Computing Almost Split Sequences: An algorithm for computing almost split sequences of finitely generated modules over a finite dimensional algebra

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Computing Almost Split Sequences

An algorithm for computing almost split sequences of finitely generated modules over a finite dimensional algebra

Tea Sormbroen Lian

Master of Science in Physics and Mathematics Supervisor: Øyvind Solberg, MATH

Department of Mathematical Sciences Submission date: June 2012

Norwegian University of Science and Technology

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Abstract

An artin algebra Λ over a commutative, local, artinian ringRwas fixed, and with this foundation some topics from representation theory were discussed. A series of functors of module categories were defined, and almost split sequences were introduced along with some results. An isomorphismωδ,X:Dδ^∗→δ∗(DTr(X)) of Γ-modules for an artin R-algebra Γ was constructed. The isomorphism ωδ,X was applied to a special case, yielding a deterministic algorithm for computing almost split sequences in the case that Ris a field.

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Norsk sammendrag

En artinsk algebra Λ over en kommutativ, lokal ring R ble fiksert, og med dette som utgangspunkt ble endel emner fra representasjonsteori diskutert. En rekke funktorer over modulkategorier ble definert, og nesten splitt-eksakte følger ble in- trodusert sammen med noen resultater. En isomorfiωδ,X:Dδ^∗→δ∗(DTr(X)) av Γ-moduler for en artinsk R-algebra Γ ble konstruert. Isomorfien ωδ,X ble anvendt ved et spesialtilfelle, og dette ga opphav til en deterministisk algoritme for ˚a regne ut nesten splitt-eksakte følger i tilfellet atRer en kropp.

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Preface

This thesis is the final part of my master’s degree in Industrial Mathematics at the Norwegian University of Science and Technology (NTNU). The thesis was written from the 23rd of January to the 18th of June. I would like to thank my supervisor Øyvind Solberg for fantastic help and feedback along the way. I would also like to thank my classmates for encouragement in the most challenging phases.

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Introduction

The aim of this thesis is to develop a method for computing almost split sequences.

Chapter 2 will serve as an introduction to the theory which will be required for our work. A series of topics will be discussed, and important result will be stated and demonstrated.

In Chapter 3 we will embark on the task of designing an algorithm for computing almost split sequences. Our approach will be divided into two main steps:

In Section 3.1 we will apply a variety of results from Chapter 2 in order to design an isomorphism which depends on certain parameters.

In Section 3.2 we will fix these parameters. Together with prior results, the isomorphism we then get will suggest a connection between identity homomorphisms and almost split sequences which motivates a deterministic algorithm for computing the latter. We will complete Chapter 3 with a presentation of this algorithm, along with a demonstration of its correctness.

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Chapter 2

Background Theory

In this chapter we will traverse a series of topics from representation theory. Section 2.1 will provide a review of basic category theory. In Section 2.2 we will give the definition of artinian rings, artinian modules and artin algebras, and, at this, set the framework for this thesis. In the subsequent sections a multitude of functors will be presented together with a survey of their respective features. A formal definition along with basic properties of almost split sequences will be presented in Section 2.9. In section 2.10 we will study a certain algebra arising from prior investigation, and we will make useful observations regarding its top and socle.

2.1 Category theory

In this thesis we shall mainly focus on modules over artin algebras, but before we introduce the framework in which we will be working most of the time, we recall the following concepts from category theory:

Definition 1.

(i) A category C consists of the following:

– a class Ob(C) of objects,

– forX,Y ∈Ob(C), a set HomC(X, Y) of morphisms fromX toY, – forX,Y,Z ∈Ob(C), a binary operation

Hom_C(Y, Z)×Hom_C(X, Y)→Hom_C(X, Z) (g, f)7→gf

such that the following holds:

∗ For all X, Y, Z, W ∈ Ob(C), f ∈ HomC(X, Y), g ∈ HomC(Y, Z) and h∈Hom_C(Z, W), then

(f g)h=f(gh).

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∗ For all X ∈ Ob(C), then there is an identity morphism 1_X ∈ Hom_C(X, X) := End_C(X) such that for allY ∈Ob(C), then

f1X = 1X

for allf ∈Hom_C(X, Y), and

1Xf =f for allf ∈Hom_C(Y, X).

For a morphism set HomC(X, Y) whereX,Y ∈Ob(C), thenX is called the source object of HomC(X, Y), and Y is called the target object of HomC(X, Y).

