Massey products generalize cup products, with the latter corresponding

to the 2-fold Massey product. This motivates the following results.

Theorem 1.0.3. In the same setting as Gold’s criterion, let n ≥ 2, and assume λ ≥ n − 1.

Then λ ≥ n ⇐⇒ the n-fold Massey product (χ, χ, . . . , χ, α) ∈ H 2(GK,S, µp) vanishes with

respect to a suitable defining system.

The above is Theorem 2.5.1, which follows as a corollary of the main result (Theorem 2.4.3) in

Chapter 2. Another corollary of Theorem 2.4.3 recovers a result of McCallum–Sharifi.

Another way to generalize Gold’s criterion is to replace the cyclotomic Zp-extension of an

imaginary quadratic field K with a different type of Zp-extension of K. The cyclotomic Zp-

extension is obtained by adjoining the pn-torsion points of the multiplicative group Gm. In contrast,

we consider the Zp-extension obtained by adjoining the pn-torsion points of an elliptic curve.

Let E/Q be an elliptic curve with complex multiplication by an imaginary quadratic field K.

Assume that the prime p splits in K as pOK = p˜p, and that E has good reduction at both primes

above p. Then,

Gal(K(E[pn])/K) ∼= (OK/pn)∗ ∼= Z/(p − 1)Z × Z/pn−1Z.

It follows that K(E[pn])/K contains a unique subextension of degree pn−1, which we denote by

Kn−1. The tower

K ⊂ K1 ⊂ · · · ⊂ Kn ⊂ · · · ⊂ K∞ :=

Kn

(cid:91)

n

2

forms a Zp-extension, which we call the elliptic Zp-extension.

It is natural to generalize Gold’s criterion and investigate the condition under which λ > 1

in the setting of the elliptic Zp-extension. However, we found that we were unable to construct

even a single example with λ ≥ 1. This led us to the realization that the behavior of the elliptic

Zp-extension of an imaginary quadratic field is analogous to that of the cyclotomic Zp-extension of

a totally real field. In this latter case, Greenberg famously conjectured that the Iwasawa invariants

µ and λ vanish. In a similar spirit, it is natural to expect that µ = λ = 0 in the context of the elliptic

Zp-extension of imaginary quadratic fields.

Subsequently, we extended this framework by generalizing the elliptic Zp-extension of imag-

inary quadratic fields to the S-ramified Zp-extensions of CM fields. Roughly speaking, such

extensions can be viewed as arising from adjoining the pn-torsion points of a CM abelian variety.

Based on both heuristic and theoretical considerations, we expect that these S-ramified extensions

of CM fields exhibit behavior analogous to that of cyclotomic Zp-extensions of totally real fields.

In the preface to Chapter 3, I summarize the key results concerning the Iwasawa λ-invariant

obtained in our joint work [QS24b; QS24a]. These papers also provide detailed comparisons

between the two settings across several arithmetic invariants, including ambiguous class groups,

norm indices, and norm-coherent sequences of units.

In the preface, I emphasize the results

concerning the λ-invariant, as they serve as a central motivation for the broader project of comparing

Greenberg’s conjecture with its natural generalization in the CM case.

This line of inquiry also leads naturally to the study of quotients of pseudo-null modules, which

forms the primary focus of Chapter 3. The motivation for this study is elaborated further in the

preface.

To study the relation between the analogue of Greenberg’s conjecture for the S-ramified Zp-

extensions of CM fields and the Generalized Greenberg Conjecture, it is essential to understand

the relationship between pseudo-null modules over the Iwasawa algebras Zp[[T1, . . . , Td]] and those

over Zp[[T1, . . . , Td−1]].

In order to explore this relationship, I established the following result

concerning the descent behavior of pseudo-null modules.

3

Theorem 1.0.4. Let R be a Noetherian ring, and let T ∈ R be a regular element. Let M be a

finitely generated R-module. Suppose that M is a pseudo-null R-module. Then M/T M is not

pseudo-null as an R/(T )-module if and only if there exists an associated prime P of M such that

T ∈ P and htR(P ) = 2.

This theorem provides a criterion for when the pseudo-null property fails to descend from M

to M/T M , and plays a key role in understanding the structure of Iwasawa modules in the context

of S-ramified Zp-extensions.

If R is a Krull domain, one can define the characteristic polynomial ChR(M ) of a finitely

generated R-module M . Theorem 1.0.4 yields the following proposition:

Proposition 1.0.5. Assume that both R and R/T are Krull domains. Let M be a finitely generated

pseudo-null R-module. Let M [T ] be the set of T -torsion elements in M . Then,

ChR/T (M/T M ) = ChR/T (M [T ]).

Theorem 1.0.4 is purely a statement in commutative algebra. Given the existence of non-

commutative Iwasawa theory, it is natural to ask whether an analogous statement holds in the

non-commutative setting. The author obtains only a partial result in this direction; the details are

presented in Chapter 3.

4

CHAPTER 2

IWASAWA λ INVARIANT AND MASSEY PRODUCT

2.1

Introduction

2.1.1 Background

Let K be a number field and K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ Kl ⊂ · · · K∞ be a Zp extension of K.

Let X = lim

←− Cl(Kl)[p∞], where Cl(Kl)[p∞] denotes the p-part of the class group of Cl(Kl). Let
µ and λ be the Iwasawa invariants of X. Our goal is to relate the value of λ with the vanishing of

Massey products under the assumption that µ = 0.

Massey products of Galois cohomology are introduced in number theory to study the structure

of Galois groups. We will introduce the definition of Massey products in section 2.2.2. One can

view Massey products as a generalization of cup products in cohomology and for example, the cup

product is a 2-fold Massey product.

The idea of using Massey products to study Iwasawa theory first appears in Sharifi’s paper

[Sha07]. McCallum and Sharifi proved that under some assumptions, we have λ ≥ 2 if and only

if a certain cup product vanishes for cyclotomic fields in [MS03, Proposition 4.2]. One can also

translate Gold’s criterion [Gol74] into group cohomology. It also has the form that λ ≥ 2 if and

only if a certain cup product vanishes for imaginary quadratic fields under some assumptions. The

two results have completely different proof. We want to find the deep reason behind it. Our main

theorem unifies these two results and when applied to these cases, we get a generalization for them.

Roughly speaking, we proved that under some assumptions, if λ ≥ n − 1, then λ ≥ n if and only

if a certain Massey product vanishes.

For readers who are familiar with Massey products, here are some differences one should

notice.

In Ján Mináč and Nguy˜ên Duy Tân’s Massey products vanishing conjecture [MT17],

Massey products vanishing means that Massey products vanish relative to all defining systems. In

our paper, Massey products vanishing means that Massey products vanish relative to a particular

defining system, which we call the proper defining system. In addition, they consider the Massey

products for absolute Galois groups and we consider the Galois groups with restricted ramifications.

5

2.1.2 The strategy and notations

The strategy of the project is divided into four steps.

(a) The Iwasawa invariant λ and λcs

Let S be the set of primes of K above p and Xcs = lim

←− ClS(Kl)[p∞]. Let µcs and λcs be
the Iwasawa invariant for the Iwasawa module Xcs. Let Dl be the subgroup of Cl(Kl)[p∞]

generated by primes in S. Then we have

0 → lim
←−

Dl → X → Xcs → 0

We can relate λ with λcs if we know lim
←−

Dl.

(b) The size of H 2(GKl,S, µp) and λcs

Let KS be the maximal extension of K unramified outside S and GKl,S = Gal(KS/Kl). We

have the following exact sequences from Kummer theory:

0 → ClS(Kl)/p → H 2(GKl,S, µp) → Br(OKl[1/p])[p] → 0

To know the information about λcs, we need information about the size of the group

ClS(Kl)[p∞]. Hence, we need information about the size of H 2(GKl,S, µp).

(c) The generalized Bockstein map and the size of H 2(GKl,S, µp)

By Lam-Liu-Sharifi-Wake-Wang’s paper [Lam+23, Theorem 2.2.4.], We have the following

formula:

I nH 2
I n+1H 2

IwGKl,S, µp)
Iw(GKl,S, µp)

∼=

H 2(GK,S, µp)
Im Ψ(n)

where I is augmentation ideal of Fp[[Gal(Kl/K)]] and Ψ(n) : H 1(GK,S, µp ⊗ I n/I n+1) →

H 2(GK,S, µp) is the generalized Bockstein map defined in [Lam+23]. We have a filtration

H 2

Iw(GKl,S, µp) ⊃ IH 2

Iw(GKl,S, µp) ⊃ I 2H 2

Iw(N, µp) ⊃ · · · ⊃ I nH 2

Iw(GKl,S, µp) · · · . Once

we know the size of H 2(GK,S, µp) and the size of Im Ψ(n), we can determine the filtration

and get the information about the size of H 2

Iw(GKl,S, µp).

6

(d) The generalized Bockstein map and Massey products

In Lam-Liu-Sharifi-Wake-Wang’s paper [Lam+23], they proved that under some circum-

stances, the image of generalized Bockstein map Im Ψ(n) is spanned by certain n-fold Massey

products.

2.1.3 Main theorem and corollaries

As one can see, our strategy does not involve the Iwasawa Main Conjecture. And the Iwasawa

λ invariant that we computed is the algebraic Iwasawa λ invariant. By following the strategy, one

of the main theorems is the following:

Theorem 2.1.1. Let K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ K∞ be a Zp extension of K and S be the set of primes

←− ClS(Kl) and
above p for K. Assume all primes in S are totally ramified in K∞/K. Let Xcs = lim
µcs, λcs be the Iwasawa invariants of Xcs. Assume Xcs has no torsion element and H 2(GK,S, µp) ∼=

Fp.

Then µcs = 0 if and only if there exists integer k such that Ψ(k) ̸= 0 for some k. If µcs = 0, then

λcs = min{n|Ψ(n) ̸= 0} − #S + 1

When we have a Galois group ∆ acts on the Iwasawa module X = lim

←− Cl(Kn) and the action
gives us a decomposition X = ⊕iεiX, the techniques used to prove Theorem 2.1.1 also apply

equivalently to calculate the Iwasawa invariant λi of εiX. See Theorem 2.4.5.

We applied the Theorem 2.1.1 in many cases in the paper. In the introduction, we list two cases

that correspond to the setting of results of Gold [Gol74] and McCallum-Sharifi [MS03].

Theorem 2.1.2. Let K be an imaginary quadratic field and assume that p ∤ hK, p splits in K as

pOK = p0˜p0. For cyclotomic Zp extension, the λ-invariants of K can be determined in terms of

Massey products as follows:

Let n ≥ 2 and suppose λ ≥ n − 1. Then λ ≥ n if and only if n-fold Massey product

(χ, χ, · · · χ, α) is zero with respect to the proper defining system.

7

Here χ is a character χ : GK,S → Gal(K∞/K) ∼= Zp and α is the generator of the principal

idea phK
0 .

Remark 2.1.3. In other words, it means

λ = min{n | n fold Massey products (χ, χ, · · · χ, α) is nonzero}

Remark 2.1.4. It is a fact that λ ≥ 1 in the case. Gold’s criterion [Gol74] said that λ ≥

2 ⇔ αp−1 ≡ 1 mod 2

0. Further calculation shows that αp−1 ≡ 1 mod 2

0 ⇔ logp(α) ≡ 0

mod p2 ⇔ χ ∪ α = 0, where logp is the p-adic logarithm. The theorem can be viewed as new proof

of Gold’s result and a generalization of Gold’s result.

Theorem 2.1.5. Let K = Q(µp) and ω : Gal(Q(µp)/Q) ∼= (Z/pZ)∗ → Zp be the Teichmüller

character. We can decompose the class group Cl(K)[p∞] as

Cl(K)[p∞] = ⊕p−2

i=0 εiCl(K)[p∞]

where εi = 1
p−1

εiCl(Kl)[p∞].

(cid:80)p−1

a=1 ωi(a)σ−1

a ∈ Zp[Gal(Q(µp)/Q]. Let λi be the λ invariant corresponding to

Fix i = 3, 5, · · · p−2 and assume that εiCl(K)[p∞] is cyclic. Let n ≥ 2 and suppose λi ≥ n−1,

then λi ≥ n if and only if n-fold Massey product εi(χ, χ, · · · χ, αi) = 0 with respect to the proper

defining system.

Remark 2.1.6. The assumption εiCl(K)[p] = Fp implies that λi ≥ 1 in the case. Note that

εiCl(K)[p∞] is cyclic if Vandiver’s conjecture holds. The theorem implies that λi ≥ 2 ⇔

εi(χ, αi) = 0 ⇔ χ ∪ αi = 0. Proposition 4.2 in McCallum and Sharifi’s paper [MS03] de-

scribes a similar result that λi ≥ 2 ⇔ χ ∪ αi = 0. The theorem can be viewed as a generalization

of McCallum and Sharifi’s results.

2.1.4 Structure of the paper

In section 2.2, we discuss the generalized Bockstein map introduced by Lam-Liu-Sharifi-Wake-

Wang[Lam+23] and recall the relation of generalized Bockestein map and Massey products. We

8

will also prove that the generalized Bockstein map preserves the group action. In section 2.3, we

prove a formula to determine the size of the second cohomology group by the generalized Bockstein

map. In section 2.4, we prove the main theorem by applying the formula into number theory. We

also list four cases in which we apply our theorem to get some interesting results. The first three

cases are cyclotomic Zp extensions of imaginary quadratic fields. The last case is the cyclotomic Zp

extension of cyclotomic fields. In the first case, we also develop a numerical criterion to determine

λ, which takes 10 pages. One can skip the numerical criterion for the first reading.

2.2 Generalized Bockstein map and Massey product

2.2.1 Generalized Bockstein map

In this section, we recall the definition and properties of the generalized Bockstein map from

[Lam+23].

Let G be a profinite group of finite p-cohomological dimension d and N be a closed normal

subgroup such that G/N is a finitely generated pro-p quotient. In the paper, we take G/N ∼= Zp

or Z/plZ. However, the definition of generalized Bockstein map works more generally. Let

Ω = Fp[[G/N ]] be the completed group algebra which is a G-module in a natural way. Let σ be

the generator of G/N and I =< σ − 1 > be the augmentation ideal in Ω. For 0 ≤ n < #G/N , we

have the following exact sequence of G−module:

0 → I n/I n+1 → Ω/I n+1 → Ω/I n → 0

After taking tensor product with a finite Fp[G]−module T , it is still an exact sequence since

every module that appeared above is a Fp module:

0 → I n/I n+1 ⊗Fp T → Ω/I n+1 ⊗Fp T → Ω/I n ⊗Fp T → 0

For 0 ≤ n < #G/N , define the generalized Bockstein map Ψ(n) to be the connecting map

Ψ(n) : H d−1(G, Ω/I n ⊗ T ) → H d(G, I n/I n+1 ⊗ T ) ∼= H d(G, T ) ⊗ I n/I n+1

9

where the last isomorphism uses the fact that I n/I n+1 is a trivial G-module. We view Ψ(0) = 0.

Recall the definition of Iwasawa cohomology groups:

H r

Iw(N, T ) = lim
←−
N ≤U ⊴◦G

H r(U, T )

where the inverse limit is taken with respect to correstriction maps and U runs over all open normal

subgroups of G containing N . Notice that if G/N is finite, then H i

Iw(N, T ) = H i(N, T ).

In

Lam-Liu-Sharifi-Wake-Wang’s paper, they proved:

Theorem 2.2.1 (Theorem A in LLSWW[Lam+23]). For each 0 ≤ n < #G/N , there is a canonical

isomorphism

H d(G, T ) ⊗ I n/I n+1
I nH d
I n+1H d
Im Ψ(n)
of Fp-modules, where d is the p-cohomological dimension of G.

Iw(N, T )
Iw(N, T )

∼=

For the remaining part of this subsection, we will show that the isomorphism above has a certain

equivalence. We can decompose the cohomological group into a direct sum of eigenspaces with

respect to a group action. For each eigenspace, we still have such isomorphism. However, the

process of checking that the generalized Bockstein map Ψ(n) preserves the group action is tedious.

One can skip the part to Remark 2.2.7.

Lemma 2.2.2. Let G/N ∼= Zp and Ul be the unique open normal subgroup of G containing N such
T := HomZUl(ZG, T ) ∼=

∼= Z/plZ. And T is a finite Fp[G]−module. Then CoIndG
Ul

that G/Ul

Fp[G/Ul] ⊗Fp T as Fp[G]− module. By Shapiro’s lemma,

H r(Ul, T ) ∼= H r(G, Fp[G/Ul] ⊗Fp T )

Proof. Let G = (cid:70)pl

i=1 Ulσi where σi are right coset representatives. Now we define a homomor-
phism α : HomZUl(ZG, T ) → Fp[G/Ul] ⊗Fp T by mapping the element ϕ ∈ HomZUl(ZG, T ) to
the element (cid:80)pl

i ϕ(σi), where ¯σi represents the image of σi in the quotient G/Ul.

i ⊗ σ−1

i=1 ¯σ−1

First, the map does not depend on the choice of right coset representatives. Let hiσi be another

set of right coset representatives where hi ∈ Ul. then

pl
(cid:88)

i=1

¯σ−1
i

¯h−1
i ⊗ σ−1

i h−1

i ϕ(hiσi) =

pl
(cid:88)

i=1

i ⊗ σ−1
¯σ−1

i ϕ(σi)

10

since ϕ(hiσi) = hiϕ(σi).

Second, the map preserves Fp[G] actions. Recall the action g ∈ G on ϕ ∈ HomZUl(ZG, T ) is

(gψ)(x) = ψ(xg).

The action g ∈ G on (cid:80)pl

i=1 ¯σi ⊗ ti ∈ Fp[G/Ul] ⊗Fp T is

g(

pl
(cid:88)

i=1

¯σi ⊗ ti) =

pl
(cid:88)

i=1

¯g¯σi ⊗ gti.

Then

α(gϕ) =

pl
(cid:88)

i=1

i ⊗ σ−1
¯σ−1

i (gϕ)(σi) =

pl
(cid:88)

i=1

i ⊗ σ−1
¯σ−1

i ϕ(σig).

Assume σig = hiσδ(i), where δ is a permutation of 1 ≤ i ≤ pl. We have

i ϕ(σig) = σ−1
σ−1

i ϕ(hiσδ(i)) = σ−1

i hiϕ(σδ(i)) = gσ−1

δ(i)ϕ(σδ(i))

and

Hence

α(gϕ) =

pl
(cid:88)

i=1

¯σ−1
i = (hiσδ(i)g−1)−1 = ¯g¯σ−1
δ(i).

¯g¯σ−1

δ(i) ⊗ gσ−1

δ(i)ϕ(σδ(i)) =

pl
(cid:88)

i=1

¯g¯σ−1

i ⊗ gσ−1

i ϕ(σi)

since δ is a permutation.

Lastly, easy to see α is bijection or one can write out the inverse map of α. Hence CoIndG
Ul

T

is isomorphice to Fp[G/Ul] ⊗Fp T as Fp[G]−modules.

