constructing endomorphism rings of large finite global...

Constructing Endomorphism Rings of Large Finite GlobalDimension

by

Ali Mousavidehshikh

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c© Copyright 2016 by Ali Mousavidehshikh

Abstract

Constructing Endomorphism Rings of Large Finite Global Dimension

Ali Mousavidehshikh

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2016

For a numerical semigroup H with generators α1, α2, ..., αs, let R be the subring of the

ring of formal power series k[[t]] (where k is a field of characteristic zero) with generators

tα1 , tα2 , ..., tαs . More precisely,

R := k[[tα1 , tα2 , ..., tαs ]] =

{∑i≥0

aiti : ai ∈ k, i ∈ H

}

For R 6= k[[t]], we construct ascending chains of rings R = R1 ( R2 ( ... ( Rl =

k[[t]], and we then consider E = EndR1

(⊕li=1Ri

). Our arguments show that the global

dimension of E depends on R1 and the way we construct our ascending chain. This

leads to an investigation of two types of constructions for our ascending chain, which

we call the “greedy” and “lazy” constructions. In the “greedy” construction we choose

Ri+1 as the endomorphism ring of the radical of Ri. In the “lazy” (or, as Iyama calls it,

saturated) construction we choose Ri+1 so that dimk(Ri+1/Ri) = 1 and the conductor of

Ri+1 is strictly larger than that of Ri. We introduce a special family of rings {Ri1 : i ∈ N},

thinking of each as the beginning of an ascending chain and we let {Ei : i ∈ N} be the

set of corresponding endomorphism rings.

This thesis consists of three main results. Firstly, if for each i our chain is constructed

via the “lazy” construction, then {gl. dim(Ei) : i ∈ N} is an unbounded set. Secondly,

under some additional assumptions on the set {Ri1 : i ∈ N} we compute the precise values

in the set {gl. dim(Ei) : i ∈ N}. Thirdly, if for each i the chain is constructed via the

“greedy” construction, then gl. dim(Ei) = 2 for all i.

ii

Dedication

Dedicated to my loving parents Zahra and Seyedalizamen whose support and constant

encouragement has gotten me here.

It is also dedicated to my sister Maryam and my brother Mahmoud who always keep

me calm and focused.

iii

Acknowledgements

I would first like to thank my supervisor, Ragnar-Olaf Buchweitz, for his patience, ex-

pertise, and encouragement. It has been an honour to be your student. I would also

like to thank my Ph.D committee members, professors Marco Gualtieri and Joe Repka,

who on a yearly basis would take time out of their busy schedule to meet with me, and

provide insight and feedback on my research. I would also like to thank Osamu Iyama

for being my external referee. In addition, I would like to thank all the members of the

homological seminar for all the fun and knowledgeable conversations.

It is very hard for me to put into words the affection that I feel for the staff at the

mathematics department at the University of Toronto. To put it kindly, their support,

generosity and smile makes the department an extremely comfortable place. I would

like to send a special thanks to Marie Bachtis, Ida Bullat, Rajni Lala, Jemima Merisca,

Patrina Seepersaud, and Aisha Sharif. Ida, you are sourly missed but never forgotten.

There are several people in the department with whom I had discussions that con-

tributed to this work or my overall understanding. In particular, I am grateful to

Robin Chhabra, Peter Crooks, Payman Eskandari, Brandon Hanson, Dave Reiss and

Ben Rifkind for their time and support. Also a big thanks to professors Shay Fuchs

and Abe Iglefeld for all the enlightening conversations. My warmest thanks also go to

anybody who took time out of their day to have a beer or two with me, the list is too

long to put here.

Finally, I would like to thank my parents, brother and sister for all their support.

I would also like to thank my close friends from high school, Jagdeep Jodha, Paul Pa-

trocinio, Ajay Sharma, and Mark Wright.

iv

Contents

1 Introduction and Background 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Frobenius Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Minimal Projective Resolution and Global Dimension . . . . . . . . . . . 4

1.4 Krull-Remak-Schmidt Categories . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Mapping Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Main Objects and Tools 11

2.1 Conventions, Definitions, and Notations . . . . . . . . . . . . . . . . . . . 11

2.2 The Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 A Presentation of a Ring, an Image and Kernel of a Map . . . . . . . . . 23

2.4 Structure of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Family of Starting Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 The symbol d e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 “Lazy” Construction 44


3.2 Special Rings I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Minor Results I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4 gl. dim(E) for l = 1, 2, 3, 4, 5 . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Constructing Endomorphism Rings of Large Global Dimension . . . . . . 69

3.5.1 Minor Results II . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5.2 Lower Bound for gl. dim(Ei) . . . . . . . . . . . . . . . . . . . . . 75

3.5.3 The Module M ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5.4 Global Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

v

4 “Greedy” Construction 116


4.2 Special Rings II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.3 Minor Results III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.4 gl. dim(Ei) = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5 Examples and Open Questions 147

5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.2 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Bibliography 156

vi

Chapter 1

Introduction and Background

1.1 Introduction

Let N0 be the set of non-negative integers. A setH ⊆ N0 is called a numerical semigroup

if it satisfies the following three properties;

(a) 0 ∈ H,

(b) If x, y ∈ H then x+ y ∈ H,

(c) H contains all but a finite number of the non-negative integers.

Given A = {α1, α2, ..., αs} ⊆ N, let

〈A〉 = 〈α1, α2, ..., αs〉 := {x1α1 + ...+ xsαs : xi ∈ N0}

We call A a generating set for 〈α1, α2, ..., αs〉. The set A is called a minimal generating

set if no proper subsets of A is a generating set. It is a standard fact that 〈A〉 forms

a numerical semigroup if and only if gcd(A) = 1, and every numerical semigroup arises

this way. Furthermore, every numerical semigroup has a unique minimal generating set,

and this set has finitely many elements (see [18] or [19]).

Fix a field k of characteristic zero and a numerical semigroup H with generators

α1, α2, . . . , αs, written in ascending order. Let R be a ring of formal power series over k

with topological generators

tα1 , tα2 , ..., tαs

1

Chapter 1. Introduction and Background 2

That is,

R := k[[tα1 , tα2 , ..., tαs ]] =

{∑i≥0

aiti : ai ∈ k, i ∈ H

}

In this case, we say that R is the ring of formal power series associated to H. There is

a 1-1 correspondence between numerical semigroups and the rings of formal power series

associated to them. Given such a ring, a natural question to ask is: can we construct an

R-module M such that the endomorphism ring of M , denoted EndR(M) has finite global

dimension? What is the minimum value (or maximum value) for the global dimension?

Can we actually compute the global dimension?

We begin with an essential property regarding reduced complete local Noetherian

rings of Krull dimension one.

Theorem 1.1.1. Let (R,m, k) be a reduced complete local Noetherian ring of (Krull)

dimension one with integral closure R̃ and total quotient ring R (obtained by inverting

all non-zero divisors of R). Then R ⊆ EndR(m) ⊆ R̃ (up to canonical identification).

Moreover, R = EndR(m) if and only if R = R̃ (see [4, 5, 21]).

Note that R is a product of finitely many fields, and R̃ is a product of finitely many

discrete valuation rings. Since R is complete and reduced, R̃ is a finitely generated

R-module (see [12], Theorem 11.7).

Given a numerical semigroupH withR being the ring of formal power series associated

to it, the preceding theorem allows us to construct an ascending chain of rings with R

being the beginning of this chain;

R = R1 ( R2 ( . . . ( Rl = k[[t]] (1.1)

We define

M :=l⊕

i=1

Ri, E := EndR1(M)

This thesis is concerned with computing the global dimension of E. If we assume

Ri+1 ⊆ EndR1(mi) in the ascending chain of rings given in (1.1), where mi is the maximal

ideal of Ri, then gl. dim(E) ≤ l (see [6], [7] example 2.2.3(2), and [11]). The following

proposition takes care of l = 1.

Proposition 1.1.2. If H = N0 and R is the ring of formal power series associated to

H, then gl. dim(R) = dim(R) = 1.


Proof. It follows that R = k[[t]], in which case R is a regular local Noetherian ring and

the result follows.

The following theorem reduces our problem to computing the projective dimension of

the simple E-modules.

Theorem 1.1.3. Let A be an associative ring with unit that is module finite over a local

Noetherian ring R in its centre. Then the global dimension of A equals the supremum of

the projective dimensions of the simple A-modules (see [3], Proposition 6.7 page 125 or

[13], 7.1.14).

The preceding theorem leads us to investigate methods for coming up with minimal

projective resolutions for the simple E-modules. The following theorem gives us a helping

hand in this matter.

Theorem 1.1.4. Let R be a complete local Noetherian commutative ring, and A be a

R-algebra which is finitely generated as an R-module. Then A = A/J(A) is a semi-simple

Artinian ring, where J(A) is the Jacobian radical of A. Suppose that 1 = e1 + ...+ en is

a decomposition of 1 ∈ A into orthogonal primitive idempotents in A. Then

A =n⊕i=1

eiA

is a decomposition in indecomposable right ideals of A and

A =n⊕i=1

eiA

is a decomposition of A into minimal right ideals. Moreover, eiA ∼= ejA if and only if

eiA ∼= ejA (see [15] Theorem 6.18, 6.21 and Corollary 6.22).

This theorem says that the indecomposable summands of A are of the form Pi = eiA.

By definition, the Pi are the indecomposable projective modules over A. The modules

Si = Pi/J(A) are the simple modules over A (as well as over the semi simple algebra A)

and Pi → Si → 0 is a projective cover.

The remainder of this chapter focuses on the necessary background required for this

thesis. Unless otherwise stated, k will be a field of characteristic zero and H will be a

numerical semigroup with H 6= N0, the set of non-negative integers. This is equivalent

to l ≥ 2 in the ascending chain of rings given in (1.1).


1.2 Frobenius Number

A Frobenius equation is an equation of the form

a1x1 + a2x2 + ...+ anxn = b

where ai ∈ N, b ∈ Z, and the solutions xi are non-negative integers.

Definition 1.2.1. Given positive integers a1, a2, ..., an with gcd(a1, a2, ..., an) = 1, the

Frobenius number of the set {a1, a2, ..., an} is the largest value b for which the Frobenius

equation

a1x1 + a2x2 + ...+ anxn = b

has no solution. The Frobenius number is denoted by F (a1, a2, ..., an).

The requirement that the greatest common divisor equal 1 is a necessary and sufficient

condition for the Frobenius number to exist. If the greatest common divisor were not 1,

every integer that is not a multiple of the greatest common divisor would be inexpressible

as a linear combination of a1, a2, ..., an (let alone a non-negative linear combination), and

therefore there would not be a largest such number. Conversely, if the greatest common

divisor is 1, Schur’s Theorem tells us that there exist a positive integer m for which

every number x ≥ m is a non-negative linear combination of a1, a2, ..., an. That is,

F (a1, a2, ..., an) ≤ m− 1 (i.e., the Frobenius number exists).

Theorem 1.2.2. (Sylvester 1884, see [20]) If a1, a2 are distinct positive integers with

gcd(a1, a2) = 1, then

F (a1, a2) = (a1 − 1)(a2 − 1)− 1 = a1a2 − (a1 + a2)

While it is possible to compute the Frobenius number F (a1, a2, ..., an) in each case, in

general, it should be noted that for n ≥ 3, no explicit formula is known for the Forbenius

number.

1.3 Minimal Projective Resolution and Global Di-

mension

The main focus of this section is to give a brief introduction to projective resolutions

and global dimension of rings. Let R be a ring with unit. All modules considered in this


thesis will be right R-modules (if R is not commutative). However, similar definitions

and results can be stated for left R-modules. The set of all right R-modules is denoted

by ModR, and the set of all left R-modules is denoted by RMod. For a more “in-depth”

look we refer the reader to [8, 9, 17].

Suppose M is a submodule of N . We say that M is a superfluous submodule of N

if for any other submodule H of N ,

M +H = N ⇒ H = N

That is, M is “extremely small” relative to N .

Let M and P be R-modules, with P a projective R-module. Then, (P, f) is called a

projective cover for M if f : P → M is a superfluous epimorphism. That is, f is an

epimorphism and ker f is a superfluous submodule of P . Another common convention is

to say P →M → 0 is a projective cover when the map P →M is understood.

A projective resolution of an R-module M is an exact sequence

· · · → P2d2−→ P1

d1−→ P0ε−→M → 0

in which each Pi is a projective R-module. A free resolution of M is a projective

resolution in which each Pi is free; a flat resolution is an exact sequence in which each

Pi is flat. A finite projective resolution of M is a projective resolution of the following

form;

0→ Pndn−→ Pn−1

dn−1−→ ...d1−→ P0

ε−→M → 0

That is, there exists a natural number n such that Pi = 0 for all i ≥ n. In this case, n

is called the length of the projective resolution. A finite projective resolution is said to

be minimal if (Pi, di) is a projective cover for Im(di) for i = 1, 2, ..., n and (P0, ε) is a

projective cover for M . The length of a minimal projective resolutions is unique.

Remark 1.3.1. It is well known that a projective resolution is minimal if and only if

Im(di) ⊆ J(Pi−1) for i = 1, 2, ..., n and P0ε→M → 0 is a projective cover.

IfM is a rightR-module andM has a finite projective resolution, the projective dimension

of M , denoted pdR(M), is defined to be the minimal length among all finite projective

resolutions of M . If no finite projective resolution exists for M then pdR(M) =∞. The

right global dimension of a ring R is

gl. dim(R) = sup{pdR(M) : M ∈ModR}


A similar definition is given for left modules and the left global dimension of a ring.

It is well known that the projective resolution of a module M is equal to the length of

any of its minimal projective resolutions, provided it has a minimal projective resolution

(recall that they all have the same length).

1.4 Krull-Remak-Schmidt Categories

In this section we introduce Krull-Remak-Schmidt Categories and give a summary of

some of the results related to them. For an “in-depth” look at these categories we refer

the reader to [10] or any book on this subject.

Definition 1.4.1. A category A is called additive if

(1) each morphism set HomA(X, Y ) is an (additive) abelian group for every X, Y ∈obj(A),

(2) the composition maps

HomA(Y, Z)× HomA(Y, Z)→ HomA(X,Z)

are bilinear, i.e., the distributive laws hold,

(3) A has a zero object,

(4) A has finite products and finite coproducts.

Definition 1.4.2. A category A is called an abelian category if it is an additive category

such that

(1) every morphism has a kernel and cokernel,

(2) every monomorphism is a kernel and every epimorphism is a cokernel.

An additive category is called Krull-Remak-Schmidt if every object decomposes

into a finite direct sum of objects having local endomorphism rings. An object is called

indecomposable if it is not isomorphic to a direct sum of two non-zero objects.

Theorem 1.4.3. (Krull-Remak-Schmidt theorem) Let A be a Krull-Remak-Schmidt Cat-

egory. Then

(1) An object is indecomposable if and only if its endomorphism ring is local.

(2) Every object is isomorphic to a finite direct sum of indecomposable objects.

(3) If

r⊕i=1

Xi∼=

s⊕j=1

Yj


where Xi, Yj are indecomposable objects in A, then r = s and there exists a permutation

σ such that Xσ(i)∼= Yi for all i.

Proposition 1.4.4. For a ring R the following are equivalent.

(1) The category of finitely generated projective right (left) R-modules is a Krull-Remak-

Schmidt category.

(2) The module R admits a decomposition R = P1 ⊕ P2 ⊕ . . .⊕ Pr such that each Pi is a

projective right (left) R-module having a local endomorphism ring.

(3) Every simple right (left) R-module admits a projective cover.

(4) Every finitely generated right (left) R-module admits a projective cover.

Proof. See Proposition 4.1 in [10].

Example 1.4.5. Here are some examples of Krull-Remak-Schmidt categories details of

which can be found in [1, 15].

(a) An abelian category in which every object has finite length.

(b) Let R be a commutative complete local Noetherian ring. The category of finitely-

generated modules over R is a Krull-Schmidt category.

(c) The category of coherent sheaves on a projective variety.

(d) The category of finitely generated modules over a finite R-algebra, where R is a

commutative Noetherian complete local ring (this is a generalization of (b)).

1.5 Mapping Cone

Let A be an additive category. A chain complex in A is a sequence of objects and

morphisms in A, called differentials,

(A•, d•) = · · · An−1 An An+1 · · ·dn dn+1

such that the composite of adjacent morphisms is zero;

dndn+1 = 0 for all n ∈ Z

We call n the homological degree of An, abbreviated H.D. If (A•, d•) and (B•, g•) are

two chain complexes, then a chain map

f = f• = (A•, d•)→ (B•, g•)


is a sequence of morphisms fn : An → Bn for all n ∈ Z making the following diagram

commute:

· · · An−1 An An+1 · · ·

· · · Bn−1 Bn Bn+1 · · ·

dn dn+1

gn gn+1

fn−1 fn fn+1

If the indices are increasing in the above complex and commutative diagram, we call the

complex and map a cochain complex and cochain map, respectively (in this case the

convention is to use superscripts instead of subscripts for the indices).

Given a chain complex (A•, d•) and an integer b, we define ((A[b])•, (d[b])•) to be the

complex with (A[b])n = An−b and (d[b])n = (−1)bdn−b. If

f = f• = (A•, d•)→ (B•, g•)

is a chain map, we define the mapping cone of f , denoted by Cone(f), to be the complex:

H.D n− 1 n n+ 1

Cone(f) =

A[1]

⊕B

•

, h•

= · · ·An−2

⊕Bn−1

An−1

⊕Bn

An

⊕Bn+1

· · ·hn hn+1

where

hn =

(−dn−1 0

fn−1 gn

)

An easy computation shows that hnhn+1 = 0, the 2× 2 zero matrix.

From here on we make the additional assumption that A is an abelian category so

that homology and cohomology of complexes is defined. We have the following triangle

A[1] Cone(f) B Af

where the maps B → Cone(f), Cone(f) → A[1] are the injection and projection maps

onto the direct summands, respectively. This gives rise to a long exact sequence of


homology groups (for more details see chapter 6 in [17] or any book on triangulated

categories such as chapter 1 in [14])

· · · Hi−1(B) Hi−1(A) Hi(Cone(f)) Hi(B) Hi(A) · · ·

Lemma 1.5.1. If (A•, d•) and (B•, g•) are two complexes which are exact at each homo-

logical degree and f : (A•, d•) → (B•, g•) is a chain map, then Cone(f) is a long exact

sequence.

Proof. Given an integer n,

{0} = Hn−1(A) Hn(Cone(f)) Hn(B) = {0}

that is, Hn(Cone(f)) = {0}.

1.6 Outline of this thesis

The beginning of each chapter contains a summary of its contents. Here we shall give a

brief outline of each chapter.

We begin chapter 2 by giving some of the definitions and notations. Then we construct

an ascending chain of rings, a module M and an endomorphism ring E. Once this is done,

we state some of the properties enjoyed by M and E. Two of the main results of this

chapter are as follows; We show that the first simple module has projective dimension

greater than or equal to one while all the other simple modules have projective dimension

greater than or equal to two (proposition 2.3.6). We also give a necessary and sufficient

condition for the projective dimension of the first simple to be one (proposition 2.3.7).

There is then a construction of a family of starting rings which will enable us to construct

endomorphism rings of large global dimension. We conclude the chapter by building some

of the theory that we will need later on.

In Chapter 3 we impose some additional hypothesis on the construction of our ascend-

ing chains in chapter 2, namely, we make the chain as long as possible and we call it the

“lazy” construction. The middle part of this chapter is devoted to the computation of the

global dimension of the endomorphism rings for some special rings and when l is small.

We also prove some additional properties enjoyed by M and E under this construction.

We then prove two of the main original results of this thesis. Firstly, we give a lower

bound for the global dimension of the endomorphism rings corresponding to the starting

rings in the family constructed in section 2.5 (theorems 3.5.8, 3.5.10, 3.5.11). Secondly,

under some additional hypothesis we compute these global dimensions (theorem 3.5.24).


In Chapter 4 we impose restrictions on the chain constructed in chapter 2 to make

the length of it as short as possible, we call it the “greedy” construction. The middle

part of this chapter is analogous to that of the preceding chapter but everything is done

under the “greedy” construction. We then prove the third main original result of this

thesis: for the family of starting rings constructed in section 2.5 the global dimension of

the endomorphism rings corresponding to the starting rings in the family is two (theorem

4.4.7).

In Chapter 5 we give an example which illustrates the possible values for the global

dimension of E when our starting ring is fixed. We conclude the chapter by discussing

some open questions related to this thesis which could be subject of future research.

Chapter 2

Main Objects and Tools

In this chapter we introduce the main objects studied in this thesis and the tools needed

to understand some of their elementary properties. More specifically, in section 2.1 we

introduce the notation and definitions used throughout this thesis. Section 2.2 focuses

on constructing endomorphism rings. We view these endomorphism rings as rings of

matrices and in section 2.4 we describe the entries of these matrices. In section 2.5 we

focus on a family of rings which will enable us to construct a set of endomorphism rings

whose global dimensions are arbitrarily large (but finite). We conclude this chapter by

introducing two of the most used tools in this thesis.

2.1 Conventions, Definitions, and Notations

We begin with some useful definitions regarding numerical semigroups and the rings

associated to them.

Definition 2.1.1. Let R be a ring of formal power series associated to a numerical

semigroup H. Recall that every numerical semigroup has a unique minimal generating

set. Let {α1, α2, ..., αs} be the minimal generating set for H. Let m be the maximal ideal

of R. We define

e(R) = min{n ∈ N| tn ∈ R}

C(R) = min{a ∈ N| tb ∈ R for all b ≥ a}

Γ(R) = {β ∈ N| tβ ∈ R and β ≤ C(R)}

Λ(R) = {β ∈ N| β < C(R) and β ∈ {α1, α2, ..., αs}}

We call e(R) the multiplicity of R. We also define e(m) = e(R),Γ(m) = Γ(R), and

11

Chapter 2. Main Objects and Tools 12

Λ(m) = Λ(R). We will always assume the elements in Γ(R) and Λ(R) are written in

ascending order. Also, all of the above definitions can be given in terms of the numerical

semigroup H;

e(H) = min{n ∈ N| n ∈ H}

C(H) = min{a ∈ N| b ∈ H for all b ≥ a}

Γ(H) = {β ∈ N| β ∈ H and β ≤ C(H)}

Λ(H) = {β ∈ N| β < C(H) and β ∈ {α1, α2, ..., αs}}

Notation 2.1.2. Given a ring R, the principal ideal generated by tn in R is denoted by

tnR.

Let R be a ring and A a subring of R such that R is integral over A. Then the

annihlator of the A-module R/A is called the conductor of A in R, denoted by c(R/A).

Explicitly, c(R/A) consists of elements a ∈ A such that aR ⊆ A. This is the largest ideal

of A that is also an ideal of R. If R is a subring of the total ring of fractions of A, then

we have the following identification:

c(R/A) = HomA(R,A)

A consequence of our definition are the following results, which we record for future

reference.

Lemma 2.1.3. Let H be a numerical semigroup with generators α1, α2, ..., αs and R be

the ring associated to H. Let m be the maximal ideal of R.

(a) 1 ≤ e(R) <∞, 1 ≤ C(R) <∞(b) e(R) = 1⇔ R = k[[t]]⇔ C(R) = 1.

(c) e(R) ≤ C(R).

(d) F (α1, α2, ..., αs) ∈ N ∪ {−1}. Moreover, F (α1, α2, ..., αs) = −1 if and only if there

exist nonnegative integers xi (with i = 1, 2, ..., s) such that

α1x1 + α2x2 + ...+ αsxs = 1

(e) If F (α1, α2, ..., αs) ≥ 1, then F (α1, α2, ..., αs) + 1 = C(R)

(f) If R 6= k[[t]], then c(R̃/R) = tC(R)R̃, where R̃ = k[[t]].

(g) Λ(R) ⊆ Γ(R). In particular, 0 ≤ |Λ(R)| ≤ |Γ(R)| < ∞ and |Γ(R)| ≥ 1. Moreover,

|Λ(R)| = 0⇔ e(R) = C(R).

Every ring of formal power series (equivalently, the semigroup associated to it) is


completely determined by the set Γ(R). As a word of warning, one cannot replace Γ(R)

by Λ(R) in the preceding sentence. For example, if R = k[[t2, t3]] and R1 = k[[t3, t4, t5]],

then Λ(R) = Λ(R1) = ∅ but R 6= R1.

Convention 2.1.4. Suppose H is a numerical semigroup and R is the ring of formal

power series associated to it with Γ(R) = {β1, β2, . . . , βr} and β1 < β2 < ... < βr. In this

case we will write

R = lead {0, β1, β2, . . . , βr}

Given a natural number n, if a1n, a2n, ..., aqn ∈ Γ(R) with 1 ≤ a1 < a2 < . . . < aq, we

write

R = lead {0, xn,� : x = a1, a2, ..., aq}

where the square consists of all the elements in Γ(R) that are not multiples of n. This

convention is naturally extended when there is more than one number with distinct

multiples of it in Γ(R). We can also use this convention for maximal ideals of a ring or

any subrings of R (see example 2.1.5).

Observe that e(R) = β1 and C(R) = βr. We use the word lead to emphasize that

these are all the powers of t that appear in R up to C(R), in a sense they are the leading

powers of R.

Example 2.1.5. (a) LetR = k[[t5, t22, t23, t26, t29]]. Then Γ(R) = {5, 10, 15, 20, 22, 23, 25}and we write

R = lead{0, 5x, 22, 23 : x = 1, 2, 3, 4, 5}

m = lead{5x, 22, 23 : x = 1, 2, 3, 4, 5}

c(R̃/R) = lead{25}

(b) Let R = k[[t5, t8, t27]]. Then C(R) = 23,

Γ(R) = {5, 8, 10, 13, 15, 16, 18, 20, 21, 23}

and we write

R = lead{0, 5x, 8y, 13, 18, 21, 23 : x = 1, 2, 3, 4, and y = 1, 2}


Definition 2.1.6. Suppose H is a numerical semigroup, 1 /∈ H and R is the ring of

formal power series associated to it with Γ(R) = {β1, β2, . . . , βr} and β1 < β2 < ... < βr

(notice that R 6= k[[t]]). We define

a1(R) := β1 − 1, and ai(R) := βi − βi−1 − 1 for 2 ≤ i ≤ r

We say that ai(R) is the i-th gap of R. Also, R is said to have r gaps, written G(R) = r.