(ii) Anadditive category is a categoryC where

– Hom_C(X, Y) is an abelian group for all X, Y ∈ Ob(C), such that the following holds:

j(f+g)h=jf h+jgh

for allW,X, Y,Z ∈Ob(C) and h∈Hom_C(W, X), f,g∈Hom_C(X, Y) andj∈Hom_C(Y, Z),

– there is a zero object 0∈Ob(C) such that

|Hom_C(X,0)|=|Hom_C(0, X)|= 1 for allX ∈Ob(C),

– for all X, Y ∈ Ob(C) there is X ⊕Y ∈ Ob(C) together with ιX ∈ Hom_C(X, X⊕Y), ιY ∈Hom_C(Y, X⊕Y), πX ∈Hom_C(X⊕Y, X) and πY ∈Hom_C(X⊕Y, Y) such that the following holds:

πXιX= 1X, πYιY = 1Y, πYιX= 0,

πXιY = 0 and

ιXπX+ιYπY = 1_X⊕Y. Theι’s andπ’s are calledinclusions andprojections.

(iii) LetCbe an additive category, and letX,Y ∈Ob(C). Supposef ∈Hom_C(X, Y).

– A kernel of f is an object Ker(f) ∈ Ob(C) together with a morphism ιf ∈HomC(Ker(f), X) such that the following holds:

∗ f ιf = 0,

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∗ For allT ∈Ob(C) andt∈Hom_C(T, X) such thatf t= 0, then there exists uniques∈Hom_C(T,Ker(f)) such that

t=ι_fs.

Ker(f) X Y

T

ιf f

t s

– Acokernel off is an object Cok(f)∈Ob(C) together with a morphism πf ∈HomC(Y,Cok(f)) such that the following holds:

∗ πff = 0,

∗ For allT ∈Ob(C) andt∈HomC(Y, T) such thattf= 0, then there exists uniques∈Hom_C(Cok(f), T) such that

t=sπ_f.

X Y Cok(f)

T

f πf

t s

(iv) Anabelian categoryis an additive categoryCsuch that for allX,Y ∈Ob(C) andf ∈Hom_C(X, Y), thenf has a kernel and a cokernel, and moreover,

Cok(ιf)'Ker(πf).

(v) LetC andDbe categories. A covariant (contravariant) functor F :C → D

consists of a map

F : Ob(C)→Ob(D) and, for allX,Y ∈Ob(C), a map

Hom_C(X, Y)→Hom_D(F(X), F(Y)) (Hom_C(X, Y)→Hom_D(F(Y), F(X))) such that the following statements hold:

– F(1X) = 1_F(X)for allX∈Ob(C),

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– IfX,Y,Z ∈Ob(C),f ∈Hom_C(X, Y) andg∈Hom_C(Y, Z), then F(gf) =F(g)F(f)

(F(gf) =F(f)F(g)).

(vi) LetC andDbe abelian categories, and let F be a covariant (contravariant) functor

F :C → D.

Moreover, let

0 A B C 0

be an exact sequence inC. (We assume that the notion of an exact sequence is familiar to the reader.) We say thatF is

– left exact if

0 F(A) F(B) F(C)

0 F(C) F(B) F(A)

is an exact sequence inD, – right exact if

F(A) F(B) F(C) 0

F(C) F(B) F(A) 0

is an exact sequence inD, – exact if

0 F(A) F(B) F(C) 0

0 F(C) F(B) F(A) 0

is an exact sequence inD.

(vi) Let C and D be categories, and let F and G be covariant (contravariant) functors fromC toD.

– Anatural transformation α:F →G consists of, for allX ∈Ob(C), a morphismαX∈Hom_D(F(X), G(X)), such that:

For anyX,Y ∈Ob(C) andf ∈Hom_C(X, Y) then F(X) G(X)

F(Y) G(Y) α_X

F(f) G(f) αY

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F(Y) G(Y)

F(X) G(X)











 αY

F(f) G(f) αX

commutes. We may denote the natural transformation by{αX}_X∈Ob(C) as well as byα, and we equivalently say thatαX is natural inX. – We say thatF and Garenaturally isomorphic if there exists a natural

transformation

α:F →G

such that, for allX ∈Ob(C), thenα_X is an isomorphism. In this case, we write

F '

nat.G.

We often writeX ∈ C in stead ofX ∈Ob(C) for an object X of a categoryC.

The following lemma states that inverses of and compositions of natural transformations, are in turn natural.

Lemma 2. Let C andD be categories.

(i) LetF and Gbe functors fromC toD, and let

{αX ∈HomD(F(X), G(X))}X∈C:F →G

be a natural transformation. Moreover, suppose that for anyX ∈ C there is ϕ_X ∈Hom_D(G(X), F(X))such that

αXϕX= 1_G(X) and

ϕXαX= 1_F(X). Then{ϕX}X∈C is a natural transformation G→F.