Here, we computed H r(Ul, T ). Next, we will compute H r

Iw(N, T ). We need the following

propositions due to Tate [Tat76].

Proposition 2.2.3. Suppose i > 0 and M = lim
←−

Ml where each Ml is a finite discrete G-module.

If H i−1(G, Ml) is finite for every l, then

H i(G, M ) = lim
←−
l

H i(G, Ml)

11

Lemma 2.2.4. Let G/N ∼= Zp and Ul be the unique open normal subgroup of G containing N such
Iw(N, T ) ∼= H r(G, Fp[[G/N ]]⊗FpT ).

∼= Z/plZ. And T is a finite Fp[G]−module. Then H r

that G/Ul

Proof. By lemma 2.2.2,

H r

Iw(N, T ) = lim
←−
N ≤Ul⊴◦G

H r(Ul, T )

= lim
←−
N ≤Ul⊴◦G

H r(G, Fp[G/Ul] ⊗Fp T )

= H r(G,

lim
←−
N ≤Ul⊴◦G

Fp[G/Ul] ⊗Fp T )

= H r(G, Fp[[G/N ]] ⊗Fp T ).

To prove the third equality is true, we need to check that it satisfies conditions in Proposition

2.2.3. First, we have that Fp[G/Ul] ⊗Fp T is finite module. Second, by [Lam+23, Proposition
2.2.2], we know that H i(Ul, T ) ∼= H i(G, Fp[G/Ul] ⊗Fp T ) is finitely generated Fp[G/Ul] module
for all i ≥ 0. Since Fp[G/Ul] is finite, we have that H i(G, Fp[G/Ul] ⊗Fp T ) is also finite.

For the last equality, note that T is a finite dimensional vector space over Fp. Hence
Fp[G/Ul] ⊗Fp T ∼= Fp[[G/N ]] ⊗Fp T as Fp module. One can see that the isomorphism

lim
←−
also preserves G action. Hence it is also a Fp[G]- isomorphism.

Remark 2.2.5. We know lim
←−

CoIndG
Ul

T := lim
←−

HomZUl(ZG, T ) ∼= Fp[[G/N ]] ⊗Fp T . But

CoIndG

N T := HomZN (ZG, T ) may not be isomorphic to lim
←−

CoIndG
Ul

T . That is one reason

why we separate the proof of lemma 2.2.2 and lemma 2.2.4 and we often write our induced module

as Ω ⊗ T instead of HomZUl(ZG, T ).

Let G be a group containing G and N as normal subgroups such that G /G = ∆ is an abelian

group and G /N is also abelian and G /N ∼= ∆ ⊕ G/N , where G/N is a finitely generated pro-p

quotient. Let T be a Fp[G ]-module and Ω = Fp[[G/N ]]. Next, We will define an action of group

∆ on H r(G, Ω ⊗ T ), H r(G, I n/I n+1 ⊗ T ), H r(G, Ω/I n ⊗ T ), and H r(G, T ). And we will show

that the generalized Bockstein map Ψ(n) preserves the action. We take G/N ∼= Zp or Z/plZ in our

paper. But the setting up works generally.

12

i ¯σi ⊗ ti ∈ Ω ⊗ T as τ ((cid:80)

i ¯σi ⊗ ti) = (cid:80)

i ¯σi ⊗ τ ti. Recall the

Define τ ∈ G acting on (cid:80)
action g ∈ G on Ω ⊗ T is g((cid:80)

i ¯g¯σi ⊗ gti. These two actions have different
effects when τ ∈ G ⊂ G . But they have the same effect when τ ∈ N ⊂ G . For every

i ¯σi ⊗ ti) = (cid:80)

τ ∈ G , we have a group homomorphism α : G → G, g → τ −1gτ and a module homomorphsim
β : Ω ⊗ T → Ω ⊗ T, (cid:80)

i ¯σi ⊗ ti → τ ((cid:80)

i ¯σi ⊗ ti). And

β(α(g)(

(cid:88)

¯σi ⊗ ti)) = β(

(cid:88)

(τ −1gτ )¯σi ⊗ τ −1gτ ti)

i

i

=

=

(cid:88)

(τ −1gτ )¯σi ⊗ τ τ −1gτ ti

i
(cid:88)

i

¯g¯σi ⊗ gτ ti

(cid:88)

= g(β(

¯σi ⊗ ti))

i

By the functorial properties of the cohomology groups [Mil20, Chapter 2, p. 66], we have a

homomorphism H r(G, Ω ⊗ T ) → H r(G, Ω ⊗ T ). This gives us an action of G on H r(G, Ω ⊗ T ).

We know the action τ ∈ G on Ω ⊗ T and the action G on Ω ⊗ T have the same effect when

τ ∈ N . By a well known fact [Mil20, Example 1.27(d), p. 67], the induced action τ ∈ N ⊂ G on

H r(G, Ω ⊗ T ) is trivial. So the action G on H r(G, Ω ⊗ T ) factors through G /N ∼= ∆ ⊕ G/N .

Hence, we get an action ∆ on H r(G, Ω ⊗ T ) by viewing ∆ as a subgroup of G /N .

Similarly, we can define the action of G on Ω/I n ⊗ T and I n/I n+1 ⊗ T in the same way. And

actions are compatible, i.e.

0

0

I n/I n+1 ⊗ T

Ω/I n+1 ⊗ T

Ω/I n ⊗ T

τ

τ

τ

I n/I n+1 ⊗ T

Ω/I n+1 ⊗ T

Ω/I n ⊗ T

0

0

Let · · · → ZG⊕k → ZG⊕k−1 → · · · → ZG⊕2 → ZG → Z → 0 be a Z[G] projective resolution

of Z. Then (α, β) defines a homomorphism of complexes

Hom(ZG⊕k, Ω ⊗ T ) → Hom(ZG⊕k, Ω ⊗ T ), ϕ → β ◦ ϕ ◦ αk

13

for any k. And it induces a homomorphism between two exact sequences of complexes:

0

0

Hom(ZG⊕k, I n/I n+1 ⊗ T )

Hom(ZG⊕k, Ω/I n+1 ⊗ T )

Hom(ZG⊕k, Ω/I n ⊗ T )

Hom(ZG⊕k, I n/I n+1 ⊗ T )

Hom(ZG⊕k, Ω/I n+1 ⊗ T )

Hom(ZG⊕k, Ω/I n ⊗ T )

0

0

By the functorial property of cohomology, we get a homomorphism between two long exact

sequences.

· · ·

· · ·

H r(G, Ω/I n+1 ⊗ T )

H r(G, Ω/I n ⊗ T )

H r+1(G, I n/I n+1 ⊗ T )

H r(G, Ω/I n+1 ⊗ T )

H r(G, Ω/I n ⊗ T )

H r+1(G, I n/I n+1 ⊗ T )

· · ·

· · ·

Each column gives an action of τ ∈ G and they are compatible with each other. In particular, since

the generalized Bockstein map is the connecting mapping, we have Ψ(n)(τ ϕ) = τ Ψ(n)(ϕ) for any

ϕ ∈ H d−1(G, T ⊗ Ω/I n). As before, all actions factor through G /N . It induces an action of ∆ on

the cohomology group. And the generalized Bockstein map Ψ(n) preserves the actions.

Remark 2.2.6. By lemma 2.2.2, we have HomZUl(ZG, T ) ∼= Fp[G/Ul] ⊗Fp T . Corresponding to
the action τ ∈ G on Fp[G/Ul] ⊗Fp T , the action τ ∈ G on ϕ ∈ HomZUl(ZG, T ) is (τ ϕ)(g) =

τ ϕ(τ −1gτ ).

Remark 2.2.7. The action ∆ on the cohomology group is a left action.

Now we introduce a new notation to express elements in Ω which will simplify our future

calculation. For simplicity, assume G/N ∼= Zp or Z/plZ and σ is the generator of G/N . Let
x = σ − 1, then I =< σ − 1 >=< x > and Ω = Fp[[G/N ]] ∼= Fp[[x]] or Fp[x]/(xpl) with respect
to G/N ∼= Zp or Z/plZ. Elements in Ω ⊗ T can be write as

(cid:88)

i

σi ⊗ ti =

(cid:88)

(1 + x)i ⊗ ti =

i

xi ⊗ ψi

(cid:88)

i

for some ψi ∈ T . Let χ be the group homomorphism χ : G → G/N ∼= Zp or Z/plZ. Then the

action g ∈ G on Ω ⊗ T can be writen as

(cid:88)

g(

xi ⊗ ψi) =

(cid:88)

(1 + x)χ(g)xi ⊗ gψi =

(cid:88)

(xi ⊗ (

k=i
(cid:88)

i

i

i

k=0

14

(cid:19)

(cid:18)χ(g)
k

gψi−k))

2.2.2 Massey product

In this section, we recall some facts of Massey products, defining systems[LM21][MT17] and

proper defining systems[Lam+23]. More reference are [Kra66][Dwy75]

For the classical definition of Massey products in group cohomology. Let A be a commutative

ring with trivial G action and discrete topology. By [NSW08], we know that inhomogeneous
continuous cochains C. = (cid:76)

k≥0 Ck(G, A) form a differential graded algebra over A. It equips with

product ∪ and differential d : Cn → Cn+1 such that: d(a ∪ b) = (d a) ∪ b + (−1)ka ∪ (d b) where

a ∈ Ck and d2 = 0. We have H n(G, A) = ker dn/ Im dn−1.

Let Un+1(A) be the group of (n + 1) × (n + 1) upper triangular matrix with diagonal entries

equal 1. Let Z(Un+1(A)) be the center of Un+1(A).
entries equal to 1 and other only possible non zero entry is in the spot (1, n + 1). Let ¯Un+1(A) =

It is a group of matrix whose diagonal

Un+1(A)/Z(Un+1(A)).

Let χ1, χ2, · · · , χn be n elements in H 1(G, A) = Hom(G, A). The defining system with
respect to χ1, χ2, · · · , χn is a homomorphism ¯ρ : G → ¯Un+1(A) with ρi,i+1 = χi, where ρi,j is the
composition of map ¯ρ : G → ¯Un+1(A) with the projection of ¯Un+1(A) to its (i, j) entries. One can
check: (cid:80)j=n

j=2 ρ1,j(g1)ρj,n+1(g2) is a cocyle inside C2 which represents an element in H 2(G, A). We

denote the element as (χ1, χ2, · · · , χn)¯ρ and call it the Massey product with respect to the defining

system ¯ρ. By [Dwy75], the Massey product (χ1, χ2, · · · , χn)¯ρ vanishing is equivalent to that ¯ρ can

be lifted to a homomorphism ρ : G → Un+1(A).

Here ¯ρ ∈ Hom(G, ¯Un+1(A)) can be think as degree one cocycle in C1(G, ¯Un+1(A)) with G
acting trivially on ¯Un+1(A). In general, the action of G on the ring A may not be trivial. We could

define defining systems as a degree one cocycle in the same philosophy and it can be applied in

general situations. References are [Lam+23]. We directly borrow definitions from Section 3 of

[Lam+23] without giving definitions again here.

But our definition of proper defining systems is a little different from the definition in [Lam+23].

We give a new definition here.

Definition 2.2.8. Let χ ∈ H 1(G, T1) and ψ0 ∈ H 1(G, T2). Then we call the defining system

15

¯ρ : G → U(A) with respect to χ, χ, · · · .χ
(cid:125)

(cid:124)

(cid:123)(cid:122)
n copies

, ψ0 as proper defining system if ¯ρ is of the following

forms:

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

∗

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−1
3
...
. . .
(cid:1)
χ (cid:0)χ

ψ2

...

2

1

0

0

0

1

0

0

χ

1

0

ψ1

ψ0

1






















Remark 2.2.9. In [Lam+23], when they define the proper defining system, they first divide the

matrix into four blocks that looks like the one in the following Lemma 2.2.10. And then they fix the

up-left block and down-right block and let the up-right block varies. Here, our definition of the

proper defining system can be viewed as a special case of them. In our definition, we fix and give a

explicit form of the first (n + 1) × (n + 1) block and we let the last column varies except 1 and ψ0.

The following Lemma 2.2.10 and Remark 2.2.11 will only be used in the section of numerical

criterion. One can skip if not interested in the numerical criterion.

Lemma 2.2.10. Let ¯ρ : G → ¯Um+n be a defining system. We can write the homomorphism ¯ρ as a

block matrix:








An

¯Bn,m

0 Dm

¯ρ =




where An is a n×n matrix and Dm is a m×m matrix. ¯Bn,m is a n×m matrix without (n, m)-entry.
Let ¯ρ′ : G → ¯Um+n be another defining system with the same first n columns and last m rows as ¯ρ,

i.e.



An

¯B′

n,m

¯ρ′ =









0 Dm

16

Then



An

¯Bn,m + ¯B′

n,m




0

Dm






is also a defining system.

Proof. The proof is a trivial calculation of matrices by the definition of defining system. We omit

here.

Remark 2.2.11. This easy observation gives us a way to generate a new defining system from old

defining systems. For proper defining system, let ¯ρn : G → ¯Un+1 be a proper defining system, i.e,

One can check that ¯ρn+m : G → ¯Um+n+1 induced by ¯ρn is still a proper defining system:

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

∗

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−2
3
...
. . .
(cid:1)
χ (cid:0)χ

ψ2

...

2

1

0

0

0

1

0

0

χ

1

0

ψ1

ψ0

1






















¯ρn =

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

χ (cid:0)χ

¯ρn+m =
































1

0
. . .

· · ·

0
...

0

0

0

0

1
. . .

0

· · ·
...

0

0

0

0

2

2

· · ·

· · ·
...
(cid:0) χ
m

(cid:1) (cid:0)χ
3

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
3

(cid:1) · · ·
χ (cid:0)χ
(cid:1) · · ·
...
. . .
. . .
. . .
(cid:1) . . .
χ (cid:0)χ
χ · · · (cid:0) χ
...
...
...
(cid:0)χ
2

1
...

0
...

χ

0

1

1

0

2

(cid:1)

(cid:1)

(cid:1)
m−1

0

0

0

1

0

0

χ

1

0

0

0

0

0

0

0

17
































∗

ψn−2
...

ψ0

0
...

0

0

0

1

If we have another proper defining system ¯ρ′

n+m : G → ¯Un+m+1,

χ (cid:0)χ

¯ρ′
n+m =
































1

0
. . .

· · ·

0
...

0

0

0

0

1
. . .

0

· · ·
...

0

0

0

0

2

2

· · ·

· · ·
...
(cid:0) χ
m

(cid:1) (cid:0)χ
3

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
3

(cid:1) · · ·
χ (cid:0)χ
(cid:1) · · ·
...
. . .
. . .
. . .
(cid:1) . . .
χ (cid:0)χ
χ · · · (cid:0) χ
...
...
...
(cid:0)χ
2

1
...

0
...

χ

1

0

0

1

2

(cid:1)

(cid:1)

(cid:1)
m−1

0

0

0

0

0

0

0

0

0

1

0

0

χ

1

0
































∗

ψ′

n+m−2

...

ψ′
m

ψ′

m−1

...

ψ′
2

ψ′
1

ψ′
0

1

Then by Lemma 2.2.10, we can produce a new proper defining system

χ (cid:0)χ
































1

0
. . .

· · ·

0
...

0

0

0

0

1
. . .

0

· · ·
...

0

0

0

0

2

2

· · ·

· · ·
...
(cid:0) χ
(cid:1)
m

(cid:1) (cid:0)χ
3

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
3

(cid:1) · · ·
χ (cid:0)χ
(cid:1) · · ·
...
. . .
. . .
. . .
χ (cid:0)χ
(cid:1) . . .
χ · · · (cid:0) χ
...
...
...
(cid:0)χ
2

0
...

1
...

χ

m−1

1

0

0

1

(cid:1)

2

(cid:1)

0

0

0

0

0

0

0

0

0

1

0

0

χ

1

0

∗

ψ′

n+m−2 + ψn−2
...

ψ′

m + ψ0

ψ′

m−1

...

ψ′
2

ψ′
1

ψ′
0

1
































The image of generalized Bockstein map Im Ψ(n) is spanned by certain Massey products relative

to a proper defining system. The next theorem is just a special case of theorem 4.3.1 in [Lam+23].

Theorem 2.2.12 ( LLSWW[Lam+23]). Let G be a group with p-cohomological dimension 2.

View Ω/I n ⊗ T as a quotient of polynomial ring in terms of variable x with coefficient in T ,

18

Let f (σ) = ψ0(σ) + ψ1(σ)x + · · · + ψn−1(σ)xn−1 be a cocycle in C1(G, Ω/I n ⊗ T ), where ψi
(cid:1) ∪ ψn−i)xn, where χ is the quotient map
is a cochain in C1(G, T ). Then Ψ(n)(f ) = ((cid:80)n
(cid:1) ∪ ψn−i is Massey product (χ(n), ψ0)
χ : G → G/N ∼= Zp or Z/plZ. And easy to see, (cid:80)n

(cid:0)χ
i

i=1

(cid:0)χ
i

i=1

relative to the proper defining system. Here χ(n) denotes n- copies of χ. More precisely, the proper

defining system is:

Remark 2.2.13. We have ((cid:0)χ

i

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

∗

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−1
3
...
. . .
(cid:1)
χ (cid:0)χ

ψ2

...

2

1

0

0

0

1

0

0

χ

1

0

ψ1

ψ0

1






















(cid:1) ∪ ψn−i)(g1, g2) = (cid:0)χ(g1)

(cid:1) ∪ g1ψn−i(g2). And this is compatible with

i

our action of G on Ω ⊗ T . And this is compatible with definition of defining system ρ to be a

cocycle: ρ(g1g2) = ρ(g1)g1ρ(g2)

2.3 Size of H 2

The philosophy of the strategy is to use Massey products to analyze the size of H 2. We first

prove some lemmas we will use later. Since they all can be derived purely from group cohomology

theory. We put them in this section. From now, we assume G has p cohomological dimension

d = 2.

Lemma 2.3.1. Let G be a profinite group with p cohomological dimension equal 2, N be a closed

normal subgroup such that G/N ∼= Zp or Z/plZ. And T is a Fp[G] module. Assume H 2(G, T ) ∼= Fp

and Ψ(k) ̸= 0 for some 0 < k < #G/N , then #H 2

Iw(N, T ) = pn where n = min{n|Ψ(n) ̸= 0}

Proof. Take n = min{n ∈ N|Ψ(n) ̸= 0}, then Ψ(n) ̸= 0. We have H 2(G, T ) ∼= Fp

∼= Im Ψ(n).

By theorem 2.2.1, I nH 2

Iw(N, T ) = I n+1H 2

Iw(N, T ). By Nakayama’s lemma, I nH 2

Iw(N, T ) = 0.

19

We have the filtration H 2

Iw(N, T ) ⊃ IH 2

Iw(N, T ) ⊃ I 2H 2
Iw(N,T )
Iw(N,T ) = Fp. Hence #H 2(N, T ) = pn.