Notice that

a1(R) > 0

ai(R) ≥ 0 for 2 ≤ i ≤ r − 1

ar(R) > 0

We can also describe the gaps of m. Since m = R/k, we define G(m) = G(R)− 1. Notice

that G(m) ≥ 0. When G(m) = 0, there are no gaps to describe, and this is equivalent to

R = lead {0, e(R)}

If G(m) ≥ 1, i.e. G(R) ≥ 2, then we define

ai(m) = ai+1(R) for i = 1, 2, ...,G(R)− 1

If R = k[[t]], we define G(R) = G(m) = 0.

Remark 2.1.7. G(R) = 1 ⇔ e(R) = C(R) ⇔ R = lead{0, e(R)}. Furthermore, since

numerical semigroups are closed under addition we have

a1(R) ≥ ai(R) for 1 ≤ i ≤ r

Notation 2.1.8. Let

g(R) =

G(R)∑i=1

ai(R)

z(R) =

G(R)∑i=2

ai(R).

We set g(R) = 0 whenever G(R) = 0 and z(R) = 0 whenever G(R) = 0 or 1. If H is

the numerical semigroup associated to R, g(R) is called the genus of H, equal to the


cardinality of the complement of H in N.

Example 2.1.9. Let H = 〈5, 8, 17, 19〉 =⇒ R = k[[t5, t8, t17, t19]] , then

Γ(R) = {5, 8, 10, 13, 15}

Λ(R) = {5, 8}

R = lead {0, 5, 8, 10, 13, 15}

C(R) = 15

F (5, 8, 17, 19) = 14

e(R) = 5

a1(R) = 4, a2(R) = 2, a3(R) = 1, a4(R) = 2, a5(R) = 1

G(R) = 5

g(R) = 10

z(R) = 6

Notice that if R1 = k[[t5, t8, t11, t12, t14]], then Λ(R) = Λ(R1) but R 6= R1.

2.2 The Construction

Suppose H is a numerical semigroup with generators α1, α2, ..., αs, F (α1, α2, ..., αs) > −1,

and let R1 be the ring of formal power series associated to H. Since R1 6= R̃1 = k[[t]], we

have R1 ( EndR1(m1) ⊆ R̃1 (Theorem 1.1.1). Moreover, m1 contains a non-zero divisor

(Proposition 2.2.1), and EndR1(m1) embeds naturally into R1 (by sending f to f(a)/a,

which is independent of the non-zero divisor a ∈ m1). It is well known that in fact

EndR1(m1) ⊆ R̃1. Furthermore, it is easy to see that EndR1(m1) is itself a ring of formal

power series. Let R2 be any ring of formal power series over k that properly contains

R1 and is contained in EndR1(m1). Notice that R2 is a local Noetherian ring of (Krull)

dimension 1. If R2 = k[[t]], then R2 = EndR1(m1) = k[[t]] in which case we define

M := R1 ⊕R2, E := EndR1(M)

If R2 6= k[[t]], pick R3 such that R2 ( R3 ⊆ EndR1(m2) ⊆ k[[t]] (this is possible by

Theorem 1.1.1). If R3 = k[[t]], define

M := R1 ⊕R2 ⊕R3, E := EndR1(M)


Notice that R1 ( R2 ( R3 = k[[t]]. If R3 6= k[[t]], repeat the process to obtain R4, and

continue in this fashion. Since R1 is missing only finitely many powers of t there exists

an l such that Rl = R̃1 = k[[t]]. Hence, we have constructed an ascending chain of rings

R1 ( R2 ( R3 ( ... ( Rl−1 ( Rl = k[[t]]

Let

M =l⊕

i=1

Ri, E = EndR1(M)

Notice that k[[t]] = Rl = EndR1(ml−1).

Proposition 2.2.1. Let (R,m, k) be a reduced local Noetherian ring with dim(R) = 1.

Then, m * Z(R) (the set of zero divisors of R).

Proof. Suppose not. Then

m ⊆ Z(R) =⋃

p minimal prime

p (since R is reduced).

By prime avoidance we have m ⊆ p for some minimal prime ideal. In particular, m =

p ⇒ dim(R) = ht(m) = ht(p) = 0, a contradiction (since dim(R) = 1). Therefore,

∃x ∈ m such that x /∈ Z(R).

Proposition 2.2.2. gl. dim(E) ≤ l (see [6] or [7] example 2.2.3(2)).

In one way our construction is more restrictive then the one built in [11]. More specifically,

the rings Ri and EndR1(mi) are always local Noetherian rings of (Krull) dimension 1.

However, it is also less restrictive since we only require Ri+1 ⊆ EndR1(mi). From here

on when we say an ascending chain of rings we mean a chain of rings with the above

restrictions imposed on it.

Given a chain of ascending rings

R1 ( R2 ( ... ( Rl = k[[t]]

we can represent E as an l × l matrix. More specifically,

Eij = HomR1(Rj, Ri).

The (Jacobian) radical of E denoted by J(E), or rad(E) is the matrix with the following


entries (see [21]):

(J(E))ij =

Eij if i 6= j

mi if i = j.

Since R1 is a complete local noetherian commutative ring and E is a finitely generated

R-module, Theorem 1.1.4 implies that the right indecomposable projective modules of

E are the matrices Pi = eiE, where ei is the l × l matrix with 1 in the ii-th entry and

zero everywhere else. We identify Pi with its non-zero row. That is, Pi is the i-th row

in E (since all other rows are zero’s). Furthermore, the simple E-modules are Si = eiD,

where D is the l × l diagonal matrix with k as its diagonal entries. We identify Si with

its non-zero row (as we did for the projective modules), that is, Si is the row matrix with

k in its i-th entry and zero everywhere else. Since R1 is in the center of E, to compute

the global dimension of E it suffices to compute the projective dimension of the simple

modules (Theorem 1.1.3).

Lemma 2.2.3. The category of finitely generated projective E-modules is a Krull-Remak-

Schmidt category.

Proof. By Theorem 1.1.3 every simple E-module has a projective cover, and Proposition

1.4.4 completes the proof.

Lemma 2.2.4. Given a simple E-module S, the objects in the projective resolution of S

are isomorphic to a finite direct sum of indecomposable objects (each of which is obviously

projective).

Proof. This follows from example 1.4.5(b) and Theorem 1.1.4.

Example 2.2.5. Let R1 = k[[t3, t4, t5]], R2 = k[[t2.t3]], R3 = k[[t]], then

E =

R1 t3R3 t3R3

R2 R2 t2R3

R3 R3 R3

Notice that Pi = eiA is a 3 × 3 matrix, for i = 1, 2, 3. But as we mentioned, for each i

we identify Pi with its non-zero row. For example,

P1 =

R1 t3R3 t3R3

0 0 0

0 0 0

, S1 =

k 0 0

0 0 0

0 0 0


In this case we simply write

P1 =(R1 t3R3 t3R3

), S1 =

(k 0 0

)as a row.

Notice that the number of simple and indecomposable projective E-modules is l. We say

that Pi is the projective module, and Si is the simple module associated to Ri.

Notation 2.2.6. Given 1 ≤ i ≤ l, if Pi is a projective E-module associated to the ring

Ri written in row notation with the zero rows taken out (example 2.2.5), we define

Eij := (Pi)j

Recall that

(J(E))ij =

Eij if j 6= i

mi if j = i.

The Jacobian radical of Pi, written in row notation with the zero rows taken out is given

by (see [21])

(J(Pi))j =

(Pi)j if j 6= i

mi if j = i.

Given a ring Ri in a chain of ascending rings

R1 ( R2 ( ... ( Rl = k[[t]]


with Γ(Ri) = {β1, β2, β3, ..., βr} (where β1 < ... < βr = C(Ri)), we define

Ri,0 = Ri = lead{0, β1, β2, ..., βr}

Ri,1 = Ri/k = mi = lead{β1, β2, ..., βr}

Ri,2 = lead{β2, ..., βr}

Ri,3 = lead{β3, ..., βr}

.

.

.

Ri,r = lead{βr} = tC(Ri)Rl

It should be noted that in general, there is no connection between Ri,j and Eij. However,

the notation Ri,j is used extensively in the computation portions of this thesis.

Example 2.2.7. Let R1 = k[[t3, t5, t7]], R2 = k[[t3, t4, t5]], R3 = k[[t]]. Then, C(R1) = 5

and Γ(R1) = {3, 5}. In particular,

R1,0 = R1 = lead{0, 3, 5}

R1,1 = m1 = lead{3, 5}

R1,2 = lead{5} = t5R3

Recall that a finitely generated R-module M is torsion-free provided the natural

map M →M ⊗R R is injective, where R is the total quotient ring of R.

Definition 2.2.8. Suppose R and S are local, Noetherian, commutative, reduced rings,

that are also complete with respect to their Jacobian radicals, respectively, and have

Krull dimension 1. We say that S is a birational extension of R provided R ⊆ S and S

is a finitely generated R-module contained in the total quotient ring R of R.

Notice that if S is a birational extension of R, then every finitely generated torsion-

free S-module is a finitely generated torsion-free R-module, but not vice versa. The

following lemma follows by clearing denominators.

Lemma 2.2.9. Suppose S is a birational extension of R. Let C and D be finitely gen-

erated torsion-free S-modules. Then HomR(C,D) = HomS(C,D). Furthermore, if M is

a finitely generated torsion-free R-module, and f : C →M is an R-linear map, then the

image of f is an S-module.


If R is a ring of formal power series associated to a numerical semigroup H, then R is

local, commutative, Noetherian, reduced, complete with respect to its Jacobian radical,

and has Krull dimension 1.

Lemma 2.2.10. Given a chain of ascending rings

R1 ( R2 ( ... ( Rl = k[[t]]

for any 1 ≤ a ≤ i ≤ l we have HomR1(Ra, Ri) = HomRa(Ra, Ri) = Ri. In particular,

Eij = HomR1(Rj, Ri) = Ri for j ≤ i

Proof. Notice that Ra is a birational extension of R1. Furthermore, for 1 ≤ a ≤ i ≤ l,

Ra and Ri are finitely generated torsion-free Ra-modules. The result follows by Lemma

2.2.9 and the fact that HomR(R,N) = N for any R-module N .

Notation 2.2.11. By Theorem 1.1.4, Pi → Si → 0 is a projective cover. We denote the

map from Pi → Si by πi. In particular, (Pi, πi) is a projective cover for Si.

Notice that

(Pi)j =

Ri if 1 ≤ j ≤ i

HomR1(Rj, Ri) if i+ 1 ≤ j ≤ l

(Si)j =

0 if i 6= j

k if i = j

We can give an explicit description of the map πi. Let (πi)j : (Pi)j → (Si)j. Then,

(πi)j =

ξi if i = j

0 if i 6= j

where ξi : Ri → Ri/mi is the quotient map. It follows that ker πi = J(Pi) for 1 ≤ i ≤ l.


R1 ( R2 ( ... ( Rl = k[[t]]

if 1 ≤ a ≤ i < b ≤ l, then HomR1(Ra, Ri) ) HomR1(Rb, Ri).


Proof. Given 1 ≤ a < b ≤ l we have

Ra ( Rb =⇒ HomR1(Ra, Ri) ⊇ HomR1(Rb, Ri) for any i

Making the additional assumption 1 ≤ a ≤ i < b ≤ l, we have k ∩HomR1(Rb, Ri) = {0},where k is the base field of R1 (in fact, of all the Ri’s in our chain) and is identified with

the set consisting of scalar multiplication. Lemma 2.2.10 yields

HomR1(Ra, Ri) = Ri ⊇ k

Hence, HomR1(Ra, Ri) ) HomR1(Rb, Ri) for 1 ≤ a ≤ i < b ≤ l.


R1 ( R2 ( ... ( Rl = k[[t]]

if 1 ≤ i < j ≤ l, then HomR1(Rj, Ri) = HomR1(Rj,mi).

Proof. Since mi ( Ri we have HomR1(Rj,mi) ⊆ HomR1(Rj, Ri) for all 1 ≤ j ≤ l. When

i < j ≤ l, then any non-zero map from Rj to Ri cannot send anything to non-zero scalars

(since k ∩HomR1(Rj, Ri) = {0} by Lemma 2.2.12). In particular, every non-zero map in

HomR1(Rj, Ri) is actually a map from Rj to mi. Since the zero map is also a map from

Rj to mi the result follows.

Proposition 2.2.14. Given a chain of ascending rings

R1 ( R2 ( ... ( Rl = k[[t]]

fix an i with 1 ≤ i ≤ l. If mi = tαRj for some α ≥ 0 and 1 ≤ j ≤ l, then

(a) α = e(Ri).

(b) Rj = EndR1(mi).

(c) i ≤ j ≤ l.

(d) If i 6= l, then i < j ≤ l.

(e) If i 6= l, then mi = HomR1(Rj, Ri).

(f) If i 6= l, then for all a with i < a ≤ j, we have HomR1(Ra, Ri) = mi.

Proof. (a) α = e(tαRj) = e(mi) = e(Ri).

(b)

EndR1(mi) = HomR1(mi,mi) = HomR1(te(Ri)Rj, t

e(Ri)Rj) = HomR1(Rj, Rj) = Rj.


(c) Since Rj = EndR1(mi) ⊇ Ri, we have i ≤ j ≤ l.

(d) If i 6= l, then Rj = EndR1(mi) ) Ri (by construction of the chain), that is i < j ≤ l.

(e) Since i 6= l, part (d) yields i < j ≤ l. By Lemmas 2.2.10 and 2.2.13 we have

HomR1(Rj, Ri) = HomR1(Rj,mi)

= HomR1(Rj, te(Ri)Rj)

= te(Ri) HomR1(Rj, Rj)

= te(Ri)Rj

= mi

(f) If i 6= l, then i < j ≤ l by part (d). For any i < a ≤ j we have Ri ( Ra ⊆ Rj. Lemma

2.2.12 yields

HomR1(Ri, Ri) ) HomR1(Ra, Ri) ⊇ HomR1(Rj, Ri)

In particular, the above chain of inclusions, part (e), and Lemma 2.2.10 yield

mi = HomR1(Rj, Ri) ⊆ HomR1(Ra, Ri) ( HomR1(Ri, Ri) = Ri

Maximality of mi implies that mi = HomR1(Ra, Ri) for all a = i+ 1, . . . , j.

Example 2.2.15. Let

R1 = lead {0, 3, 4, 6}

R2 = EndR1(m1) = lead {0, 3}

R3 = lead {0, 2}

R4 = EndR1(m3) = k[[t]]

Notice that m1 6= t3R2. That is, the converse of part (b) in Proposition 2.2.14 is false.

Lemma 2.2.16. Given an ascending chain of rings

R1 ( R2 ( · · · ( Rl = k[[t]]

then

(a) e(Rl) = C(Rl) = 1, and e(Rl−1) = C(Rl−1).

(b) dimk(Rl/R1) = g(R1)


Proof. (a) Since Rl = k[[t]] we have e(R1) = 1 = C(R1). Moreover,

Rl = EndR1(ml−1) ⇒ ml−1 = lead {e(Rl−1)}

⇒ Rl−1 = lead {0, e(Rl−1)}

⇒ e(Rl−1) = C(Rl−1)

(b) Let H be the numerical semigroup associated to R1. Suppose b1, b2, ..., br are the

natural numbers missing from H, then Rl/R1 has the set {tbi + R1 : i = 1, 2, . . . , r} as

its basis over k. Hence, dimk(Rl/R1) = g(R1).

Lemma 2.2.17. Let

R1 ( R2 ( ... ( Rl = k[[t]]

be an ascending chain of rings. Then

(a) a1(R1) = e(R1)− 1

(b) 1 ≤ l ≤ g(R1) + 1

(c) e(Ri) = 1⇔ C(Ri) = 0⇔ Ri = k[[t]]⇔ mi = tRi

Proof. (a) a1(R1) = β1 − 1 = e(R1)− 1.

(b) Notice that l = 1 if and only if R1 = k[[t]]. If R1 6= k[[t]], then l ≥ 2. Moreover,

g(R1) is the number of powers of t which are missing from R1, and we atleast put one

power of t in at each stage in our construction, thus, l ≤ g(R1) + 1.

(c) e(Ri) = 1⇔ t ∈ Ri ⇔ C(Ri) = 1⇔ Ri = k[[t]]⇔ mi = tRi = lead {0, 1}.

2.3 A Presentation of a Ring, an Image and Kernel

of a Map

Our aim in this section is to give an elegant way of determining the image and kernel of

a map. We begin with a useful method for describing a ring associated to a numerical

semigroup.

Definition 2.3.1. A presentation of a ring of formal power series R associated to a

numerical semigroup H is a table with the top row consisting of the non-negative integers

upto C(R), and in the bottom row we place a x under a natural number n if tn ∈ R and

a zero if tn /∈ R. We will sometimes shorten the top row when necessary by omitting

the non-negative integers n for which tn /∈ R. There is also an alternate way to shorten


the table and it is as follows: given integers a, b with a ≤ b, we will use the shorthand

notation a . . . b in the top row to mean all the integers between a and b. We put an x in

the second row if those powers are in R and a zero if they are not.

Since every ring of formal power series R associated to a numerical semigroup H is

uniquely determined by Γ(R), each such ring gives rise to a unique presentation and vice

versa.

Example 2.3.2. Let R = k[[t4, t11, t13, t14]]. Then, R has the following presentation:

Powers of t 0 1 2 3 4 5 6 7 8 9 10 11

R x 0 0 0 x 0 0 0 x 0 0 x

The two shorter versions are

Powers of t 0 4 8 11

R x x x x

Powers of t 0 . . . 4 . . . 8 . . . 11

R x 0 x 0 x 0 x

The power of these presentations is that they enable us to find the image and kernel

of maps, as the following example illustrates.

Example 2.3.3. Let

R1 = k[[t4, t11, t13, t14]] = lead {0, 4, 8, 11}

R2 = EndR1(m1) = k[[t4, t7, t9, t10]] = lead {0, 4, 7}

R3 = EndR1(m2) = k[[t3, t4, t5]] = lead {0, 3}

R4 = EndR1(m3) = k[[t]] = lead {0, 1}

Then

E =

R1,0 R1,1 R1,2 R1,3

R2,0 R2,0 R2,1 R2,2

R3,0 R3,0 R3,0 R3,1

R4,0 R4,0 R4,0 R4,0

=

P1

P2

P3

P4


Consider the map P1

⊕P3

(1,−t4)−→ P2

Suppose we want to compute

∆ := (1,−t4)

P1

⊕P3

= Im(1,−t4)

We use linear algebra to define the image of a map. That is,

(1,−t4)

P1

⊕P3

= (1,−t4)

(R1,0 R1,1 R1,2 R1,3

R3,0 R3,0 R3,0 R3,1

)= (1, 0,−t4, 0)E

Since projective modules have many entries (in this case they have four entries) we

compute ∆ entry by entry. Let ∆j be the j-th entry of ∆, then

∆j = (1,−t4)

(P1)j

⊕(P3)j


We say ∆j has the following presentation:

Value of j ∆j 0 1 2 3 4 5 6 7 8 9 10 11

j = 1 R1,0 x 0 0 0 x 0 0 0 x 0 0 x

⊕(−t4)R3,0 0 0 0 0 x 0 0 x x x x x

j = 2 R1,1 0 0 0 0 x 0 0 0 x 0 0 x

⊕(−t4)R3,0 0 0 0 0 x 0 0 x x x x x

j = 3 R1,3 0 0 0 0 0 0 0 0 x 0 0 x

⊕(−t4)R3,0 0 0 0 0 x 0 0 x x x x x

j = 4 R1,4 0 0 0 0 0 0 0 0 0 0 0 x

⊕(−t4)R3,1 0 0 0 0 0 0 0 x x x x x

(2.1)

Computing each row gives us the following presentation

Value of j ∆j 0 1 2 3 4 5 6 7 8 9 10 11

j = 1 x 0 0 0 x 0 0 x x x x x

j = 2, 3 0 0 0 0 x 0 0 x x x x x

j = 4 0 0 0 0 0 0 0 x x x x x

Thus, Im(1,−t4) =(R2,0 R2,1 R2,1 R2,2

). A power of t induces a non-trivial kernel if

there is more than one x in the column corresponding to that power. More precisely, if(w1 w2 w3 w4

z1 z2 z3 z4

)∈

(R1,0 R1,1 R1,2 R1,3

R3,0 R3,0 R3,0 R3,1

)

and

(1,−t4)

(w1 w2 w3 w4

z1 z2 z3 z4

)= (0 0 0 0)


then wi − t4zi = 0 for i = 1, 2, 3, 4. This can be given the following presentation:

Value of j 0 1 2 3 4 5 6 7 8 9 10 11 12

j = 1 0 0 0 0 x11 0 0 0 x1

2 0 0 x13 x1

4 ⊆ R1,0

x11 0 0 0 x1

2 0 0 x13 x1

4 x15 x1

6 x17 x1

8 ⊆ R3,0

j = 2 0 0 0 0 x21 0 0 0 x2

2 0 0 x23 x2

4 ⊆ R1,1

x21 0 0 0 x2

2 0 0 x23 x2

4 x25 x2

6 x27 x2

8 ⊆ R3,0

j = 3 0 0 0 0 0 0 0 0 x31 0 0 x3

2 x33 ⊆ R1,2

0 0 0 0 x31 0 0 x3

2 x33 x3

4 x35 x3

6 x37 ⊆ R3,0

j = 4 0 0 0 0 0 0 0 0 0 0 0 x41 x4

2 ⊆ R1,3

0 0 0 0 0 0 0 x41 x4

2 x43 x4

4 x45 x4

6 ⊆ R3,1

(2.2)

Where x11 says the coefficient of t4 in (P1)1 = R1,0 must equal the scalar in (P3)1 = R3,0.

Similar idea works for the other xji . As one can see, notationally this becomes very messy,

as such we abuse notation and simply give the following presentation for the ker(1,−t4):

Value of j 0 1 2 3 4 5 6 7 8 9 10 11

j = 1 0 0 0 0 x 0 0 0 x 0 0 x

x 0 0 0 x 0 0 x x x x x

j = 2 0 0 0 0 x 0 0 0 x 0 0 x

x 0 0 0 x 0 0 x x x x x

j = 3 0 0 0 0 0 0 0 0 x 0 0 x

0 0 0 0 x 0 0 x x x x x

j = 4 0 0 0 0 0 0 0 0 0 0 0 x

0 0 0 0 0 0 0 x x x x x

(2.3)

Since the presentation for j = 1 and j = 2 are the same we shorten the presentation as

follows:

Value of j 0 1 2 3 4 5 6 7 8 9 10 11

j = 1, 2 0 0 0 0 x 0 0 0 x 0 0 x

x 0 0 0 x 0 0 x x x x x

j = 3 0 0 0 0 0 0 0 0 x 0 0 x

0 0 0 0 x 0 0 x x x x x

j = 4 0 0 0 0 0 0 0 0 0 0 0 x

0 0 0 0 0 0 0 x x x x x

(2.4)


It thus follows that

ker(1,−t4) =

(t4y1 t4y2 t4y3 t4y4

y1 y2 y3 y4

)

where y1, y2 ∈ R2,0, y3 ∈ R2,1, y4 ∈ R2,2. In particular,

ker(1,−t4) =

(t4

1

)P2

The presentation in (2.4) can be obtained much quicker if we look at the original presen-

tation in (2.1) and reverse the action of (1,−t4) on each row on the columns that have

more than one x in them. For example,

Value of j ∆j 0 1 2 3 4 5 6 7 8 9 10 11

j = 1 R1,0 x 0 0 0 x 0 0 0 x 0 0 x

⊕(−t4)R3,0 0 0 0 0 x 0 0 x x x x x

Notice that columns 4, 8, and 11 and on have more than one x appearing in them. The

action of (1,−t4) on the first row is multiplication by 1, the reverse of this process is

to leave things as they are in the columns with more than one x. However, the action

of (1,−t4) on the second row is multiplication by −t4, the reverse of this action is to

decrease the powers by 4 in the columns with more than one x. Which gives the following

presentation;

Value of j 0 1 2 3 4 5 6 7 8 9 10 11

j = 1 0 0 0 0 x 0 0 0 x 0 0 x

x 0 0 0 x 0 0 x x x x x

Same as the result in (2.3). The same idea works for other maps, however, the more

columns and rows in the map, the more relations that the kernel will have (the xji ’s that

appear in (2.2)).

Remark 2.3.4. Suppose Af−→ B, where f can be expressed as a row matrix. Then the

preceding discussion shows that f is injective if and only if in the presentation of image

of f every column has at most one x in it.


Lemma 2.3.5. Let

R1 ( R2 ( . . . ( Rl

be an ascending chain of rings, {P1, P2, . . . , Pl} be the set consisting of the indecomposable

projective modules. Suppose

P

Q1

Q2

...

Qn

f := (ta1 , ta2 , . . . , tan)

where ai ∈ N0, and P,Qj ∈ {P1, P2, . . . , Pl}. Then, f is injective if and only if n = 1.

Proof. If n = 1, then f is obviously injective. Conversely, suppose f is injective. We can

assume Qj = Pij with ij ∈ {1, 2, . . . , l}. Notice that (Qj)1 = Rij for all j = 1, 2, . . . n.

Let w = max{α1 + C(Ri1), α2 + C(Ri2), . . . , αn + C(Rin)}. Then, in the presentation for

the image of f the column corresponding to w has an “x” appearing in that column n

times. Hence, by the preceding remark n = 1.

For any 1 ≤ i, j ≤ l,

HomE(Pi, Pj) = HomE(eiE, ejE) ∼= ejEei ⊆ k[[t]].

Therefore, any non-zero morphism Pi → Pj is of the form utα for some α ∈ N0 and

u a unit. Adjusting the morphism by multiplication by u−1, an automorphism of Pj,

we can assume without loss of generality that the non-zero morphisms from Pi to Pj

are multiplication with some tα. The next proposition gives us a lower bound for the

projective dimension of the simple modules.


R1 ( R2 ( ... ( Rl = k[[t]]

pdE(Si) ≥ 1 for 1 ≤ i ≤ l. Moreover, pdE(Si) ≥ 2 for 2 ≤ i ≤ l.

Proof. Since none of the Si are projective E-modules, we have pdE(Si) ≥ 1 for 1 ≤ i ≤ l.