(ii) LetF,GandH be functors fromC toD. Let α:F →Gandβ :G→H be natural transformations. Thenβα:F →H defined by

(βα)_X:=β_X◦α_X is a natural transformation.

Proof.

(i) SupposeF andGare covariant (contravariant) functors. By the definition of a natural transformation, then{ϕX}_X∈C:G→Fis a natural transformation

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if and only if the following diagram commutes:

G(X) F(X)

G(Y) F(Y) ϕ_X

G(f) F(f) ϕ_Y

G(Y) F(Y)

G(X) F(X)











 ϕY

G(f) F(f) ϕX

(2.1) That is, we must show that

F(f)ϕX =ϕYG(f) (2.2)

(F(f)ϕ_Y =ϕ_XG(f)) (2.3)

for allX ∈ C. Since{αX}X∈C is a natural transformation, then G(f)αX =αYF(f)

(G(f)αX =αYF(f))

for allX ∈ C. Composing withϕY (ϕX) from the left andϕX (ϕY) from the right, we get

ϕYG(f)αXϕX=ϕYαYF(f)ϕX

(ϕ_XG(f)α_Yϕ_Y =ϕ_Xα_XF(f)ϕ_Y,)

hence (2.2) ((2.3)) holds.

(ii) We assume F, G and H are covariant functors. Let X, Y ∈ C and f ∈ Hom_C(X, Y). Then

(βα)YF(f) =βYαYF(f) =βYG(f)αX=H(f)βXαX=H(f)(βα)X, hence the following diagram is commutative:

F(X) F(Y)

G(X) G(Y)

H(X) H(Y) F(f)

G(f)

H(f) α_X

βX

α_Y

βY

(βα)X (βα)Y

Thusβαis a natural transformation. The proof is similar ifF,GandH are contravariant.

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We will now look at two important types of functors, namely the covariant and the contravariant hom functors.

Lemma 3. Let C be a category, and letX ∈ C.

(i) There is a covariant functor

Hom_C(X,−) :C →Set defined by

Hom_C(X,−)(Y) := Hom_C(X, Y) forY ∈ C, and for anyY,Z∈ C andf ∈Hom_C(Y, Z), then

Hom_C(X,−)(f) : Hom_C(X, Y)→Hom_C(X, Z) g7→f g.

(ii) There is a contravariant functor

Hom_C(−, X) :C →Set defined by

Hom_C(−, X)(Y) := Hom_C(Y, X) forY ∈ C, and for anyY,Z∈ C andf ∈Hom_C(Y, Z), then

Hom_C(−, X)(f) : Hom_C(Z, X)→Hom_C(Y, X) g7→gf.

Proof.

(i) It is evident that HomC(X, Y)∈Set, and that

Hom_C(X,−)(1Y) = [g7→1Yg=g] = 1_Hom_C_(X,−)(Y₎

for allY ∈ C. ForY,ZandW ∈ C,f1∈HomC(Y, Z) andf2∈HomC(Z, W), then

(Hom_C(X,−)(f2f1))(g) = (f2f1)g

=f₂(f₁g)

= (Hom_C(X,−)(f2))(f1g)

= Hom_C(X,−)(f₂) Hom_C(X,−)(f₁)(g) for allg∈Hom_C(X, Y), hence

Hom_C(X,−)(f₂f₁) = Hom_C(X,−)(f₂) Hom_C(X,−)(f₁).

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(ii) Similar to (i).

We hereby introduce a compact, yet informative way of writing the resulting morphism when applying a hom functor to a morphism.

Definition 4. Let C be a category, and let X ∈ C. We denote Hom_C(X,−)(f) by (f◦ −)_X,since it takes a morphism and composes it withf from the left hand side. Similarly, we denote Hom_C(X,−)(f) by (− ◦f)_X, since it takes a morphism and composes it with f from the right hand side.

We observe that the hom functors are in fact left exact. That is, in the envi- ronment where left exactness is defined, namely for for abelian categories.

Lemma 5. IfCis an abelian category andX ∈ C, thenHomC(X,−)andHomC(−, X) are left exact functors

C →Ab.

Proof. By the very definition of an abelian category, then Hom_C(X,−) and HomC(−, X) are functors

C →Ab. We show the left exactness of Hom_C(X,−). Let

0 A f B g C 0

be an exact sequence inC. We need to show that

0 Hom_C(X, A)(f◦ −)XHom_C(X, B)(g◦ −)XHom_C(X, C) is an exact sequence in Ab.

We first show that (f◦ −)X is a monomorphism. In Ab, this is the same as being injective. Supposeg∈HomC(X, A) such that

(f ◦ −)X(g) =f g= 0.