I iH 2
I i+1H 2

when i < n,

Iw(N, T ) ⊃ · · · ⊃ I nH 2

Iw(N, T ) = 0 and

Theorem 2.3.2. Let G be a profinite group, N be a closed normal subgroup such that G/N ∼= Zp

or Z/plZ and T be a Fp[G] module. Fix an integer n < #G/N . Assume the generalized Bockstein

map Ψ(i) = 0 for all 0 ≤ i < n, then the value Ψ(n)(f ), where f (σ) = ψ0(σ) + ψ1(σ)x + · · · +

ψn−1(σ)xn−1 is a cocycle in C1(G, Ω/I n ⊗ T ), only depends on the cohomology class of ψ0 and

does not depend on other coefficients ψi, 0 < i < n.

Proof. Let f ′ = ψ′

0 + ψ′

1x + · · · + ψ′

n−1xn−1 be another cocycle in C1(G, Ω/I n ⊗ T ) such that

ψ0 and ψ′

0 are in the same cohomology class. Then (ψ0 − ψ′
(cid:1) ∪ m)x2 + · · · + ((cid:0) χ

0)(σ) = σm − m for some m ∈ T .
0) + (χ ∪ m)x + ((cid:0)χ
(cid:1) ∪ m)xn−1. One can check that
Let δ = (ψ0 − ψ′
Ψ(n)(δ) = d((cid:0)χ
(cid:1) ∪ m)xn = 0 ∈ H 2(G, T ) ⊗ I n/I n+1. By adding δ to f ′, we can assume that
0 = ψ0. Then f − f ′ ∈ C1(G, I/I n ⊗ T ). Since Ω/I n−1 ∼= Fp[x]/(xn−1) is isomorphic to
ψ′
I/I n ∼= (x)/(xn) by a multiplication of x, Ψ(n)(f − f ′) = Ψ(n−1)( f −f ′

x ) = 0 by assumption.

n−1

n

2

Hence Ψ(n)(f ) only depends on the cohomology class of ψ0.

Under the conditions Ψ(i) = 0 for all 0 ≤ i < n, for convenience, we use Ψ(n)([ψ0]) to denote

Ψ(n)(ψ0 + ψ1x + · · · + ψn−1xn−1) since the value only depends on the cohomology class of ψ0.

And in the language of Massey product, Ψ(n)([ψ0]) equals to the Massey product (χ(n), ψ0) relative

to a proper defining system.

Notice from the following exact sequence:

H 1(G, Ω/I n+1 ⊗ T ) → H 1(G, Ω/I n ⊗ T ) → H 2(G, I n/I n+1 ⊗ T )

Hence Ψ(n)(f ) is zero for a cocycle f (σ) = ψ0(σ)+ψ1(σ)x+· · ·+ψn−1(σ)xn−1 in C1(G, Ω/I n⊗T )

if and only if we can lift f to C1(G, Ω/I n+1 ⊗ T ), i.e. there is a cocycle in C1(G, Ω/I n+1 ⊗ T ) in

the form of ˜f (σ) = ψ0(σ) + ψ1(σ)x + · · · + ψn−1(σ)xn−1 + ψn(σ)xn.

Definition 2.3.3. Let ψ be an element in H 1(G, T ) = H 1(G, Ω/I ⊗ T ). We say ψ has p cyclic

Massey product vanishing property, if we can lift ψ to an element in H 1(G, Ω ⊗ T ), i.e. there is an

20

element in C1(G, Ω ⊗ T ) in the form (cid:80)

i ψi(σ)xi with ψ0 = ψ, where the sum over all 0 ≤ i < ∞

or 0 ≤ i < pk with respect to G/N ∼= Zp or Z/pkZ.

Remark 2.3.4. Assume Ψ(i) = 0 for all 0 ≤ i < n and ψ has p cyclic Massey product vanishing

property. Then we have Ψ(n)([ψ]) = 0.

Theorem 2.3.5. The element ψ ∈ H 1(G, T ) has p cyclic Massey product vanishing property if and

olny if ψ ∈ Im(H 1

Iw(N, T ) Cor−−→ H 1(G, T )) .

Proof. By lemma 2.2.4, we have H 1

Iw(N, T ) ∼= H 1(G, Ω ⊗ T ). And the map H 1(G, Ω ⊗ T ) →
H 1(G, Ω/I ⊗ T ) induced by the quotient Ω → Ω/I corresponding to the correstriction map

H 1

Iw(N, T ) → H 1(G, T ) by definition.

2.4 Applications to number theory

Now we apply previous sections to number theory. Though all notations are standard, we list

them here for convenience:

(a) p: odd prime.

(b) K: number field.

(c) µpl: group of pl-th roots of unity.

(d) S: set of primes of K above p.

(e) KS: maximal algebraic extension of K which is unramified outside primes above p and

infinite primes.

(f) GK,S = Gal(KS/K).

(g) OK: ring of integers of K.

(h) OK,S: ring of S -integers of K.

21

(i) Cl(K): class group of K.

(j) hK: the size of Cl(K).

(k) ClS(K): S -class group of K.

(l) Br(OK[1/p]): The subgroup of the Brauer group Br(K) of K consisting of all classes

of central simple K-algebras, which are split outside of primes above p. Hence, we have

Br(OK[1/p]) ∼= (Q/Z)#S−1 as abelian group.

(m) A[n]: subgroup of elements of abelian group A that annihilated by n.

(n) Let K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ K∞ be a Zp extension of K.

(o) X: the Iwasawa module X = lim

←− Cl(Kl).

(p) λ: the Iwasawa invariant λ for X.

(q) χ: character that equal the restriction GK,S

Res−−→ Gal(K∞/K) ∼= Zp.

Theorem 2.4.1. Let Kl/K be a Z/plZ extension of K inside KS. Take G = GK,S = Gal(KS/K),

N = GKl,S = Gal(KS/Kl), T = µp. Let α ∈ NrmKl/K(OKl,S), then α has p cyclic Massey

product vanishing property.

Let K∞/K be a Zp extension of K inside KS. Take G = GK,S = Gal(KS/K), N = GK∞,S =
Nrm−−→ K) where the inverse limit is taking over all

Gal(KS/K∞), T = µp. Let α ∈ Im(lim
←−

OKl,S

sub-extension of K∞/K with respect to norm map. Then α has p cyclic Massey product vanishing

property.

Proof. By Kummer theory, H 1(G, µp) ∼= K ∗∩(K ∗

S)p/(K ∗)p and H 1(N, µp) = K ∗

l ∩(K ∗

S)p/(K ∗

l )p.

The corestriction map from H 1(N, µp) to H 1(G, µp) corresponds to Norm map[Mil20]. Notice

that OKl,S ∈ (K ∗

S)p. And by theorem 2.3.5, the theorem holds. The second conclusion follows by

taking the inverse limit.

22

Remark 2.4.2. The map lim
←−

OKl,S

lim
←−

OKl,S to the first factor.

Nrm−−→ OK,S (cid:44)→ K is the projection map from the inverse limit

From Kummer theory, we have the following two exact sequences:

(good references are

[Kol04][Lam+23][NSW08])

0 → O∗

K,S/p := O∗

K,S/(O∗

K,S)p → H 1(GK,S, µp) → ClS(K)[p] → 0

0 → ClS(K)/p → H 2(GK,S, µp) → Br(OK[1/p])[p] → 0

(2.1)

(2.2)

map,i.e.: O∗

The map O∗

K,S/p → K ∗ ∩ (K ∗

K,S/p → H 1(GK,S, µp) is the composite of the natural inclusion and Kummer
S)p/(K ∗)p ∼= H 1(GK,S, µp). The map H 1(GK,S, µp) → ClS(K)[p]
is the map α ∈ K ∗ ∩ (KS)p/(K ∗)p ∼= H 1(GK,S, µp) → [I] ∈ ClS(K)[p] where I is an ideal such
that I p = αOK,S. Since Br(OK[1/p]) ∼= (Q/Z)#S−1, so Br(OK[1/p])[p] ∼= F#S−1

p

Now, we can state and prove our main theorem.

Theorem 2.4.3. Let K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ K∞ be a Zp extension of K and S be the set of primes

above p for K. Assume all primes in S are totally ramified in K∞/K. Let Xcs = lim
←− ClS(Kl) and
µcs, λcs be the Iwasawa invariant of Xcs. Assume Xcs has no torsion element and H 2(GK,S, µp) ∼=

Fp.

Then µcs = 0 if and only if there exists k such that Ψ(k) ̸= 0 for some k. If µcs = 0, then

λcs = min{n|Ψ(n) ̸= 0} − #S + 1.

Proof. We have the following exact sequence:

0 → ClS(Kl)/p → H 2(GKl,S, µp) → Br(OKl[1/p])[p] → 0

for every l. since ClS(Kl)/p is finite group, it satisfies Mittag-Leffler condition. Thus the above

exact sequence remains exact after taking inverse limit.

0 → lim

←− ClS(Kl)/p → lim
←−

H 2(GKl,S, µp) → lim
←−

Br(OKl[1/p])[p] → 0

23

Since all primes in S are totally ramified K∞/K, we have

Br(OKl[1/p])[p] ∼= Br(OKl+1[1/p])[p] : [A] → [A ⊗Kl Kl+1]

for every l, where [A] is a class of central simple Kl-algebra represented by A . And we know the

composite map

Cor ◦ Res : Br(OKl[1/p])[p] ∼= Br(OKl+1[1/p])[p] → Br(OKl[1/p])[p]

is the multiplication by p. Hence the composite map is 0. Therefore, the correstriction map

Br(OKl+1[1/p])[p] → Br(OKl[1/p])[p] is zero map. And take inverse limit with respect to the

correstriction map, we have lim
←−

Br(OKl[1/p])[p] ∼= Br(OK[1/p])[p] ∼= F#S−1

p

.

We have the following exact sequence:

0 → pClS(Kl) → ClS(Kl) → ClS(Kl)/p → 0

Take inverse limit,

0 → pXcs → Xcs → lim

←− ClS(Kl)/p → 0

Hence lim

←− ClS(Kl)/p = Xcs/p. We have µcs = 0 if and only if Xcs/p is a finite group if and only
H 2(GKl,S, µp) is a finite group if and only if there exists k such that Ψ(k) ̸= 0 for some k by
lemma 2.3.1 . Now assume µcs = 0. It is well known that the Zp rank of Xcs is λcs. Since we

if lim
←−

assume that Xcs has no torsion element, we have lim

←− ClS(Kl)/p = Xcs/p ∼= Fλcs
p .

Recall that p cohomological dimension of GK,S is 2. We have

# lim
←−

H 2(GKl,S, µp) = #H 2

Iw(GK∞,S, µp) = pn,

where n = min{n|Ψ(n) ̸= 0} by lemma 2.3.1. Therefore, by the exact sequence, we have

λcs − min{n|Ψ(n) ̸= 0} + #S − 1 = 0

Hence λcs = min{n|Ψ(n) ̸= 0} − #S + 1

24

In the proof of the theorem, to use the lemma 2.3.1, we take G = GK,S, N = GK∞,S,
χ : G → G/N ∼= Zp and Ω = Fp[[G/N ]]. And these are the set up that we use for most of the

section. The purpose of the setup is purely for theoretical consistency. In practice, to calculate the

Massey product, we would like Ω = Fp[[G/N ]] to be small. If we know λcs < pl for some l in
advance, we can take G = GK,S, N = GKpl ,S, χ : G → G/N ∼= Z/plZ and Ω = Fp[[G/N ]].

Theorem 2.4.4. Let K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ K∞ be a Zp extension of K and S be the set of primes

←− ClS(Kl) and
above p for K. Assume all primes in S are totally ramified in K∞/K. Let Xcs = lim
µcs, λcs be the Iwasawa invariant of Xcs. Assume Xcs has no torsion element and H 2(GK,S, µp) ∼=

Fp and µcs = 0. Assume λcs < pl, Then λcs = min{n|Ψ(n) ̸= 0} − #S + 1. Here the definition of
Ψ(n) is with respect to G = GK,S, N = GKl,S, χ : G → G/N ∼= Z/plZ and Ω = Fp[[G/N ]]

Proof. We have the following exact sequence:

0 → ClS(Kl)/p → H 2(GKl,S, µp) → Br(OKl[1/p])[p] → 0

We will use Lemma 13.15 in [Was97] and the same notation as [Was97, Lemma 13.15]. We
have ClS(Kl)/p = Xcs/( (1+T )pl

Y0, pXcs) = Xcs/(T pl−1Y0, pXcs). Let f be the characteristic

−1

T

polynomial of Xcs. Since Xcs has no torsion element, we have f Xcs = 0. We have

ClS(Kl)/p = Xcs/(T pl−1Y0, pXcs) = Xcs/(T pl−1Y0, pXcs, f Xcs)

= Xcs/(T pl−1Y0, pXcs, T λcsXcs) = Xcs/(pXcs, T λcsXcs)

= Xcs/(pXcs, f Xcs) = Xcs/pXcs

∼= Fλcs
p

since λcs < pl.

The remaining proof is similar to the proof of Theorem 2.4.3

Next, we will discuss the situation when we have a group ∆ acting on our Iwasawa module. We

can decompose our Iwasawa module as a direct sum of eigenspace with respect to the action. For

each direct sum, we can also define the Iwasawa λ invariant. We will show that we can compute

the Iwasawa λ invariant in the same strategy since we have proved that the generalized Bockstein

map preserves the group action in section 2.2.1.

25

Let k be a number field and K/k be an abelian extension. Denote Gal(K/k) = ∆. Let

K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ K∞ be a Zp extension of K and K∞/k is an abelian extension. Suppose
all the field extensions Kl/k, K∞/k, KS/k are Galois extensions. Assume Gal(K∞/k) ∼= ∆ ⊕ Zp.

Let G = Gk,S = Gal(KS/k), G = GK,S = Gal(KS/K), N = GK∞,S = Gal(KS/K∞) and

T = µp. By subsection 2.2.1, the Galois group Gal(K/k) = ∆ can act on H i(GK,S, Ω/I n ⊗ µp)

and Ψ(n)(τ ϕ) = τ Ψ(n)(ϕ) for any ϕ ∈ H 1(G, Ω/I n ⊗ µp) and τ ∈ ∆.

Let ˆ∆ := Hom(∆, Zp) be the character group. Let ω ∈ ˆ∆ and define

εω =

1
#∆

(cid:88)

σ∈∆

ω(σ)σ−1 ∈ Zp[∆].

Let X be any Zp[∆] module. Then the standard process gives us a decomposition of X:

X =

εωX.

(cid:77)

ω∈ ˆ∆

totally ramified starting K. Let Xcs = lim

Theorem 2.4.5. Let K∞/K be the Zp extension as set up above. Assume all primes in S begin
←− ClS(Kl). Assume p ∤ #∆. The action of ∆ on
Xcs gives a decomposition Xcs = ⊕ωεωXcs. Let µω,cs, λω,cs be the Iwasawa invariant of εωXcs.
Assume εωXcs has no torsion element and εωH 2(GK,S, µp) ∼= Fp. Then µω,cs = 0 if and only

if there exist k such that εωΨ(k) ̸= 0 for some k.

If µω,cs = 0 then λω,cs = min{n|εωΨ(n) ̸=

0} − dimFpεωBr(OK[1/p])[p]

Proof. The proof is almost the same as Theorem 2.4.3, we give a sketch of proof here.

We have the following exact sequence:

0 → ClS(Kl)/p → H 2(GKl,S, µp) → Br(OKl[1/p])[p] → 0

for every l. The action of ∆ gives us:

0 → εωClS(Kl)/p → εωH 2(GKl,S, µp) → εωBr(OKl[1/p])[p] → 0

Take inverse limit,

0 → εω lim

←− ClS(Kl)/p → εω lim
←−

H 2(GKl,S, µp) → εω lim
←−

Br(OKl[1/p])[p] → 0

26

We have µω,cs = 0 if and only if εω lim

←− ClS(Kl)/p is finite if and only if εω lim
←−

H 2(GKl,S, µp) is

finite if and only if there exist k such that εωΨ(k) ̸= 0 for some k.

Assume µω,cs = 0 now. Similar argument as proof of Theorem 2.4.3, we have

εω lim
←−

Br(OKl[1/p])[p] = εωBr(OK[1/p])[p]

and

and

εω lim

←− ClS(Kl)/p ∼= Fλω,cs

p

εω lim
←−

H 2(GKl,S, µp) = min{n|Ψ(n)|εωH 2(GK,S ,Ω/I n⊗µp) ̸= 0},

where Ψ(n)|εωH 2(GK,S ,Ω/I n⊗µp) denotes Ψ(n) restricting on εωH 2(GK,S, Ω/I n ⊗ µp).

Since Ψ(n)(τ ϕ) = τ Ψ(n)(ϕ) for any ϕ ∈ H 1(G, Ω/I n ⊗ µp) and τ ∈ ∆, we have

Ψ(n)|εωH 2(GK,S ,Ω/I n⊗µp) = 0

holds if and only if εωΨ(n) = 0. Hence the conclusion follows from the exact sequence.

Remark 2.4.6. We can have a similar theorem as theorem 2.4.4 with the action of ∆. We omit it

here since it is essentially the same.

Before we apply the theorem to a specific field, we first recall some well-known lemmas in the

classical Iwasawa theory. They will be used in the next section. The next lemma is Proposition

13.26 in the book [Was97].

Lemma 2.4.7. Let p be odd. Suppose K is a CM-field and K∞/K is the cyclotomic Zp exten-

sion. The complex conjugation action gives us the decomposition Cl(Kl)[p∞] = Cl(Kl)[p∞]+ ⊕

Cl(Kl)[p∞]−. Assume all primes which are ramified in K∞/K are totally ramified. Then the map

is injective.

Cl(Kl)[p∞]− → Cl(Kl+1)[p∞]−

27

By using the lemma 2.4.7, one can get the following lemma. It is Proposition 13.28 in [Was97].

Lemma 2.4.8. The same assumption as Lemma 2.4.7, then X − = lim

←− Cl(Kl)[p∞]− contains no

finite Λ-submodules.

In the paper [Gol74], Gold refined the Lemma 2.4.7 to the following theorem.

Theorem 2.4.9. The same assumption as lemma 2.4.7, let Dl be the subgroup of Cl(Kl)[p∞] gener-

ated by primes of Kl above p. Hence we have ClS(Kl)[p∞] = Cl(Kl)[p∞]/Dl and ClS(Kl)[p∞]− =

Cl(Kl)[p∞]−/D−

l by definition. Then the map

ClS(Kl)[p∞]− → ClS(Kl+1)[p∞]−

is injective. Moreover #D−

m = ps(m−l)#D−
l

for any integer m ≥ l, where s is the number of primes

of K +
l

lying over p which split in Kl/K +
l .

The conclusion about the size of Dl is hidden in Gold’s proof of his theorem.

Theorem 2.4.10. The same assumption as lemma 2.4.7, then X −

cs = lim

←− ClS(Kl)[p∞]− contains

no finite Λ-submodules.