Suppose pdE(Si) = 1 for some 2 ≤ i ≤ l. Then, by Theorem 1.1.4 and Lemma 2.3.5 we


have the following exact sequence for some α ≥ 0, 1 ≤ j ≤ l;

0 Si Pi Pj 0πi tα

(2.5)

The entries of Si are zero except for the i-th entry which is k. Since i 6= 1, (2.5) yields

the following exact sequence;

0 (Si)1 = 0 (Pi)1 = Ri (Pj)1 = Rj 0(πi)1 tα

In particular, tαRj = Ri, this implies that α = 0 (since k is a subset of Ri) which in turn

yields i = j. Furthermore, since the i-th entry of Si is k and the i-th entry of Pi is Ri,

(2.5) gives us the following exact sequence;

0 k = Ri/mi Ri Ri 0id

That is, mi = Ri, a contradiction. Hence, pdE(Si) ≥ 2 for 2 ≤ i ≤ l, completing the

proof.

The next proposition gives a necessary and sufficient condition for pdE(S1) = 1.


R1 ( R2 ( ... ( Rl = k[[t]]

pdE(S1) = 1 if and only if m1 = te(R1)Rj for some 1 < j ≤ l.

Proof. Suppose pdE(S1) = 1, then, by Theorem 1.1.4 and Lemma 2.3.5 we have the

following exact sequence for some α ≥ 0, 1 ≤ j ≤ l;

0 S1 P1 Pj 0π1 tα

Therefore,

0 k = R1/m1 R1 Rj 0ξ tα


is a short exact sequence, where ξ is the quotient map. Therefore, m1 = tαRj where,

α = e(R1), 1 < j ≤ l (Proposition 2.2.14).

Conversely, suppose m1 = te(R1)Rj for some 1 < j ≤ l. Then, for 2 ≤ a ≤ j we have

R1 ( Ra ⊆ Rj and Lemma 2.2.12 implies that

R1 = HomR1(R1, R1) ) HomR1(Ra, R1) ⊇ HomR1(Rj, R1)

Since m1 ( R1 we have

HomR1(Ra,m1) ⊆ HomR1(Ra, R1)

In particular,

R1 = HomR1(R1, R1) ) HomR1(Ra, R1)

= HomR1(Ra,m1) (Lemma 2.2.13)

= HomR1(Ra, te(R1)Rj)

= te(R1) HomR1(Ra, Rj)

= te(R1)Rj (Lemma 2.2.10)

= m1

Maximality of m1 implies that m1 = HomR1(Ra, R1) = HomR1(Ra,m1). Moreover, for

j + 1 ≤ a ≤ l, using Proposition 2.2.13 yields

HomR1(Ra, R1) = HomR1(Ra,m1) = HomR1(Ra, te(R1)Rj) = te(R1) HomR1(Ra, Rj)

It follows that

(P1)a =

R1 if a = 1

m1 if 2 ≤ a ≤ j

HomR1(Ra, R1) if j + 1 ≤ a ≤ l

=

R1 if a = 1

te(R1)Rj if 2 ≤ a ≤ j

te(R1) HomR1(Ra, Rj) if j + 1 ≤ a ≤ l


and

(Pj)a = Eja =

Rj if 1 ≤ a ≤ j

HomR1(Ra, Rj) if j + 1 ≤ a ≤ l

That is,

0 S1 P1 Pj 0π1 te(R1)

is a projective resolution for S1, and it is minimal by Proposition 2.3.6, completing the

proof.

In example 2.2.15, m2 = t3R4 , however, pdE(S2) ≥ 2 by Proposition 2.3.6. In

particular, Proposition 2.3.7 fails to hold if we replace S1 by Si for some 1 < i ≤ l.

2.4 Structure of E

We now turn our attention to the entries of the matrix E and how the entries in a given

row are related to the entries in the rows preceding it and succeeding it.

Lemma 2.4.1. Let

R1 ( R2 ( ... ( Rl = k[[t]]

be an ascending chain of rings. The entries of the matrix E satisfy the following proper-

ties;

(a) If 1 ≤ j ≤ i ≤ l,, then Eij = Ri.

(b) Eij ⊇ Ei(j+1) (that is, there is a descending chain of rings or modules as we go across

a given row. Note that Eij could equal Ei(j+1), for example, E21 = E22 = R2 always).

Furthermore, Eij ⊆ E(i+1)j (there is an ascending chain of rings or modules as we go

down a given column).

(c) If 1 ≤ i ≤ l − 1, then Eii ) Ei(i+1).

(d) HomR1(EndR1(ml), Rl) = Rl. If i 6= l, then HomR1(EndR1(mi), Ri) = mi.

(e) If 1 ≤ i ≤ l − 1, then Ei(i+1) = mi.

(f) Eil = tC(Ri)Rl for 1 ≤ i ≤ l − 1 and Ell = Rl

Proof. (a) This follows from Lemma 2.2.10.

(b) This follows from the fact that HomR1(�, Ri) is a contravariant functor and HomR1(Ri,�)

is a covariant functor.


(c) This is a consequence of part (b) and Lemma 2.2.12.

(d) The first part follows from part (a) and the fact that EndR1(ml) = Rl. If i 6= l then

by Theorem 1.1.1, Ri ( EndR1(mi). That is, k ∩ HomR1(EndR1(mi), Ri) = {0} (where

k is the base field of R1, in fact, of all the Ri, and it is identified with scalar multiplica-

tion) and Ri = HomR1(Ri, Ri) ) HomR1(EndR1(mi), Ri). Given a non-negative integer

b, tb ∈ EndR1(mi) if and only if tbtx = tb+x ∈ mi for any tx ∈ mi, in particular,

HomR1(EndR1(mi), Ri) ⊇ mi.

Thus,

Ri ) HomR1(EndR1(mi), Ri) ⊇ mi,

the maximality of mi gives the desired result.

(e) Given 1 ≤ i ≤ l − 1, we have Ri ( Ri+1 ⊆ EndR1(mi). Parts (a), (c) and (d) imply

that

Ri = Eii ) Ei(i+1) ⊇ HomR1(EndR1(mi), Ri) = mi.

Maximality of mi gives Ei(i+1) = mi.

(f) Given 1 ≤ i ≤ l − 1, since Rl = k[[t]]

Eil = HomR1(Rl, Ri) = tC(Ri)Rl

The second part follows form part (a).

The preceding proposition gives us a very nice description of the entries of E on its

main diagonal, below it, the entries right above the main diagonal (Ei(i+1)), and the

entries in column l. In particular,

E =

R1 m1 ∗ ∗ ∗ ∗ ∗ ∗ tC(R1)Rl

R2 R2 m2 ∗ ∗ ∗ ∗ ∗ tC(R2)Rl

R3 R3 R3 m3 ∗ ∗ ∗ ∗ tC(R3)Rl

R4 R4 R4 R4 m4 ∗ ∗ ∗ tC(R4)Rl

......

......

. . . . . . ∗ ∗ ...

Rl−1 Rl−1 Rl−1 Rl−1 . . . tC(Rl−1)Rl

Rl Rl Rl Rl . . . Rl


The ∗ entries are unknown and must be computed on a base by base cases. Notice that

ml−1 = tC(Rl−1)Rl by Lemma 2.2.16(a). We conclude this section by looking at the entries

of E when the ring R has the property e(R) = C(R).

Lemma 2.4.2. Let

R1 ( R2 ( ... ( Rl = k[[t]]

be an ascending chain of rings, and suppose e(Ri) = C(Ri) for some 1 ≤ i ≤ l. Then

Eij =

Ri if 1 ≤ j ≤ i

te(Ri)Rl if i+ 1 ≤ j ≤ l

Proof. By Proposition 2.4.1 Eij = Ri for 1 ≤ j ≤ i. Moreover, e(Ri) = C(Ri) implies

that

Ri = lead {0, e(Ri)}

Then, for any j with i+ 1 ≤ j ≤ l

HomR1(Rj, Ri) = lead {e(Ri)} = te(Ri)Rl

Proposition 2.4.3. Let

R1 ( R2 ( ... ( Rl = k[[t]]

be an ascending chain of rings. Then, pdE(Sl) = 2.

Proof. By Lemma 2.2.16(a) we have Rl−1 = lead {e(Rl−1)} with e(Rl−1) = C(Rl−1). By

Lemma 2.4.2

E(l−1)j =

Ri if 1 ≤ j ≤ l − 1

te(Rl−1)Rl if j = l

Let

∆ = (1, t)

Pl−1⊕Pl


The image of (1, t) has the following presentation:

Value of j ∆j 0 1 . . . e(Rl−1) . . .

1 ≤ j ≤ l − 1 Rl−1 x 0 0 x x

⊕tRl 0 x x x x

j = l Rl−1 0 0 0 x x

⊕tRl 0 x x x x

That is, Im(1, t) = ker πl = J(Pl). The kernel of (1, t) has the following presentation:

Value of j e(Rl−1)− 1 e(Rl−1) . . .

1 ≤ j ≤ l 0 x x

x x x

Thus,

ker(1, t) =

(te(Rl−1)

−te(Rl−1)−1

)Pl

Hence,

0 Sl Pl

Pl−1

⊕Pl

Pl 0πl (1, t)

(te(Rl−1)

−te(Rl−1)−1

)

is a projective resolution for Sl. Furthermore, it is minimal by Proposition 2.3.6, com-

pleting the proof.

2.5 Family of Starting Rings

In this section we construct a family of starting rings recursively and state some of the

properties of these starting rings. These rings will play a fundamental role in constructing

endomorphism rings of large global dimension.


Definition 2.5.1. Let n ≥ 6 be an even integer. Define

R11 := lead

{0, n,

3n

2

}and

R21 := lead

{0, n,

3n

2, C(R2

1)

}

where3n

2+ 2 ≤ C(R2

1) ≤ 2n. For each i ≥ 3, let

Ri1 := lead

{0,jn

2, C(Ri

1) : j = 2, 3, ..., i+ 1

}

where C(Ri1) = C(Ri−1

1 ) +n

2for i ≥ 3.

The following results are a direct consequence of our construction, we record them

here for future reference.

Lemma 2.5.2. G(Ri1) = i+ 1, e(Ri

1) = n for i = 1, 2, 3, ...

Lemma 2.5.3. C(Ri1) = C(R2

1) + (i− 2)n

2for all i ≥ 2.

Lemma 2.5.4. Λ(R11) = {n}, and Λ(Ri

1) =

{n,

3n

2

}for all i ≥ 2.

Lemma 2.5.5.

a1(R11) = n− 1, a2(R

11) =

n

2− 1

and for i ≥ 2,

aj(Ri1) =

n− 1 if j = 1n

2− 1 for 2 ≤ j ≤ i

C(Ri1)− (i+ 1)

n

2− 1 if j = i+ 1

Notice that C(Ri1) = C(Ri−1

1 ) +n

2for i ≥ 3. This implies that

C(Ri1)− (i+ 1)

n

2− 1 = C(R2

1)−3n

2− 1 for i ≥ 3


Notice the above equality is also true when i = 2. Hence, for i ≥ 2 we have

aj(Ri1) =

n− 1 if j = 1n

2− 1 for 2 ≤ j ≤ i

C(R21)−

3n

2− 1 if j = i+ 1

Lemma 2.5.6. Γ(R11) =

{n,

3n

2

}, and for i ≥ 2 we have Γ(Ri

1) ={βi1, β

i2, ..., β

ii+1

}where

βij =

n+ (j − 1)n

2if 1 ≤ j ≤ i

C(Ri1) if j = i+ 1

Definition 2.5.7. Suppose

R1 ( R2 ( . . . ( Rl = k[[t]]

is a chain of ascending rings. Fix i, and suppose Γ(Ri) = {β1, β2, ..., βr} with β1 < β2 <

... < βr = C(Ri). Let γ1 be the number of positive powers of t between tC(Ri)−β1 and

tC(Ri) (inclusive) which are missing from Ri. Let γ2 be the number of positive powers of

t between tC(Ri)−β2 and tC(Ri)−β1−1 (inclusive) which are missing from Ri, continuing this

process, this stops at γr, the number of positive powers of t between tC(Ri)−βr = t0 and

tC(Ri)−βr−1−1 (inclusive) which are missing from Ri. Define

Φ(Ri) = {γj | j = 1, 2, ..., r}

Lemma 2.5.8. Φ(R11) = {γ11 , γ12} =

{n− 1,

n

2− 1}

, and for i ≥ 2 we have Φ(Ri1) ={

γi1, γi2, ..., γ

ii+1

}where

γij =

n− 2 if j = 1n

2− 1 if 2 ≤ j ≤ i− 1

n

2if j = i

C(Ri1)− βii − 1 if j = i+ 1

Notice that

C(Ri1)− βii − 1 = C(R2

1) + (i− 2)n

2−(n+ (i− 1)

n

2

)− 1 = C(R2

1)−3n

2− 1


That is, for i ≥ 2 we have

γij =

n− 2 if j = 1n

2− 1 if 2 ≤ j ≤ i− 1

n

2if j = i

C(R21)−

3n

2− 1 if j = i+ 1

Suppose

R1 ( R2 ( . . . ( Rl = k[[t]]

is a chain of ascending rings. Given 1 ≤ i ≤ l, if e(Ri) < C(Ri) then Λ(Ri) is not empty.

Let

Λ(Ri) = {α1, ..., αs}, Γ(Ri) = {β1, ..., βr},

where the elements are listed in ascending order. Since Λ(Ri) ⊆ Γ(Ri), for each αa ∈Λ(Ri) there exists a βja ∈ Γ(Ri) such that αa = βja . Since α1 = β1 we have a = 1 = j1.

However, for a ≥ 2 this need not be the case. This leads us to the following useful

definition.

Definition 2.5.9. Given 1 ≤ i ≤ l, let

Λ(Ri) = {α1, ..., αs}

Γ(Ri) = {β1, ..., βr}

Φ(Ri) = {γ1, ..., γr}

For each a ∈ {1, 2, ..., s}, define

λa = i+

ja∑h=1

γh

Since j1 = 1 we have λ1 = i+ γ1. We define

χ(Ri) = {λ1, ..., λs}

If e(Ri) = C(Ri), we define χ(Ri) = ∅. If follows that |χ(Ri)| = |Λ(Ri)|.


Example 2.5.10. Let R1 = k[[t5, t11, t14, t17, t18]], then

C(R1) = 14

Λ(R1) = {5, 11}

Γ(R1) = {5, 10, 11, 14}

Φ(R1) = {γ1, γ2, γ3, γ4} = {3, 4, 1, 2}

Using the notation above we have α1 = 5 = β1, α2 = β3, that is, a = 1 = j1 and j2 = 3.

Moreover,

λ1 = 1 + γ1 = 4

λ2 = 1 +3∑

h=1

γh = 1 + 3 + 4 + 1 = 9

which yields χ(R1) = {4, 9}.

Lemma 2.5.11. χ(R11) = {λ11} = {γ11 + 1} = {(n− 1) + 1 = n}, and for i ≥ 2 we have

χ(Ri1) = {λi1, λi2} where

λij =

1 + γi1 = n− 1 if j = 1

1 + γi1 + γi2 if j = 2

Moreover,

λ2j =

1 + γ21 = n− 1 if j = 1

1 + γ21 + γ22 =3n

2− 1 if j = 2

and for i ≥ 3

λij =

1 + γi1 = n− 1 if j = 1

1 + γi1 + γi2 =3n

2− 2 if j = 2

2.6 The symbol d e

In this section we focus on the symbol d e and its properties.

Definition 2.6.1. Let X be a 1 × l row matrix, and Xj be its j-th entry. Given an


integer a ≥ 0, we define Xdae to be a 1× (l + a) row matrix with the following entries:

(Xdae)j =

X1 if 1 ≤ j ≤ a

Xj−a if a+ 1 ≤ j ≤ l + a

That is, we are putting a string of X1’s (in fact, a of them) at the beginning of X to

obtain Xdae. Notice that

Xd0e = X

Given integers a, b ≥ 0,

(Xdae)dbe = Xda+ be = (Xdbe)dae

Notation 2.6.2. Let p ∈ N, and X i be a 1× l row matrix for i = 1, 2, ..., p. We define

X =

p⊕i=1

X i =

X1

X2

...

Xp

It follows that X is a p× l matrix.

Definition 2.6.3. Given

X =

p⊕i=1

X i

where X i are 1× l row matrices, we define

Xdae =

p⊕i=1

X idae =

X1daeX2dae

...

Xpdae

Given an integer a ≥ 0, if f : X → Y is a map given in matrix form, where

X =

p⊕i=1

X i, Y =

p⊕i=1

Y i


and X i, Y i are 1 × l row matrices, then f has p columns. Since the number of rows of

Xdae, Y dae is also p, we define fdae : Xdae → Y dae by setting fdae = f (the only thing

we have done is change the domain and co-domain). We abuse notation and we use f in

place of fdae.

Let

R1 ( R2 ( R3 ( ... ( Rl−1 ( Rl = k[[t]]

be an ascending chain constructed in section 2.2. The maps πi : Pi → Si are not given

by a matrix . We define

πidae : Pidae → Sidae

as follows;

(π1dae)j =

ξi if 1 ≤ j ≤ a+ 1

0 if a+ 2 ≤ j ≤ l + a

and for 2 ≤ i ≤ l,

(πidae)j =

ξi if j = i+ a

0 if j 6= i+ a

where ξi : Ri → Ri/mi is the quotient map, Pi, Si are 1× l row matrices, and

(S1dae)j =

k if 1 ≤ j ≤ a+ 1

0 if a+ 2 ≤ j ≤ l + a

(P1dae)j =

R1 if 1 ≤ j ≤ a+ 1

(P1)j−a if a+ 2 ≤ j ≤ l + a,

for 2 ≤ i ≤ l,

(Sidae)j =

k if j = i+ a

0 if j 6= i+ a

(Pidae)j =

Ri if 1 ≤ j ≤ a+ 1

(Pi)j−a if a+ 2 ≤ j ≤ l + a


Example 2.6.4. Suppose R1 = k[[t3, t4, t5]], R2 = k[[t2, t3]], R3 = k[[t]]. Then

E =

R1 t3R3 t3R3

R2 R2 t2R3

R3 R3 R3

and

S1d2e =(k k k 0 0

)S2d2e =

(0 0 0 k 0

)P1d2e =

(R1 R1 R1 t3R3 t3R3

)= (P1d1e)d1e

P2d2e =(R2 R2 R2 R2 t2R3)

)P1

⊕P2

d2e =

(P1d2eP2d2e

)=

(R1 R1 R1 t3R3 t3R3

R2 R2 R2 R2 t2R3

)=

P1d2e⊕

P2d2e

Moreover,

π1d2e : P1d2e → S1d2e and (π1d2e)j =

ξ1 if 1 ≤ j ≤ 3

0 if 4 ≤ j ≤ 5

π2d2e : P2d2e → S2d2e and (π2d2e)j =

ξ2 if j = 4

0 if j 6= 4

The following two results are an immediate consequence of our definitions above and

we record them here for future reference.

Lemma 2.6.5. Let p ∈ N. Suppose

X =

p⊕i=1

X i, Y =

p⊕i=1

Y i, Z =

p⊕i=1

Zi

where X i, Y i, Zi are 1× l rows. If

Xdae f→ Y dae g→ Zdae


is exact at Y dae (i.e. ker g = Im(f)) for some a ∈ N0, then

Xdbe f→ Y dbe g→ Zdbe

is exact at Y dbe for every b ∈ N0.

Lemma 2.6.6. Using the notation in Proposition 2.6.5,

Im(X

f→ Y)⊆ J(Y )

if and only if

Im(Xdae f−→ Y dae

)⊆ J(Y dae) for any a ∈ N0

Chapter 3

“Lazy” Construction

In this chapter we concentrate on a construction of our chain which maximizes its length,

called the “lazy” construction. More specifically, in section 3.1 we give the precise defi-

nition of this construction and introduce some of the necessary notation. In section 3.2

we compute the global dimension of endomorphism rings for specific starting rings. In

Section 3.3 we give some of the results which are a consequence of this construction.

Section 3.4 focuses on computing the global dimension of endomorphism rings when the

length of the chain is small. In section 3.5 we combine this construction with the family

of starting rings constructed in section 2.5 to obtain a set of endomorphism rings whose

global dimensions are arbitrarily large (but finite).


Given a numerical semigroup H, let R be the ring of formal power series associated to

H. Then, H has a minimal generating set, say {α1, α2, ..., αs} written in ascending order.

That is,

H = 〈α1, α2, ..., αs〉 ⇔ R = k[[tα1 , tα2 , ..., tαs ]]

Given a non-negative integer b with b 6= αi, we define

H[[b]] = 〈α1, α2, ..., αs, b〉

Since gcd(α1, α2, ..., αs) = 1 implies that gcd(α1, α2, ..., αs, b) = 1, the set H[[b]] is a

numerical semigroup. We define R[[tb]] to be the ring of formal power series associated

to H[[b]]. It should be noted that H ⊆ H[[b]], and equality holds if and only if b ∈ H.

44

Chapter 3. “Lazy” Construction 45

We are now in position to describe the “lazy” construction.

Let H be a numerical semigroup with minimal generating set {α1, α2, ..., αs}, and

F (α1, α2, ..., αs) ≥ 1 (i.e. 1 /∈ H). Let R1 be the ring of formal power series associated

to H. We define

Ri = Ri−1[[tC(Ri−1)−1]] for i ≥ 2

Since only finitely many powers of t are missing from R1, there exists an l ≥ 2 such that

Rl = k[[t]]. In particular, we have constructed the following ascending chain of rings:

R1 ( R2 ( · · · ( Rl = k[[t]] (3.1)

Let

M :=

(l⊕

i=1

Ri

), E := EndR1(M)

We say the ascending chain in (3.1), M , and E are constructed via the “lazy” construc-

tion.

For 1 ≤ i ≤ l − 1 we have Ri 6= k[[t]]. Lemma 2.1.3(e) yields tC(Ri)−1 /∈ Ri and

tx ∈ Ri for all x ≥ C(Ri). In particular, Ri ( Ri+1 ⊆ EndR1(mi). Hence, gl. dim(E) ≤ l

(Proposition 2.2.2).

Lemma 3.1.1. Let

R1 ( R2 ( ... ( Rl

be an ascending chain of ring constructed via the ”lazy” construction. Then,

(a) 1 ≤ l = g(R1) + 1, e(Rl) = C(Rl) = 1. If l ≥ 2, then e(Rl−1) = C(Rl−1) = 2.

(b) e(Ri) ≤ e(Ri−1) for i = 2, 3, ..., l. Moreover, if e(Ri−1) = C(Ri−1) then e(Ri) =

e(Ri−1)− 1.

(c) C(Ri) ≤ C(Ri−1)− 1 for i = 2, 3, ..., l. Moreover, if e(Ri−1) = C(Ri−1), then C(Ri) =

C(Ri−1)− 1.

(d) If e(Ri) = C(Ri) for some i = 1, 2, ..., l, then e(Rj) = C(Rj) for all i ≤ j ≤ l.

Proof. (a) Notice that l = 1⇔ R1 = k[[t]]. If R1 6= k[[t]], then l ≥ 2. Since g(R1) is the

number of powers of t which are missing from R1 and we put them in one at a time to

construct our chain, we have l = g(R1) + 1. Also, Rl = k[[t]] ⇒ e(Rl) = C(Rl) = 1. If

l ≥ 2, then Rl−1 = k[[t2, t3]]⇒ e(Rl−1) = C(Rl−1) = 2.


(b) Since Ri−1 ( Ri we have e(Ri) ≤ e(Ri−1). If e(Ri−1) = C(Ri−1), then Ri−1 =

lead {0, e(Ri−1)} ⇒ Ri = lead {0, e(Ri−1)− 1} ⇒ e(Ri) = e(Ri−1)− 1.

(c) Since Ri−1 ( Ri we have C(Ri) ≤ C(Ri−1) − 1. If e(Ri−1) = C(Ri−1), then Ri−1 =

lead {0, C(Ri−1)} ⇒ Ri = lead {0, C(Ri−1)− 1} ⇒ C(Ri) = C(Ri−1)− 1.

(d) By part (b) e(Ri+1) = e(Ri)− 1, and by part (c) C(Ri+1) = C(Ri)− 1. In particular,

e(Ri+1) = e(Ri)− 1 = C(Ri)− 1 = C(Ri+1). A similar proof shows the result is true for

i+ 2, i+ 3, ..., l.

The projective modules under the lazy construction have a very nice description. We

state this as a lemma for future reference.

Lemma 3.1.2. Suppose

R1 ( R2 ( . . . ( Rl = k[[t]]

is a chain of ascending rings constructed via the “lazy” construction. Fix i, and let

Φ(Ri) = {γj | j = 1, 2, ..., r}, then the i-th projective module Pi has the following entries;

(Pi)j = Eij =

Ri,0 = Ri if 1 ≤ j ≤ i

Ri,1 = mi if i+ 1 ≤ j ≤ i+ γ1

Ri,2 if i+ γ1 + 1 ≤ j ≤ i+ γ1 + γ2·...

Ri,r = tC(Ri)Rl = c(Ri) if l − γr + 1 ≤ j ≤ l

Moreover, |Φ(Ri)| = |Γ(Ri)| = G(Ri) and

l = i+r∑j=1

γj

In particular,

l − γr + 1 = i+ 1 +r−1∑h=1

γh

If e(Ri) = C(Ri), then Φ(Ri) = {γ1 = C(Ri)− 1 = e(Ri)− 1}. More specifically, the

above formula for Pi coincides with Lemma 2.4.2.


Example 3.1.3. Let R1 = k[[t5, t8, t17, t19]] = lead{0, 5, 8, 13, 15}, then

l = 11

C(R1) = 15

Γ(R1) = {5, 8, 10, 13, 15}

G(R1) = 5

γ1 = 3, γ2 = 2, γ3 = 1, γ4 = 3, γ5 = 1

The first row of E is

(P1)j = E1j =

R1,0 = R1 if j = 1

R1,1 = m1 if 2 ≤ j ≤ 4

R1,2 if 5 ≤ j ≤ 6

R1,3 if j = 7

R1,4 if 8 ≤ j ≤ 10

R1,5 = t15R11 if j = 11

3.2 Special Rings I

In this section we compute the global dimension for some special starting rings. All the

constructions in this section are via the lazy construction.


R1 = lead {0, n}

with n > 1. Then, gl. dim(E) = 2.