Then sincef is a monomorphism, it follows thatg= 0.

Forh∈Hom_C(X, B), then

(f◦ −)X(g◦ −)X(h) = (f◦ −)X(gh) =f gh= (f g◦ −)X(h), hence

(f ◦ −)X(g◦ −)X = (f g◦ −)X. That is,

Im((f◦ −)X)⊆Ker((g◦ −)X).

We finally show that

Ker((g◦ −)X)⊆Im((f◦ −)X.

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Leth∈Ker((g◦ −)X).Thengh= 0, and sincef is the kernel ofg, thenhfactors throughf. That is, there isj∈Hom_C(X, A) such that

h=f j= (f◦ −)_X(j), henceh∈Im((f◦ −)_X).

By similar arguments, Hom_C(−, X) is a left exact functor.

Definition 6. We let

(i) Set := the category of sets, where the morphisms are maps,

(ii) Ab := the category of abelian groups, where the morphisms are abelian group homomorphisms,

(iii) Mod(S) := the category of leftS-modules for a ringS, where the morphisms areS-module homomorphisms.

Note that the categories of (ii) and (iii) of the above definition are abelian categories. The abelian group structure on the hom sets originates from the abelian group structure on the objects themselves; in particular, from that on the target objects: For example, for X,Y ∈Ab andf,g∈Hom_C(X, Y), then

(f+g)(x) :=f(x) +g(x) for allx∈X.

SupposeC is an abelian category, and consider the following diagram inC:

0 Ker(f) A B Cok(f) 0

0 Ker(g) C D Cok(g) 0

ιf f πf

ιg g πg

u_Ker u v v_Cok

(2.4) It can be shown that there exist uniqueC-homomorphismsuKer∈HomC(Ker(f),Ker(g)) andvCok∈HomC(Cok(f),Cok(g)) such that the above diagram commutes.

Definition 7. The kernel map of u and the cokernel map of v (with respect to Diagram 2.4)are the C-homomorphisms u_Ker∈Hom_C(Ker(f),Ker(g)) andv_Cok∈ Hom_C(Cok(f),Cok(g)) making Diagram 2.4 commutative. We will stick to the subscripts Ker and Cok for kernel maps and cokernel maps throughout this thesis.

The fact thatu_Kerandv_Cokfrom Definition 7 are dependent on all of Diagram 2.4 and not only on uand v, respectively, suggests a more clarifying notation for these morphisms. However, whenever this notation is used it will be clear from the context which diagram the kernel or cokernel morphism originates from, and we will omit specifying this explicitly.

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We now turn our attention to module categories. For a ring S then the left S^op-modules are the rightS-modules. Throughout this thesis we will be referring to the left S-modules as merely S-modules, and to the right S-modules as S^op- modules. Also, for a homomorphism set Hom_Mod(S)(A, B) whereA,B∈Mod(S), we will instead write HomS(A, B).

The hom functors conveniently commute with direct sums, as stated by Lemma 8.

Lemma 8. Let ⊕ⁿ_i=1Xi be a direct sum in Mod(S) for some ring S, with given inclusions

νi:Xi→ ⊕ⁿ_i=1Xi

and projections

ρi :⊕ⁿ_i=1Xi→Xi

for1≤i≤n. Then the following holds for anyY ∈Mod(S).

(i) There is an isomorphism of sets

ξ: HomS(⊕ⁿ_i=1Xi, Y)→ ⊕ⁿ_i=1HomS(Xi, Y) f 7→ {f ν_i}ⁿ_i=1,

whose inverse is given by

ξ⁻¹:⊕ⁿ_i=1HomS(Xi, Y)→HomS(⊕ⁿ_i=1Xi, Y) {fi}ⁿ_i=17→

n

X

i=1

fiρi.

(ii) If either Y =S or Xi = S for 1 ≤i ≤n, then the homomorphism sets in question areS^op-modules, andξ andξ⁻¹ are isomorphisms ofS^op-modules.

Proof.

(i) Supposef ∈HomS(⊕ⁿ_i=1Xi, Y). Then ξ⁻¹ξ(f) =ξ⁻¹({f νi}ⁿ_i=1) =

n

X

i=1

f νiρi=f

n

X

i=1

νiρi

| {z }

=1_⊕n i=1Xi

=f,

hence

ξ⁻¹ξ= 1_Hom_S_(⊕n i=1X_i,Y). Suppose{f_i}ⁿ_i=1∈ ⊕ⁿ_i=1Hom_S(X_i, Y). Then

ξξ⁻¹({fi}ⁿ_i=1) =ξ

n

X

i=1

fiρi

!