Proof. One can use a similar argument as the proof that lemma 2.4.7 implies lemma 2.4.8 in

[Was97]. I omit the proof here since it is almost the same argument.

2.5 Application to concrete examples

In this section, we will apply the theorems that we get in the previous section 2.4 to some

concrete examples. In the following examples, they all are the cyclotomic Zp extension. The same

strategy can apply to other Zp extensions as long as it satisfies the conditions in the theorem. The

first three cases are about imaginary quadratic fields. The last case is about cyclotomic fields. In

the first case of the imaginary quadratic field, we also develop a numerical criterion, which can

help us to compute the λ invariant through the computer. The section of numerical criterion is

independent from other cases. It is suggested to skip the numerical criterion subsection for the first

time reading.

28

2.5.1

Imaginary Quadratic field

2.5.1.1

case 1

Let K be an imaginary quadratic field and hK = #Cl(K), p ∤ hK, p splits in K. Let

K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ K∞ be the cyclotomic Zp extension of K. Let pOKl = pl˜pl where ˜pl is
the complex conjugation of pl . Let (α) = phK

0 , ˜α is the conjugation of α. By Kummer

0 , (˜α) = ˜phK

√
theory, α corresponds to an element in H 1(G, µp), i.e. σ → σ( p

√
α)/ p

α. Recall χ is a character

χ : GK,S

Res−−→ Gal(K∞/K) ∼= Zp.

Theorem 2.5.1. Let K be an imaginary quadratic field, p ∤ hK, p splits in K and n ≥ 2. Assume

λ ≥ n − 1, then λ ≥ n ⇔ n-fold Massey product (χ, χ, · · · χ, α) vanishes with respect to a proper

defining system.

Remark 2.5.2. It is well known that λ ≥ 1 is always true. Hence λ ≥ 2 ⇔ the cup product

χ ∪ α = 0. Later, in the section on numerical criterion, we will show that it is easy to see that

χ ∪ α = 0 ⇔ logp α ≡ 0 mod p2, where logp is the p-adic log. And this is a version of Gold’s

criterion [Gol74]. Our theorem gives a new proof of Gold’s criterion and generalizes the criterion.

proof of theorem 2.5.1.

We have that K is an imaginary quadratic field and K∞/K is a cyclotomic Zp extension. Hence,

it satisfies the assumption in theorem 2.4.9 and theorem 2.4.10 and we will use the same notation

as them. By using Proposition 13.22 in [Was97], we know Cl(Kl)[p∞]+ = 0. Hence D−

l = Dl and

Cl(Kl)[p∞]− = Cl(Kl)[p∞]. By definition, we have that D−
l

is generated by one element pl/˜pl.

Hence Dl = D−
l

is a cyclic group. There is only one prime in K +
l

lying over p that splits in Kl/K +
l .

Thus #Dl = pl#D0 = pl. We have Dl = Z/plZ. We have the following exact sequence:

0 → Dl → Cl(Kl)[p∞] → ClS(Kl)[p∞] → 0

Take the inverse limit, we have

0 → Zp → X → Xcs → 0

29

By lemma 2.4.8 and theorem 2.4.10, we know that X and Xcs have no finite Λ-module. View them

as Zp module, we have RankZp

Zp − RankZpX + RankZpXcs = 0. So λ = λcs + 1.

By the exact sequence (2.2), we have

0 → ClS(K)/p = 0 → H 2(GK,S, µp) → Br(OK[1/p])[p] = Fp → 0

Hence H 2(GK,S, µp) ∼= Fp. Thus, our case satisfies all conditions in the theorem 2.4.3. We

have λcs = min{n|Ψ(n) ̸= 0} − 2 + 1. Therefore, we have λ = min{n|Ψ(n) ̸= 0}. Hence

λ ≥ n ⇔ Ψ(i) = 0 for all 0 ≤ i < n.

By exact sequence (2.1) and Cl(K)[p] = 0, H 1(GK,S, µp) ∼= O∗
KS

/p =< α, ˜α >. Assume

Ψ(i) = 0 for all 0 ≤ i < n − 1 now, then by theorem 2.3.2, Im Ψ(n−1) is generated by Ψ(n−1)([α])

and Ψ(n−1)([˜α]).

Let ζpl be the primitive pl-th root of unit. Let Ql−1 be the unique subfield of Q(µpl) such that

Gal(Ql−1/Q) = Z/pl−1Z. Then Kl = KQl. We have

and

Let

then

p = NrmQ(µpn )/Q(1 − ζpl)

1 − ζpl = NrmQ(µpl+1 )/Q(µpl )(1 − ζpl+1).

ηl := NrmQ(µpl+1 )/Ql(1 − ζpl+1),

p = NrmQl/Q(ηl), ηl = NrmQl+1/Ql(ηl+1).

Since Kl = KQl and K ∩ Ql = Q, then

Gal(Kl/K) = Gal(Ql/Q), Gal(Kl+1/Kl) = Gal(Ql+1/Ql).

So p = NrmKl/K(ηl) and ηl = NrmKl+1/Kl(ηl+1). We know

ηl = NrmQ(µpl+1 )/Ql(1 − ζpl+1) ∈ OQl,S ⊂ OKl,S.

30

Hence the sequence (ηl)l ∈ lim
←−

OKl,S. By theorem 2.4.1, we know p has p cyclic Massey product

vanishing property.

On the other hand, α ˜α = ±phK . Hence,

Ψ(n−1)([α]) + Ψ(n−1)([˜α]) = hKΨ(n−1)([±p]) = 0.

So Im Ψ(n−1) is generated by Ψ(n−1)([α]).

We have Ψ(n−1) = 0 if and only if Ψ(n−1)([α]) = 0 i.e. the Massey product (χ(n−1), α) relative

to a proper defining system vanishes by theorem 2.2.12.

Remark 2.5.3. Suppose we know λ < pl in advance. Assume λ ≥ n − 1, then λ ≥ n ⇔ n-fold

Massey product (χ, χ, · · · χ, α) = 0 with respect to a proper defining system, where we could take

χ : G → Gal(Kl/K) ∼= Z/plZ instead of χ : G → Gal(K∞/K) ∼= Zp as above. The proof is

similar by using Theorem 2.4.4 instead of Theorem 2.4.3. We will use this idea to develop numerical

criterion.

2.5.1.2 numerical criterion

In this subsection, we will translate the Massey product language in the previous case 2.5.1.1

to numerical criterion. One is suggested to skip this subsection for the first time reading. It does

not influence the reading for other cases.

Let K be an imaginary quadratic field, p ∤ hK, p splits in K. Assume we know λ < p in
advance. By the remark 2.5.3, we can take G = GK,S, N = GK1,S, χ : G → G/N ∼= Fp,
Ω = Fp[G/N ] through this subsection. Now, we can state our numerical criterion:

(a) The Iwasawa invariant λ ≥ 1, always true.

(b) The Iwasawa invariant λ ≥ 2 ⇔ logp α ≡ 0 mod p2, where logp is the p-adic log.

(c) Assume λ ≥ 2 is true, then χ ∪ α = 0 ⇐⇒ ∃β ∈ K ∗

1 s.t. NrmK1/K(β) = α. Define
1 , where σ is the generator of the group G/N = Gal(K1/K) ∼= Fp

1 = (cid:81)p−1
A′

i=0 σi(βi) ∈ K ∗

such that χ(σ) = 1.

31

Claim: There exists α1 ∈ K ∗ s.t. vp(α1A′

1) ≡ 0 mod p for all primes p in K such that p ∤ p,

where vp is the valuation corresponding to the prime ideal p.

Then λ ≥ 3 ⇐⇒ χ ∪ α1 = 0 ⇐⇒ logp α1 ≡ 0 mod p2.

(d) Assume λ ≥ 3, then χ ∪ α1 = 0 ⇐⇒ ∃β1 ∈ K ∗

1 s.t. NrmK1/K(β1) = α1, Define

2 = (cid:81)p−1
A′

i=0 σi(βi(i−1)/2) ∈ K ∗

1 and B′

1 = (cid:81)p−1

i=0 σi(βi

1) ∈ K ∗
1 .

Claim: There exists α2 ∈ K ∗ s.t. vp(α2A′

2B′

1) ≡ 0 mod p for all primes p in K such that

p ∤ p.

Then λ ≥ 4 ⇐⇒ χ ∪ α2 = 0 ⇐⇒ logp α2 ≡ 0 mod p2.

(e) continues in a similar way · · ·

In the remaining subsection, we will explain the reason for the numerical criterion. Here is the

idea. To prove the Massey product relative to the proper defining system ¯ρn : GK → ¯Un+1(Fp), n ≤

p vanishes,

¯ρn =

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

∗

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−2
3
...
. . .
(cid:1)
χ (cid:0)χ

ψ2

...

2

1

0

0

0

1

0

0

χ

1

0

ψ1

ψ0

1






















we would like to construct the cochain ψn−1 explicitly to fill the ∗ spot. We first construct ψ′

n−1

when we view the Massey product over GK = Gal(K sep/K) instead of GK,S. In this case, it does

not matter if we assume that K has p-th root of units. Hence, we can exploit the classical results

and generalize them. Then we compare the Massey product when we view it over GK and GK,S

differently. The difference of ψn−1 and ψ′

n−1 is an element αn−1 in K. Finally, to check the Massey

product is zero over GK,S, we restrict the Massey product to GKp0

, where Kp0 is the completion of

K at prime p0.

32

Next, we introduce some classical results. Let K be any field (not necessarily a number field)

such that char(K) ̸= p. Let GK = Gal(K sep/K) be the absolute Galois group. Let ζp be the p-th

root of the unit. Then [K(ζp) : K] has degree which is prime to p. Hence

Cor ◦ Res : H i(GK, µp) → H i(GK(ζp), µp) → H i(GK, µp)

is an isomorphism. Therefore, Res : H i(GK, µp) → H i(GK(ζp), µp) is an injective map. To

consider the Massey product relative to certain defining system vanishing in H 2(GK, µp), we

just need to restrict all cochains to GK(ζp) and consider the Massey product vanishing inside

H 2(GK(ζp), µp). It does not matter if we assume ζp ∈ K. Now we can assume that ζp ∈ K. In

other words, the action GK on µp is trivial. We can view µp as Fp by identifying ζp ∈ µp with

1 ∈ Fp.

As mentioned before, to give a defining system is the same to give a homomorphism ¯ρ : GK →
¯Un+1(µp) = ¯Un+1(Fp). The Massey product vanishing is equivalent to that ¯ρ can be lifted to a

homomorphism ρ : GK → Un+1(µp). Let K¯ρ be the subfield fixed by ker ¯ρ. Then the Massey

product (χ1, χ2, · · · , χn)¯ρ vanishing is equivalent to that we can extend the Galois extension K¯ρ/K

to a Galois extension Kρ/K such that

Gal(Kρ/K)

Un+1(µp)

Gal(K¯ρ/K)

¯Un+1(µp)

is commutative.

To understand this better, we first consider 2-fold Massey products i.e. cup products. Let

a, b be two linearly independent elements in F ∗/(F ∗)p corresponding to χa, χb ∈ H 1(GK, µp) ∼=

F ∗/(F ∗)p. Then K(a1/p) is the subfield fixed by group ker χa and K(b1/p) is the subfield fixed

by group of ker χb. The cup product χa ∪ χb = 0 is equivalent that ∃β ∈ K(a1/p) such that

NrmK(a1/p)/K(β) = b. Translating in the language of Massey product, we have a defining system

¯ρ : GK → ¯U3(µp) ∼= Fp ⊕ Fp, σ → (χa(σ), χb(σ))

33

The 2-fold Massey product vanishing is equivalent to that we can extend the Galois extension

K(a1/p, b1/p)/K to a Heisenberg extension K(a1/p, b1/p, A1/p)/K such that

Gal(K(a1/p, b1/p, A1/p)/K) ∼= U3(Fp).

This is a theorem proved by Romyar Sharifi[Sha99]. Another reference is [MT16]:

Theorem 2.5.4 (Sharifi). The same notation as before. Assume χa ∪χb = 0. Let A1 = (cid:81)p−1

i=0 σj(βj)

where σ is a generator of Gal(K(a1/p/K) such that χ(σ) = 1, then σ(A1) = A1

βp
b .

Theorem 2.5.5 (Sharifi). The same notation as before. Assume χa ∪ χb = 0. Let A = f A1 where

f ∈ K ∗, then the homomorphism

¯ρ : GK → ¯U3(µp) ∼= Fp ⊕ Fp, σ → (χa(σ), χb(σ))

can be lifted to a Heisenberg extension ρ : GK → U3(µp) such that Resker χa(ρ1,3) = χA.

Remark 2.5.6. When we have another lifting corresponding to A′, the difference between A and

A′ is an element in K ∗. If we have another lifting ρ′, then −dρ1,3 = χa ∪ χb and −dρ′

1,3 = χa ∪ χb.

So d(ρ1,3 − ρ′

1,3) = 0. Hence ρ1,3 − ρ′

1,3 is a cocycle in C1. There exists f ∈ K such that

ρ1,3 − ρ′

1,3 = χf . We have A = f A′ up to multiplication by an element of K(a1/p)∗p.

The theorem gives us an explicit description of ρ1,3. And the converse is also true. If we can

find such Heisenberg extension K(a1/p, b1/p, A1/p)/K then the cup product χa ∪ χb is zero.

Now, we generalize the idea to give a similar description for Massey product (χ(n), ψ0) relative

to a proper defining system ¯ρn+1 : GK → ¯Un+2(Fp).

Because we use the proper defining system, the Im ¯ρn+1 is not the whole group ¯Un+2(Fp). We

need the following definition and lemma to describe Im ¯ρn+1.

Definition 2.5.7. We define group

Mn := ⟨ s, t0, t1, . . . , tn | sp = 1, tp

0 = 1, tp

1 = 1, . . . , tp

n = 1,

titjt−1

i t−1

j = 1, st0s−1t−1

0 = t1, st1s−1t−1

1 = t2,

. . . , stn−1s−1t−1

n−1 = tn, stns−1t−1

n = 1 ⟩

34

Remark 2.5.8. One can check that Mn is a semiproduct. We have Mn

∼= Fp ⋉ Fn+1

p

, where

Fp =< s > and Fn+1

p =< t0, t1, · · · , tn >.

Lemma 2.5.9. Let ρn+1 : GK → Un+2(Fp) be the homomorphism defined by

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

ψn

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−1
3
...
. . .
(cid:1)
χ (cid:0)χ

ψ2

...

2

1

0

0

1

0

χ

1

ψ1

ψ0






















ρn+1 =

0
where ker χ ̸= ker ψ0. Then Im ρn+1 is isomorphic to Mn. If we view Im ρn+1 ⊂ Un+2(Fp) as a

0 0

0

1

0

0

subgroup, then we can take

1 1 0 0 0 · · · 0

1 0 0 0 0 · · · 0






















0 1 1 0 0
...
...
...

...

...

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0





0



...




0



0




0


1

0
. . .

0

1

1

0






















0 1 0 0 0
...
...
...

...

...

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0





0



...




0



0




1


1

0
. . .

0

0

1

0

s =

0 0 0 1 1

, t0 =

0 0 0 1 0

,

1 0 0 0 0 · · · 0

1 0 0 0 0 · · · 1






















0 1 0 0 0
...
...
...

...

...

t1 =

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

0 1 0 0 0
...
...
...

...

...

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

0
. . .

0

0

1

0





0



...




0



0




0


1

0
. . .

0

0

1

0





0



...




0



1




0


1






















, · · · , tn =

35

Proof. Since ker χ ̸= ker ψ0, [GK : ker χ] = p and [GK : ker ψ0] = p , so ker χ and ker ψ0 do not

contain each other. Hence Im ρn+1 contains

1 1 0 0 0 · · · ∗

1 0 0 0 0 · · · ∗






















0 1 1 0 0
...
...
...

...

...

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0





∗



...




∗



∗







0

1

0
. . .

0

1

1

0






















0 1 0 0 0
...
...
...

...

...

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0





∗



...




∗



∗







1

1

0
. . .

0

0

1

0

A =

0 0 0 1 1

, B0 =

0 0 0 1 0

,

One can define B1 = AB0A−1B−1

0 , · · · , Bn = ABn−1A−1B−1

n−1. We can directly map s to A and

t0 to B0. This gives us an isomorphism between Mn and ρn+1. To check the group relations, it is a

trivial calculation of matrices. We omit the calculation here.

On the other hand, we can use A, B0 generating the following two matrices:

1 0 0 0 0 · · · 0

1 0 0 0 0 · · · 1






















t1 =

0 1 0 0 0
...
...
...

...

...

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

0
. . .

0

0

1

0





0



...




0



1




0


1

, · · · , tn =

0 0 0 1 0






















0 1 0 0 0
...
...
...

...

...

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0





0



...




0



0




0


1

0
. . .

0

0

1

0

This gives us the description of Im ρn+1 as in the lemma.

36

Remark 2.5.10. If we take subgroup Vn+1 ⊂ Un+2(Fp) whose elements are of the form:

1 0 0 0 0 · · · ∗






















0 1 0 0 0
...
...
...

...

...

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

0
. . .

0

0

1

0





∗



...




∗



∗




∗



1

then we have an exact sequence,

0 → Vn+1 → Im ρn+1 → Im ρn+1/Vn+1 → 0

Easy to see that Vn+1

by s. And Im ρn+1

∼= Fn+1
p

is generated by t0, t1, · · · , tn and Im ρn+1/Vn+1

∼= Fp is generated
∼= Fp ⋉ Vn+1. Hence, to describe the presentation of Im ρn+1, all we need is to

describe the presentations of Vn+1 and Fp and the action of Fp on Vn+1. These are how we define

Mn.

Remark 2.5.11. By the lemma 2.5.9, if we can find a Galois extension L/K such that Gal(L/K) ∼=

Mn, then we can determine a proper defining system ¯ρn+2 : GK → ¯Un+3. And the converse is also

true.

To determine whether a group is isomorphic to Mn, we only need to determine what elements

are mapped to s, t0 and check all relations if we know the group has the same size as Mn.

As before, Assume K is a field that contains p-th root of the unit. Let χ ∈ H 1(GK, µp) ∼=
Hom(GK, Fp) and ψ0 ∈ H 1(GK, µp) ∼= Hom(GK, Fp). Let K(a1/p) be the fixed subfield of ker χ

and K(b1/p) be the fixed subfield of ker ψ0 and assume they are different field. Let σa and σb be

the generator of Gal(K(a1/p, b1/p)/K) such that

σa(a1/p) = ζpa1/p

σb(a1/p) = a1/p

σa(b1/p) = b1/p

σb(b1/p) = ζpb1/p

37

Lemma 2.5.12. Assume χ ∪ ψ0 = 0,i.e. there exists β ∈ K(a1/p) such that NrmK(a1/p)/K(β) = b.