Proof. Let mi be the maximal ideal of Ri. Notice that l = n by Lemma 3.1.1(a), and

the rings in our chain are

Ri = lead {0, n− i+ 1} where 1 ≤ i ≤ n


For a fixed i, where 1 ≤ i ≤ n− 1, we have

(Pi)j = Eij =

Ri = Ri,0 if 1 ≤ j ≤ i

tn−i+1Rn = Ri,1 if i+ 1 ≤ j ≤ n,

(Pn)j = Enj = Rn = Rn,0 for 1 ≤ j ≤ n

Moreover, Ri,1 = mi = tn−i+1Rn for i = 1, 2, ..., n. We compute the minimal projective

resolutions of Si. For i = 1 we have

(kerπ1)j =

tnRn if j = 1

(P1)j if j 6= 1= tnRn if 1 ≤ j ≤ n

In particular,

0 S1 P1 Pn 0π1 tn

is a projective resolution for S1, and it is minimal by Proposition 2.3.6. For i = 2, 3, ..., n,

let

∆i = (1, tn−i+1)

Pi−1⊕Pn

Then

Value of j (∆i)j 0 n− i+ 1 n− i+ 2 . . .

1 ≤ j ≤ i− 1 Ri−1,0 x 0 x x

⊕tn−i+1Rn,0 0 x x x

i ≤ j ≤ n Ri−1,1 0 0 x x

⊕tn−i+1Rn,0 0 x x x

That is,

∆i = (1, tn−i+1)

Pi−1⊕Pn

= J(Pi) = ker πi


Moreover, ker(1, tn−i+1) has the following presentation:

Value of j 0 1 . . . n− i+ 1 n− i+ 2 . . .

1 ≤ j ≤ n 0 0 0 0 x x

0 x x x x x

Therefore,

0 Si Pi

Pi−1

⊕Pn

Pn 0πi (1, tn−i+1)

(tn−i+2

−t

)

is a projective resolutions for Si, and it is minimal by Proposition 2.3.6. The result

follows by Theorem 1.1.3.

An immediate consequence of Lemma 3.2.1 is: if e(R1) = C(R1), then gl. dim(E) =

2. Moreover, combining Lemma 3.1.1(d) with the proof in Lemma 3.2.1 shows that if

Ri−1 = lead {0, e(Ri−1)} for some 2 ≤ i ≤ l, then the minimal projective resolution of Si

is given by

0 Si Pi

Pi−1

⊕Pl

Pl 0πi (1, te(Ri))

(te(Ri−1)

−t

)

That is, pdE(Sj) = 2 for all j with i ≤ j ≤ l. This leads us to the following which we

present as a lemma for future reference.

Lemma 3.2.2. Suppose 2 ≤ i ≤ l with C(Ri−1) = e(Ri−1). Then,

(a) If e(R1) = C(R1) then gl. dim(E) = 2

(b) pdE(Sj) = 2 for all i ≤ j ≤ l.

(c) pdE(Sj) = 2 for z(R1) + 2 ≤ j ≤ l.

Proof. The only parts we need to prove is parts (c). This follows from the fact that

e(Rz(R1)+1)) = C(Rz(R1)+1)).

Lemma 3.2.3. Suppose b ∈ N, n > 1 with

R1 = lead {0, xn : x = 1, 2, ..., b}


Then gl. dim(E) = 2.

Lemma 3.2.1 is a special case of this lemma (by setting b = 1).

Proof. The rings in our chain are

R2 = lead{0, xn, bn− 1 : x = 1, 2, ..., b− 1}

R3 = lead{0, xn, bn− 2 : x = 1, 2, ..., b− 1}...

Rb(n−1)+1 = lead {0, 1} = k[[t]]

with l = b(n− 1) + 1 by Lemma 3.1.1(a). For a fixed i with 1 ≤ i ≤ (b− 1)(n− 1) + 1,

we have

(Pi)j = Eij =

Ri if 1 ≤ j ≤ i

tnRn+i−1 if i+ 1 ≤ j ≤ n+ i− 1

t2nR2n+i−2 if n+ i ≤ j ≤ 2n+ i− 2

· ·

· ·

· ·

tbn−i+1Rb(n−1)+1 if (b− 1)(n− 1) + i+ 1 ≤ j ≤ b(n− 1) + 1.

If (b− 1)(n− 1) + 2 ≤ i ≤ b(n− 1), we have

Eij =

Ri if 1 ≤ j ≤ i

tb(n−1)−i+2Rb(n−1)+1 if i+ 1 ≤ j ≤ b(n− 1) + 1.

If i = b(n− 1) + 1 then Eij = Rb(n−1)+1 for 1 ≤ j ≤ b(n− 1) + 1. Furthermore,

mi =

tnRn+i−1 if 1 ≤ i ≤ (b− 1)(n− 1) + 1

tb(n−1)−i+2Rb(n−1)+1 if (b− 1)(n− 1) + 2 ≤ i ≤ b(n− 1) + 1

A similar proof to the one given in Lemma 3.2.1 show that the minimal resolutions for

the simple modules are as follows;

0 S1 P1 Pn 0π1 tn


For 2 ≤ i ≤ (b− 1)(n− 1) + 1

0 Si Pi

Pi−1

⊕Pn+i−1

Pn+i−2 0πi (1, tn))

(tn

−1

)

Let ρi = b(n− 1)− i+ 2. For (b− 1)(n− 1) + 2 ≤ i ≤ b(n− 1) + 1

0 Si Pi

Pi−1

⊕Pb(n−1)+1

Pb(n−1)+1 0πi (1, tρi)

(tρi+1

−t

)

The result follows by Theorem 1.1.3.

Lemma 3.2.4. Suppose b ∈ N, n > 1 with

R1 = lead {0, xn, bn+ c : x = 1, 2, ..., b}

where 1 < c ≤ n. Then gl. dim(E) = 2.

The case c = 1 is the preceding lemma (we don’t allow it here due to notational purposes).

Proof. Notice that l = b(n− 1) + c by Lemma 3.1.1(a), and the rings in our chain are

R2 = lead {0, xn, bn+ c− 1 : x = 1, 2, ..., b}

R3 = lead {0, xn, bn+ c− 2 : x = 1, 2, ..., b}...

Rb(n−1)+c = lead {0, 1} = k[[t]]

We give the minimal projective resolutions for the simple modules (the proof is similar

to the one given in Lemma 3.2.1);

0 S1 P1 Pn 0π1 tn

For 2 ≤ i ≤ (b− 1)(n− 1) + c


0 Si Pi

Pi−1

⊕Pn+i−1

Pn+i−2 0πi (1, tn)

(tn

−1

)

Let ρi = b(n− 1) + c− i+ 1. For (b− 1)(n− 1) + c+ 1 ≤ i ≤ b(n− 1) + c

0 Si Pi

Pi−1

⊕Pb(n−1)+c

Pb(n−1)+c 0πi (1, tρi)

(tρi+1

−t

)


The next lemmas gives us the first view of a starting ring that leads to an endomor-

phism ring with global dimension 3.

Lemma 3.2.5. Suppose n ≥ 3 and

R1 = lead {0, n, n+ 1, n+ 3}

then gl. dim(E) = 3.

Note that the result is false for n = 2. If n = 2 then n + 2 = 4 = 2n and tn+2 would be

in R1, which would give gl. dim(E) = 2 (Proposition 3.2.1).

Proof. Notice that l = n+1 by Lemma 3.1.1(a), and the rings in our chain are as follows:

R2 = lead {0, n}

R3 = lead {0, n− 1}...

Rn+1 = lead {0, 1} = k[[t]]

If i = n+ 1, then (Pi)j = Eij = Rn+1 for 1 ≤ j ≤ n+ 1. If 2 ≤ i ≤ n, then

(Pi)j = Eij =

Ri = Ri,0 if 1 ≤ j ≤ i

tn−i+2Rn+1 = mi = Ri,1 if i+ 1 ≤ j ≤ n+ 1,


and

(P1)j = E1j =

R1 = R1,0 if j = 1

m1 = R1,1 if 2 ≤ j ≤ n− 1

tn+1Rn = R1,2 if j = n

tn+3Rn+1 = R1,3 if j = n+ 1,

Since z(R1) = 1, Lemma 3.2.2(c) yields pdE(Si) = 2 for 3 ≤ i ≤ n + 1. We show that

the minimal projective resolution of S1 is

0 S1 P1

Pn−1

⊕Pn

Pn+1 0π1 (tn, tn+1)

(t3

−t2

)

(3.2)

First, notice that Rn−1 = lead {0, 3}, Rn = lead {0, 2} and

(Pn−1)j =

Rn−1 = Rn−1,0 if 1 ≤ j ≤ n− 1

t3Rn+1 = Rn−1,1 if n ≤ j ≤ n+ 1,

(Pn)j =

Rn = Rn,0 if 1 ≤ j ≤ n

t2Rn+1 = Rn,1 if j = n+ 1,

Let

∆ = (tn, tn+1)

Pn−1

⊕Pn


Then the image of (tn, tn+1) has the following presentation:

Value of j ∆j n n+ 1 n+ 2 n+ 3 . . .

1 ≤ j ≤ n− 1 tnRn−1,0 x 0 0 x x

⊕tn+1Rn,0 0 x 0 x x

j = n tnRn−1,1 0 0 0 x x

⊕tn+1Rn,0 0 x 0 x x

j = n+ 1 tnRn−1,1 0 0 0 x x

⊕tn+1Rn,1 0 0 0 x x

That is, Im(tn, tn+1) = J(P1) = kerπ1. Furthermore, ker(tn, tn+1) has the following

presentation:

Value of j 0 1 2 3 . . .

1 ≤ j ≤ n 0 0 0 x x

0 0 x x x

That is

ker(tn, tn+1) =

(t3

−t2

)Pn+1 (

J(Pn−1)

⊕J(Pn)

= J

Pn−1⊕Pn

Therefore, (3.2) is a minimal projective resolution for S1. Now we show that the minimal

projective resolution for S2 is


0 S2 P2

P1

⊕Pn+1

0 Pn+1

Pn−1

⊕Pn

π2 (1, tn)

(tn tn+1

−1 −t

)(t3

−t2

)

(3.3)

Let

∆′ = (1, tn)

P1

⊕Pn+1

Then the image of (1, tn) has the following presentation:

Value of j ∆′j 0 n n+ 1 n+ 2 n+ 3 . . .

j = 1 R1,0 x x x 0 x

⊕tnRn+1,0 0 x x x x

2 ≤ j ≤ n− 1 R1,1 0 x x 0 x


j = n R1,2 0 0 x 0 x


j = n+ 1 R1,3 0 0 0 0 x



That is Im(1, tn) = J(P2) = ker π2. Moreover, ker(1, tn) has the following presentation:

Value of j 0 1 2 . . . n n+ 1 n+ 2 n+ 3 . . .

1 ≤ j ≤ n− 1 0 0 0 0 x x 0 x x

x x 0 x x x x x x

j = n 0 0 0 0 0 x 0 x x

0 x 0 x x x x x x

j = n+ 1 0 0 0 0 0 0 0 x x

0 0 0 x x x x x x

A similar calculation to the ones done above show that

ker(1, tn) =

(tn tn+1

−1 −t

)Pn−1⊕Pn

(

J(P1)

⊕J(Pn+1)

= J

P1

⊕Pn+1

An element is in the kernel of a matrix if and only if it is in the kernel of each of its rows.

Since (tn, tn+1) = −tn(−1,−t), we have ker(tn, tn+1) = ker(−1,−t). But, we already

know ker(tn, tn+1) from the projective resolution of S1. In particular,

ker

(tn tn+1

−1 −t

)=

(t3

−t2

)Pn+1 (

J(Pn−1)

⊕J(Pn)

= J

Pn−1⊕Pn

In particular, (3.3) is a minimal projective resolution for S2, and the result follows by

Theorem 1.1.3.

Lemma 3.2.6. Suppose n ≥ 3 with

R1 = lead {0, n, n+ 1, 2n}

then gl. dim(E) = 3

.

Proof. The case n = 3 is taken care of by Lemma 3.2.5. Suppose n ≥ 4. Lemma 3.1.1(a)


yields l = 2n− 2, and the rings in our chain are as follows:

R2 = lead {0, n, n+ 1, 2n− 1}

R3 = lead {0, n, n+ 1, 2n− 2}...

R2n−2 = leat {0, 1} = k[[t]]

For 1 ≤ i ≤ n− 2

Eij =

Ri,0 if 1 ≤ j ≤ i

Ri,1 if i+ 1 ≤ j ≤ i+ (n− 2)

Ri,2 = tn+1Rn+i−1 if j = i+ (n− 2) + 1

Ri,3 = t2n−i+1R2n−2 if i+ n ≤ j ≤ 2n− 2

and for n− 1 ≤ i ≤ 2n− 2, we have

Eij =

Ri if 1 ≤ j ≤ i

t2n−i−1R2n−2 if i+ 1 ≤ j ≤ 2n− 2

Since z(R1) = 2n − (n + 1) − 1 = n − 2, Lemma 3.2.2(c) implies that pdE(Si) = 2 for

n ≤ i ≤ 2n − 2. Furthermore, it also gives the minimal projective resolution of Si for

n ≤ i ≤ 2n− 2;

0 Si Pi

Pi−1

⊕P2n−2

P2n−2 0π1 (1, te(Ri))

(te(Ri−1)

−t

)

Notice that for n ≤ i ≤ 2n− 2 we have e(Ri) = e(Ri−1)− 1.

The rings Rn−1 and Rn in our chain are

Rn−1 = lead {0, n}, Rn = lead {0, n− 1}


This yields

(J(P1))j =

R1,1 if 1 ≤ j ≤ n− 1

R1,2 if j = n

R1,3 if n+ 1 ≤ j ≤ 2n

(Pn−1)j =

Rn−1,0 if 1 ≤ j ≤ n− 1

Rn−1,1 if n ≤ j ≤ 2n− 2

(Pn)j =

Rn,0 if 1 ≤ j ≤ n

Rn,1 if n+ 1 ≤ j ≤ 2n− 2

Let

∆ = (tn, tn+1)

Pn−1⊕Pn

then the image of (tn, tn+1) has the following presentation:

Values of j ∆j n n+ 1 2n . . .

1 ≤ j ≤ n− 1 tnRn−1,0 x 0 x . . .

⊕tn+1Rn,0 0 x x . . .

j = n tnRn−1,1 0 0 x . . .

⊕tn+1Rn,0 0 x x . . .

n+ 1 ≤ j ≤ 2n− 2 tnRn−1,1 0 0 x . . .

⊕tn+1Rn+1,1 0 0 x . . .

That is, Im(tn, tn+1) = J(P1) = ker π1. The ker(tn, tn+1) has the following presentation:

Values of j n− 1 n . . .

1 ≤ j ≤ 2n− 2 0 x . . .

x x . . .


That is,

ker(tn, tn+1) =

(tn

−tn−1

)P2n−2 ( J

Pn−1

⊕P2n−2

Hence,

0 S1 P1

Pn−1

⊕Pn

P2n−2 0π1 (tn, tn+1)

(tn

−tn−1

)

is a minimal projective resolution of S1.

A similar computation to the one above gives the minimal resolutions of Si for 2 ≤ i ≤n− 1, and they are as follows:

0 Sn−1 Pn−1

Pn−2

⊕P2n−2

0 P2n−2

P2n−4

⊕P2n−3

πn−1 (1, tn)

(tn tn+1

−1 −t

)(t3

−t2

)

and for 2 ≤ i ≤ n− 2


0 Si Pi

Pi−1

⊕Pn−i+2

⊕Pn−i+3

0 P2n−2

Pn−i+1

⊕Pn−i+2

⊕P2n−2

πi (1, tn, tn+1)

−tn −tn+1 0

1 0 −tn−i+1

0 1 tn−i

tn−i+2

−tn−i+1

t


3.3 Minor Results I

Since Ri+1 = Ri[[tC(Ri)−1]] for 1 ≤ i ≤ l − 1, the largest powers of t missing from Ri

is “forced” in to construct Ri+1 and no other powers of t are “forced” in. This process

starts by “forcing” in the largest power of t missing from R1. This proves the following

lemma.

Lemma 3.3.1. If e(Rj) < e(R1) for some 2 ≤ j ≤ l, then G(Rj) = 1.

The following proposition shows that the global dimension of the endomorphism ring

is completely determined when the projective dimension of the first simple module is one.

Proposition 3.3.2. Suppose pdE(S1) = 1, then gl. dim(E) = 2 = pdE(Sj), for any

2 ≤ j ≤ l.

Proof. Let e(R1) = n > 1 and G(R1) = r. Consider the intervals a1(R1), a2(R1), ..., ar(R1)

for R1. Recall that r ≥ 1 . If r = 1, then

R1 = lead {0, n}

and Lemma 3.2.1 gives us the desired result.

If r ≥ 2, then a1(R1) = n − 1 ≥ a2(R1). Since pdE(S1) = 1 we have the following

projective resolution for S1;


0 S1 P1 Pj 0π1 tn

for some 1 ≤ j ≤ l. This gives the following exact sequence;

0 k = R1/m1 R1 Rj 0ξ1 tn

Thus, tnRj = m1. Notice that 1 < j ≤ l by Proposition 2.2.14(d). Moreover,

tb ∈ Rj ⇔ tb+n ∈ m1

In particular,

ai(Rj) = ai+1(R1) and G(Rj) = r − 1

Suppose a1(R1) > a2(R1). Then

a1(Rj) = a2(R1) < n− 1 < e(R1)⇒ e(Rj) < e(R1)

and G(Rj) = 1 by Lemma 3.3.1, which in turn yields r = 2. That is,

R1 = lead {0, n, n+ c} with 1 < c < n,

and the result follows by Lemma 3.2.4.

If a1(R1) = a2(R1), then r ≥ 2. If r = 2 the result follows by Lemma 3.2.3. Suppose

r > 2, then we have a2(R1) = n− 1 ≥ a3(R1). If a3(R1) < n− 1 we show that r = 3. Let

Γ(R1) = {β1, β2, . . . , βr} and Γ(Rj) = {α1, α2, . . . , αr−1}

then β1 = n, β2 = 2n, and β3 = 2n + c for some 1 < c < n. If r ≥ 4, then 2n + c + 2 ≤β4 ≤ 3n and this implies that

α1 = β2 − n = n, α2 = β3 − n = n+ c, α3 = β4 − n

where n + c + 2 ≤ α3 ≤ 2n. That is, the first four powers of t that appear in Rj are

0, n, n + c,and α3. However, since the rings were constructed via the lazy construction

no such ring exists in our chain, a contradiction. Hence, r = 3 and

R1 = lead {0, n, 2n, 2n+ c},


and the result follows by Lemma 3.2.4. If a1(R1) = a2(R1) = a3(R1), we repeat the

preceding argument and continue this until we get to ar(R1). In particular, we have the

following two possible cases;

Case 1. a1(R1) = a2(R1) = ... = ar(R1). In this case

R1 = lead {0, xn : x = 1, 2, ..., r},

and the result follows by Lemma 3.2.3,

or

Case 2. a1(R1) = a2(R1) = ... = ar−1(R1) > ar(R1). In this case

R1 = lead {0, xn, (r − 1)n+ c : x = 1, 2, ..., r − 1}

where 1 < c < n, and the result follows by Lemma 3.2.4.

Next we focus on how the projective resolutions of the simple modules begin under

the lazy construction.

Proposition 3.3.3. Let

R1 ( R2 ( . . . ( Rl = k[[t]]

be a chain of ascending rings constructed via the lazy construction with e(R1) = n > 1.

(a) If C(R1) = e(R1) := n > 1, then the minimal projective resolution of S1 is given by;

0 S1 P1 Pn 0π1 tn

(b) If 2 ≤ i ≤ l and e(Ri) = C(Ri), then the minimal projective resolution of Si has the

following beginning;

0 Si Pi

Pi−1

⊕Pl

· · ·πi (1, te(Ri))

Proof. (a) If C(R1) = n then

R1 = lead {0, n},


the result follows by Lemma 3.2.1.

(b) For i = l the result follows from Proposition 2.4.3. Suppose 2 ≤ i ≤ l − 1. Since

e(Ri) = C(Ri) then Ri = lead{0, e(Ri)} with e(Ri) ≥ 2. and

Ri−1 = lead{0, e(Ri), e(Ri) + 1, . . . , e(Ri) + c, e(Ri) + c+ 2},

where 0 ≤ c ≤ e(Ri)− 2. In particular,

(Pi)j =

Ri if 1 ≤ j ≤ i

mi if i+ 1 ≤ j ≤ l

(Pi−1)j =

Ri−1 if 1 ≤ j ≤ i− 1

mi−1 if j = i (Lemma 2.4.1(e))

HomR1(Rj, Ri−1) if i+ 1 ≤ j ≤ l

By Lemma 2.4.1(b), (Pi−1)j ⊆ (Pi)j for 1 ≤ j ≤ l, and combining this with mi = te(Ri)Rl

yields

(1, te(Ri))

Pi−1⊕Pl

j

=

Ri if 1 ≤ j ≤ i− 1

mi if i ≤ j ≤ l= (J(Pi))j = (ker πi)j,

completing the proof.

Conjecture 1. If C(R1) > n, then the minimal projective resolution of S1 has the

following beginning;

0 S1 P1

⊕sa=1 Pλa . . .

π1 ζ

where Λ(R1) = {α1, ..., αs}, χ(R1) = {λ1, ..., λs}, and ζ = (tα1 , ..., tαs).

Proposition 3.3.4. Suppose conjecture 1 is true and 2 ≤ i ≤ l. If e(Ri) < C(Ri), then

the projective resolution of Si has the following beginning;

0 Si Pi

Pi−1

⊕⊕sa=1 Pλa

· · ·πi τi

where Λ(Ri) = {α1, ..., αs}, χ(Ri) = {λ1, ..., λs}, and τi = (1, tα1 , ..., tαs).


Proof. Fix 2 ≤ i ≤ l. If e(Ri) < C(Ri), then 2 ≤ i ≤ l − 2. Let

Λ(Ri) = {α1, ..., αs}

Γ(Ri) = {β1, ..., βr}

Φ(Ri) = {γ1, ..., γr}

Notice that the above sets only depend on Ri (that is, they do not depend on where Ri

appears is in our chain). Given our ascending chain of rings

R1 ( R2 ( . . . ( Ri ( Ri+1 ( . . . ( Rl, (3.4)

then,

E := EndR1

(l⊕

j=1

Rj

)=

P1

P2

...

Pi−1

Pi

Pi+1

...

Pl

The following set depends where Ri is in the ascending chain;

χ(R1) = {λ1, ..., λs}

where

λa = i+

ja∑h=1

γh

If we cut our ascending chain so that the chain starts at Ri, that is, we think of Ri as

the starting ring;

Ri ( Ri+1 ( . . . ( Rl, (3.5)


then,

E ′ := EndR1

(l⊕j=i

Rj

)= EndRi

(l⊕j=i

Rj

)=

P ′i

P ′i+1...

P ′l

is a l − (i− 1)× l − (i− 1) matrix with P ′jdi− 1e = Pj for i ≤ j ≤ l. Furthermore,

(P ′i )j =

Ri if j = 1

HomR1(Rj, Ri) if 2 ≤ j ≤ l − (i− 1)

=

Ri if j = 1

HomRi(Rj, Ri) if 2 ≤ j ≤ l − (i− 1)

By conjecture 1, the beginning of the minimal projective resolution of P ′i is given by;

0 S ′i P ′i⊕s

a=1 P′λ′a

· · ·π′i ζ

where

ζ = (tα1 , tα2 , . . . , tαs)

λ′a = 1 +

ja∑h=1

γh

(S ′i)j =

k if j = 1

0 if 2 ≤ j ≤ l − (i− 1)

(π′i)j =

ξi if j = 1

0 if 2 ≤ j ≤ l − (i− 1)

with ξi : Ri → k = Ri/mi is the natural map. This yields the following exact sequence

0 J(P ′i )⊕s

a=1 P′λ′a

· · ·ζ


Lemma 2.6.5 yields the following exact sequence;

0 (J(P ′i ))di− 1e⊕s

a=1 P′λ′adi− 1e · · ·

ζ

(3.6)

Notice that λ′a are obtained from the ascending chain given in (3.5), which is the one

with Ri as the starting ring. To convert these subscripts when R1 is the starting ring

(the ascending chain given in (3.4)) we need to add i− 1 to these subscripts (since i− 1

rings are removed when Ri is the starting ring). More precisely, the exact sequence in

(3.6) becomes the following when R1 is the starting ring;

0 (J(P ′i ))di− 1e⊕s

a=1 P′λ′a+(i−1)di− 1e · · ·

ζ

(3.7)

Since

λ′a + (i− 1) = 1 +

ja∑h=1

γh + i− 1 = λa

the sequence in (3.7) becomes

0 (J(P ′i ))di− 1e⊕s

a=1 P′λadi− 1e · · ·

⊕sa=1 Pλa

ζ

(3.8)

Observe that

Im(ζ) = ((J(P ′i )di− 1e)j =

mi if 1 ≤ j ≤ i

HomR1(Rj, Ri) if i+ 1 ≤ j ≤ l

By Lemma 2.4.1(b), (Pi−1)j ⊆ (Pi)j for 1 ≤ j ≤ l. Combining this with the sequence in

(3.8) gives us the following beginning to the minimal projective resolution of Si;


0 Si Pi

Pi−1

⊕⊕sa=1 Pλa

· · ·πi τi = (1, ζ)

as desired.

3.4 gl. dim(E) for l = 1, 2, 3, 4, 5

Our goal in this section is to compute the global dimension of E for small values of l.

More specifically, we compute gl. dim(E) for l = 1, 2, 3, 4, 5. If l = 1, then R1 = k[[t]]

and E = EndR1(R1) = R1. In particular, gl. dim(E) = 1 by Proposition 1.1.2.

Lemma 3.4.1. If l = 2, then gl. dim(E) = 2.

Proof. Since R2 = k[[t]] we have R1 = k[[t2, t3]] and the result follows by Lemma 3.2.1.

Lemma 3.4.2. If l = 3, then gl. dim(E) = 2.

Proof. There are two possible cases for R1;

R1 = k[[t3, t4, t5]] ⇒ gl. dim(E) = 2 (Lemma 3.2.1)

R1 = k[[t2, t5]] ⇒ gl. dim(E) = 2 (Lemma 3.2.3)

Lemma 3.4.3. If l = 4, then gl. dim(E) = 2 or 3.