Define

A1 =

p−1
(cid:89)

i=0

a(βi), A2 =
σi

p−1
(cid:89)

i=0

σi
a(β

i(i−1)
2

), A3 =

p−1
(cid:89)

i=0

σi
a(β

i(i−1)(i−2)
3

), · · · , An =

p−1
(cid:89)

i=0

a(β( i
σi

n))

where n < p, Then

σa(A1)
A1

b = βp,

σa(A2)
A2

σa(A1) = β

p(p−1)
2

, · · · ,

σa(An)
An

n)
σa(An−1) = β(p

Proof. Easy to check the following equation:

p−1
(cid:88)

(σa − 1)(

iσi

a) +

i=0

p−1
(cid:88)

i=0

σi
a = p

p−1
(cid:88)

(σa − 1)(

i=0

i(i − 1)
2

σi
a) + σa

p−1
(cid:88)

i=0

iσi

a =

p(p − 1)
2

· · ·

p−1
(cid:88)

(σa − 1)(

i=0

(cid:19)

σi
a) + σa

(cid:18) i
n

p−1
(cid:88)

(cid:18) i

(cid:19)

n − 1

i=0

σi
a =

(cid:19)

(cid:18)p
n

Lemma 2.5.13. Notations as before and n < p, then the field extension

K(a1/p, b1/p, A1/p

1

, A1/p
2

, · · · , A1/p

n )/K

is a Galois extension. And

Gal(K(a1/p, b1/p, A1/p

1

, A1/p
2

, · · · , A1/p

n )/K) ∼= Mn

. Therefore, it corresponds a proper defining system ¯ρn+2 : GK → ¯Un+3. And Resker χψi = χAi

for 1 ≤ i ≤ n < p − 1.

Proof. We have

and

σa(A1)/A1 = βp/b ∈ K(a1/p, b1/p)∗p

σb(A1)/A1 = 1 ∈ K(a1/p, b1/p)∗p.

38

Hence K(a1/p, b1/p, A1/p

1 )/K is a Galois extension.

Lift σa and σb to Gal(K(a1/p, b1/p, A1/p

1 )/K). We denote them as ˜σa and ˜σb.

As said in remark 2.5.11, to prove Gal(K(a1/p, b1/p, A1/p

1 )/K) ∼= M1, we only need to determine

what is mapped to s, t0 and check that it satisfies the relations in the presentation of group M1. Lift

σa and σb to Gal(K(a1/p, b1/p, A1/p

1 )/K). We denote them as ˜σa and ˜σb. We would like to map ˜σa,

˜σb to s, t0 respectively. Next, we will check it satisfies the relations.

By lemma 2.5.12,

for some j1 ∈ Fp. Then

˜σa(A1/p

1 ) = ζ j1

p A1/p

1 β/b1/p

a(A1/p
˜σ2

1 ) = ζ j1

p A1/p

1 β/b1/pζ j1

p ˜σa(β)/b1/p

· · ·

a(A1/p
˜σp

1 ) = ζ j1p

p A1/p

1 Nrm(β)/b = A1/p

1

Hence ˜σp

a = 1.
We have ˜σb(A1/p

1 ) = ζ i1

p A1/p

1

for some i1 ∈ Fp. Hence ˜σp

b (A1/p

1 ) = A1/p

1

(2.3)

. So ˜σp

b = 1 .

We have

˜σa˜σb(A1/p

1 ) = ˜σa(ζ i

pA1/p

1 ) = ζ i1+j1

p

A1/p

1 β/b1/p

˜σb˜σa(A1/p

1 ) = ˜σb(ζ j1

p A1/p

1 β/b1/p) = ζ j1+i1−1

p

A1/p

1 β/b1/p

˜σa˜σb˜σ−1

a ˜σ−1

b (A1/p

1 ) = ζpA1/p

1

Define ˜σA1 = ˜σa˜σb˜σ−1

a ˜σ−1
b

. We have

˜σp
A1

= 1, ˜σa˜σA1 ˜σ−1

a ˜σ−1
A1

= 1, ˜σb˜σA1 ˜σ−1

b ˜σ−1
A1

= 1

Hence, ˜σA1 plays the role t1 in the presentation of M1 and we checked that the relations are

satisfied. Therefore, we have

Gal(K(a1/p, b1/p, A1/p

1 )/K) ∼= M1.

39

And

GK → Gal(K(a1/p, b1/p, A1/p

1 )/K) ∼= M1 ⊂ U3(Fp)

gives us Resker χψ1 = χA1.

For next step: Similarly, we have

˜σa(A2)/A2 = βp(p−1)/2/˜σa(A1) = βp(p−1)/2b/(A1βp) ∈ K(a1/p, b1/p, A1/p

1 )∗p

and

˜σb(A2)/A2 = 1 ∈ K(a1/p, b1/p, A1/p

1 )∗p.

And we know from last paragraph, Gal(K(a1/p, b1/p, A1/p

1 )∗p)/K) is generated by ˜σa, ˜σb. Hence

for any σ ∈ Gal(K(a1/p, b1/p, A1/p

1 )∗p/K), we have σ(A2)/A2 ∈ K(a1/p, b1/p, A1/p

1 )∗p. Therefore,

K(a1/p, b1/p, A1/p

1

, A1/p

Lift ˜σa, ˜σb, ˜σA1 to Gal(K(a1/p, b1/p, A1/p

1

2 )/K is a Galois extension.
, A1/p

2 )/K). And we still denote the lifting as

˜σa, ˜σb, ˜σA1

by a little abusing notation. Similarly, we will check that ˜σa, ˜σb are mapped to s, t0 ∈ M2 and they

satisfy the relations in the definition of M2.

By lemma 2.5.12

˜σa(A1/p

2 ) = ζ j2

p A1/p

2 b1/pβ

p−3

2 /A1/p

1

for some j2 ∈ Fp. By equation (2.3), we have

p−1
(cid:89)

i=0

a(A1/p
˜σi

1 ) = A1

p−1
(cid:89)

i=0

a(βp−1−i)/b
˜σi

p−1
2 = b

p−1
2

a(A1/p
˜σp

2 ) = A1/p
2 b

p−1
(cid:89)

i=0

˜σi
a(β

p−3
2 )/

p−1
(cid:89)

i=0

Hence ˜σp

a = 1.

a(A1/p
˜σi

1 ) = A1/p

2

We have ˜σb(A2) = A2, so ˜σb(A1/p

2 ) = ζ i2

p A1/p

2

for some i2 ∈ Fp. So ˜σp

b = 1

We have

a (A1/p
˜σ−1

2 ) = ζ −j2−j1
p

A1/p

2 A1/p

1 /˜σ−1

a (β

p−1

2 )

40

˜σb(A1/p

2 ) = ζ −i2

p A1/p

2

˜σA1(A1/p

2 ) = ˜σa˜σb˜σ−1

a ˜σ−1

b (A1/p

2 ) = ζ i1

p A1/p

2

Hence ˜σp
A1

= 1.

˜σa˜σA1 ˜σ−1

a ˜σ−1
A1

(A1/p

2 ) = ζpA1/p

2

Define ˜σA2 = ˜σa˜σA1 ˜σ−1

a ˜σ−1

A1 . Then ˜σp

A2

= 1. And one can check

˜σa˜σA2 ˜σ−1

a ˜σ−1
A2

= 1, ˜σb˜σA2 ˜σ−1

b ˜σ−1
A2

= 1, ˜σA1 ˜σA2 ˜σ−1
A1

˜σ−1
A2

= 1

These imply that ˜σa, ˜σb satisfies the relations in the definition of M2. And ˜σA1, ˜σA2 plays the role
2 )/K) ∼= M2. It determines a
as t1, t2. Hence we have an isomorphism Gal(K(a1/p, b1/p, A1/p
2 )/K) ∼= M2 ⊂ U4(Fp) ⊂ ¯U5(Fp). And

proper defining system GK → Gal(K(a1/p, b1/p, A1/p

, A1/p

, A1/p

1

1

Resker χψ2 = χA2.

The general case can be checked by the same process and the calculation is tedious. We omit

here.

Remark 2.5.14. The proof for the case M1 is the same as the proof of theorem 2.5.5 in [MT16]

and [Sha99]. We just imitate the proof and get the generalized result. But the calculation becomes

more and more tedious.

Now, if we have χ ∪ ψ0 = 0, we can construct a Galois field extension and it corresponds to

a proper defining system. How do we get all proper defining systems? We need the following

definition and lemma.

Definition 2.5.15. The proper defining system we get by the way in lemma 2.5.13 is called the

standard proper defining system.

By previous lemmas, fix χ and ψ0, the standard proper defining system depends on the choice

of a, b, β and choices of the lifting of σa and σb.

41

Lemma 2.5.16. All proper defining systems with respect to χ can be obtained from the standard

proper defining system by operations in lemma 2.2.10 and remark 2.2.11.

Proof. Assume we have a proper defining system ¯ρn : GK → ¯Un+1(Fp), n ≤ p.

Then χ ∪ ψ0 = 0. By lemma 2.5.13, we can get a standard proper defining system:

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

∗

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−2
3
...
. . .
(cid:1)
χ (cid:0)χ

ψ2

...

2

1

0

0

0

1

0

0

χ

1

0

ψ1

ψ0

1

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

∗

(cid:1)

(cid:1)

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψ′
3
...
. . .
χ (cid:0)χ
(cid:1)

...

ψ′
2

2

n−2

1

¯ρn =

¯ρ′
n =

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0











































0

0

0

1

0

0

χ

1

0

ψ′
1

ψ′
0

1

42

where ψ0 = ψ′

0. Assume ψi = ψ′

i for 0 ≤ i ≤ m. Then by lemma 2.2.10

ψ′

n−2 − ψn−2
...

m+1 − ψm+1
































1

0
. . .

· · ·

0
...

0

0

0

0

1
. . .

0

· · ·
...

0

0

0

0

χ (cid:0)χ

2

2

· · ·

(cid:1) (cid:0)χ
3

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
3

(cid:1) · · ·
(cid:1) · · ·
χ (cid:0)χ
· · ·
...
...
. . .
. . .
. . .
(cid:1) . . . (cid:0) χ
χ (cid:0)χ
(cid:1) ψ′
m+1
(cid:0) χ
(cid:1)
m
...
(cid:0)χ
2

χ · · ·
...
...

1
...

0
...

χ

0

1

1

0

(cid:1)

2

0

0

0

0

0

0

0

0

0

1

0

0

χ

1

0

∗

0
...

0

0

0

1
































is a proper defining system that is induced by the following defining system:

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

∗

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψ′
3
...
. . .
χ (cid:0)χ
(cid:1) ψ′

2

1

n−2 − ψn−2
...

m+3 − ψm+3

0

0

0

1

0

0

χ ψ′

m+2 − ψm+2

1 ψ′

m+1 − ψm+1

0

1






















Then the lemma is followed by induction.

In the lemma 2.5.13, we can construct a standard proper defining system from a field extension.

Conversely, we can get a field extension from a proper defining system.

Lemma 2.5.17. Given a proper defining system ¯ρn : GK → ¯Un+1, the fixed subfield by ker ¯ρn can

be writen in the form

43






K

a1/p, b1/p

0

, (A0,1b1)1/p, (A0,2A1,1b2)1/p, (A0,3A1,2A2,1b3)1/p,

· · · , (A0,n−2A1,n−3 · · · An−2,1bn−2)1/p






where a, b0, b1, · · · , bn−2 ∈ K and there exist βi ∈ K(a1/p) satisfying that NrmK(a1/p)/K(βi) = bi
for 0 ≤ i ≤ n − 3 and Ai,j := (cid:81)p−1

j)
(k
a(β
i

k=0 σk

)

Proof. The result directly follows from lemma 2.5.16 by using the correspondence between the

standard proper defining system and Mn-field extension.

Another method to check the lemma is by using the same method in the proof of lemma 2.5.13.

Check that the field extension is Galois extension and ˜σa and ˜σb0 are generators of Mn and satisfy

the relations. The calculation will become tedious, we omit it here.

Now, back to our case that K is a number field. Instead consider the group GK = Gal(K sep/K),
we consider the group GK,S = Gal(K S/K). For a defining system ¯ρn : GK,S → ¯Un+1, the subfield

L fixed by ker ¯ρn is a field extension that is unramified outside S. Conversely, if we have a field
extension L/K that is unramified outside S and Gal(L/K) is isomorphic to a subgroup of ¯Un+1,
then we have a defining system ¯ρn : GK,S → Gal(L/K) ⊂ ¯Un+1. We need the following known

fact.

Lemma 2.5.18. Let K be a number field containing p-th root of the unit. Let a ∈ K ∗ and p be

a prime that does not divide p, then K(a1/p)/K is unramified at p if and only if the valuation

vp(a) ≡ 0 mod p

Now, we apply all we have to the case 2.5.1.1 where K is an imaginary quadratic field and p splits

in K and the size of the class group hK is prime to p. Notations as the beginning of case 2.5.1.1 and
the beginning of this subsection 2.5.1.2, recall that the character χ is GK,S → Gal(K1/K) ∼= Fp

where K1 = KQ1.

Lemma 2.5.19. Let ψ0 ∈ H 1(GK,S, µp) ∼= K ∗ ∩ K ∗p
χ ∪ ψ0 = 0 if and only if that logp(α) ∼= 0 mod p2 where logp is the p-adic log.

S /K ∗p correspond α ∈ K ∗ ∩ K ∗p

S /K ∗p. Then

44

Proof. Let pOK = p0˜p0 and pOK1 = p1˜p1. Let Kp0 and K˜p0 be the completion of K at prime

p0 and ˜p0 respectively. Similarly for the definition K1,p1 and K1,˜p1. We will use Poitou-Tate

Duality which is the theorem 8.6.7 in [NSW08]. We use the same notation as in chapter 8.6
S) ∼= Fp as GK,S module. Lemma 8.6.3
of [NSW08]. Take A = µp, then A′ = Hom(µp, O∗
in [NSW08] tells us X1(GK,S, Fp) ∼= Hom(ClS(K), Fp) = 0. By theorem 8.6.7 in [NSW08],
We have X2(GK,S, µp) ∼= X1(GK,S, Fp)∨ = 0. This implies that the map H 2(GK,S, µp) →

, µp) is injective. The cup product χ ∪ ψ0 vanishes in H 2(GK,S, µp)

, µp) ⊕ H 2(GK˜p0

H 2(GKp0
if and only if Resχ ∪ Resψ0 vanishes both in H 2(GKp0

, µp) and H 2(GK˜p0
1,p1). One can check that Kp0 = Qp
(K ∗
Kp0, χ ∪ ψ0 vanishes if and only if α ∈ NrmK1,p1 /Kp0
and K1,p1 = Q1,p which is the completion of Q1 at p. And Q1,p/Qp is totally ramified degree p
p ⊕ (1 + pZp) ∼=
extension. We can decompose Q∗
p ⊕ Zp is given by t → (t mod p, logp(t)/ logp(1 + p)). By local class field theory, we have
F∗

p ⊕ (1 + pZp) since p is odd and Z∗

p = pZ ⊕ F∗

In local field

∼= F∗

, µp).

p

(K ∗

NrmK1,p1 /Kp0
if and only if logp(α) = 0 mod p2. And similar story happened when we complete at ˜p0. Hence

p. We have α ∈ NrmK1,p1 /Kp0

p⊕(1+p2Zp) as a subgroup of Q∗

1,p1) ∼= Z⊕F∗

1,p1)

(K ∗

logp(α) = 0 mod p2 if and only if χ ∪ ψ0 = 0

Remark 2.5.20. The value logp α depends on the embedding K → Qsep

p . But the criterion

logp(α) ≡ 0 mod p2 does not depend on the embedding.

Here is the relation between our result and the classical result Gold criterion [Gol74]

Theorem 2.5.21. Let K be a imaginary quadratic field, p ∤ hK, p split in K i.e. pOK = p0 ˜p0.

Take (α) = phK

0 , then λ ≥ 2 ⇔ the cup product χ ∪ α = 0 ⇔ logp(α) ≡ 0 mod p2 ⇔ αp−1 ≡ 1

mod ˜p2

0 where logp is p-adic log.

Proof. We only need to prove the last equivalent relation. Complete K at ˜p0, We have K˜p0

∼= Qp

and α ∈ Z∗
p

∼= F∗

p ⊕ 1 + pZp. And logp(α) ≡ 0 mod p2 ⇐⇒ α ∈ F∗

p ⊕ 1 + p2Zp ⇐⇒ αp−1 ≡ 1

mod p2 in Qp ⇐⇒ αp−1 ≡ 1 mod ˜p2

0 in K.

45











































Let ¯ρn : GK,S → ¯Un+1 be a proper defining system.

¯ρn =

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

∗

· · ·

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−2
3
...
. . .
(cid:1)
χ (cid:0)χ

ψ2

...

2

1

0

0

0

1

0

0

χ

1

0

ψ1

ψ0

1

When restricted on GK1,S = ker χ, the cochain ψi ∈ C1(GK,S, µp) becomes a cocycle in

C1(GK1,S, µp). So ResGK1,S ψi corresponds to a cocycle (σ → σ(A1/p
K ∗

S and the cocycle depends on the choice of A1/p

since we do not have µp ∈ K. Assume
that the Massey product relative to the proper defining system ¯ρn : GK,S → ¯Un+1 vanishes, i.e.

)/A1/p
i

) for some Ai ∈

1 ∩ K ∗p

i

i

there exists a cochain ψn−1 ∈ C1(GK,S, µp) fitting in the lifting ρn : GK,S → Un+1.

ρn =

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

· · · ψn−1

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−2
3
...
. . .
χ (cid:0)χ
(cid:1)

ψ2

...

2

1

0

0

0

1

0

0

χ

1

0

ψ1

ψ0

1

Similarly, ResGK1,S ψn−1 is a cocycle and corresponds to a element An−1 ∈ K ∗
¯ρn−→ ¯Un+1 be the composition of GK → GK,S and ¯ρn. Then ¯ρ′
¯ρ′
n : GK → GK,S

1 ∩ K ∗p

S . Let

n is a proper

defining system over GK. Then the Massey product relative to ¯ρ′

n also vanishes. Then there exists

a cochain ψ′

n−1 : GK → µp fitting in the cochain ρ′

n : GK → Un+1.

46

ρ′
n =

2

0 1
...
...


1 χ (cid:0)χ



















0 0

0 0

0 0

0

0

0

0 0

0

(cid:1) (cid:0)χ
3

χ (cid:0)χ
...
...

2

(cid:1)

(cid:1)

n−1

· · · ψ′

(cid:1) (cid:0)χ
4
(cid:1) (cid:0)χ
· · · ψn−2
3
...
. . .
(cid:1)
χ (cid:0)χ

ψ2

...

2

1

0

0

0

1

0

0

χ

1

0

ψ1

ψ0

1






















And ResGK1
Massey product, we have d(ψ′

n−1 is a cocycle in C1(GK1, µp) and corresponds to A′
ψ′

n−1 ∈ K ∗

1 . By definition of

n−1−ψn−1) = 0. Hence ψ′

n−1−ψn−1 is a cocycle in C1(GK, µp) which

corresponds to σ → σ(f 1/p)
f 1/p

for some f ∈ K ∗. When restricting on GK1, we have f An−1 = A′

n−1.

We remark here that we have to choose the p-th root of An−1, A′

n−1, f properly so that cocycles

are compatible. A different choice of p-th root of Ai changes the corresponding cocycle a multiple

of (σ → σ(ζp)
ζp

). And χ ∪ (σ → σ(ζp)
ζp

) = 0. For our case now, we care about when the Massey

products vanish. Therefore, we do not need to care too much about the choice of the p-th root of

the element. For our purpose, the key is that there exists f ∈ K ∗ such that f An−1 = A′

n−1 where

An−1 ∈ K ∗

1 ∩ K ∗p

S and A′

n−1 ∈ K ∗

1 . By lemma 2.5.18, we have an element A ∈ K ∗

1 ∩ K ∗p

S if and

only if the valuation vp(A) ≡ 0 mod p where p does not divide p.

Now we combine all we have and explain how our numerical criterion works:

(a) The Iwasawa invariant λ ≥ 1, always true.

(b) The Iwasawa invariant λ ≥ 2 ⇔ logp α ≡ 0 mod p2, where logp is the p-adic log (Reason:

lemma 2.5.19).

(c) Assume λ ≥ 2 is true, then χ ∪ α = 0 ⇐⇒ ∃β ∈ K ∗

1 s.t. NrmK1/K(β) = α. Define
1 , where σ is the generator of the group G/N = Gal(K1/K) ∼= Fp

1 = (cid:81)p−1
A′

i=0 σi(βi) ∈ K ∗

Claim: There exists α1 ∈ K ∗ s.t. vp(α1A′

1) ≡ 0 mod p for all p ∤ p, where vp is the

valuation corresponding to prime ideal p. (Reason: by previous argument, the difference

47

between "correct" A1 and A′

1 we constructed is an element in α1 ∈ K ∗. We want α1A′

1 to be

our A1.)

Then λ ≥ 3 ⇐⇒ χ ∪ α1 = 0 ⇐⇒ logp α1 ≡ 0 mod p2(Reason: Let ψ′

1, ψ′

2 be the

cochain in C1(GK, µp) corresponding to A′

1 and A′

2 = (cid:81)p−1

i=0 σi(βi(i−1)/2). Over GK, we

2

2

χ ∪ ψ′

= −dψ′

2 |GKp0

have χ ∪ ψ′

1 + (cid:0)χ
+(cid:0)χ

2. Restrict on GKp0

(cid:1) ∪ ψ0 = −dψ′
(cid:1) ∪ ψ0 |GKp0
, the Massey product χ ∪ ψ1 + (cid:0)χ

through GK → GKp0
. Let ψ1 ∈ C1(GK,S, µp) correspond to the A1.
(cid:1) ∪ ψ0 + χ ∪ (ψ1 − ψ′
1)

1 |GKp0
Restrict to GKp0
vanishes in H 2(GKp0
H 2(GKp0
if and only if logp(α1) ≡ 0 mod p2. Notice that we can not directly use lemma 2.5.19 since

, µp). A similar argument as lemma 2.5.19, we have the Massey product vanishing

, µp) if and only if χ ∪ (ψ1 − ψ′

= 0,i.e. χ ∪ α1 = 0 in

(cid:1) ∪ ψ0 = χ ∪ ψ′

1) |GKp0

, we have

1 + (cid:0)χ

2

2

α1 may not be in H 1(GK,S, µp). However, since we have restricted ψ1 − ψ′

1 on GKp0

, we can

, µp). And similarly, restricting on GK˜p0

directly work in H 2(GKp0
Since H 2(GK,S, µp) → H 2(GKp0
product vanishing if and only if logp(α) ≡ 0 mod p2.)

, µp) ⊕ H 2(GK˜p0

, we get the same result.

, µp) is injective, we conclude the Massey

(d) Assume λ ≥ 3, then χ ∪ α1 = 0 ⇐⇒ ∃β1 ∈ K ∗

1 s.t. NrmK1/K(β1) = α1, (Reason: the cup

product χ ∪ α1 should be viewed in H 2(GK, µp)). Define A′

2 = (cid:81)p−1

i=0 σi(βi(i−1)/2) ∈ K ∗
1

and B′

1 = (cid:81)p−1
So we construct A′

i=0 σi(βi

1) ∈ K ∗

1 . (Reason: we want the Massey product vanishing on GK first.

2 and B′

1 correspond to the cocycle in this way. See lemma 2.5.13 and

lemma 2.5.16.)

Claim: There exists α2 ∈ K ∗ s.t. vp(α2A′

2B′

1) ≡ 0 mod p for all p ∤ p.(Reason: similarly

to the previous case, the difference between "correct" A2 and A′

2B′

1 is an element α2 ∈ K ∗.)

Then λ ≥ 4 ⇐⇒ χ ∪ α2 = 0 ⇐⇒ logp α2 ≡ 0 mod p2 .(Reason: similar as previous case.)

(e) continues in a similar way · · ·

Remark 2.5.22. The numerical criterion is not perfect, we only know the existence of βi and α1

and we do not have a logarithm to compute them.

48

I hope the following lemma can inspire people to come up with explicit numerical criterion

though I can not do it.

Lemma 2.5.23. Let K be a imaginary quadratic field, p ∤ hK, p split in K i.e. pOK = p0 ˜p0. Let
¯ρn : GK,S → ¯Un+1 be a defining system. Then the Massey product (χ1, χ2, · · · , χn)¯ρn relative to a

defining system ¯ρn vanishes over GK,S if and only if it vanishes over GK

Proof. We have the following commutative diagram.

GKp0

GK

GK,S

It induces the following diagram.

H 2(GK,S, µp)

H 2(GKp0

, µp) ⊕ H 2(GK˜p0

, µp)

H 2(GK, µp)

The row is an injective map by Poitou-Tate Duality (See proof of lemma 2.5.19). The column is an

inflation map. If the Massey product vanishes over GK,S, then after the inflation map, it vanishes

over GK. If the Massey product vanishes over GK, we restrict on local field, then it vanishes over

GKp0

and GK˜p0

. Since the row map is injective, it vanishes in GK,S.

Remark 2.5.24. By the following well-known exact sequence:

H 1(GK, µp)

χ∪−

−−→ H 2(GK, µp) Res−−→ H 2(GK1, µp)

The Massey product (cid:80)n−1
exists χb ∈ H 1(GK, µp) such that χ ∪ χb = (cid:80)n−1

i=0

(cid:0)χ
i

(cid:0)χ
i

(cid:1) ∪ ψi.

i=0

(cid:1) ∪ ψi in our case will be zero when restrict on GK1. Therefore, there

2.5.1.3

case 2

Let K be an imaginary quadratic field and Cl(K)[p∞] = Z/plZ. Assume p remains prime over

K/Q, i.e. pOK = p0. Then there is only one prime that is ramified in the Zp cyclotomic extension

K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ K∞. Let I be an ideal in K such that [I] ̸= 0 ∈ Cl(K)[p]. Let α be the

generator of the principal ideal I p.

49

Theorem 2.5.25. Let K be an imaginary quadratic field and Cl(K)[p∞] = Z/pnZ. Assume p

remains prime over K/Q and n ≥ 2. Assume λ ≥ n − 1, then λ ≥ n ⇔ n-fold Massey product

(χ, χ, · · · χ, α is zero with respect to a proper defining system.

Remark 2.5.26. It is well known that λ ≥ 1 is always true in the case.

proof of theorem 2.5.25.

In this case, we have Cl(Kl)[p∞]+ = 0 by Proposition 13.22 in [Was97]. We have Dl is a

subgroup of Cl(Kl)[p∞] generated by pl. Hence Dl = D+

l = 0 by definition. Therefore, we have

ClS(Kl)[p∞] = Cl(Kl)[p∞] = Cl(Kl)[p∞]−. Our case satisfies conditions in lemma 2.4.8. We
have X ∼= Xcs and λ = λcs.

By the exact sequence (2.2),

0 → ClS(K)/p = Fp → H 2(GK,S, µp) → Br(OK[1/p])[p] = 0 → 0

We have H 2(GK,S, µp) = Fp. The theorem 2.4.3 implies

λ = λcs = min{n|Ψ(n) ̸= 0} − 1 + 1 = min{n|Ψ(n) ̸= 0}.

By the exact sequence (2.1),

0 → O∗

K,S/p ∼= Fp → H 1(GK,S, µp) → ClS(K)[p] ∼= Fp → 0

we know that H 1(GK,S, µp) is generated by p and α. A similar argument as in theorem 2.5.1,

we know p has p cyclic Massey product vanishing property. Similarly, the Iwasawa invariant

λ ≥ n ⇔ Ψ(i) = 0 for all 0 ≤ i < n. Assume Ψ(i) = 0 for all 0 ≤ i < n − 1, then by theorem

2.3.2, Im Ψ(n−1) is generated by Ψ(n−1)([α]). So Ψ(n−1) = 0 if and only if Ψ(n−1)([α]) = 0 i.e. the

Massey product (χ(n−1), α) relative to a proper defining system vanishes by theorem 2.2.12.

2.5.1.4

case 3

Let K be an imaginary quadratic field and Cl(K)[p∞] = Z/plZ. Assume p is ramified in K/Q,

i.e. pOK = p2

0. Then there is only one prime that is ramified in the Zp cyclotomic extension

50

K ⊂ K1 ⊂ K2 ⊂ · · · ⊂ K∞. Let I be an ideal in K such that [I] ̸= 0 ∈ Cl(K)[p]. Let α be the

generator of the principal ideal I p.

Theorem 2.5.27. Let K be an imaginary quadratic field and Cl(K)[p∞] be a cyclic group and

p is ramified in K/Q and n ≥ 2. Assume λ ≥ n − 1, then λ ≥ n ⇔ n-fold Massey product

(χ, χ, · · · χ, α) is zero with respect to a proper defining system.

Remark 2.5.28. It is well known that λ ≥ 1 is always true in the case.

proof of theorem 2.5.27 .

Similar as Theorem 2.5.25, we have Cl(Kl)[p∞]+ = 0 and Dl = D+

l = 0 by definition. These

imply ClS(Kl)[p∞]− = Cl(Kl)[p∞]− = Cl(Kl)[p∞]. By the exact sequence (2.2),

0 → ClS(K)/p ∼= Fp → H 2(GK,S, µp) → Br(OK[1/p])[p] = 0 → 0

we have H 2(GK,S, µp) ∼= Fp. Our case satisfies conditions in Theorem 2.4.3. The same argument

as Theorem 2.5.25 gives us the conclusion.

2.5.2 Cyclotomic field

Let K = Q(µp) where µp is the group of p-th roots of unit as before. Let ω : Gal(Q(µp)/Q) ∼=

(Z/pZ)∗ → Zp be the Teichmüller character. Let X be the Iwasawa module which is the inverse

limit of the p-part of the class group of Q(µpl) with respect to the norm map. By Corollary 10.15

in [Was97], Gal(Q(µp)/Q) acts on X and X is decomposed as direct sum of eigenspace, i.e.

X = ⊕p−2

i=0 εiX where εi = 1
p−1

(cid:80)p−1

a=1 ωi(a)σ−1

a ∈ Zp[Gal(Q(µp)/Q]. Fix i = 3, 5, · · · , p − 2.

Assume that εiCl(K)[p∞] is cyclic, in other words, we have εiX = Λ/(fi). Notice that by Theorem

10.16 in [Was97], εiCl(K)[p∞] is cyclic if Vandiver’s conjecture holds. Let λi = deg(fi). Let

Ii be an ideal in K such that [Ii] ̸= 0 ∈ εiCl(K)[p]. Let αi be the lift of [Ii] by the map

εiH 1(GK,S, µp) → εiCl(K)[p].

51

Theorem 2.5.29. Let K = Q(µp). Fix i = 3, 5, · · · p − 2 and assume that εiCl(K)[p∞] is cyclic.

Let n ≥ 2. Assume λi ≥ n − 1, then λi ≥ n ⇔ n-fold Massey product εi(χ, χ, · · · χ, αi) is zero

with respect to a proper defining system.

proof of theorem 2.5.29 .

Our case satisfies the setting up in Theorem 2.4.5, where k = Q, K = Q(µp), ∆ =

Gal(Q(µp)/Q). We have Cl(Kl) = ClS(Kl). By the exact sequence:

0 → εiClS(K)/p = Fp → εiH 2(GK,S, µp) → εiBr(OK[1/p])[p] = 0 → 0

we have εiH 2(GK,S, µp) ∼= Fp. By Theorem 2.4.5, we have

λi = λi,cs = min{n|εiΨ(n) ̸= 0} − dimFpεiBr(OK[1/p])[p] = min{n|εiΨ(n) ̸= 0}.

By the exact sequence:

0 → εiO∗

K,S/p = 0 → εiH 1(GK,S, µp) → εiCl(K)[p] = Fp → 0

we know that εiH 1(GK,S, µp) is generated by αi by our definition. Assume λi ≥ n − 1, then

Ψ(j)|εiH 2(GK,S ,Ω/I n⊗µp) = 0 for 0 ≤ j < n − 1. So Im Ψ(n−1)|εiH 2(GK,S ,Ω/I n−1⊗µp) is generated by

Ψ(n−1)|εiH 2(GK,S ,Ω/I n−1⊗µp)([αi]). And Ψ(n−1)|εiH 2(GK,S ,Ω/I n−1⊗µp)([αi]) = εiΨ(n−1)(f ) for certain
cocycle f = (cid:80)n−2

k=0 ψkxk ∈ C1(G, Ω/I n−1 ⊗ µp) such that ψ0 = αi. Hence εiΨ(n−1) = 0 if and only

if the n-fold Massey product εi(χ, χ, · · · χ, αi) = 0 with respect to a proper defining system.

Remark 2.5.30. This is a generalization of McCallum and Sharifi’s result of proposition 4.2 in

[MS03].

52

Preface to Chapter 3

The preface serves as a motivation for the material presented in Chapter 3. It provides a summary

of joint work by Matt Stokes and the author, reproduced from the papers [QS24b; QS24a]. In the

preface, I focus primarily on the results related to Greenberg’s conjecture for totally real fields. The

aforementioned papers establish more comparison theorems between the S-ramified Zp-extensions

of CM fields and the cyclotomic Zp-extensions of totally real fields.

In Gold’s criterion, we consider the cyclotomic Zp-extension of imaginary quadratic fields. This

extension is constructed by adjoining the pn-torsion points of Gm, the multiplicative group scheme.

A natural generalization is to consider the Zp-extension obtained by adjoining the pn-torsion points

of an elliptic curve E, where E is an elliptic curve with complex multiplication by K. More

generally, one can also consider adjoining the pn-torsion points of a CM abelian variety.

All such constructions can be described uniformly via class field theory. Let K be a CM field

and let K + denote its maximal totally real subfield, i.e., the subfield fixed by complex conjugation.

Let S+ = {P1, P2, . . . , Ps} be the set of primes of K + lying above p. Assume that each prime
˜Pi for 1 ≤ i ≤ s, where ˜Pi is the complex conjugate

above p in K + splits in K. Write PiOK = Pi

of Pi, and define S = {P1, P2, . . . , Ps}.

Theorem A.31. There exists a Zp-extension of K unramified outside S and it is unique if Leopoldt’s

conjecture holds for K.

We refer to such a Zp-extension as the S-ramified Zp-extension of the CM field K.

Interestingly, when Matt Stokes and I studied the Iwasawa λ-invariant for the S-ramified Zp-

extension of a CM field and attempted to generalize Gold’s criterion, we found that the behavior

diverges from that of the cyclotomic Zp-extension of an imaginary quadratic field. Instead, the

S-ramified Zp-extension of a CM field K behaves analogously to the cyclotomic Zp-extension of

its maximal totally real subfield K +. In the spirit of Greenberg’s conjecture, the Iwasawa µ and λ

invariants for S-ramified Zp-extension of CM field should all be zero!

Using Gold’s criterion, Sands [San93] constructed infinitely many imaginary quadratic fields

for which the λ-invariant is greater than 1 for the cyclotomic Zp-extension. Motivated by the

53

analogy described above, we adapted Sands’ construction to produce infinitely many real quadratic

fields that are not p-rational. Moreover, we demonstrated that, for a subfamily of real quadratic

fields, both the Iwasawa µ and λ invariants vanish in the cyclotomic Zp-extension.

In light of this analogy, Matt and I propose the following conjecture as an analogue of Green-

berg’s conjecture.

Conjecture A.32. Let K∞/K be a S-ramified Zp-extension. Let An denote the p-primary part

of the class group of Kn. Then An will be bounded as n → ∞. In other words, the Iwasawa

invariants µ and λ vanish.

Although we cannot prove the conjecture in its entirety, we can establish that it holds under

certain assumptions. From now on, we assume that all primes in S are totally ramified in the

S-ramified Zp-extension.

Let in,m : Cl(Kn) → Cl(Km) for m ≥ n be the natural map between class groups induced by

inclusion. Define Hn = ∪m≥n Ker(in,m).

Theorem A.33. Assume that p is inert in K +/Q and that Leopoldt’s conjecture holds for K. Then

the following statements are equivalent.

(a) A0 = H0.

(b) |An| is bounded as n → ∞.

Let Bn be the subgroup of An fixed by Gal(Kn/K), and let Dn be the subgroup of An generated

by prime ideals lying above S.

Theorem A.34. Assume that p splits completely in K +/Q and that Leopoldt’s conjecture holds for

K. Then the following statements are equivalent.

(a) Bn = Dn for all sufficiently large n.

(b) |An| is bounded as n → ∞.

54

These two theorems are, in fact, analogous to Greenberg’s results [Gre76] for the cyclotomic

Zp-extension of totally real fields. Moreover, the proofs are also analogous to Greenberg’s proofs.

We also proved other analogous theorems, see details in paper [QS24a].

Greenberg [Gre01] also proposed the “Generalized Greenberg Conjecture".

Conjecture A.35. [Gre01, Conjecture 3.5] Suppose that F is any number field and p is a prime.

Let ˜F denote the composition of all Zp-extensions of F . Let ˜L denote the pro-p Hilbert class field
of ˜F and let ˜X = Gal( ˜L/ ˜F ), regarded as a module over the ring ˜Λ = Zp[[Gal( ˜F /F )]]. Then ˜X is
a pseudo-null ˜Λ-module.

We now have three conjectures under consideration:

the original Greenberg conjecture for

the cyclotomic Zp-extension of totally real fields, the analogous Greenberg conjecture stated in

Conjecture A.32 for the S-ramified Zp-extension of CM fields, and the Generalized Greenberg

conjecture stated in Conjecture A.35. It is natural to ask how these conjectures are related.