Proof. There are three possibilities for R1;

R1 = k[[t4, t5, t6, t7]] ⇒ gl. dim(E) = 2 (Lemma 3.2.1)





Proof. There are five possibilities for R1;

R1 = k[[t5, t6, t7, t8, t9]] ⇒ gl. dim(E) = 2 (Lemma 3.2.1)


R1 = k[[t4, t6, t7, t9]] ⇒ gl. dim(E) = 2 (Lemma 3.2.4)


The fifth case is when R1 = k[[t4, t5, t6, t8]]. In this case

E =

R1 R1,1 R1,2 R1,3 R1,4

R2 R2 R2,1 R2,1 R2,1

R3 R3 R3 R3,1 R3,1

R4 R4 R4 R4 R4

R5 R5 R5 R5 R5

Similar calculations to the ones done in section 3.2 show that the minimal projective

resolutions of S1 and S2 are as follows;

0 S1 P1

P2

⊕P3

⊕P4

0

P5

⊕P5

π1 (t4, t5, t6)

−t4 −t4

t3 0

0 t2


0 S2 P2

P1

⊕P5

P2

⊕P3

⊕P4

0

P5

⊕P5

π1 (1, t4)

(t4 t5 t6

−1 −t −t2

)

−t4 −t4

t3 0

0 t2

Moreover, since z(R1) = 1 we have pdE(Si) = 2 for 3 ≤ i ≤ 5 (Lemma 3.2.2). Hence,

gl. dim(E) = 3 by Theorem 1.1.3.

3.5 Constructing Endomorphism Rings of Large Global

Dimension

Throughout this section {Ri1 : i ∈ N} is a set of starting rings constructed in section 2.5

and for each natural number i, the ascending chain

Ri1 ( Ri

2 ( . . . ( Rili

= k[[t]]

is constructed via the lazy construction. We define

M i =

li⊕j=1

Rij and Ei = EndRi

1(M i)

In section 3.5.1 we investigate the lengths of the chains, projective modules, and the

projective dimension of the first simple modules for i = 1, 2. Once this is done, this

forms the backbone of the proofs of the main results in this chapter. In section 3.5.2

the first main result of this thesis is proved: we establish a lower bound for the global

dimension of Ei for each i ∈ N. Section 3.5.3 focuses on the module we get when we

remove the starting ring from our chain and instead start our chain from the second ring.

We conclude this section by proving the second main result of this thesis in section 3.5.4.

More precisely, we compute the global dimensions of Ei for a given set of starting rings.


3.5.1 Minor Results II

This section is devoted to building the machinery needed in the next three sections. The

following proposition gives us a formula for li, the length of the chain with starting ring

Ri1.

Lemma 3.5.1. Given an even integer n ≥ 6, then l1 =3n

2− 1 and for i ≥ 2

li = i(n

2− 1)

+ C(R21)− n = C(R2

1) + (i− 2)n

2− i

Moreover,

li+1 = li +n

2− 1 for i ≥ 2

Proof.

l1 = 1 + g(R11) (Lemma 3.1.1)

= 1 + a1(R11) + a2(R

11)

= 1 + (n− 1) +(n

2− 1)

(Lemma 2.5.5)

=3n

2− 1

and for i ≥ 2,

li = 1 + g(Ri1) (Lemma 3.1.1)

= 1 +

G(Ri1)∑

j=1

aj(Ri1)

= 1 +i+1∑j=1

aj(Ri1) (Lemma 2.5.2)

= 1 + a1(Ri1) +

i∑j=2

aj(Ri1) + ai+1(R

i1)

= 1 + (n− 1) + (i− 1)(n

2− 1)

+ C(R21)−

3n

2− 1 (Lemma 2.5.5)

= i(n

2− 1)

+ C(R21)− n

= C(R21) + (i− 2)

n

2− i


In particular, for i ≥ 2

li+1 = (i+ 1)(n

2− 1)

+ C(R21)− n

= i(n

2− 1)

+ C(R21)− n+

n

2− 1

= li +n

2− 1

Notice that l2 is not necessarily equal to l1 + n2− 1. To see this, the preceding

proposition yields

l2 = C(R21)− 2 and l1 +

n

2− 1 =

3n

2− 1 +

n

2− 1 = 2n− 2.

That is, the two are equal if and only if C(R21) = 2n (which is not true in general). The

following proposition allows us to go back and forth between the projective Ei modules

and projective Ei+1 modules. It plays a fundamental role in the main theorems we prove

later on in this thesis.

Proposition 3.5.2. Given an even integer n ≥ 6, let {Ri1 | i ∈ N} be a set of starting

rings constructed in section 2.5. Then

(a) For a given i ≥ 2 and j = 1, 2, ..., li,

(P i+1n2+j−1)a = (P i

j )a−n2+1 if n

2≤ a ≤ li+1

In particular, for i ≥ 2,

(P i+1n2

)a = (P i1)a−n

2+1 if n

2≤ a ≤ li+1

(b) For a given i ≥ 2 and j = 1, 2, ..., li,

(P i+1n2+j−1)a =

Rij if 1 ≤ a ≤ n

2− 1

(P ij )a−n

2+1 if n

2≤ a ≤ li+1

=((P ij

⌈n2− 1⌉))

a

In particular, for i ≥ 2,

(P i+1n/2 )a =

Ri1 if 1 ≤ a ≤ (n/2)− 1

(P i1)a−(n/2)+1 if (n/2) ≤ a ≤ li+1

=((P i1

⌈n2− 1⌉))

a


Proof. (a) Fix an i ≥ 2 and j = 1, 2, ..., li. Given a with n2≤ a ≤ li+1,

(P ij )a−n

2+1 = Ei

j(a−n2+1) = HomRi

1(Ri

a−n2+1, R

ij)

= HomRi+1n2

(Ri+1n2+a−n

2+1−1, R

i+1n2+j−1)

= HomRi+1n2

(Ri+1a , Ri+1

n2+j−1)

= HomRi+11

(Ri+1a , Ri+1

n2+j−1)

= Ei+1(n2+j−1)a

= (P i+1n2+j−1)a

For i ≥ 2, setting j = 1 yields

(P i+1n2

)a = (P i1)a−n

2+1 if n

2≤ a ≤ li+1

(b) Given i ≥ 2 and j = 1, 2, ..., li,

(P i+1n2+j−1)a =

Ri+1n2+j−1 if 1 ≤ a ≤ n

2+ j − 1

(P i+1n2+j−1)a if n

2+ j ≤ a ≤ li+1

=

Ri+1n2+j−1 if 1 ≤ a ≤ n

2− 1

(P in2+j−1)a if n

2≤ a ≤ li+1

=

Rij if 1 ≤ a ≤ n

2− 1

(P ij )a−n

2+1 if n

2≤ a ≤ li+1

=((P ij

⌈n2− 1⌉))

a

For i ≥ 2, setting j = 1 gives

(P i+1n2

)a =

Ri1 if 1 ≤ a ≤ n

2− 1

(P i1)a−n

2+1 if n

2≤ a ≤ li+1

=((P i1

⌈n2− 1⌉))

a

Remark 3.5.3. Since (Sij)1 = 0 for 2 ≤ j ≤ li and li+1 = li +n

2− 1 for i ≥ 2, we have

Sij

⌈n2− 1⌉

= Si+1j+n

2−1 for i ≥ 2, 2 ≤ j ≤ li


A consequence of Proposition 3.5.2 is

P 3q+2n−1 = P 3q+1

n2

⌈n2− 1⌉

= P 3q1 [n− 2]

P 3q+23n2−2 = P 3q+1

n−1

⌈n2− 1⌉

= P 3qn2dn− 2e = P 3q−1

1

⌈3n

2− 3

⌉for q ≥ 1. We conclude this section by computing the projective dimension of S1

1 as an

E1-module and S21 as an E2-module.

Lemma 3.5.4. pdE1(S11) = 1 and gl. dim(E1) = 2.

Proof. By Lemma 3.2.4

0 S11 P 1

1 P 1n 0

π11 tn

is the minimal projective resolution of S11 . That is, pdE1(S1

1) = 1. The second part

follows by Proposition 3.3.2.

The following notation will be very useful throughout this thesis.

Notation 3.5.5. Let

ε = C(R21)−

3n

2, ε1 = C(R2

1)− n, ε2 = C(R21)−

n

2

ζ = (tn, t3n2 ), τ =

(t3n2 t2n

−tn −t 3n2

):=

(τ1

τ2

), φ =

(tε1

−tε

)

Notice that

ζτ = 0, τφ = 0, ζφ = 0, τ1φ = 0, τ2φ = 0

Lemma 3.5.6. The minimal projective resolution of S21 is as follows;

0 S21 P 2

1

P 2n−1

⊕P 2

3n2−1

P 2l2

0π21 ζ φ

(3.9)

In particular, pdE2(S21) = 2.


Proof. Since

R21 = lead

{0, n,

3n

2, C(R2

1)

}R2n−1 = lead {0, ε1}

R23n2−1 = lead {0, ε}

R2l2

= k[[t]]

Lemma 3.1.2 yields

(P 21 )j =

R21,0 = R2

1 if j = 1

R21,1 = m2

1 if 2 ≤ j ≤ n− 1

R21,2 if n ≤ j ≤ 3n

2− 1

R21,3 if 3n

2≤ j ≤ l2

(P 2n−1)j =

R2n−1,0 = R2

n−1 if 1 ≤ j ≤ n− 1

R2n−1,1 = m2

n−1 = tε1R2l2

if n ≤ j ≤ l2

(P 23n2−1)j =

R23n2−1,0 = R2

3n2−1 if 1 ≤ j ≤ 3n

2− 1

R23n2−1,1 = m2

3n2−1 = tεR2

l2if 3n

2≤ j ≤ l2

and (P 2l2

)j = R2l2

= k[[t]] for 1 ≤ j ≤ l2. Let

∆ = (tn, t3n2 )

P 2n−1

⊕P 2

3n2−1


This gives us the following table;

Value of j ∆j n 3n2C(R2

1) . . .

1 ≤ j ≤ n− 1 tnR2n−1 x 0 x x

⊕t3n2 R2

3n2−1 0 x x x

n ≤ j ≤ 3n2− 1 tnm2

n−1 0 0 x x

⊕t3n2 R2

3n1−1 0 x x x

3n2≤ j ≤ l2 tnm2

n−1 0 0 x x

⊕t3n2 m2

3n2−1 0 0 x x

Therefore, ker ζ has the following representation;

Value of j ε ε+ 1 . . . ε1 . . .

1 ≤ j ≤ l2 0 0 0 x x

x x x x x

That is,

(tn, t3n2 )

P 2n−1

⊕P 2

3n2−1

= J(P 21 ) = ker π2

1

and

φ(P 2l2

) = ker

P 2n−1

⊕P 2(3n/2)−1

ζ−→ P 21

(

J(P 2n−1)

⊕J(P 2

(3n/2)−1)

= J

P 2n−1

⊕P 2(3n/2)−1

Hence, (3.9) is a projective resolution for S2

1 (it is minimal by Proposition 2.3.6), com-

pleting the proof.

3.5.2 Lower Bound for gl. dim(Ei)

In this section we prove the first main result of this thesis. More precisely, we obtain a

lower bound for the global dimension of Ei. We begin with a useful definition.


Definition 3.5.7. Let n be a positive even integer and a ≤ b be non-negative integers,

we define

Aba(n) ={xn

2: x = a, a+ 1, ..., b

}

Now we are in position to prove the first main result of this thesis.

Theorem 3.5.8. Let {Ri1|i ∈ N} be a set of starting rings constructed in section 2.5.

(a) If q ≥ 1, then

0 S3q+21 P 3q+2

1

P 3q+2n−1

⊕P 3q+2

3n2−2

(J(P 3q−11 ))

⌈3n2− 3⌉

0π3q+21 ζ µ

(3.10)

is an exact sequence, where

µ =

(tn2

−1

), Im(ζ) = J(P 3q+2

1 )

(b) If q ≥ 0, then

0 S3q+21 W0 W1 W2 · · · Wq+1 Wq+2 0

d0 d1 d2 d3 dq+1 dq+2

(3.11)


is a minimal projective resolution for S3q+21 , where

Wj =

P 3q+21 if j = 0

P 3q+2(n−1)+3(j−1)(n

2−1)

⊕

P 3q+2(n−1)+3(j−1)(n

2−1)+(n

2−1)

if j = 1, 2, ..., q

P 3q+2(n−1)+3q(n

2−1)

⊕

P 3q+2(n−1)+3q(n

2−1)+n

2

if j = q + 1

P 3q+2l3q+2

if j = q + 2

and

dj =

π3q+21 if j = 0

ζ if j = 1

τ for j = 2, ..., q + 1

φ if j = q + 2

In particular, pdE3q+2(S3q+21 ) = q + 2 for q ∈ N0 ⇒ gl. dim(E3q+2) ≥ q + 2 for q ∈ N0.

Notice that if q = 0 the second row for Wj above is omitted.

Proof. (a) Fix q ≥ 1, definition 2.5.1 and the lazy construction yield

R3q+21 = lead

{0,jn

2, C(R3q+2

1 ) : j = 2, 3, ..., 3q + 3

}R3q+2n−1 = lead

{0,jn

2, C(R3q+2

n−1 ) : j = 2, 3, ..., 3q + 1

}R3q+2

3n2−2 = lead

{0,jn

2, C(R3q+2

3n2−2

): j = 2, 3, ..., 3q

}


where

C(R3q+2n−1 ) = C(R3q+2

1 )− n

C(R3q+2

3n2−2

)= C(R3q+2

1 )− 3n

2

and

G(R3q+21 ) = 3q + 3

G(R3q+2n−1 ) = G(R3q+2

1 )− 2 = 3q + 1

G(R3q+2

3n2−2

)= G(R3q+2

1 )− 3 = 3q

Lemmas 3.1.2 and 2.5.8 yield

(P 3q+21 )j =

R3q+21,0 if j = 1

R3q+21,1 if 2 ≤ j ≤ n− 1

R3q+21,2 if n ≤ j ≤ 3n

2− 2

R3q+21,3 if 3n

2− 1 ≤ j ≤ 2n− 3

R3q+21,a if an

2− a+ 2 ≤ j ≤ (a+1)n

2− a for 2 ≤ a ≤ 3q + 1

R3q+21,3q+2 if (3q+2)n

2− 3q ≤ j ≤ (3q+3)n

2− (3q + 1)

R3q+21,3q+3 if (3q+3)n

2− 3q ≤ j ≤ l3q+2

(P 3q+2n−1 )j =

R3q+2n−1,0 if 1 ≤ j ≤ n− 1

R3q+2n−1,1 if n ≤ j ≤ 2n− 3

R3q+2n−1,2 if 2n− 2 ≤ j ≤ 5n

2− 4

R3q+2n−1,3 if 5n

2− 3 ≤ j ≤ 3n− 5

R3q+2n−1,a if an

2+ n− a ≤ j ≤ (a+3)n−2(a+2)

2for 2 ≤ a ≤ 3q − 1

R3q+2n−1,3q if (3q+2)n

2− 3q ≤ j ≤ (3q+3)n

2− (3q + 1)

R3q+2n−1,3q+1 if (3q+3)n

2− 3q ≤ j ≤ l3q+2


(P 3q+2

3n2−2

)j

=

R3q+23n2−2,0 if 1 ≤ j ≤ 3n

2− 2

R3q+23n2−2,1 if 3n

2− 1 ≤ j ≤ 5n

2− 4

R3q+23n3−2,2 if 5n

2− 3 ≤ j ≤ 3n− 5

R3q+23n3−2,3 if 3n− 4 ≤ j ≤ 7n

2− 6

R3q+23n2−2,a if (a+3)n

2− (a+ 1) ≤ j ≤ (a+4)n−2(a+3)

2for 2 ≤ a ≤ 3q − 2

R3q+23n2−2,3q−1 if (3q+2)n

2− 3q ≤ j ≤ (3q+3)n

2− (3q + 1)

R3q+23n2−2,3q if (3q+3)n

2− 3q ≤ j ≤ l3q+2

Using Lemma 3.5.1 we get

l3q+2 −(3q + 3)n

2− 3q + 1 = C(R2

1)−3n

2− 1

Let

∆ = (tn, t3n2 )

P 3q+2n−1

⊕P 3q+2

3n2−2

f(a) =

(a+ 3)n

2− (a+ 1)

h(a) =(a+ 4)n

2− (a+ 3)

ρ(a) = C(R3q+21 )− an

2


The image of (tn, t3n2 ) has the following presentation:

Value of j ∆j n 3n2

2n A3q+35 (n) C(R3q+2

1 ) · · ·1 ≤ j ≤ n− 1 tnR3q+2

n−1,0 x 0 x x x x

⊕t3n2 R3q+2

3n2−2,0 0 x 0 x x x

n ≤ j ≤ 3n2− 2 tnR3q+2

n−1,1 0 0 x x x x

⊕t3n2 R3q+2

3n2−2,0 0 x 0 x x x

3n2− 1 ≤ j ≤ 2n− 3 tnR3q+2

n−1,1 0 0 x x x x

⊕t3n2 R3q+2

3n2−2,1 0 0 0 x x x

2n− 2 ≤ j ≤ 5n2− 4 tnR3q+2

n−1,2 0 0 0 x x x

⊕t3n2 R3q+2

3n2−2,1 0 0 0 x x x

For 2 ≤ a ≤ 3q − 2 we have

Value of j ∆j A3q+3a+4 (n) C(R3q+2

1 ) · · ·f(a) ≤ j ≤ h(a) tnR3q+2

n−1,a+1 x x x

⊕t3n2 R3q+2

3n2−2,a x x x

and (note that h(3q − 2) + 1 = f(3q − 1))

Value of j ∆j(3q+3)n

2C(R3q+2

1 ) · · ·f(3q − 1) ≤ a ≤ h(3q − 2) + n

2tnR3q+2

n−1,3q x x x

⊕t3n2 R3q+2

3n2−2,3q−1 x x x

h(3q − 2) + n2

+ 1 ≤ a ≤ l3q+2 tnR3q+2n−1,3q+1 0 x x

⊕t3n2 R3q+2

3n2−2,3q 0 x x

In particular, Im(ζ) = J(P 3q+21 ) = ker π3q+2

1 . The kernel of ζ has the following presenta-


tion:

Value of j n A3q3 (n) ρ(3) · · · (3q+1)n

2· · · ρ(2) · · ·

1 ≤ j ≤ 5n2− 4 0 x 0 0 x 0 x x

x x x x x x x x

For 2 ≤ a ≤ 3q − 2 the kernel of ζ has the following presentation:

Value of a (a+1)n2

A3qa+2(n) ρ(3) · · · (3q+1)n

2· · · ρ(2) · · ·

f(a) ≤ j ≤ h(a) 0 x 0 0 x 0 x x

x x x x x x x x

and

Value of a 3qn2

ρ(3) · · · (3q+1)n2

· · · ρ(2) · · ·f(3q − 1) ≤ a ≤ h(3q − 2) + n

20 0 0 x 0 x x

x x x x x x x

h(3q − 2) + n2

+ 1 ≤ a ≤ l3q+2 0 0 0 0 0 x x

0 x x x x x x

Since

R3q−11 = lead

{0,xn

2, C(R3q−1

1 ) : x = 2, 3, ..., 3q}

with G(R3q−11 ) = 3q (Lemma 2.5.2) and C(R3q−1

1 ) = C(R3q+21 )− 3n

2(by construction of the

starting rings). By Lemma 3.1.2 the projective module P 3q−11 has the following entries;

(P 3q−11 )j =

R3q−11,0 if j = 1

R3q−11,1 if 2 ≤ j ≤ n− 1

R3q−11,a if an

2− a+ 2 ≤ j ≤ (a+1)n

2− a for 2 ≤ a ≤ 3q − 2

R3q−11,3q−1 if (3q−1)n

2− (3q − 3) ≤ j ≤ 3qn

2− (3q − 2)

R3q−11,3q if 3qn

2− (3q − 3) ≤ j ≤ l3q−1

By Lemma 3.5.1 we have

l3q+2 = l3q−1 +3n

2− 3


and

an

2− a+ 2 +

3n

2− 3 = f(a)

(a+ 1)n

2− a+

3n

2− 3 = h(a)

In particular,

((J(P 3q−1

1 ))

⌈3n

2− 3

⌉)j

=

R3q−11,1 if 1 ≤ j ≤ 5n

2− 4

R3q−11,a if f(a) ≤ j ≤ h(a) for 2 ≤ a ≤ 3q − 2

R3q−11,3q−1 if f(3q − 1) ≤ j ≤ h(3q − 2) + n

2

R3q−11,3q if h(3q − 2) + n

2+ 1 ≤ j ≤ l3q+2

which yields ker ζ = µ((J(P 3q−11 ))

⌈3n2− 3⌉). Therefore, the sequence in (3.10) is an exact

sequence, as desired.

(b) We proceed by induction on q. The case q = 0 is Lemma 3.5.6. Assume the result

holds for q− 1 (with q ≥ 1). By Theorem 1.1.4 and part (a) of this theorem the minimal

projective resolution of S3q+21 has the following beginning;

0 S3q+21 P 3q+2

1

P 3q+2n−1

⊕P 3q+2

3n2−2

π3q+21 ζ

(3.12)

Moreover, part (a) gives the following exact sequence;

0 S3q+21 P 3q+2

1

P 3q+2n−1

⊕P 3q+2

3n2−2

(J(P 3q−11 ))

⌈3n2− 3⌉

0π3q+21 ζ φ

(3.13)

By induction, pdE3q−1(S3q−11 ) = (q − 1) + 2 = q + 1 (since S

3(q−1)+21 = S3q−1

1 ) and

0 S3q−11 L0 L1 L2 · · · Lq Lq+1 0

f0 f1 f2 f3 fq fq+1

(3.14)


is a minimal projective resolution for S3q−11 , where

Lj =

P 3q−11 if j = 1

P 3q−1(n−1)+3(j−1)(n

2−1)

⊕

P 3q−1(n−1)+3(j−1)(n

2−1)+(n

2−1)

if j = 1, 2, ..., q − 1

P 3q−1(n−1)+3(q−1)(n

2−1)

⊕

P 3q−1(n−1)+3(q−1)(n

2−1)+n

2

if j = q

P 3q−1l3q−1

if j = q + 1

and

fj =

π3q−11 if j = 0

ζ if j = 1

τ for j = 2, ..., q

φ if j = q + 1

Since Im(f1) = ker(f0) = J(P 3q−11 ), the exact sequence in (3.14) yields the following

exact sequence;

0 J(P 3q−11 ) L1 L2 · · · Lq Lq+1 0

f1 f2 f3 fq fq+1

(3.15)

Lemma 2.6.5 and (3.15) imply that the following sequence is exact;

0 J(P 3q−11 )

⌈3n2− 3⌉

L1

⌈3n2− 3⌉

L2

⌈3n2− 3⌉

0 Lq+1

⌈3n2− 3⌉

Lq⌈3n2− 3⌉

· · ·

f1 f2

f3fqfq+1

(3.16)


Splicing (3.13) and (3.16) yields the following exact sequence;

0 S3q+21 P 3q+2

1

P 3q+2n−1

⊕P 3q+2

3n2−2

L1

⌈3n2− 3⌉

0 Lq+1

⌈3n2− 3⌉

Lq⌈3n2− 3⌉

· · · L2

⌈3n2− 3⌉

π3q+21 ζ τ = µζ

f2

fqfq+1 f3

(3.17)

Let

Wj =

P 3q+21 if j = 0

P 3q+2n−1 ⊕ P

3q+23n2−2 if j = 1

Lj−1[3n2− 3] if j = 2, ..., q + 2

and

dj =

π3q+21 if j = 0

ζ if j = 1

τ if j = 2

fj−1 if j = 3, ..., q + 2

In particular, (3.17) becomes the following exact sequence;

0 S3q+21 W0 W1 W2 · · · Wq+1 Wq+2 0

d0 d1 d2 d3 dq+1 dq+2

(3.18)


For j = 2, ..., q we have

Wj = Lj−1

⌈3n

2− 3

⌉

=

P 3q−1n−1+3((j−1)−1)(n

2−1)⌈3n2− 3⌉

⊕P 3q−1n−1+3((j−1)−1)+(n

2−1)⌈3n2− 3⌉

=

P 3q+2n−1+3(j−2)(n

2−1)+3(n

2−1)

⊕P 3q+2n−1+3(j−2)(n

2−1)+(n

2−1)+3(n

2−1)

(Proposition 3.5.2)

=

P 3q+2n−1+3(j−1)(n

2−1)

⊕P 3q+2n−1+3(j−1)(n

2−1)+(n

2−1)

and

Wq+1 = Lq

⌈3n

2− 3

⌉

=

P 3q−1n−1+3(q−1)(n

2−1)⌈3n2− 3⌉

⊕P 3q−1n−1+3(q−1)(n

2−1)+n

2

⌈3n2− 3⌉

=

P 3q+2

n−1+3(q−1)(n2−1)+3( 3n

2−1)

⊕P 3q+2n−1+3(q−1)(n

2−1)+n

2+3(n

2−1)

(Proposition 3.5.2)

=

P 3q+2n−1+3q(n

2−1)

⊕P 3q+2n−1+3q(n

2−1)+n

2

Wq+2 = Lq+1

⌈3n

2− 3

⌉= P 3q−1

l3q−1

⌈3n

2− 3

⌉= P 3q+2

l3q−1+n−1+(n2−2) (Proposition 3.5.2)

= Pl3q−1+3n2−3

= Pl3q+2 (Lemma 3.5.1)


That is, (3.18) is a projective resolution for S3q+21 . By Theorem 1.1.4

0 S3q+21 W0

d0

is a projective cover for S3q+21 . Since (3.12) is the start of the minimal projective resolution

for S3q+21 and (3.14) is the minimal projective resolution for S3q−1

1 , we have

Im(d1) = Im(ζ) = ker π3q+21 = J(P 3q+2

1 )

Im(d2) = Im(τ) = ker ζ ⊆ J(W1)

and for 3 ≤ j ≤ q + 2,

Im(Lj−1

fj−1−→ Lj−2

)⊆ J(Lj−2) by minimality of (3.14).

This implies that

Im(dj) = Im(fj−1)

⊆ J

(Lj−2

⌈3n

2− 3

⌉)(Lemma 2.6.6)

= J(Wj−1)

Thus, (3.18) is a minimal projective resolution for S3q+21 , as desired. The second part is

a consequence of what we just proved.