Given the observed similarities between the cyclotomic Zp-extensions of totally real fields and

the S-ramified Zp-extensions of CM fields, it is reasonable to expect that the original Greenberg

conjecture is equivalent to the analogous Greenberg conjecture A.32. In [Fuj17], Fuji established

that the analogous Greenberg conjecture A.32 implies the Generalized Greenberg conjecture, under

certain conditions.

Moreover, the Generalized Greenberg conjecture implies the original Greenberg conjecture,

assuming the validity of Vandiver’s conjecture, since the cyclotomic Zp-extension is the only

Zp-extension of a totally real field under this assumption. These implications suggest a circular

relationship among the three conjectures, and naturally lead to the expectation that they may, in

fact, be equivalent.

The first two conjectures can also be restated in the language of pseudo-null modules. For

instance, in Conjecture A.32, let

X = lim

←− Cl(Kn)[p∞],

where Kn denotes the n-th layer of the S-ramified Zp-extension of a CM field K. Then Conjec-

55

ture A.32 predicts that X is a pseudo-null module over the Iwasawa algebra Zp[[Gal(K∞/K)]] ∼=

Zp[[T ]], a formal power series ring in one variable.

In the case of the Generalized Greenberg conjecture A.35, the relevant Iwasawa algebra is

˜Λ = Zp[[Gal( ˜F /F )]] ∼= Zp[[T1, . . . , Td]],

which is a power series ring in several variables. To compare Conjecture A.32 with the Generalized

Greenberg conjecture A.35, it becomes essential to analyze how the pseudo-nullity of modules be-

haves under specialization—specifically, how a pseudo-null module over a multi-variable Iwasawa

algebra relates to its image over a quotient algebra in fewer variables.

For example, if M is a pseudo-null module over Zp[[T, S]], we would like to understand whether

the quotient M/T M is pseudo-null over Zp[[S]]. This question lies at the heart of the structural

relationships between the conjectures, and constitutes one of the main themes of Chapter 3.

56

CHAPTER 3

QUOTIENT OF PSEUDO-NULL MODULE

3.1

Introduction

Let Λd = Zp[[T1, . . . , Td]] be the Iwasawa algebra. In Iwasawa theory, we say that a Λd-module

M is pseudo-null if its annihilator is not contained in any prime ideal of height at most one. We

now consider the notion of pseudo-nullity to modules over any ring R.

Definition 3.1.1. Let R be a ring. An R-module M is pseudo-null if and only if MP = 0 for any

prime ideal P with ht(P ) ≤ 1, or equivalently, if ht(AnnR(M )) ≥ 2.

Here, htR(P ) denotes the height of the ideal P in R, and MP is the localization of M at the

prime ideal P . The annihilator of M in R is denoted by AnnR(M ).

We recall some additional definitions and notations. Let AssR(M ) be the set of associated

primes of the R-module M . For an element T ∈ R, we sometimes write (M )T := M/T M and

M [T ] := {m ∈ M | T m = 0}.

Theorem 3.1.2. Let R be a Noetherian ring, and let T be a regular element in R. Suppose M is a

finitely generated pseudo-null R-module. Then M/T M is not pseudo-null as an R/(T )-module if

and only if there exists an associated prime P of M with htR(P ) = 2 such that T ∈ P .

The proof of Theorem 3.1.2 is inspired by [Oza]. This theorem provides a criterion for

determining when M/T M is pseudo-null as an R/(T )-module.

In practice, consider Λd−1 =

Zp[[T1, . . . , Td−1]] = Λd/(Td) and let M = Xd be the unramified Iwasawa module associated

with a certain Zd

p-extension. Theorem 3.1.2 helps determine when Xd/(Td) is a pseudo-null Λd−1-

module. Moreover, the unramified Iwasawa module associated with a Zd−1

p

-extension can be related

to the module Xd/(Td) via an exact sequence. This allows for an inductive approach, reducing

statements from Zd

p-extensions to Zd−1

p

-extensions.

If we further assume that R is a Krull domain, we can also talk about the characteristic ideal

ChR(M ) of the finitely generated R-module M . We can prove the following proposition by

57

applying Theorem 3.2.1. This result can be viewed as a generalization of [BBL14, Proposition

2.7].

Proposition 3.1.3. Assume that R and R/(T ) are Krull domains. Let M be a finitely generated

pseudo-null R-module. Then, we have

ChR/T (M/T M ) = ChR/T (M [T ]).

Theorem 3.1.2 can be viewed as a purely commutative algebra statement. It is natural to wonder

if there is a similar statement in non-commutative algebra, and one can apply it to non-commutative

Iwasawa theory. However, the proof of Theorem 3.1.2 relies heavily on commutative algebra.

While there is a general notion of associated primes in non-commutative algebra, the author can

only prove partial results, see Theorem 3.4.9.

3.2 Proof of Main Theorem in Commutative algebra

We prove the following theorem in this section.

Theorem 3.2.1. Let R be a noetherian ring and T is a regular element in R. Let M be a finitely

generated R-module. Assume M is a pseudo-null R module. Then M/T M is not pseudo-null as

R/(T ) module if and only if T ∈ P where P is an associated prime of M and htR(P ) = 2.

We begin with some definitions and lemmas.

Definition 3.2.2. Define Dn

R as the subcategory of the category of R-modules consisting of modules

M such that ht(AnnR(M )) ≥ n + 1.

Dn

R := {M |R module M such that MP = 0 for all prime ideal P with ht(P ) ≤ n}

:= {M |R module M such that ht(AnnR(M )) ≥ n + 1}.

We see that an R-module M belongs to D1

R if and only if M is a pseudo-null module. The

following property follows easily from the fact that localization is an exact functor.

Lemma 3.2.3. Let 0 → M ′ → M → M ′′ → 0 be an exact sequence of R-modules. Then M is in

Dn

R if and only if both M ′ and M ′′ are in Dn
R.

58

Lemma 3.2.4. Let T be a regular element of R, and let M be an R-module. Then

M/T M ∈ Dn

R/T

if and only if M/T M ∈ Dn+1
R .

Proof. Let P be a prime ideal in R containing T , and let ¯P be its image in R/(T ). We have

ht(P ) = dim(RP ) and ht( ¯P ) = dim((R/(T )) ¯P ) = dim(RP /(T )). By [Mat70, (15.F) Lemma 4],

since T is a regular element of R, we have dim(RP /(T )) = dim(RP ) − 1. This implies

htR(P ) = htR/T ( ¯P ) + 1.

The conclusion now follows from the definition.

We primarily use Lemma 3.2.4 in the case n = 1. Now we begin the proof of Theorem 3.2.1.

Let R be a Noetherian ring, and let M be a finitely generated R-module. Consider the shortest

primary decomposition of the zero submodule in the Noetherian R-module M :

0 =

r
(cid:92)

i=1

Yi.

In other words, AssR(M/Yi) = {Pi}, where Pi ̸= Pj for i ̸= j. By [Mat70, Lemma 8.E], we have

AssR(M ) = {P1, P2, . . . , Pr}.

Lemma 3.2.5. Let R be a Noetherian ring and M be a finitely generated R-module. Assume

M ∈ D1

R and consider the following short exact sequence.

0 → M

ϕ
−→

r
(cid:77)

i=1

M/Yi → D → 0

where ϕ is induced by the natural projection and D is the cokernel of the map ϕ. Then D is in D2
R.

Proof. An element in (cid:76)r

i=1(∩j̸=iYj + Yi)/Yi can be written as (¯a1, ¯a2, · · · , ¯ar) for elements

ai ∈ ∩j̸=iYj. Notice that

Hence,

ϕ(a1 + a2 + · · · + ar) = (¯a1, ¯a2, · · · , ¯ar)

r
(cid:77)

(∩j̸=iYj + Yi)/Yi ⊂ Im(ϕ)

i=1

59

By lemma3.2.3, it is enough to show

M/(∩j̸=iYj + Yi) ∈ D2
R

Notice that AssR(M/Yi) = {Pi} gives us (cid:112)AnnR(M/Yi) = Pi by [Mat70, Proposition 8.B ].

Since M ∈ D1

R, then M/Yi ∈ D1

R, which means htR(AnnR(M/Yi)) ≥ 2. Hence htR(Pi) ≥ 2.

Put Ii := AnnR(M/(∩j̸=iYj + Yi)). Since AnnR(M/Yi) ⊂ Ii,

Pi = (cid:112)AnnR(M/Yi) ⊂

(cid:112)

Ii

which tells us htR(Ii) ≥ 2. If we assume that htR(Ii) = 2, then Pi =

√

Ii and htR(Pi) = 2. Hence,

∩j̸=i AnnR(M/Yj) ⊂ Ii ⊂ Pi

By [Ati18, Proposition 1.11],

for some j ̸= i. We therefore have

AnnR(M/Yj) ⊂ Pi

(cid:113)

Pj =

AnnR(M/Yj) ⊂ Pi

It tells us Pj = Pi since htR(Pi) = 2 and htR(Pj) ≥ 2. It is a contradiction because Pj ̸= Pi by

definition. Therefore, we have ht(Ii) ≥ 3 for 1 ≤ i ≤ r, which implies that M/(∩j̸=iYj + Yi) ∈

D2
R.

Since D ∈ D2

R, by Lemma 3.2.3, we have D/T D ∈ D2

R and D[T ] ∈ D2

R. By Lemma 3.2.4,

we have D/T D ∈ D1

R/T and D[T ] ∈ D1

R/T . We have the following exact sequence,

D[T ] → M/T M →

r
(cid:77)

i=1

M/(Yi + T M ) → D/T D → 0

By Lemma 3.2.3, we know M/T ∈ D1

R/T if and only if M/(Yi + T M ) is in D1

R/T for all 1 ≤ i ≤ r.

Lemma 3.2.6. The same setup as previously, then M/(Yi + T M ) ̸∈ D1

R/T if and only if T ∈ Pi

and ht(Pi) = 2.

60

Proof. Assume M/(Yi + T M ) ̸∈ D1

R/T , then M/Yi is not in D2

R by Lemma 3.2.3 and Lemma

3.2.4. Since M/Yi is finitely generated R-module, there exists a surjective map

(R/Ji)n ↠ M/Yi

where Ji = AnnR(M/Yi). We have

(R/(Ji + T R))n ↠ M/(Yi + T M )

Hence R/(Ji + T R) ̸∈ D1

R/T by Lemma 3.2.3.

If htR(Ji + T R) ≥ 3, then R/(Ji + T R) ∈ D2

R, which implies R/(Ji + T R) ∈ D1

R/T by

Lemma 3.2.4. Contradiction! Therefore, htR(Ji + T R) ≤ 2.

Pi ⊂ Pi + T R =

(cid:112)

Ji + T R ⊂

(cid:112)

Ji + T R

Since htR(Pi) ≥ 2, we have Pi = Pi + T R =

√

Ji + T R and htR(Pi) = 2. Hence, T ∈ Pi and

ht(Pi) = 2.

Conversely, assume T ∈ Pi and ht(Pi) = 2. Then T n ∈ AnnR(M/Yi) for some integer n since

Pi = (cid:112)AnnR(M/Yi). Assume M/(Yi + T M ) ∈ D1

R/T . Then M/(Yi + T M ) ∈ D2

R by Lemma

3.2.4. We have the following exact sequence,

M/(Yi + T M ) ×T−−→ M/(Yi + T 2M ) → M/(Yi + T M ) → 0

By Lemma 3.2.3, we have M/(Yi + T 2M ) ∈ D2

R. Continuing the same process, we have M/(Yi +

T kM ) ∈ D2

R for any integer k ≥ 1. In particular, M/Yi = M/(Yi + T nM ) ∈ D2

R. It contradicts

to that htR(Pi) = 2.

3.3 Characteristic ideal

If R is a Krull domain, we can define the characteristic ideal of a finitely generated R-module

M . Recall that if R is a Krull domain, then for any finitely generated torsion R-module M , there

exists a short exact sequence:

0 → A → M →

R/pei

i → B → 0,

n
(cid:77)

i=1

61

where A and B are pseudo-null R-modules and pi is a prime ideal of height 1. The R-module

E(M ) :=

n
(cid:77)

i=1

R/pei
i

is called the elementary module attached to M . The characteristic ideal of M is then defined as

ChR(M ) :=

n
(cid:89)

i=1

pei
i .

If M is a finitely generated R-module but not torsion, we set ChR(M ) := 0. Hence, a finitely

generated R-module M is pseudo-null if and only if ChR(M ) = R. If M ′, M, M ′′ are finitely

generated R-modules and

0 → M ′ → M → M ′′ → 0,

then we have ChR(M ) = ChR(M ′) ChR(M ′′).

There is another way to describe the characteristic ideal of a finitely generated torsion R-module

M :

ChR(M ) =

plp(M ⊗RRp),

(cid:89)

p

where the product runs over all prime ideals p of height 1, and lp(M ⊗R Rp) denotes the length of

the Rp-module M ⊗R Rp.

Proposition 3.3.1. Let R and R/(T ) both be Krull domains. Let M be a finitely generated

pseudo-null R-module. Then

ChR/T (M/T M ) = ChR/T (M [T ]).

This result can be viewed as a generalization of [BBL14, Prop 2.7]. Before proving the

proposition, we first establish the following lemma. Throughout this section, we assume that R and

R/(T ) are both Krull domains so that characteristic ideals are well-defined. Let π : R → R/(T )

be the quotient map.

Lemma 3.3.2. Let N be a finitely generated pseudo-null R-module and assume that AssR(N ) =

{P }.

62

If T ∈ P and ht(P ) = 2, then

ChR/T (N/T N ) = ChR/T (N [T ]) = π(p)lp((N/T )⊗RRp).

Otherwise, ChR/T (N/T N ) = ChR/T (N [T ]) = R.

Proof. Let ¯Q be a prime ideal of height 1 in R/T , and let Q = π−1( ¯Q) be the preimage of ¯Q. Then

Q is a prime ideal in R of height 2. Consider the short exact sequence:

0 → N [T ] → N ×T−−→ N → N/T N → 0.

Taking the localization at the prime ideal Q, we obtain:

0 → N [T ] ⊗R RQ → N ⊗R RQ

×T−−→ N ⊗R RQ → N/T N ⊗R RQ → 0.

We have N ⊗R RQ ̸= 0 if and only if AnnR(N ) ⊆ Q. Since P = (cid:112)AnnR(N ) ⊆ Q and

htR(P ) ≥ 2 while htR(Q) = 2, it follows that htR(P ) = 2 and T ∈ Q = P .

Since N is finitely generated, there exists an integer n such that Qn ⊆ AnnR(N ). Thus,

N ⊗R RQ can be viewed as a module over the Artinian ring RQ/Qn, which implies that the length

lQ(N ⊗R RQ) is finite. Consequently, the first and last terms in the exact sequence have the same

finite length as RQ-modules.

The conclusion follows from the following observation: If an R-module M is annihilated by

T , then

M ⊗R RQ = M ⊗R/T (R/T ⊗R RQ) = M ⊗R/T (R/T ) ¯Q.

Any RQ-submodule of M ⊗R RQ is also an (R/T ) ¯Q-module. Hence, the length of M ⊗R RQ as

an RQ-module is the same as the length of M ⊗R/T (R/T ) ¯Q as an (R/T ) ¯Q-module.

Proof of Proposition 3.3.1. Recall that we have the following exact sequence:

0 → M [T ] →

(cid:77)

i

(M/Yi)[T ] → D[T ] → M/T →

(cid:77)

i

M/(Yi + T M ) → D/T D → 0.

We know that D/T D and D[T ] are pseudo-null as R/T -modules. By Lemma 3.3.2, we also

have

ChR/T ((M/Yi)[T ]) = ChR/T (M/(Yi + T M )).

63

Hence, it follows that

ChR/T (M [T ]) = ChR/T (M/T M ).

As in [BBL14], we can deduce the following propositions from Proposition 3.3.1. We will

record these propositions here, and the reader can refer to [BBL14] for their proofs.

Proposition 3.3.3 (Prop 2.10 in [BBL14]). Let A be Krull domain and B = A[[T ]]. Let π : B → A

be the projection given by T → 0 and let M be a finitely generated torsion B-module. Then

ChA(M [T ])π(ChB(M )) = ChA(M/T M ).

Moreover,

ChA(M [T ]) = 0 ⇐⇒ π(ChB(M )) = 0 ⇐⇒ ChA(M/T M ) = 0

and in this case M [T ] and M/T M are A-modules of the same rank.

Proposition 3.3.4 (Cor 2.11 in [BBL14]). In the above setting, assume that M/T M is a finitely

generated torsion A-module. Then M is a pseudo-null B-module if and only if ChA(M [T ]) =

ChA(M/T M ). Moreover, if M/T M is pseudo-null A-module, then M is pseudo-null B-module.

3.4 Pseudo-null in non-commutative Iwasawa algebra

In the previous section, we say a R-module M is pseudo-null if Mp = 0 for any prime ideal p

of height 1. In non-commutative Iwasawa algebra, there is another generalization of the definition

of the pseudo-null module. We will recall the definition and its relation with the classical one. We

will try to extend the results in previous sections to the general case. However, we only can give

partial results.

Let R be associated algebra and M be a R-module. Let E(M ) be the injective envelope of M

[Ste12, Chap.V §2]. One can think E(M ) is the "minimal" injective R-module containing M in

some sense. View R as a R module. Consider the "minimal" injective resolution of R,

0 → R

µ0−→ E0

µ1−→ E1

µ2−→ · · ·

64

In other words, E0 = E(R) and Ei = E(coker µi). Define the full subcategory Cn

R of the category

of R modules consisting of all R modules M such that HomR(M, E0 ⊕ E1 ⊕ E2 · · · En) = 0. The

subcategory Cn

R is studied in [Ste12, Chap.VI] and related to Hereditary Torsion Theory.

Proposition 3.4.1 (Chap.VI, Prop. 6.9 in [Ste12]). An R-module M lies in Cn

R if and only if

Exti

R(M ′, R) = 0 for any i ≤ n and any submodule M ′ ⊂ M .

We can relate Cn

R with Dn

R when R is a commutative noetherian ring. Let R be a commutative

noetherian ring. We that R satisfies Serre’s condition (Sn) if

depth(Rp) ≥ min{n, ht(p)}

Proposition 3.4.2 (Chap.VII, Prop 6.8 in [Ste12]). If commutative noetherian ring R satisfied

Serre’s condition (Sn), then a R torsion module M lies in Cn

R if and only if Mp = 0 for all primes

p with ht(p) ≤ n.

The integrally closed noetherian commutative integral domain satisfies Serre’s condition (S1).

The Cohen–Macaulay ring satisfies Serre’s condition (Sn) for all n.

The proof of Theorem 3.2.1 depends on the Lemma 3.2.3 and Lamma 3.2.4.

It is easy to

see, the analogue of Lemma 3.2.3 also holds for Cn

R. In other words, for a short exact sequence

0 → M ′ → M → M ′′ → 0, we have M ∈ Cn

R if and only if M ′ ∈ Cn

R and M ′′ ∈ Cn

R. For the

analogue of Lemma 3.2.4, there is an easy proof (and stronger results) when R is a commutative

noetherian ring.