Notation 3.5.9. Let

η =

(tC(R

21)−(n/2)

−tC(R21)−n

), σ =

(t3n/2

−tn

), ε = C(R2

1)−3n

2

The following two theorems cover the cases when i is congruent to zero or 1.


(a) The minimal projective resolution of S31 is as follows;

0 S31 P 3

1

P 3n−1

⊕P 3

3n2−2

P 3l3

0π31 ζ η


In particular, pdE3(S31) = 2.


0 S3q1 P 3q

1

P 3qn−1

⊕P 3q

3n2−2

(J(P 3q−31 ))

⌈3n2− 3⌉

0π3q1 ζ µ


µ =

(tn2

−1

)

(c) If q ≥ 1, then

0 S3q1 W0 W1 W2 · · · Wq Wq+1 0

d0 d1 d2 d3 dq dq+1

is a minimal projective resolution for S3q1 , where

Wj =

P 3q1 if j = 0

P 3q(n−1)+3(j−1)(n

2−1)

⊕

P 3q(n−1)+3(j−1)(n

2−1)+(n

2−1)

if j = 1, 2, . . . , q

P 3ql3q

if j = q + 1

and

dj =

π3q1 if j = 0

ζ if j = 1

τ if j = 2, . . . , q

η if j = q + 1

In particular, pdE3q(S3q1 ) = q + 1 for q ∈ N⇒ gl. dim(E3q) ≥ q + 1 for q ∈ N.

Proof. The proof of part (a) is similar to the proof given in Lemma 3.5.6 and proofs of

parts (b) and (c) is similar to the proof given in Theorem 3.5.8.



(a) The minimal projective resolution of S11 and S4

1 are as follows;

0 S11 P 1

1 P 1n 0

π11 tn

0 S41 P 4

1

P 4n−1

⊕P 4

3n2−2

P 4l4−(ε−1) 0

π41 ζ σ

In particular, pdE1(S11) = 1 and pdE4(S4

1) = 2.


0 S3q+11 P 3q+1

1

P 3q+1n−1

⊕P 3q+1

3n2−2

(J(P 3q−21 ))

⌈3n2− 3⌉

0π3q+11 ζ µ


µ =

(tn/2

−1

)

(c) If q ≥ 1, then

0 S3q+11 W0 W1 W2 · · · Wq Wq+1 0

d0 d1 d2 d3 dq dq+1

is a minimal projective resolution for S3q+11 , where

Wj =

P 3q+11 if j = 0

P 3q+1(n−1)+3(j−1)(n

2−1)

⊕

P 3q+1(n−1)+3(j−1)(n

2−1)+(n

2−1)

if j = 1, 2, . . . , q

P 3q+1l3q+1−(ε−1) if j = q + 1


and

dj =

π3q+11 if j = 0

ζ if j = 1

τ if j = 2, . . . , q

σ if j = q + 1

In particular, pdE3q+1(S3q+11 ) = q + 1 for q ∈ N0 ⇒ gl. dim(E3q+1) ≥ q + 1 for q ∈ N0.

Proof. The proof of part (a) is similar to the proof given in Lemmas 3.5.4 and 3.5.6, and

proofs of parts (b) and (c) is similar to the proof given in Theorem 3.5.8.

3.5.3 The Module M ′

Let

R1 ( R2 ( · · · ( Rl = k[[t]]

be an ascending chain of rings constructed via the lazy construction. We define

M ′ =l⊕

i=2

Ri and E ′ = EndR2(M′)

Since E is an l× l matrix, the matrix E ′ is (l−1)× (l−1) matrix. Furthermore, R1 ( R2

implies that EndR2(M′) = EndR1(M

′). The matrix E in block form has the following

form;

E =

(R1 (M ′)∗

HomR1(R1,M′) E ′

)

where

HomR1(R1,M′) = HomR1

(R1,

l⊕i=2

Ri

)=

l⊕i=2

HomR1(R1, Ri) =l⊕

i=2

Ri =

R2

R3

...

Rl


and

(M ′)∗ = HomR1(M′, R1)

= HomR1

(l⊕

i=2

Ri, R1

)

=l⊕

i=2

HomR1(Ri, R1)

=(

HomR1(R2, R1) HomR1(R3, R1) . . . HomR1(Rl, R1))

Let S ′2, S′3, . . . , S

′l be the simple E ′-modules and P ′2, P

′3, . . . , P

′l be the indecomposable

projective E ′-modules. A consequence of this construction is the following result which

we state as a Lemma for future reference.

Lemma 3.5.12. The simple modules S ′i and projective modules P ′i satisfy the following;

S ′id1e =

(k, k, 0, . . . , 0) if i = 2

Si if 3 ≤ i ≤ l

P ′id1e = Pi for 2 ≤ i ≤ l

Since l′i = li − 1 the map π′i : P ′i → S ′i for 2 ≤ i ≤ li has li − 1 coordinates, and it is

given by

(π′i)j =

ξi if i = j + 1

0 if i 6= j + 1

where ξi : Ri → k is the natural map. This leads to the following corollary.

Corollary 3.5.13. pdE(Si) = pdE′(S′i) for 3 ≤ i ≤ l.

Proof. This follows from Lemmas 3.5.12, 2.6.5, and 2.6.6.

In fact, given a minimal projective resolution of S ′i for 3 ≤ i ≤ l, when we apply d1eto its minimal projective resolution we get a minimal projective resolution for Si. We

can also do this in the reverse direction (for this we would need to remove the first row

and column of E). Hence, knowing the minimal resolution of one gives us the minimal

resolution of the other (for 3 ≤ i ≤ l).


3.5.4 Global Dimension

In this section we focus on computing the global dimension of Ei for a specific set of

starting rings. All the construction are via the lazy construction. To begin, we introduce

some notation.

Notation 3.5.14. Given positive inters a, b with a ≤ b, let

Vi(a, b) = max{pdEi(Sij) : j = a, a+ 1, ..., b}

Lemma 3.5.15. Suppose {Ri1 : i ∈ N} is a family of starting rings constructed in section

2.5. Then,

pdEi(Sij) = pdEi+1

(Si+1j+n

2−1

)for i ≥ 2, 2 ≤ j ≤ li. In particular,

Vi+1

(n2

+ 1, li+1

)= Vi(2, li)

for i ≥ 2.

Proof. Since Rij = Ri+1

j+n2−1 for i ≥ 2, 1 ≤ j ≤ li, we have

Sij

⌈n2− 1⌉

= Si+1j+n

2−1 for i ≥ 2, 2 ≤ j ≤ li (not true if j = 1)

P ij

⌈n2− 1⌉

= P i+1j+n

2−1 for i ≥ 2, 1 ≤ j ≤ li,

and li+1 = li +n

2− 1. In particular, given i ≥ 2 and 2 ≤ j ≤ li, if

0 Sij P ij W i

1 · · · W ia 0

πij f1 f2 fa

is a minimal projective resolution of Sij, then by Lemmas 2.6.5, 2.6.6 and Proposition


3.5.2,

0 Sij⌈n2− 1⌉

P ij

⌈n2− 1⌉

W i1

⌈n2− 1⌉

0 W ia

⌈n2− 1⌉

· · ·

πij⌈n2− 1⌉

f1

f2

fa

is a minimal projective resolution of Sijdn2−1e = Si+1j+n

2−1. Hence,pdEi(Sij) = pdEi+1

(Si+1j+n

2−1

)for i ≥ 2, 2 ≤ j ≤ li. The second part is a consequence of what we just proved.

The matrix Ei+1 can be written as follows:

Ei+1 =

P i+11

P i+12

P i+13

···

P i+1n2−1

P i+1n2

P i+1n2+1

···

P i+1li+1

=

P i+11

P i+12

P i+13

···

P i+1n2−1

P i1

⌈n2− 1⌉

P i2

⌈n2− 1⌉

···

P ili

⌈n2− 1⌉

Lemma 3.5.16. Suppose {Ri

1 : i ∈ N} is a family of starting rings constructed in section

2.5, then

gl. dim(Ei+1) = max{Vi+1

(1,n

2

), Vi(2, li)

}for i ≥ 2

Proof. By Theorem 1.1.3 and Lemma 3.5.15,

gl. dim(Ei+1) = maxVi+1(1, li+1)

= max{Vi+1

(1,n

2

), Vi+1

(n2

+ 1, li+1

)}= max

{Vi+1

(1,n

2

), Vi(2, li)

}


Lemma 3.5.17. Suppose n ≥ 6 is an even number and

R11 = lead

{0, n,

3n

2

},

then gl. dim(E1) = 2, V1(1, l1) = {1, 2} and V1(2, l1) = {2} where l1 =3n

2− 1.

Proof. This follows from Lemma 3.2.4 and its proof.

For the rest of this section

R11 = lead {0, 6, 9} := B1

1

Ri1 = lead {0, 3b, 11 + 3(i− 2) : b = 2, 3, ..., i+ 1} for i ≥ 2

Bi1 = lead {0, 3b, 12 + 3(i− 2) : b = 2, 3, ..., i+ 1} for i ≥ 2

Notice that these two sets of starting rings are of the type which was introduced in section

2.5 with n = 6, C(R21) = 11 and C(B2

1) = 12. We define

M i =

li⊕j=1

Rij and Ei = EndR1(M

i).

Lemma 3.5.18. gl. dim(E2) = 3 and V2(1, l2) = V2(2, l2) = {2, 3} where l2 = 9.

Proof. Notice that l2 = l1 + 1 = 9 and

S1j−1d1e = S2

j for 3 ≤ j ≤ l2

P 1j−1d1e = P 2

j for 2 ≤ j ≤ l2

Lemmas 2.6.5, 2.6.6, and 3.5.17 imply that pdE2(S2j ) = pdE1(S1

j−1) = 2 for 3 ≤ j ≤ 9.


Moreover, using the resolutions of S1j−1, the minimal resolutions of S2

j are as follows:

0 S2j P 2

j

P 2j−1

⊕P 25+j

P 24+j 0 for 3 ≤ j ≤ 4

π2j (1, t6)

(t6

−1

)

0 S2j P 2

j

P 2j−1

⊕P 29

P 29 0 for 5 ≤ j ≤ 9

π2j (1, t10−j)

(t11−j

−t

)

By Theorem 3.5.8, the minimal projective resolution of S21 is

0 S21 P 2

1

P 25

⊕P 28

P 29 0

π21 (t6, t9)

(t5

−t2

)

A simple computation (like the ones done in section 3.2) shows that that the minimal

projective resolution of S22 is

0 S22 P 2

2 P 21 ⊕ P 2

7 P 25 ⊕ P 2

8

0 P 29

π22 (1, t6)

(−t6 −t9

1 t3

)(−t5

t2

)



Notation 3.5.19. Let

1 = inclusion map

In is the n× n identity matrix

ζ = (t6, t9)

ϑ = (1, t3)

θ =

(t9

−t6

)

φ =

(t8

−t5

)

φ1 =

(t5

−t2

)

η =

(t6

−t3

)

τ =

(t9 t12

−t6 −t9

)


Proof. By Lemma 3.5.15, V3(4, 11) = V2(2, 9) = {2, 3}. By Theorem 3.5.10, the minimal

projective resolution of S31 is

0 S31 P 3

1

P 35

⊕P 37

P 311 0

π31 ζ φ

(3.19)

We now show pdE3(S32) = pdE3(S3

3) = 3, the proofs of which are given in great detail for

future reference. Using the definition introduced in section 3.5.3 we let

(M3)′ =

l3⊕j=2

R3j , (E3)′ = EndR3

2((M3)′) = EndR3

1((M3)′)

Since R32 = B2

1 , by Theorem 3.5.8 the minimal projective resolution of (S32)′ after renum-


bering the subscripts is

0 (S32)′ (P 3

2 )′(P 3

6 )′

⊕(P 3

9 )′(P 3

11)′ 0

(π32)′ ζ η

(3.20)

where the above modules are (E3)′-modules. Applying d1e to the above exact sequence

makes the modules above into E3-modules, and Lemmas 2.6.5 and 2.6.6 yield the follow-

ing minimal projective resolution:

0 (S32)′d1e (P 3

2 )′d1e(P 3

6 )′d1e⊕

(P 39 )′d1e

(P 311)′d1e 0

(π32)′d1e ζ η

where

((π32)′d1e)j = 0 for 3 ≤ j ≤ l3,

and it is the natural map when j = 1, 2. More specifically,

0 (S32)′d1e P 3

2

P 36

⊕P 39

P 311 0

(π32)′d1e ζ η

(3.21)

is a minimal projective resolution. We can think of the sequences in (3.19) and (3.21)

as complexes by extending by zero’s on both sides. We have the following commutative


diagram

0 S31 P 3

1

P 35

⊕P 37

P 311 0

0 (S32)′d1e P 3

2

P 36

⊕P 39

P 311 0

π31 ζ φ

(π32)′d1e ζ η

1 1 I2 t2

(3.22)

Taking the mapping cone gives us the following exact sequence (Lemma 1.5.1);

0

0

⊕(S3

2)′d1e

S31

⊕P 32

P 31

⊕P 36

⊕P 39

P 35

⊕P 37

⊕P 311

0

P 311

⊕0

(0 0

1 (π32)′d1e

) (−π3

1 0

1 ζ

) (−ζ 0

I2 η

)

(−φ 0

t2 0

)


which in turn yields the exact sequence

0 (S32)′d1e

S31

⊕P 32

P 31

⊕P 36

⊕P 39

P 35

⊕P 37

⊕P 311

0 P 311

(1, (π32)′d1e)

(−π3

1 0

1 ζ

) (−ζ 0

I2 η

)

(−φt2

)

Let

γ0 = (1, (π32)′d1e)

γ1 =

(−π3

1 0

1 ζ

)

γ2 =

(−ζ 0

I2 η

)

γ3 =

(−φt2

)

δj =

0 if j = 1

identity if 2 ≤ j ≤ l3


We have the following commutative diagram with exact columns;

0 0 0 0 0

0 S31 S3

1 0 0 0 0

0 (S32)′d1e

S31

⊕P 32

P 31

⊕P 36

⊕P 39

P 35

⊕P 37

⊕P 311

P 311 0

0 S32 P 3

2

P 31

⊕P 36

⊕P 39

P 35

⊕P 37

⊕P 311

P 311 0

0 0 0 0 0

id

γ0 γ1 γ2 γ3

π32 (1, ζ) γ2 γ3

1

(1

0

)

(0, 1)δ I3 I3 id

(3.23)

Since the top two rows are exact the bottom row is exact. Moreover, since the sequences

in (3.19) and (3.21) are minimal and the third, fourth and fifth columns in (3.22) map

into the Jacobian radical of their codomains (or target space), the bottom row in (3.23)

is a minimal projective resolution of S32 .

By Lemma 3.5.18 or Theorem 3.5.8 the minimal projective resolution of S21 is (where

all modules are E2-modules);

0 S21 P 2

1

P 25

⊕P 28

P 29 0

π21 ζ φ1


By Lemmas 2.6.5 and 2.6.6 the sequence

0 S21d1e P 2

1 d1eP 25 d1e⊕

P 28 d1e

P 29 d1e 0

π21d1e ζ φ1

(3.24)

is a minimal projective resolution of S21d1e. Since l3 = l2 + 2 the modules in the above

sequence are (E3)′-modules. Then (3.20) and (3.24) give the following commutative

diagram with exact rows;

0 (S32)′ (P 3

2 )′(P 3

6 )′

⊕(P 3

9 )′(P 3

11)′ 0

0 S21d1e P 2

1 d1eP 25 d1e⊕

P 28 d1e

P 29 d1e 0

(π32)′ ζ η

π21d1e ζ φ1

1 1 I2 t

Taking the mapping cone and using a similar argument given for the bottom row of (3.23)

shows that the minimal projective resolution for (S33)′ is as follows;

0 (S33)′ (P 2

1 )d1e

(P 32 )′

⊕P 25 d1e⊕

P 28 d1e

(P 36 )′

⊕(P 3

9 )′

⊕P 29 d1e

0 (P 311)′

(π33)′ (1, ζ)

(−ζ 0

I2 φ1

)

(−ηt

)

(3.25)

Applying d1e to (3.25), Lemmas 2.6.5, 2.6.6 and 3.5.12, Proposition 3.5.2, and the fact


that (π33)′d1e = π3

3 shows that

0 S33 P 3

3

P 32

⊕P 37

⊕P 310

P 36

⊕(P 3

9

⊕P 311

0 P 311

π33 (1, ζ)

(−ζ 0

I2 φ1

)

(−ηt

)

is the minimal projective resolution of S33 . Hence, gl. dim(E3) = 3 and V3(1, l3) =

V3(2, l3) = {2, 3}.


Proof. By Lemma 3.5.15, V4(4, 13) = V3(2, 11) = {2, 3}. A similar proof to the one given


in Lemma 3.5.20 shows that the minimal resolution of S41 , S

42 , and S4

3 is as follows;

0 S41 P 4

1

P 45

⊕P 47

P 412 0

π41 ζ θ

0 S42 P 4

2

P 41

⊕P 46

⊕P 48

P 45

⊕P 47

⊕P 413

0 P 412

π42 (1, ζ)

(−ζ 0

I2 θ

)

(−θ1

)

0 S43 P 4

3

P 42

⊕P 47

⊕P 49

P 46

⊕P 48

⊕P 413

0 P 413

π43 (1, ζ)

(−ζ 0

I2 φ

)

(−θt

)

and the result follows.

Lemma 3.5.22. gl. dim(E5) = 4 and V5(1, l5) = V5(2, l5) = {2, 3, 4} where l5 = 15.

Proof. By Lemma 3.5.15, V5(4, 15) = V4(2, 13) = {2, 3}. A similar proof to the one given


in Lemma 3.5.20 shows that the minimal resolution of S51 , S

52 , and S5

3 is as follows;

0 S51 P 5

1

P 55

⊕P 57

P 511

⊕P 514

P 515 0

π51 ζ τ φ1

0 S52 P 5

2

P 51

⊕P 56

⊕P 58

P 55

⊕P 57

⊕P 513

0 P 515

P 511

⊕P 514

π52 (1, ζ)

(−ζ 0

I2 θ

)

(−τϑ

)

−φ1

0 S53 P 5

3

P 52

⊕P 57

⊕P 59

P 56

⊕P 58

⊕P 514

0 P 513

π43 (1, ζ)

(−ζ 0

I2 θ

)

(−θ1

)


Notation 3.5.23. Let

ε = C(R21)−

3n

2, ε1 = C(R2

1)− n, ε2 = C(R21)−

n

2

We are now in position to prove the second main result of this thesis.


Theorem 3.5.24. Let

R11 = lead {0, 6, 9}

Ri1 = lead {0, 3b, 11 + 3(i− 2) : b = 2, 3, ..., i+ 1} for i ≥ 2

where for each i the chain

Ri1 ( Ri

2 ( . . . ( Rili

= k[[t]],

the module M i and the ring Ei are constructed via the lazy construction. Then,

gl. dim(Ei) =

q + 2 if i = 3q or i = 3q + 1

q + 3 if i = 3q + 2,

V1(1, l1) = {1, 2}, V1(2, l1) = {2}, and Vi(1, li) = Vi(2, li) = {2, 3, . . . , gl. dim(Ei)} for

i ≥ 2.

Proof. We proceed by induction on i. The result holds for 1 ≤ i ≤ 5 by Lemmas 3.5.17,

3.5.18, 3.5.20, 3.5.21, and 3.5.22. Suppose the result holds for i where i ≥ 5. We consider

three cases.

Case 1. Suppose i = 3q + 1. By the induction hypothesis gl. dim(Ei) = q + 2 and

Vi+1(4, li+1) = Vi(2, li) = {2, 3, . . . , q + 2}

Since i+ 1 = 3q + 2 by Theorem 3.5.8 the minimal projective resolution of Si+11 is

H.D 0 1 . . . q q + 1 q + 2

0 Si+11 P i+1

1

P i+15

⊕P i+17

· · ·P i+16q−1

⊕P i+16q+1

P i+16q+5

⊕P i+16q+8

P i+1li+1

0πi+11 ζ τ τ τ φ1

That is, pdEi+1(Si+11 ) = q + 2. By Theorem 3.5.11 the minimal projective resolution of


(Si+12 )′ (as an (Ei+1)′-module) is:

H.D 0 1 . . . q q + 1

0 (Si+12 )′ (P i+1

2 )′(P i+1

6 )′

⊕(P i+1

8 )′· · ·

(P i+16q )′

⊕(P i+1

6q+2)′

(P i+1li+1−2)

′ 0(πi+1

2 )′ ζ τ τ θ

(3.26)

Applying d1e to (3.26) and using a similar reasoning as done in the proof of Lemma

3.5.20 yields the following commutative diagram with exact rows (for convenience we

have removed the superscript i+ 1);

H.D −1 0 1 . . . q q + 1 q + 2

0 S1 P1

P5

⊕P7

· · ·P6q−1

⊕P6q+1

P6q+5

⊕P6q+8

Pli+10

0 S ′2[1] P2

P6

⊕P8

· · ·P6q

⊕P6q+2

Pli+1−2 0 0

π1 ζ τ τ τ φ1

π′2d1e ζ τ τ θ

δ−1 δ0 δ1 δqδq+1

δq+2

where

δ−1 = inclusion := 1

δ0 = inclusion := 1

δi = I2 for 1 ≤ i ≤ q

δq+1 = (1, t3)

δq+2 = 0


Notice that

δ0(P1) ⊆ P2

δi

P6i−1

⊕P6i+1

⊆

J(P6i−1)

⊕J(P6i+2)

for 1 ≤ i ≤ q

δq+1

P6q+5

⊕P6q+8

⊆ J(Pli+1−2)

δq+2(Pli+1) = 0

The first through to the third inclusions follow from Lemma 2.4.1(b) and the fourth one

is obvious. Taking the mapping cone and using a similar argument as in the proof of


Lemma 3.5.20 shows that the minimal projective resolution of Si+12 is as follows;

0 Si+12 P i+1

2

P i+11

⊕P i+16

⊕P i+18

P i+15

⊕P i+17

⊕P i+112

⊕P i+114

P i+16q−1

⊕P i+16q+1

⊕P i+1li+1−2

P i+16(q−1)−1

⊕P i+16(q−1)+1

⊕P i+16q

⊕P i+16q+2

. . .

P i+111

⊕P i+113

⊕P i+118

⊕P i+120

P i+16q+5

⊕P i+16q+8

P i+1li+1

0

πi+12 (1, ζ)

(−ζ 0

δ1 τ

)

(−τ 0

δ2 τ

)

(−τ 0

δq−1 τ

)(−τ 0

δq θ

)

(−τδq+1

)

−φ1

That is, pdEi+1(Si+12 ) = pdEi(Si+1

1 ) + 1 = q + 3.

By Theorem 3.5.11 the minimal projective resolution of Si1 is given by

H.D 0 1 . . . q q + 1

0 Si1 P i1

P i5

⊕P i7

· · ·P i5+6(q−1)

⊕P i7+6(q−1)

P ili−1 0

π1 ζ τ τ θ


The above modules are Ei-modules. Applying d1e to the above exact sequence makes

the modules into (Ei+1)′-modules (since li+1 = li + 2), by Lemmas 2.6.5, 2.6.6 and (3.26)

we get the following commutative diagram;

H.D −1 0 1 . . . q q + 1

0 (Si+12 )′ (P i+1

2 )′(P i+1

6 )′

⊕(P i+1

8 )′· · ·

(P i+16q )′

⊕(P i+1

6q+2)′

(P i+1li+1−2)

′ 0

0 Si1d1e P i1d1e

P i5d1e⊕

P i7d1e

· · ·P i5+6(q−1)d1e⊕

P i7+6(q−1)d1e

P ili−1d1e 0

(πi+12 )′ ζ τ τ θ

πi1[1] ζ τ τ θ

σ−1 σ0 σ1 σq σq+1

where

σ−1 = inclusion := 1

σ0 = inclusion := 1

σi = I2 for 1 ≤ i ≤ q

σq+1 = inclusion := 1

Notice that

σ0((Pi+12 )′) ⊆ P i

1d1e

σi

(P i+16i )′

⊕(P i+1

6i+2)′

⊆

J(P i5+6(i−1))d1e)⊕

J(P i7+5(i−1))d1e)

for 1 ≤ i ≤ q

σq+1((Pi+1li+1−2)

′ ⊆ J(P ili−1d1e)

Taking the mapping cone and using a similar argument as in the proof of Lemma 3.5.20


shows that the minimal projective resolution of (Si+13 )′ is as follows;

0 (Si+13 )′ P i

1d1e

(P i+12 )′

⊕P i5d1e⊕

P i7d1e

(P i+16 )′

⊕(P i+1

8 )′

⊕P i11d1e⊕

P i13d1e

(P i+16q )′

⊕(P i+1

6q+2)′

⊕P ili−1d1e

(P i+16(q−1))

′

⊕(P i+1

6(q−1)+2)′

⊕P i5+6(q−1)d1e⊕

P i7+6(q−1)d1e

. . .

(P i+112 )′

⊕(P i+1

14 )′

⊕P i17d1e⊕

P i19d1e

(P i+1li+1−2)

′ 0

(πi+13 )′ (1, ζ)

(−ζ 0

σ1 τ

)

(−τ 0

σ2 τ

)

(−τ 0

σq−1 τ

)(−τ 0

σq θ

)

(−θ1

)


Applying d1e gives us the minimal projective resolution of Si+13 ;

0 Si+13 P i+1

3

P i+12

⊕P i+17

⊕P i+19

P i+16

⊕P i+18

⊕P i+113

⊕P i+115

P i+16q

⊕P i+16q+2

⊕P ili+1−1

P i+16(q−1)

⊕P i+16(q−1)+2

⊕P i+17+6(q−1)

⊕P i+19+6(q−1)

. . .

P i+112

⊕P i+114

⊕P i+119

⊕P i+121

P i+1li+1−2 0

πi+13 (1, ζ)

(−ζ 0

σ1 τ

)

(−τ 0

σ2 τ

)

(−τ 0

σq−1 τ

)(−τ 0

σq θ

)

(−θ1

)

Hence, pdEi+1(Si+13 ) = q + 2, and the result follows.

Case 2. Suppose i = 3q. By the induction hypothesis gl. dim(Ei) = q + 2 and

Vi+1(4, li+1) = Vi(2, li) = {2, 3, . . . , q + 2}


Since i+ 1 = 3q + 1 by Theorem 3.5.11 the minimal projective resolution of Si+11 is

H.D 0 1 . . . q q + 1

0 Si+11 P i+1

1

P i+15

⊕P i+17

· · ·P i+15+6(q−1)

⊕P i+17+6(q−1)

P i+1li+1−1 0

πi+11 ζ τ τ θ

That is, pdEi+1(Si+11 ) = q + 1. A similar argument as the one in case 1 shows that the

minimal projective resolution of Si+12 and Si+1

3 are as follows;

0 Si+12 P i+1

2

P i+11

⊕P i+16

⊕P i+18

P i+15

⊕P i+17

⊕P i+112

⊕P i+114

P i+15+6(q−1)

⊕P i+17+6(q−1)

⊕P i+1li+1

P i+15+6(q−2)

⊕P i+17+6(q−2)

⊕P i+16q

⊕P i+16q+2

. . .

P i+111

⊕P i+113

⊕P i+118

⊕P i+120

P i+1li+1−1 0

πi+12 (1, ζ)

(−ζ 0

I2 τ

)

(−τ 0

I2 τ

)

(−τ 0

I2 τ

)(−τ 0

I2 θ

)

(−θ1

)


0 Si+13 P i+1

3

P i+12

⊕P i+17

⊕P i+19

P i+16

⊕P i+18

⊕P i+113

⊕P i+115

P i+16q

⊕P i+16q+2

⊕P ili+1

P i+16(q−1)

⊕P i+16(q−1)+2

⊕P i+17+6(q−1)

⊕P i+19+6(q−1)

. . .

P i+112

⊕P i+114

⊕P i+119

⊕P i+121

P i+1li+1

0

πi+13 (1, ζ)

(−ζ 0

I2 τ

)

(−τ 0

I2 τ

)

(−τ 0

I2 τ

)(−τ 0

I2 φ

)

(−θt

)

That is, pdEi+1(Si+12 ) = pdEi+1(Si+1

3 ) = q + 2 and the result follows.

Case 3. Suppose i = 3q + 2. By the induction hypothesis gl. dim(Ei) = q + 3 and

Vi+1(4, li+1) = Vi(2, li) = {2, 3, . . . , q + 3}

Since i + 1 = 3q + 3 = 3(q + 1) by Theorem 3.5.10 the minimal projective resolution of

Si+11 is

H.D 0 1 . . . q q + 1 q + 2

0 Si+11 P i+1

1

P i+15

⊕P i+17

· · ·P i+15+6(q−1)

⊕P i+17+6(q−1)

P i+15+6q

⊕P i+17+6q

P i+1li+1

0πi+11 ζ τ τ τ φ


That is, pdEi+1(Si+11 ) = q + 2. A similar argument as the one in case 1 shows that the

minimal projective resolution of Si+12 and Si+1

3 are as follows;

0 Si+12 P i+1

2

P i+11

⊕P i+16

⊕P i+18

P i+15

⊕P i+17

⊕P i+112

⊕P i+114

P i+15+6q

⊕P i+17+6q

⊕P i+1li+1

P i+15+6(q−1)

⊕P i+17+6(q−1)

⊕P i+16(q+1)

⊕P i+13+6(q+1)

P i+15+6(q−2)

⊕P i+17+6(q−2)

⊕P i+16q

⊕P i+12+6q

· · ·

P i+111

⊕P i+113

⊕P i+118

⊕P i+120

P i+1li+1

0

πi+12 (1, ζ)

(−ζ 0

I2 τ

)

(−τ 0

I2 τ

)

(−τ 0

I2 τ

)(−τ 0

I2 τ

)(−τ 0

I2 η

)

(−φt2

)


0 Si+13 P i+1

3

P i+12

⊕P i+17

⊕P i+19

P i+16

⊕P i+18

⊕P i+113

⊕P i+115

P i+16(q+1)

⊕P i+13+6(q+1)

⊕P i+1li+1

P i+16q

⊕P i+12+6q

⊕P i+11+6(q+1)

⊕P i+14+6(q+1)

P i+16(q−1)

⊕P i+12+6(q−1)

⊕P i+11+6q

⊕P i+13+6q

· · ·

P i+112

⊕P i+114

⊕P i+119

⊕P i+121

P i+1li+1

0

πi+13 (1, ζ)

(−ζ 0

I2 τ

)

(−τ 0

I2 τ

)

(−τ 0

I2 τ

)(−τ 0

I2 τ

)(−τ 0

I2 φ1

)

(−ηt

)

That is, pdEi+1(Si+12 ) = pdEi+1(Si+1

3 ) = q+ 3 (recall that i+ 1 = 3(q+ 1)) and the result

follows.

The preceding theorem says that the projective dimension of the simple Ei-modules

are as far as they could be from being homogeneous. Furthermore, part of the proof in

Theorem 3.5.24 can be generalized to any even integer n ≥ 6.

Corollary 3.5.25. Suppose the hypotheses of the previous theorem holds, then

{pdEi(Sij) : j = 1, 2, . . . , li} =

{1, 2} if i = 1

{2, 3, . . . , gl. dim(Ei)} if i ≥ 2


Corollary 3.5.26. Given an even integer n ≥ 6, let

R11 = lead

{0, n,

3n

2

}Ri

1 = lead

{0,bn

2,n(i+ 1)

2+ 2 : b = 2, 3, ..., i+ 1

}for i ≥ 2

where for each i the chain

Ri1 ( Ri

2 ( . . . ( Rili

= k[[t]],

the module M i and the ring Ei are constructed via the lazy construction. Then,

pdEi(Si2) = pdEi(Si1) + 1.

Proof. The proof in Theorem 3.5.24 shows this for n = 6. The value of n effects the

indices of the projective modules that appear in our projective resolutions, and when we

apply Lemma 3.5.15. However, the part of the proof to do with projective resolutions for

Si2 and Si3 only depends on ε = 2. Of course, for a given n, the indices of the projective

modules appearing in the minimal projective resolution of Si1 are given by Theorems

3.5.8, 3.5.10 and 3.5.11, and a similar argument in the proof of 3.5.24 gives the indices of

the projective modules that appear in the minimal projective resolution of Si2 and Si3.

Corollary 3.5.27. If ε = 2, then

pdE2(S2j ) =

3 if j = 2

2 if j 6= 2

In particular, gl. dim(E2) = 3.

Proof. Since ε = 2, R1j = R2

j+1 for 1 ≤ j ≤ l1 =3n

2− 1 and l2 = l1 + 1. More specifically,

P 1j d1e = P 2

j+1 for 1 ≤ j ≤ l1

S1j d1e = S2

j+1 for 2 ≤ j ≤ l1

Lemmas 2.6.5 and 2.6.6 imply that pdE2(S2j+1) = pdE1(S1

j ) for 2 ≤ j ≤ l1. By Lemma

3.2.4, pdE1(S1j ) = 2 for 2 ≤ j ≤ l1. By Theorem 3.5.8, pdE2(S2

1) = 2 and by Corollary

3.5.26, pdE2(S22) = 3.

Chapter 4

“Greedy” Construction

In this chapter we concentrate on a construction of our chain which minimizes its length,

called the “greedy” construction. More specifically, in section 4.1 we give the precise

definition of this construction. Section 4.2 focuses on computing the global dimension of

endomorphism rings for specific starting rings. In Section 4.3 we prove some of the results

which are a consequence of this construction. We conclude this chapter by proving the

third main theorem of this thesis.


Given a ring of formal power series R1 6= k[[t]] associated to a numerical semigroup H,

define R2 = EndR1(m1) ) R1 (Theorem 1.1.1). If R2 = k[[t]], then stop. If not, let

R3 = EndR1(m2) ) R2 ) R1. If R3 = k[[t]], then stop. Otherwise, continue the process.

Since only finitely many positive powers of t are missing from R1, there exist a natural

number l such that Rl = k[[t]]. In particular,

Ri = EndR1(mi−1) for 2 ≤ i ≤ l

Since R1 is associated to a numerical semigroup, Ri are rings of formal power series. We

have constructed an ascending chain of rings;

R1 ( R2 ( ... ( Rl = k[[t]] (4.1)

Let

M :=

(l⊕

i=1

Ri

), E := EndR1(M)

116

Chapter 4. “Greedy” Construction 117

We say that the ascending chain (4.1), M and E are constructed via the ”greedy”.

Moreover, gl. dim(E) ≤ l by Proposition 2.2.2. This is the construction given in [11].

Example 4.1.1. Suppose R1 = lead {0, 5, 8}. Then,

R2 = EndR1(m1) = lead {0, 3}

R3 = EndR1(m2) = lead {0, 1} = k[[t]]

4.2 Special Rings II

In this section we compute the global dimension for some special starting rings. The

rings are analogous to the ones in section 3.2, however, the chains are constructed via

the greedy construction.


R1 = lead {0, n}

with n > 1. Then, gl. dim(E) = 2.

Proof. Notice that

R2 = EndR1(m1) = lead {0, 1} = k[[t]]

and

E =

(R1 tnR2

R2 R2

)

Since

(kerπ1)j = tnR2 for 1 ≤ j ≤ 2

= J(P1)

the minimal projective resolution for S1 is:

0 S1 P1 P2 0π1 tn

Also pdE(S2) = 2 by Proposition 2.4.3, the result follows by Theorem 1.1.3.


Lemma 4.2.2. Suppose b ≥ 1, n ≥ 2, and

R1 = lead {0, xn : x = 1, 2, ..., b}

Then, gl. dim(E) = 2.

This is a generalization of Lemma 4.2.1 (by setting b=1).

Proof. The rings in our chain are as follows;

R2 = lead {0, xn : x = 1, 2, ..., b− 1}

R3 = lead {0, xn : x = 1, 2, ..., b− 2}...

Rb = lead {0, n}

Rb+1 = lead {0, 1} = k[[t]]

This enables us to find the entries of E. For 1 ≤ i ≤ b,

Eij =

Ri if 1 ≤ j ≤ i

tnRi+1 if j = i+ 1...

...

t(b−i+1)nRb+1 if j = b+ 1.

If i = b+ 1, then Eij = Rb+1 for 1 ≤ j ≤ b+ 1. Moreover,

mi = tnRi+1 for 1 ≤ i ≤ b

and mb+1 = tRb+1. The minimal projective resolutions for the simple modules are as

follows (proof is similar to the one given in Lemma 4.2.1);

0 S1 P1 P2 0π1 tn

For 2 ≤ i ≤ b,


0 Si Pi

Pi−1

⊕Pi+1

Pi 0πi (1, tn)

(tn

−1

)

and

0 Sb+1 Pb+1

Pb

⊕Pb+1

Pb+1 0πb+1 (1, t)

(tn

−tn−1

)


Example 4.2.3. Let b = n = 3, then

R1 = lead {0, 3, 6, 9}

R2 = lead {0, 3, 6}

R3 = lead {0, 3}

R4 = lead {0, 1}

and

E =

R1 t3R2 t6R3 t9R4

R2 R2 t3R3 t6R4

R3 R3 R3 t3R4

R4 R4 R4 R4

,

The minimal projective resolutions of the simple modules are as follows;

0 S1 P1 P2 0π1 t3

0 S2 P2

P1

⊕P3

P2 0π2 (1, t3)

(t3

−1

)


0 S3 P3

P2

⊕P4

P3 0π3 (1, t3)

(t3

−1

)

0 S4 P4

P3

⊕P4

P4 0π4 (1, t)

(t3

−t2

)

Therefore, gl. dim(E) = 2.

Lemma 4.2.4. Suppose b ≥ 1, n ≥ 2, and

R1 = lead {0, xn, bn+ c : x = 1, 2, ..., b}


The case c = 1 is taken care of in Lemma 4.2.2.

Proof. The rings in our chain are as follows

R2 = EndR1(m1) = lead {0, xn, (b− 1)n+ c : x = 1, 2, ..., b− 1}

R3 = EndR1(m2) = lead {0, xn, (b− 2)n+ c : x = 1, 2, ..., b− 2}...

Rb+1 = EndR1(mb) = lead {0, c}

Rb+2 = EndR1(mb+1) = lead {0, 1} = k[[t]]

For 1 ≤ i ≤ b+ 1 the entries of E are as follows;

Eij =

Ri if 1 ≤ j ≤ i

t(j−i)nRj if i+ 1 ≤ j ≤ b+ 1

t(b+1−i)n+cRb+2 if j = b+ 2

and E(b+2)j = Rb+2 for all 1 ≤ j ≤ b + 2. Moreover, for 1 ≤ i ≤ b we have mi = tnRi+1,

mb+1 = tcRb+2, and mb+2 = tRb+2. A similar proof as the one given in Lemma 4.2.1


shows that the minimal projective resolution of the simple modules are as follows;

0 S1 P1 P2 0π1 tn

0 Si Pi

Pi−1

⊕Pi+1

Pi 0 for 2 ≤ i ≤ bπi (1, tn)

(tn

−1

)

0 Sb+1 Pb+1

Pb

⊕Pb+2

Pb+1 0πb+1 (1, tc)

(tn

−tn−c

)

0 Sb+2 Pb+2

Pb+1

⊕Pb+2

Pb+2 0πi (1, t)

(tc

−tc−1

)



R1 = lead {0, n, n+ 1, 2n}

Then gl. dim(E) = 3.

Proof. The rings in our chain are as follows:

R2 = lead {0, n}

R3 = k[[t]]

The entries of E are:

E =

R1 R1,1 t2nR3

R2 R2 tnR3

R3 R3 R3


Let

∆ = (tn, tn+1)

P2

⊕P2

The image of (tn, tn+1) has the following presentation:

Value of j ∆j n n+ 1 . . . 2n . . .

j = 1, 2 tnR2 x 0 0 x x

⊕tn+1R2 0 x 0 0 x

j = 3 tn(tnR3) 0 0 0 x x

⊕tn+1(tnR3) 0 0 0 0 x

That is, Im(tn, tn+1) = ker π1 = J(P1). The kernel of (tn, tn+1) has the following presen-

tation:

Value of j n n+ 1 . . .

1 ≤ j ≤ 3 0 x x

x x x

That is,

ker(tn, tn+1) =

(tn+1

−tn

)P3 ( J

P2

⊕P2

Thus, the minimal projective resolution for S1 is

0 S1 P1

P2

⊕P2

P3 0π1 (tn, tn+1)

(tn+1

−tn

)


Let

∆′ = (1, tn)

P1

⊕P3

The image of (1, tn) has the following presentation:

Value of j ∆′j 0 n n+ 1 . . . 2n . . .

j = 1 R1 x x x 0 x x

⊕tnR3 0 x x x x x

j = 2 R1,1 0 x x 0 x x

⊕tnR3 0 x x x x x

j = 3 t2nR3 0 0 0 0 x x

⊕tnR3 0 x x x x x

That is, Im(1, tn) = ker π2 = J(P2). The kernel of (1, tn) has the following presentation:

Value of j 0 1 . . . n n+ 1 . . . 2n . . .

j = 1, 2 0 0 0 x x 0 x x

x x 0 x x x x x

A similar argument to the one given in the proof of Lemma 3.2.5 shows that

ker(1, tn) =

(tn tn+1

−1 −t

)P2

⊕P2

( J

P1

⊕P3

and

ker

(tn tn+1

−1 −t

)=

(tn+1

−tn

)P3 ( J

P2

⊕P2

Thus, the minimal projective resolution for S2 is


0 S2 P2

P1

⊕P3

P2

⊕P2

P3 0π2 (1, tn)

(tn tn+1

−1 −t

) (tn+1

−tn

)

By Proposition 2.4.3 we have pdE(S3) = 2, and the result follows by Theorem 1.1.3.


R1 = lead {0, n, n+ 1, n+ 3}


Proof. The rings in our chain are as follows;

R2 = lead {0, 3}

R3 = lead {0, 1} = k[[t]]

and

E =

R1 m1 tn+3R3

R2 R2 t3R3

R3 R3 R3

A similar argument to the one given in the proof of Lemma 4.2.5 shows that the minimal

projective resolutions of the simple modules are as follows;

0 S1 P1

P2

⊕P2

P3 0π1 (tn, tn+1)

(t4

−t3

)


0 S2 P2

P1

⊕P3

P2

⊕P2

0 P3

π2 (1, t3)

(tn tn+1

−tn−3 −tn−2

)

(t4

−t3

)

0 S3 P3

P2

⊕P3

P3 0π3 (1, t)

(t3

−t2

)


4.3 Minor Results III

In this section we concentrate on some of the results that follow when our chain

R1 ( R2 ( ... ( Rl = k[[t]]

is constructed via the greedy construction.

Lemma 4.3.1. (a) If e(Ri) = C(Ri) then i = l − 1 or i = l.

(b) If e(Ri) < C(Ri) then C(Ri+1) = C(Ri)− e(Ri).

Proof. (a) If i = l, nothing to prove. If i < l, then e(Ri) = C(Ri) > 1. This implies

that Ri = lead{0, e(Ri)}. In particular, Ri+1 = EndR1(mi) = lead{0, 1} = k[[t]]. Hence,

i+ 1 = l, as desired.

(b) Let Γ(Ri) = {β1, β2, · · · , βr}, where e(Ri) = β1 < β2 < · · · < βr = C(Ri) (notice

that r ≥ 2). Then, tx ∈ Ri+1 = EndR1(mi) for all x ≥ βr − β1, which implies that

C(Ri+1) ≤ C(Ri) − e(Ri). However, tβr−β1−1 /∈ Ri+1. To see this, if tβr−β1−1 ∈ Ri+1,

then tβr−1 = tβ1tβr−β1−1 ∈ mi ⊆ Ri, a contradiction. Hence, C(Ri+1) ≥ C(Ri) − e(Ri),


If l = 1, that is, R1 = k[[t]], then Proposition 1.1.2 implies that gl. dim(E) = 1.

Meanwhile, if l = 2 then Lemma 4.2.1 yields gl. dim(E) = 2.



Proof. This follows from Lemmas 4.2.2, 4.2.4, 4.2.5, 4.2.6 and Propositions 2.2.2, 2.3.6.

Lemma 2.2.17 gives us l ≤ g(R1) + 1, however, under the greedy construction this

upper bound is usually too large.

Conjecture 2. l ≤ G(R1) + 1.

The following two examples illustrate that the inequality can be an equality or strict.

Example 4.3.3. Suppose R1 = lead {0, 10, 20, 30}. Then G(R1) = 3, and

R2 = lead {0, 10, 20}

R3 = lead {0, 10}

R4 = lead {0, 1} = k[[t]]

That is, l = G(R1) + 1.

Example 4.3.4. Suppose R1 = lead {0, 10, 16, 20, 22, 24, 26, 30}. Then G(R1) = 7,

however,

R2 = lead {0, 10, 14, 16, 20}

R3 = lead {0, 6, 10}

R4 = lead {0, 4}

R5 = lead {0, 1} = k[[t]]

That is, l < G(R1) + 1.

This construction is more erratic then the lazy construction. One of the main problems

that arises is the erratic behaviour of the multiplicity of Ri in our chain. The following

examples show how some of the results that hold under the lazy construction fail under

the greedy construction.

Example 4.3.5. If R1 = lead {0, 7, 10, 13}, then R2 = lead {0, 3, 6}. That is,

e(R2) = 3 < 7 = e(R1) but G(R2) = 2.

In particular, Lemma 3.3.1 does not hold for the greedy construction.


Example 4.3.6. Let

R1 = lead{0, 3, 6, 7, 9}

R2 = EndR1(m1) = lead{0, 3, 4, 6}

R3 = EndR1(m2) = lead{0, 3}

R4 = EndR1(m3) = lead{0, 1}

Then

E =

R1,0 R1,1 R1,2 R1,4

R2,0 R2,0 R2,1 R2,3

R3,0 R3,0 R3,0 R3,1

R4,0 R4,0 R4,0 R4,0

=

P1

P2

P3

P4

A simple calculation shows that the minimal projective resolution of the simple modules


are as follows;

0 S1 P1 P2 0π1 t3

0 S2 P2

P1

⊕P3

⊕P3

P2

⊕P4

0π2 (1, t3, t4)

−t3 0

1 −t4

0 t3

0 S3 P3

P2

⊕P4

P3

⊕P3

0 P4

π3 (1, t3)

(t3 t4

−1 −t

)

(t4

−t3

)

0 S4 P4

P3

⊕P4

P4 0π4 (1, t)

(t3

−t2

)

In particular, gl. dim(E) = 3. This shows that Proposition 3.3.2 fails under the greedy

construction.

4.4 gl. dim(Ei) = 2

Throughout this section {Ri1| i ∈ N} is the set of starting rings constructed in section

2.5 with M i and Ei constructed via the ”greedy” construction. The third main result of

this thesis is that gl. dim(Ei) = 2 for all i ∈ N (Theorem 4.4.7). In particular, when the


endomorphism rings are constructed via the “greedy” construction their global dimension

stays constant.

To begin, we describe the rings

Ri1 ( Ri

2 ( ... ( Rili

= k[[t]]

when the chain is constructed via the “greedy” construction. Fix i, let Rij = Rj and

write

R1 = lead{0, β11 , β

21 , ..., β

r1}

where βj1 ∈ Γ(R1), C(R1) = βr1 and G(R1) = r. The “greedy” construction yields

R2 = EndR1(m1) = lead{0, β21 − β1

1 , β31 − β1

1 , ..., βr1 − β1

1}

where C(R2) = βr1 − β11 . Let βa2 = βa+1

1 − β11 for a = 1, 2, ..., r − 1. In particular,

R2 = lead{0, β12 , β

22 , ..., β

r−12 }

Notice that G(R2) = r − 1 = G(R1) − 1. Now we repeat the process for R3. More

precisely,

R3 = EndR1(m2)

= lead{0, β22 − β1

2 , β32 − β1

2 , ..., βr−12 − β1

2}

= lead{0, β31 − β2

1 , β41 − β2

1 , ..., βr1 − β2

1}

Letting βa3 = βa+12 − β1

2 for a = 1, 2, ..., r − 2, we get

R3 = lead{0, β13 , β

23 , ..., β

r−23 }

Notice that G(R3) = r − 2 and C(R3) = βr2 − β12 . In general,

Rj = EndR1(mj−1) = lead{0, βj1 − βj−11 , βj+1

1 − βj−11 , ..., βr1 − βj−11 } for 2 ≤ j ≤ r

For j = r we get

Rr = lead{0, β1r}


where β1r = βr1 − βr−11 . Notice that Rr+1 = k[[t]] = lead{0, 1}. This proves the following

lemma.

Lemma 4.4.1. For each i ∈ N the following hold;

(a) li = G(Ri1) + 1 = i+ 2. Moreover, G(Ri

j+1) = G(Rij)− 1 for j = 1, 2, ..., li − 1

(b) As a matrix, the entries of Ei are as follows;

(P ij )a = Ei

ja =

Rij,0 if 1 ≤ a ≤ j

Rij,a−j if j + 1 ≤ a ≤ li

(c) For a fixed i,

Rij,a = te(R

ij)Ri

j+1,a−1 for 1 ≤ a ≤ G(Rij)

(d) For a fixed i,

R11 = lead

{0, n,

3n

2

}Ri

1 = lead

{0, n,

3n

2, ...,

(i+ 1)n

2, C(Ri

1)

}for i ≥ 2

Rij = lead

{0,n

2, n, ...,

(i− j + 1)n

2, C(Ri

1)−jn

2

}for 2 ≤ j ≤ li − 2

Rili−1 = lead

{0, C(R2

1)−3n

2

}Rili

= k[[t]]

(e) For i ≥ 3 and 3 ≤ j ≤ li we have Rij = Ri−1

j−1. In particular,

P i−1j−1d1e = P i

j and (J(P i−1j−1))d1e = J(P i

j )

Part (d) of the previous proposition tells us e(Ri1) = n for all i ∈ N. The next

proposition gives us the minimal projective resolution of the first simple module.

Lemma 4.4.2. The minimal projective resolution of Si1 is:

0 Si1 P i1 P i

2 0πi1 tn

In particular, pdEi(Si1) = 1 for all i ∈ N.


Proof. By Proposition 4.4.1 (b)

(P i1)a = Ri

1,a−1 for 1 ≤ a ≤ G(Ri1) + 1

In particular,

(kerπi1)a = (J(P i1))a =

Ri1,1 if a = 1, 2

Ri1,a−1 if 3 ≤ a ≤ G(Ri

1) + 1

=

te(Ri1)Ri

2,0 if a = 1, 2

te(Ri1)Ri

2,a−2 if 3 ≤ a ≤ G(Ri1) + 1

(Lemma 4.4.1(c))

=

tnRi2,0 if a = 1, 2

tnRi2,a−2 if 3 ≤ a ≤ G(Ri

1) + 1


The following results use the notations introduced in section 3.5.4.

Lemma 4.4.3. gl. dim(E1) = gl. dim(E2) = 2.

Proof. Since R11 = lead

{0, n,

3n

2

}, the greedy construction yields

R12 = lead

{0,n

2

}R1

3 = lead{0, 1} = k[[t]]

In particular,

E1 = EndR11(R1

1 ⊕R12 ⊕R1

3) =

R11,0 R1

1,1 R11,2

R12,0 R1

2,0 R12,1

R13,0 R1

3,0 R13,0

By Lemma 4.4.2 the minimal projective resolution of S1

1 is as follows;

0 S11 P 1

1 P 12 0

π11 tn


We show that

0 S12 P 1

2

P 11

⊕P 13

P 12 0

π12 (1, t

n2 )

(tn

−tn2

)

(4.2)

0 S13 P 1

3

P 12

⊕P 13

P 13 0

π13 (1, t)

(tn2

−tn2−1

)

(4.3)

are the minimal projective resolutions for S12 and S1

3 , respectively. Let

∆ = (1, tn2 )

P11

⊕P 13

The image of (1, t

n2 ) has the following presentation:

Value of j ∆j 0 n2

. . . n 3n2

. . .

j = 1 R11,0 x 0 0 x x x

⊕tn/2R1

3,0 0 x x x x x

j = 2 R11,1 0 0 0 x x x

⊕tn/2R1

3,0 0 x x x x x

j = 3 R11,2 0 0 0 0 x x

⊕tn/2R1

3,0 0 x x x x x

This shows that Im(1, tn2 ) =

(R1

2,0 R12,1 R2,1

)= J(P 1

2 ) = kerπ12. The ker(1, t

n2 ) has the


following presentation;

Value of j n2

. . . n . . . 3n2

. . .

j = 1, 2 0 0 x 0 x x

x 0 x x x x

j = 3 0 0 0 0 x x

0 0 x x x x

This shows that

ker(1, tn2 ) =

(tn

−tn2

)P 12 (

(J(P 1

1 )

J(P 13 )

)=

J(P 11 )

⊕J(P 1

3 )

= J

P11

⊕P 13

That is, (4.2) is the minimal projective resolution for S1

2 . Now we show that (4.3) is a

minimal projective resolution for S13 . Let

∆′ = (1, t)

P12

⊕P 13

The image of (1, t) has the following presentation:

Value of j ∆′j 0 . . . n2

. . .

j = 1, 2 R12,0 x 0 x x

⊕tR1

3,0 0 x x x

j = 3 R12,1 0 0 x x

⊕tR1

3,0 0 x x x

In particular, Im(1, t) =(R1

3,0 R13,0 R3,1

)= J(P 1

3 ) = kerπ13. The ker(1, t) has the

following presentation;

Value of j n2− 1 n

2. . .

j = 1, 2, 3 0 x x

x x x


This shows that

ker(1, t) =

(tn/2

−tn2−1

)P 13 (

(J(P 1

2 )

J(P 13 )

)=

J(P 12 )

⊕J(P 1

3 )

= J

P12

⊕P 13

That is, (4.3) is a minimal projective resolution for S1

3 . By Theorem 1.1.3 gl. dim(E1) = 2.