Lemma 3.4.3. Assume R is a commutative noetherian ring and let T be a regular element of R.

Let M be a finitely generated R-module. Then M/T M ∈ Cn

R/T if and only if M/T M ∈ Cn+1
R .

Proof. By the Proposition 3.4.1, we only need to show that Exti

R/T (M/T M, R/T ) = 0 for i ≤ n

if and only if Exti

R(M/T M, R) = 0 for i ≤ n + 1. By the (15.C) Theorem 28 in [Mat70], we

have Exti

R/T (M/T M, R/T ) = 0 for i ≤ n if and only if there exists a R/T regular-sequence

¯a0, ¯a1, · · · , ¯an of length (n + 1) in AnnR/T (M/T M ). Then T, a0, · · · , an is a R-regular sequence

of length (n + 2) in AnnR(M/T M ). Since Exti

R(M/T M, R) = 0 for i ≤ n + 1 if and only if

65

there exists a R regular-sequence of length (n + 1) in AnnR(M/T M ), we have M/T M ∈ Cn
R

implies M/T M ∈ Cn+1

R . The converse holds because every regular sequence can be extended to a

maximal regular sequence and every maximal regular sequence has the same length.

Next, we will prove the analogue of Lemma 3.2.4 for non-commutative case. First, we need to

clarify the meaning of the regular element T in a non-commutative ring R. We take the definition

(5.13) in [Lam12, Chapter 5B]. Say T is regular element in a non-commutative ring R if T is a

central element in R and (T ) ̸= R. To simplify the notation ,we use R/T denoting the quotient

R/(T ).

Theorem 3.4.4. Assume R is non-commutative algebra and T is a regular element in R. Let M

be the right R-module. Then M/T M ∈ Cn

R/T if and only if M/T M ∈ Cn+1
R .

Let N be a submodule of M . Consider the following short exact sequence.

0 → R ×T−−→ R → R/T R → 0

It gives us that

Exti

R(N, R) ×T−−→ Exti

R(N, R) → Exti

R(N, R/T ) → Exti+1

R (N, R) ×T−−→ Exti+1

R (N, R).

The first arrow and the last arrow should be zero map since N T = 0. For all i ≥ 0, we have

0 → Exti

R(N, R) → Exti

R(N, R/T ) → Exti+1

R (N, R) → 0.

If

Exti

R(N, R/T ) = Exti

R/T (N, R/T ) for all 0 ≤ i ≤ n,

(3.1)

(3.2)

then the conclusion follows directly from the definitions. However, in general, Exti

R(N, R/T ) is

not isomorphic to Exti

R/T (N, R/T )! For counterexamples, see Pavel Čoupek’s comments in the

StackExchange discussion [NB].

Fortunately, our situation is highly specialized, as we already know that the Ext groups vanish

for all 0 ≤ i ≤ n. This allows us to apply induction and dimension shifting to establish the desired

equality (3.2).

66

We begin by proving a lemma analogous to results found in [Sta24, Tag 087M] and [NB]. This

lemma will play a key role in proving both directions of Theorem 3.4.4.

Lemma 3.4.5. Let R be an associated ring and T be a regular element of R. Let M be a R-module.

Assume T is a regular element for M , i.e. M [T ] = 0. Then Exti

R(M, R/T ) = Exti

R/T (M/T, R/T )

for all i ≥ 0.

Proof. Take a projective R-module resolution of M ,

· · · → P2 → P1 → P0 → M → 0

We claim that

· · · → P2/T → P1/T → P0/T → M/T → 0

is a projective R/T -module resolution of M/T . Thus

Exti

R/T (M/T, R/T ) = H i(HomR/T (P·/T, R/T ))

= H i(HomR(P·, R/T ))

= Exti

R(M, R/T )

Now we prove the claim. Assume that 0 → S0 → P0 → M → 0 is a short exact sequence.

Then

TorR

1 (M, R/T ) = M [T ] = 0 → S0/T → P0/T → M/T M → 0

and TorR

i (S0, R/T ) = 0 for i ≥ 1. Assume that 0 → S1 → P1 → S0 → 0. Then

0 = TorR

i (S0, R/T ) → S1/T → P1/T → S0/T → 0

and TorR

i (S1, R/T ) = 0. Continue the process and put all short exact sequences together, we have

· · · → P2/T → P1/T → P0/T → M/T → 0 is a projective R/T -module resolution of M/T .

Lemma 3.4.6. Assume R is non-commutative algebra and T is a regular element in R. Let N be

the right R/T -module. Assume that n ≥ 1. If Extn−1

R/T (N, R/T ) = Extn

R/T (N, R/T ) = 0, then

Extn+1

R (N, R/T ) = Extn+1

R/T (N, R/T ).

67

Proof. View N as an R-module first. We consider the following exact sequence of R-modules,

0 → S0 → P0 → N → 0

(3.3)

where P0 is a projective R-module. By the dimension shifting argument, we have

Exti

R(S0, R/T ) ∼= Exti+1

R (N, R/T )

for all i ≥ 1. In particular, Extn

R(S0, R/T ) ∼= Extn+1

R (N, R/T ). Since P0 is a projective R-module,

we have P0[T ] = TorR

1 (P0, R/T ) = 0. Since S0 is a submodule of P0, we have S0[T ] = 0 as well.

By Lemma3.4.5, we have Extn

R(S0, R/T ) ∼= Extn

R/T (S0/T, R/T ).

Tensor R/T with the above short exact sequence (3.3) , we get the following exact sequence.

0 → N [T ] = N → S0/T → P0/T → N → 0

One can also see this by snake lemma. This is exact sequence of R/T -modules and we can split it

into two short exact sequences.

0 → N → S0/T → K → 0

0 → K → P0/T → N → 0

(3.4)

(3.5)

Here K is an R/T -module. Applying Hom(−, R/T ) to the exact sequence (3.4), we get a long

exact sequence.

0 = Extn−1

R/T (N, R/T ) → Extn

R/T (K, R/T ) → Extn

R/T (S0/T, R/T ) → Extn

R/T (N, R/T ) = 0

Thus, we have Extn

R/T (K, R/T ) ∼= Extn

R/T (S0/T, R/T ).

Notice that the module P0/T is free as R/T -module. By the dimension shifting argument for

exact sequence (3.5), we have Exti
R/T (K, R/T ) ∼= Extn+1

we have Extn

R/T (N, R/T ).

R/T (K, R/T ) ∼= Exti+1

R/T (N, R/T ) for all i ≥ 1. In particular,

Combining all, we have

68

Extn+1

R (N, R/T ) ∼= Extn

R(S0, R/T ) ∼= Extn

R/T (S0/T, R/T )

∼= Extn

R/T (K, R/T ) ∼= Extn+1

R/T (N, R/T ).

Lemma 3.4.7. Assume R is non-commutative algebra and T is a regular element in R. Let M be

the right R-module. If M/T M ∈ Cn

R/T , then we have M/T M ∈ Cn+1
R .

Proof. Let N be a submodule of M/T M . Since HomR(N, R/T ) = HomR/T (N, R/T ) always

holds, the desired equation (3.2) holds when n = 0.

If n = 1, to show that the desired equation (3.2) holds, we only need to show that

Ext1

R(N, R/T ) = 0.

Applying HomR(−, R/T ) to the exact sequence (3.3), we get the following exact sequence,

HomR(N, R/T ) → HomR(P0, R/T ) → HomR(S0, R/T )

→ Ext1

R(N, R/T ) → Ext1

R(P0, R/T ) = 0.

We can rewrite it as the follows,

0 = HomR/T (N, R/T ) → HomR/T (P0/T, R/T ) → HomR/T (S0/T, R/T )

(3.6)

→ Ext1

R(N, R/T ) → 0.

If we can prove the middle arrow is an isomorphism, then we can get Ext1

R(N, R/T ) = 0.

Applying the HomR/T (−, R/T ) to the exact sequence (3.4), we get the following exact sequence,

0 → HomR/T (K, R/T ) → HomR/T (S0/T, R/T ) → HomR/T (N, R/T ) = 0.

Hence, we have HomR/T (K, R/T ) ∼= HomR/T (S0/T, R/T ).

Applying the HomR/T (−, R/T ) to the exact sequence (3.5), we get the following exact sequence,

HomR/T (N, R/T ) = 0 → HomR/T (P0/T, R/T ) → HomR/T (K, R/T )

→ Ext1

R/T (N, R/T ) = 0.

69

Hence, we have HomR/T (P0/T, R/T ) ∼= HomR/T (K, R/T ). One can check that the composite
of the isomorphic map HomR/T (P0/T, R/T ) ∼= HomR/T (K, R/T ) ∼= HomR/T (S0/T, R/T ) is the

same as the middle arrow in the exact sequence (3.6). Hence, we get Ext1

R(N, R/T ) = 0 =

Ext1

R/T (N, R/T ). The desired equation 3.2 holds.

If n ≥ 2, then we can use the Lemma 3.4.6 and induction on n to deduce the desired equation

(3.2). Hence, we proved one direction of the Theorem 3.4.4.

The proof of another direction of Theorem 3.4.4 is basically reversing the argument.

Lemma 3.4.8. Assume R is non-commutative algebra and T is a regular element in R. Let M be

the right R-module. If M/T M ∈ Cn+1

R , then we have M/T M ∈ Cn

R/T .

Proof. Let N be a submodule of M/T M . If n = 0, the desired equation (3.2) HomR(N, R/T ) =

HomR/T (N, R/T ) holds without proof.

If n = 1, we have HomR(N, R/T ) = Ext1

R(N, R/T ) = 0 by exact sequence (3.1). We only

need to show Ext1

R/T (N, R/T ) = 0. Applying HomR(−, R/T ) to the exact sequence (3.3), we get

the following exact sequence.

HomR(N, R/T ) = 0 → HomR(P0, R/T ) → HomR(S0, R/T )

(3.7)

→ Ext1

R(N, R/T ) = 0

Hence, we have

HomR/T (P0, R/T ) = HomR(P0, R/T ) ∼= HomR(S0, R/T ) = HomR/T (S0, R/T ).

Applying HomR/T (−, R/T ) to the exact sequence (3.4), we get the following exact sequence.

0 → HomR/T (K, R/T ) → HomR/T (S0/T, R/T ) → HomR/T (N, R/T ) = 0

Hence, we have HomR/T (K, R/T ) ∼= HomR/T (S0/T, R/T ).

Applying HomR/T (−, R/T ) to the exact sequence (3.5), we get the following exact sequence.

70

HomR/T (N, R/T ) → HomR/T (P0/T, R/T ) → HomR/T (K, R/T )

→ Ext1

R/T (N, R/T ) → Ext1

R/T (P0/T, R/T )

Since HomR/T (N, R/T ) = 0 and Ext1

R/T (P0/T, R/T ) = 0, we have

0 → HomR/T (P0/T, R/T ) → HomR/T (K, R/T ) → Ext1

R/T (N, R/T ) → 0

The composite of the map

and the map

HomR/T (P0/T, R/T ) → HomR/T (K, R/T )

HomR/T (K, R/T ) ∼= HomR/T (S0/T, R/T )

is exactly the middle arrow in the exact sequence (3.7). Therefore, we have Ext1

R/T (N, R/T ) = 0.

If n ≥ 2, by induction on n, we have N ∈ Cn−1

Extn−1

R/T (N, R/T )0. By Lemma 3.4.6, we have Extn

R/T . Therefore, we have Extn−2
R(N, R/T ) = Extn

R/T (N, R/T ). Hence, the

R/T (N, R/T ) =

desired equation (3.2) holds by induction.

Next, we would like to get an analogue of Theorem 3.2.1. In other words, assuming M ∈ C1
R,

we want to find a necessary and sufficient condition for M/T M ∈ C1

R/T . The author only can show

the following theorem. The proof is due to Jie Yang when the author discusses questions with him.

Theorem 3.4.9. Let R be an associated ring. Let T be a regular element and M be a finitely

generated R-module. If Exti

R(M, R) = 0 for i = 0, 1 and Ext2

R(M, T )[T ] = 0, then

Exti

R/T (M/T, R/T ) = 0

for i = 0, 1.

Proof. We will use the Grothendieck spectral sequence.

{R module }op

F−→ {R/T module }op G−→ { abelian group }

M

→

M ⊗R R/T

→ HomR/T (S, R/T )

S

71

By Hom-tensor adjunction, we have

G ◦ F (M ) = HomR/T (M ⊗R R/T, R/T )

= HomR(M, HomR/T (R/T, R/T ))

= HomR(M, R/T )

Let M be an injective module in {R module }op. Then M is a projective R-module. Hence, M/T M

is a projective R/T R module. Therefore, M/T is G-acyclic since Exti

R/T (M/T, R/T ) = 0 for

i ≥ 1. We have the spectral sequence.

Extp

R/T (TorR

q (M, R/T ), R/T ) ⇒ Extp+q

R (M, R/T )

By the five-term exact sequence,

0 → Ext1

R/T (M/T, R/T ) → Ext1
= HomR/T (M [T ], R/T ) → Ext2

R(M, R/T ) → HomR/T (TorR
R/T (M/T, R/T ) → Ext2

R(M, R/T )

1 (M, R/T ), R/T )

By the short exact sequence 0 → R ×T−−→ R → R/T → 0, we have

0 → HomR(M, R) → HomR(M, R) → HomR(M, R/T ) = HomR/T (M/T, R/T )

→ Ext1

R(M, R) → Ext1

R(M, R) → Ext1

R(M, R/T ) → Ext2

R(M, R) ×T−−→ Ext2

R(M, R)

If Ext2

R(M, T )[T ] = 0, then Ext1

R(M, R/T ) = 0. Hence Ext1

R/T (M/T, R/T ) = 0 by the five-term

exact sequence.

72

BIBLIOGRAPHY

[Ati18]

M. Atiyah. Introduction to commutative algebra. CRC Press, 2018.

[BBL14] A. Bandini, F. Bars, and I. Longhi. Characteristic ideals and Iwasawa theory. 2014.

arXiv: 1310.0680.

[Dwy75] W. G. Dwyer. “Homology, Massey products and maps between groups”. In: J. Pure

Appl. Algebra 6.2 (1975), pp. 177–190.

[Fuj17]

S. Fujii. “On Greenberg’s generalized conjecture for CM-fields”. In: Crelle’s Journal
2017 (2017), pp. 259–278.

[Gol74]

[Gre01]

R. Gold. “The nontriviality of certain Zl-extensions”. In: J. Number Theory 6 (1974),
pp. 369–373.

“Iwasawa theory—past and present”.

In: Class field theory—its
R. Greenberg.
centenary and prospect (Tokyo, 1998). Vol. 30. Adv. Stud. Pure Math. Math. Soc.
Japan, Tokyo, 2001, pp. 335–385.

[Gre76]

R. Greenberg. “On the Iwasawa invariants of totally real number fields”. In: Amer. J.
Math. 98.1 (1976), pp. 263–284.

[Kol04]

M. Kolster. “K-theory and arithmetic”. In: Contemporary developments in algebraic
K-theory. ICTP Lect. Notes, XV. Abdus Salam Int. Cent. Theoret. Phys., Trieste,
2004, pp. 191–258.

[Kra66]

D. Kraines. “Massey higher products”. In: Trans. Amer. Math. Soc. 124 (1966),
pp. 431–449.

[Lam+23] Y. H. J. Lam, Y. Liu, R. Sharifi, P. Wake, and J. Wang. “Generalized Bockstein maps

and Massey products”. In: Forum Math. Sigma 11 (2023), Paper No. e5, 41.

[Lam12]

T.-Y. Lam. Lectures on modules and rings. Vol. 189. Springer Science & Business
Media, 2012.

[LM21]

I. Limonchenko and D. Millionshchikov. “Higher order Massey products and appli-
cations”. In: Topology, geometry, and dynamics—V. A. Rokhlin-Memorial. Vol. 772.
Contemp. Math. Amer. Math. Soc., [Providence], RI, 2021, pp. 209–240.

[Mat70]

H. Matsumura. Commutative algebra. Vol. 120. WA Benjamin New York, 1970.

[Mil20]

J. Milne. Class Field Theory (v4.03). Available at www.jmilne.org/math/. 2020.

[MS03]

W. G. McCallum and R. T. Sharifi. “A cup product in the Galois cohomology of

73

number fields”. In: Duke Math. J. 120.2 (2003), pp. 269–310.

[MT16]

J. Mináč and N. ê. D. Tân. “Triple Massey products vanish over all fields”. In: J.
Lond. Math. Soc. (2) 94.3 (2016), pp. 909–932.

[MT17]

J. Mináč and N. ê. D. Tân. “Triple Massey products and Galois theory”. In: J. Eur.
Math. Soc. (JEMS) 19.1 (2017), pp. 255–284.

[NB]

N.B. Flat base exchange for Ext groups. Mathematics Stack Exchange. https://math.
stackexchange.com/q/3821520.

[NSW08]

J. Neukirch, A. Schmidt, and K. Wingberg. Cohomology of number fields. Second.
Vol. 323. Grundlehren der mathematischen Wissenschaften [Fundamental Principles
of Mathematical Sciences]. Springer-Verlag, Berlin, 2008, pp. xvi+825.

[Oza]

M. Ozaki. “Iwasawa invariants of Zp-extensions over an imaginary quadratic field.
Class field theoryits centenary and prospect (Tokyo, 1998), 387–399”. In: Adv. Stud.
Pure Math 30 ().

[QS24a]

P. Qi and M. Stokes. An Analogue of Greenberg’s Conjecture for CM Fields. 2024.
arXiv: 2410.05706.

[QS24b]

P. Qi and M. Stokes. On the Non p-Rationality and Iwasawa Invariants of Certain
Real Quadratic Fields. 2024. arXiv: 2408.03836.

[San93]

J. W. Sands. “On the nontriviality of the basic Iwasawa λ-invariant for an infinitude
of imaginary quadratic fields”. In: Acta Arith. 65.3 (1993), pp. 243–248.

[Sha07]

R. T. Sharifi. “Massey products and ideal class groups”. In: J. Reine Angew. Math.
603 (2007), pp. 1–33.

[Sha99]

R. T. Sharifi. Twisted Heisenberg representations and local conductors. Thesis
(Ph.D.)–The University of Chicago. ProQuest LLC, Ann Arbor, MI, 1999, p. 65.

[Sta24]

T. Stacks project authors. The Stacks project. https://stacks.math.columbia.edu. 2024.

[Ste12]

[Tat76]

B. Stenström. Rings of quotients: an introduction to methods of ring theory. Vol. 217.
Springer Science & Business Media, 2012.

J. Tate. “Relations between K2 and Galois cohomology”. In: Invent. Math. 36 (1976),
pp. 257–274.

[Was97]

L. C. Washington. Introduction to cyclotomic fields. Second. Vol. 83. Graduate Texts
in Mathematics. Springer-Verlag, New York, 1997, pp. xiv+487.

74