Now we turn our attention to E2. Since

R21 = lead

{0, n,

3n

2, C(R2

1)

}R2

2 = lead{

0,n

2, ε1

}R2

3 = lead {0, ε}

R24 = lead {0, 1} = k[[t]]

we have

E2 = EndR21(R2

1 ⊕R22 ⊕R2

3 ⊕R24) =

R2

1,0 R21,1 R2

1,2 R21,3

R22,0 R2

2,0 R22,1 R2

2,2

R23,0 R2

3,0 R23,0 R2

3,1

R24,0 R2

4,0 R24,0 R2

4,0

By Lemma 4.4.2 the minimal projective resolution of S2

1 is as follows;

0 S21 P 2

1 P 22 0

π21 tn

Now we compute the minimal projective resolution for S22 . Let

∆ = (1, tn2 )

P21

⊕P 23


The image of (1, tn2 ) has the following presentation:


ε1 . . . n . . . 3n2

. . . C(R21) . . .

j = 1 R21,0 x 0 0 0 x 0 x 0 x x

⊕tn/2R2

3,0 0 x x x x x x x x x

j = 2 R21,1 0 0 0 0 x 0 x 0 x x

⊕tn/2R2

3,0 0 x x x x x x x x x

j = 3 R21,2 0 0 0 0 0 0 x 0 x x

⊕tn/2R2

3,0 0 x x x x x x x x x

j = 4 R21,3 0 0 0 0 0 0 0 0 x x

⊕tn/2R2

3,1 0 0 x x x x x x x x

That is, Im(1, tn2 ) = J(P 2

2 ) = ker π22. The ker(1, t

n2 ) has the following presentation:

Value of j n2

n ε2 . . . 3n2

. . . C(R21) . . .

j = 1, 2 0 x 0 0 x 0 x x

x x x x x x x x

j = 3 0 0 0 0 x 0 x x

0 x x x x x x x

j = 4 0 0 0 0 0 0 x x

0 0 x x x x x x

This shows that ker(1, tn2 ) =

(tn

−tn2

)P 22 . Hence,

0 S22 P 2

2

P 21

⊕P 23

P 22 0

π22 (1, t

n2 )

(tn

−tn2

)


is a projective resolution for S22 and it is minimal by Proposition 2.3.6. Let

∆′ = (1, tε)

P22

⊕P 24

Then

Value of j ∆′j 0 ε . . . n2

. . . ε1 . . .

j = 1, 2 R22,0 x 0 0 x 0 x x

⊕tεR2

4,0 0 x x x x x x

j = 3 R22,1 0 0 0 x 0 x x

⊕tεR2

4,0 0 x x x x x x

j = 4 R22,2 0 0 0 0 0 x x

⊕tεR2

4,0 0 x x x x x x

That is, Im(1, tε) = J(P 23 ) = ker π2

3. The ker(1, tε) has the following presentations:

Value of j n2− ε n

2. . . ε1 . . .

j = 1, 2, 3 0 x 0 x x

x x x x x

j = 4 0 0 0 x x

0 x x x x

That is,

ker(1, tε) =

(tn2

−tn2−ε

)P 23

Hence,

0 S23 P 2

3

P 22

⊕P 24

P 23 0

π23 (1, tε)

(tn2

−tn2−ε

)

is a projective resolutions for S23 . Proposition 2.4.3 gives pdE2(S2

4) = 2, and the result


follows by Theorem 1.1.3.

Notation 4.4.4. For i ≥ 3, let

εi = C(Ri1)−

3n

2, εi1 = C(Ri

1)− n, εi2 = C(Ri1)−

n

2

Lemma 4.4.5. Suppose {Ri1 : i ∈ N} is a family of starting rings constructed in section

2.5.

(a) If i ≥ 2, then the minimal projective resolutions of Sili−1 and Sili are as follows;

0 Sili−1 P ili−1

P ili−2

⊕P ili

P ili−1 0

πili−1 (1, tε)

(tn2

−tn2−ε

)

(4.4)

0 Sili P ili

P ili−1

⊕P ili

P ili

0πili (1, t)

(tε

−tε−1

)

(4.5)

(b) For all i ∈ N we have

0 Si2 P i2

P i1

⊕P i3

P i2 0

πi2 (1, tn2 )

(tn

−tn2

)

(4.6)

(c) For i ≥ 3, the minimal projective resolution of Si3 is as follows;

0 Si3 P i3

P i2

⊕P i4

P i3 0

πi3 (1, tn2 )

(tn2

−1

)

(4.7)

Warning that parts (a) and (c) are false for i = 1 (see Lemma 4.4.3).


Proof. (a) Fix i ≥ 2. Lemma 4.4.1(d) yields

Rili−2 = lead

{0,n

2, ε1

}Rili−1 = lead {0, ε}

Rili

= k[[t]]

In particular,

(P ili−2)j =

Rili−2,0 if 1 ≤ j ≤ li − 2

Rili−2,1 if j = li − 1

Rli−2,2 if j = li

(P ili−1)j =

Rili−1,0 if 1 ≤ j ≤ li − 1

Rili−1,1 if j = li

=⇒ J(P ili−1)j =

Rili−1,0 if 1 ≤ j ≤ li − 2

Rili−1,1 if j = li − 1, li

,

(P ili)j = Ri

li,0for 1 ≤ j ≤ li =⇒ (J(P i

li))j =

Rili,0

if 1 ≤ j ≤ li − 1

Rli,1 if j = li

Let

∆ = (1, tε)

Pili−2

⊕P ili

Then

Value of j ∆j 0 1 . . . ε . . . n2

. . . ε1 . . .

1 ≤ j ≤ li − 2 Rili−2,0 x 0 0 0 0 x 0 x x

⊕tεRi

li,00 0 0 x x x x x x

j = li − 1 Rili−2,1 0 0 0 0 0 x 0 x x

⊕tεRi

li,00 0 0 x x x x x x

j = li Rili−2,2 0 0 0 0 0 0 0 x x

⊕tεRi

li,00 0 0 x x x x x x

That is, Im(1, tε) = J(P ili−1) = kerπili−1. Moreover, ker(1, tε) has the following presenta-


tion;

Value of j n2− ε . . . n

2. . . ε1 . . .

1 ≤ j ≤ li − 1 0 0 x 0 x x

x 0 x x x x

j = li 0 0 0 0 x x

0 0 x x x x

That is,

ker(1, tε) =

(tn2

−tn2−ε

)P ili−1

Hence, the sequence in (4.4) is a projective resolution for Sili−1 and it is minimal by

Proposition 2.3.6. The minimal projective resolution for Sili given in (4.5) follows from

Proposition 2.4.3, where e(Rl−1) = ε.

(b) Lemma 4.4.3 gives the desired result for i = 1, 2. Fix i ≥ 3. Then Lemma 4.4.1 (a),

(b) and (d) yields (notice that li = i+ 2 ≥ 5)

Ri1 = lead

{0, n,

3n

2, ...,

(i+ 1)n

2, C(Ri

1)

}and (P i

1)j = Ri1,j−1 for 1 ≤ j ≤ li

Ri2 = lead

{0,n

2, n, ...,

(i− 1)n

2, εi1

}and (P i

2)j =

Ri2,0 if j = 1, 2

Ri2,j−2 if 3 ≤ j ≤ li

Ri3 = lead

{0,n

2, n, ...,

(i− 2)n

2, εi2

}and (P i

3)j =

Ri3,0 if 1 ≤ j ≤ 3

Ri3,j−3 if 4 ≤ j ≤ li

Let

∆ = (1, tn2 )

Pi1

⊕P i3


The image of (1, tn2 ) has the following presentation:


Ai−12 (n) εi1 . . . in2

. . . (i+1)n2

. . . C(Ri1) . . .

j = 1 Ri1,0 x 0 x 0 0 x 0 x 0 x x

⊕tn/2Ri

3,0 0 x x x x x x x x x x

j = 2 Ri1,1 0 0 x 0 0 x 0 x 0 x x

⊕tn/2Ri

3,0 0 x x x x x x x x x x

Value of j ∆j Aj−1j−2(n) Ai−1j (n) εi1 . . . in2

. . . (i+1)n2

. . . C(Ri1) . . .

3 ≤ j ≤ i+ 1 Ri1,j−1 0 x x 0 x 0 x 0 x x

⊕tn/2Ri

3,j−3 x x x x x x x x x x

Value of j ∆j εi1 . . . C(Ri1) . . .

j = i+ 2 Ri1,i+1 0 0 x x

⊕tn/2Ri

3,i−1 x x x x

Hence, Im(1, tn2 ) = J(P i

2) = ker πi2. The kernel of (1, tn2 ) has the following presentation:

Value of j n2

Ai2(n) εi2 . . . (i+1)n2

. . . C(Ri1) . . .

j = 1, 2 0 x 0 0 x 0 x x

x x x x x x x x

Value of j (j−1)n2

Aij(n) εi2 . . . (i+1)n2

. . . C(Ri1) . . .

3 ≤ j ≤ i+ 1 0 x 0 0 x 0 x x

x x x x x x x x

Value of j εi2 . . . C(Ri1) . . .

j = i+ 2 0 0 x x

x x x x


Therefore,

ker(1, tn2 ) =

(tn

−tn2

)P i2

That is, (4.6) is a projective resolution for Si2, and they are minimal by Proposition 2.3.6.

(c) Fix i ≥ 3, then li = i+ 2 ≥ 5. Lemma 4.4.1 (a), (b) and (d) yield

Ri1 = lead

{0, n,

3n

2, ...,

(i+ 1)n

2, tC(R

i1)

}and (P i

1)j = Ri1,j−1 for 1 ≤ j ≤ li

Ri2 = lead

{0,n

2, n, ...,

(i− 1)n

2, εi1

}and (P i

2)j =

Ri2,0 if j = 1, 2

Ri2,j−2 if 3 ≤ j ≤ li

Ri3 = lead

{0,n

2, n, ...,

(i− 2)n

2, εi}

and (P i3)j =

Ri3,0 if 1 ≤ j ≤ 3

Ri3,j−3 if 4 ≤ j ≤ li

Ri4 = lead

{0,n

2, n, ...,

(i− 3)n

2, C(Ri

1)− 2n

}and (P i

4)j =

Ri4,0 if 1 ≤ j ≤ 4

Ri4,j−4 if 5 ≤ j ≤ li

Let a ≤ b be natural numbers and define

∆ = (1, tn2 )

Pi2

⊕P i4

Then

Value of j ∆j 0 Ai−21 (n) εi . . . (i−1)n2

. . . εi1 . . .

j = 1, 2 Ri2,0 x x 0 0 x 0 x x

⊕tn/2Ri

4,0 0 x x x x x x x

j = 3 Ri2,1 0 x 0 0 x 0 x x

⊕tn/2Ri

4,0 0 x x x x x x x


Value of j ∆j(j−3)n

2Ai−2j−2(n) εi . . . (i−1)n

2. . . εi1 . . .

4 ≤ j ≤ i+ 1 Ri2,j−2 0 x 0 0 x 0 x x

⊕tn/2Ri

4,j−4 x x x x x x x x

Value of j ∆j εi . . . εi1 . . .

j = i+ 2 Ri1,i+1 0 0 x x

⊕tn/2Ri

3,i−1 x x x x

Hence, Im(1, tn2 ) = J(P i

3) = ker πi3. The kernel of (1, tn2 ) has the following presentation:

Value of j 0 Ai−21 (n) εi . . . (i−1)n2

. . . εi1 . . .

1 ≤ j ≤ 3 0 x 0 0 x 0 x x

x x x x x x x x

Value of j (j−3)n2

Ai−2j−2(n) εi . . . (i−1)n2

. . . εi1 . . .

4 ≤ j ≤ i+ 1 0 x 0 0 x 0 x x

x x x x x x x x

Value of j εi . . . εi1 . . .

j = i+ 2 0 0 x x

x x x x

Therefore,

ker(1, tn2 ) =

(tn2

−1

)P i3

That is, (4.7) is a projective resolution for Si3, and it is minimal by Proposition 2.3.6.

Lemma 4.4.6. The minimal projective resolutions of the simple E3-modules are as fol-

lows;

0 S31 P 3

1 P 32 0

π31 tn


0 S32 P 3

2

P 31

⊕P 33

P 32 0

π32 (1, t

n2 )

(tn

−tn2

)

0 S33 P 3

3

P 32

⊕P 34

P 33 0

π33 (1, t

n2 )

(tn2

−1

)

0 S34 P 3

4

P 33

⊕P 35

P 34 0

π34 (1, tε)

(tn2

−tn2−ε

)

0 S35 P 3

5

P 34

⊕P 35

P 35 0

π35 (1, t)

(tε

−tε−1

)

Proof. By Lemma 4.4.1(a) l3 = 5. The minimal projective resolutions follow from Lem-

mas 4.4.2 and 4.4.5.

We now prove the third main result of this thesis.

Theorem 4.4.7. Let {Ri1| i ∈ N} be a set of starting rings constructed in section 2.5.


(a) For i ≥ 3, the minimal projective resolutions of the simple Ei-modules are as follows:

0 Si1 P i1 P i

2 0πi1 tn

(4.8)

0 Si2 P i2

P i1

⊕P i3

P i2 0

πi2 (1, tn2 )

(tn

−tn2

)

(4.9)

0 Sili−1 P ili−1

P ili−2

⊕P ili

P ili−1 0

πili−1 (1, tε)

(tn2

−tn2−ε

)

(4.10)

0 Sili P ili

P ili−1

⊕P ili

P ili

0πili (1, t)

(tε

−tε−1

)

(4.11)

and for 3 ≤ j ≤ li − 2,

0 Sij P ij

P ij−1

⊕P ij+1

P ij 0

πij (1, tn2 )

(tn2

−1

)

(4.12)

(b) gl. dim(Ei) = 2 for all i ∈ N.

Proof. (a) We proceed by induction on i. The result is true for E3 by Lemma 4.4.6.

Assume the result is true for i − 1 ≥ 3. Lemmas 4.4.2 and 4.4.5 gives us the exact

sequences (4.8), (4.9), (4.10), (4.11), and the minimal projective resolution of Si3. If

4 ≤ j ≤ li − 2, the minimal projective resolution of Sij has the following beginning by

Theorem 1.1.4;

0 Sij P ij

πij


More specifically, we have the following short exact sequence;

0 Sij P ij J(P i

j ) 0πij inc

(4.13)

where inc is the inclusion map. Since i, j ≥ 4, Proposition 4.4.1(e) yields (J(P i−1j−1))d1e =

J(P ij ). By the induction hypothesis, the following sequence is exact:

0 Si−1j−1 P i−1j−1

P i−1j−2

⊕P i−1j

P i−1j−1 0

πi−1j−1 (1, tn2 )

(tn2

−1

)

Since Im(1, tn2 ) = ker(πi−1j−1) = J(P i−1

j−1), we get the following short exact sequence;

0 J(P i−1j−1)

P i−1j−2

⊕P i−1j

P i−1j−1 0

(1, tn2 )

(tn2

−1

)

By Lemma 2.6.5, the following sequence is exact:

0 J(P i−1j−1)d1e

P i−1j−2d1e⊕

P i−1j d1e

P i−1j−1d1e 0

(1, tn2 )

(tn2

−1

)

By Lemma 4.4.1(e) the above short exact sequence is the short exact sequence

0 J(P ij )

P ij−1

⊕P ij+1

P ij 0

(1, tn2 )

(tn2

−1

)

(4.14)

Splicing sequences (4.13) and (4.14) gives the following projective resolution for Sij;


0 Sij P ij

P ij−1

⊕P ij+1

P ij 0

πij (1, tn2 )

(tn2

−1

)

By Proposition 2.3.6, the projective resolutions are minimal.

(b) This is a direct consequence of part (a), Theorem 1.1.3 and Lemma 4.4.3.

Chapter 5

Examples and Open Questions

In this chapter we give an example illustrating the possible values for global dimension

of E when the only restriction on its construction is the one given in section 2.2. We

conclude with some open questions which could be used for future research.

We start by recalling the restriction in section 2.2. Suppose H is a numerical semi-

group with generators α1, α2, ..., αs, F (α1, α2, ..., αs) > −1, and let R1 be the ring of for-

mal power series associated to H. Since R1 6= R̃1 = k[[t]], we have R1 ( EndR1(m1) ⊆ R̃1

(Theorem 1.1.1). Moreover, m1 contains a non-zero divisor (Proposition 2.2.1), and

EndR1(m1) embeds naturally into R1 (by sending f to f(a)/a, which is independent of

the non-zero divisor a ∈ m1). It is well known that in fact EndR1(m1) ⊆ R̃1. Further-

more, it is easy to see that EndR1(m1) is itself a ring of formal power series. Let R2

be any ring of formal power series over k that properly contains R1 and is contained

in EndR1(m1). Notice that R2 is a local Noetherian ring of (Krull) dimension 1. If

R2 = k[[t]], then R2 = EndR1(m1) = k[[t]] in which case we define

M := R1 ⊕R2, E := EndR1(M)

If R2 6= k[[t]], pick R3 such that R2 ( R3 ⊆ EndR1(m2) ⊆ k[[t]] (this is possible by

Theorem 1.1.1). If R3 = k[[t]], define

M := R1 ⊕R2 ⊕R3, E := EndR1(M)

Notice that R1 ( R2 ( R3 = k[[t]]. If R3 6= k[[t]], repeat the process to obtain R4, and

continue in this fashion. Since R1 is missing only finitely many powers of t there exists

an l such that Rl = R̃1 = k[[t]]. Hence, we have constructed an ascending chain of rings

R1 ( R2 ( R3 ( ... ( Rl−1 ( Rl = k[[t]]

147

Chapter 5. Examples and Open Questions 148

Let

M =l⊕

i=1

Ri, E = EndR1(M)

Notice that k[[t]] = Rl = EndR1(ml−1).

5.1 Examples

In this section we give an example which illustrate the range of values for gl. dim(E) for

a fixed starting ring.

Example 5.1.1. Let R1 = k[[t3, t7, t8]] = lead{0, 3, 6}. The arrows below indicate the

ways in which we can construct M .

lead{0, 3, 6}

lead{0, 3, 4, 6} lead{0, 3, 5}

lead{0, 3}

lead{0, 2}

lead{0, 1}

In total, there are 7 different ways of constructing M .

(1) The ”greedy” construction of M has the following presentation using the arrows:


R1 = lead{0, 3, 6}

R2 = lead{0, 3}

R3 = lead{0, 1}

Then, gl. dim(E) = 2 by Lemma 4.2.2.

(2) The ”lazy” construction of M has the following presentation using the arrows:

R1 = lead{0, 3, 6}

R2 = lead{0, 3, 5}

R3 = lead{0, 3}

R4 = lead{0, 2}

R5 = lead{0, 1}

Then, gl. dim(E) = 2 by Lemma 3.2.3.

(3) The third construction of M has the following presentation using the arrows;


R1 = lead{0, 3, 6}

R2 = lead{0, 3}

R3 = lead{0, 2}

R4 = lead{0, 1}

Then

E =

R1 t3R2 t6R4 t6R4

R2 R2 t3R4 t3R4

R3 R3 R3 t2R4

R4 R4 R4 R4

The minimal projective resolutions for the simple modules are as follows;

0 S1 P1 P2 0π1 t3

0 S2 P2

P1

⊕P4

P2 0π2 (1, t3)

(t3

−1

)

0 S3 P3

P2

⊕P4

P4 0π3 (1, t2)

(t3

−t2

)

0 S4 P4

P3

⊕P4

P4 0π4 (1, t)

(t2

−t

)


Hence, gl. dim(E) = 2.

(4) The fourth construction of M has the following presentation using the arrows:

R1 = lead{0, 3, 6}

R2 = lead{0, 3, 5}

R3 = lead{0, 3}

R4 = lead{0, 1}

Then

E =

R1 t3R3 t3R3 t6R4

R2 R2 m2 t5R4

R3 R3 R3 t3R4

R4 R4 R4 R4


0 S1 P1 P3 0π1 t3

0 S2 P2

P1

⊕P4

P4 0π2 (1, t5)

(t6

−1

)

0 S3 P3

P2

⊕P4

P3

⊕P4

P4 0π3 (1, t3)

(t3 t5

−1 −t2

) (t3

−t

)


0 S4 P4

P3

⊕P4

P4 0π4 (1, t)

(t3

−t2

)


(5) The fifth construction of M has the following presentation using the arrows:

R1 = lead{0, 3, 6}

R2 = lead{0, 3, 4, 6}

R3 = lead{0, 3}

R4 = lead{0, 2}

R5 = lead{0, 1}

Then

E =

R1 t3R3 t3R3 t6R5 t6R5

R2 R2 m2 t4R4 t6R5

R3 R3 R3 t3R5 t3R5

R4 R4 R4 R4 t2R5

R5 R5 R5 R5 R5


0 S1 P1 P3 0π1 t3

0 S2 P2

P1

⊕P4

P5 0π2 (1, t4)

(t6

−t2

)


0 S3 P3

P2

⊕P5

P3

⊕P4

P5 0π3 (1, t3)

(t3 t4

−1 −t

) (t3

−t2

)

0 S4 P4

P3

⊕P5

P5 0π4 (1, t2)

(t3

−t

)

0 S5 P5

P4

⊕P5

P5 0π5 (1, t)

(t2

−t

)


(6) The sixth construction of M has the following presentation using the arrows:

R1 = lead{0, 3, 6}

R2 = lead{0, 3, 4, 6}

R3 = lead{0, 3}

R4 = lead{0, 1}

Then

E =

R1 t3R3 t3R3 t6R4

R2 R2 m2 t6R4

R3 R3 R3 t3R4

R4 R4 R4 R4



0 S1 P1 P3 0π1 t3

0 S2 P2

P1

⊕P3

P4 0π2 (1, t4)

(t7

−t3

)

0 S3 P3

P2

⊕P4

P3

⊕P3

P4 0π3 (1, t3)

(t3 t4

−1 −t

) (t4

−t3

)

0 S4 P4

P3

⊕P4

P4 0π4 (1, t)

(t3

−t2

)


(7) The seventh construction of M has the following presentation using the arrows:

R1 = lead{0, 3, 6}

R2 = lead{0, 3, 5}

R3 = lead{0, 2}

R4 = lead{0, 1}

Then

E =

R1 m1 t6R4 t6R4

R2 R2 t3R3 t5R4

R3 R3 R3 t2R4

R4 R4 R4 R4


The minimal projective resolution of the simple modules are as follows;

0 S1 P1

P2

⊕P4

P3 0π1 (t3, t6)

(t3

−1

)

0 S2 P2

P1

⊕P3

P2

⊕P4

P3 0π2 (1, t3)

(t3 t6

−1 −t3

) (t3

−1

)

0 S3 P3

P2

⊕P4

P4 0π3 (1, t2)

(t3

−t

)

0 S4 P4

P3

⊕P4

P4 0π4 (1, t)

(t2

−t

)

Hence, gl. dim(E) = 3. This shows that the possible values for gl. dim(E) are two or

three.

5.2 Open Questions

The construction in section 2.2 (also stated at the beginning of this chapter) and the

results in this thesis give rise to some open questions for future research. Two such

questions have already been mentioned in sections 3.3 and 4.3.

• The upper bound for global dimension of E. The upper bound

gl. dim(E) ≤ l

seems to be high. In fact, throughout this thesis the only time this upper bound

was achieved was when l = 1, 2, 3. It would be interesting to see if there is a smaller

upper bound for gl. dim(E) when l 6= 1, 2, 3.


• Possible values for gl. dim(E). Suppose we fix a starting ring R associated to

a numerical semigroup H. The question which arises is what are the possible

values for gl. dim(E) when R is the starting ring and the only restriction on the

construction of our ascending chain is the one given in section 2.2? For example, in

section 5.1 we showed that if R = k[[t3, t7, t8]] then the possible values for gl. dim(E)

are 2 or 3.

• Other families of starting rings. The family of starting rings constructed in

section 2.5 leads to the question of whether it is possible to construct other families

of starting rings with analogous results given in this thesis. Also, it would be inter-

esting to construct other families of starting rings with new results for gl. dim(E).

• Projective dimension of the first and second simple module. The results

in this thesis suggest that

pdE(S1) ≤ pdE(S2).

Furthermore, under the lazy construction they suggest that

pdE(S1) ≤ pdE(S2) ≤ pdE(S1) + 1.

• Values of l. Fix a starting ring R. Let Rchain be the set of all ascending chains

such that R is the starting ring in the chain, and the chain satisfies the condition

given in section 2.2. Moreover, let Rlength be the set consisting of the lengths of the

chains that appear in Rchain. Since R is associated to some semigroup, Rchain is a

finite set, hence, Rlength is a finite set. We know that Rlength is bounded above by

g(R) + 1. It would be interesting to know all the elements of Rlength or to obtain a

lower bound for Rlength. For example, if R = k[[t3, t7, t8]], then example 5.1.1 shows

that Rlength = {3, 4, 5}.

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