= ( _n

X

i=1

fiρi

! νj

)ⁿ

j=1

= ( _n

X

i=1

fi(ρiνj) )ⁿ

j=1

.

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Note thatρ_iν_j = 1_X_j fori=j and 0 otherwise, hence

n

X

i=1

fi(ρiνj) =fj, implying that

ξξ⁻¹= 1_⊕ⁿ

i=1HomS(Xi,Y). (ii) We leave this as an exercise.

We can identify a module over a ring S with the set of S-module homomorphisms from S to the module, because of the following result.

Lemma 9. Let S be a ring, and let M ∈Mod(S). Then

(i) Hom_S(S, M)is an S-module with the following multiplication S×HomS(S, M)→HomS(S, M) : Fors∈S andf ∈HomS(S, M), then

(s·f)(a) :=f(as) fora∈S.

(ii) The map

ξM : HomS(S, M)→M f 7→f(1_S) is an isomorphism ofS-modules.

(iii) In the case thatM =S, then

ξ_S : End_S(S)→S^op is a ring isomorphism.

Proof.

(i) Fors∈S, we must check thatsf ∈Hom_S(S, M). For s⁰,a,a⁰∈S, then (sf)(s⁰a+a⁰) =f((s⁰a+a⁰)s)

=f(s⁰(as)) +f(a⁰s)

=s⁰f(as) +f(a⁰s)

=s⁰(sf)(a) + (sf)(a⁰).

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Lets, s⁰∈S andf,f⁰∈Hom_S(S, M). It is obvious that (s+s⁰)f =sf+s⁰f

and

s(f+f⁰) =sf+sf⁰. For anya∈S, then

((ss⁰)f)(a) =f(a(ss⁰)) =f((as)s⁰) = (s⁰f)(as) = (s(s⁰f))(a), hence

(ss⁰)f =s(s⁰f).

(ii) We first show that ξ_M is an S-module homomorphism. Let s ∈ S and f, f⁰∈Hom_S(S, M). Then

ξ_M(sf+g) = (sf+g)(1_S) =sf(1_S) +g(1_S) =sξ_M(f) +ξ_M(g).

We now show that that ξM is bijective. If ξM(f) = 0, then f(1S) = 0, implying that

f(s) =sf(1S) = 0 for alls∈S, hencef = 0. Thenξ_M is injective.

For anym∈M, we leave it up to the reader to check that fm(s) :=sm

fors∈S defines anS-module homomorphismfm∈HomS(S, M). Then ξM(fm) =fm(1S) =m.

ThusξM is surjective.

(iii) Letf₁, f₂∈End(S).Then

ξ_S(f₁f₂) = (f₁f₂)(1_S)

=f1(f2(1S))

=f1(f2(1S)1S)

=f2(1S)f1(1S)

=ξS(f2)ξS(f1),

henceξ_S is a ring isomorphism.

We finally make a useful observation regarding exactness of sequences in Mod(S) (for a ringS):

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Lemma 10. LetS be a ring, and let

0 A B C

B⁰ C⁰ D⁰ 0

f g

α '

u v

β '

be a commutative diagram of exact rows inMod(S). Then

0 A f B g C vβ D⁰ 0

is an exact sequence ofS-modules.

Proof. The exactness inAandB follows from the exactness of the given diagram.

Sincev and β are epimorphisms then so is the compositionvβ, so the sequence is exact in D⁰. Since (vβ)g = (vu)α= 0, then Im(g)⊆Ker(vβ). We need to show that Ker(vβ)⊆Im(g).

Letc ∈Ker(vβ), that is, vβ(c) = 0. Then β(c)∈Ker(v) = Im(u), so there is b⁰∈B⁰ such that

u(b⁰) =β(c).

Thenα⁻¹(b⁰)∈B such that

βgα⁻¹(b⁰) =u(b⁰) =β(c).

Sinceβ is a monomorphism, this implies that g(α⁻¹(b⁰)) =c, hencec∈Im(g).

The reader should be familiar with the notions of projective and injective modules as well as the material traversed in this section. For a definition of these concepts along with some basic results, we refer to [1, Ch. 5]. Note especially the relation between projective and injective modules and exactness of hom functors of Proposition 16.9.

2.2 Our framework

In this thesis we will be studying modules over a fixed R-algebra Λ,¹ where R is a given ring. A lot of the results will depend on certain properties exhibited by R and Λ, which we will dedicate this section to be familiarized with. First of all, we shall assume that Ris commutative. We now give the definition of the second condition which we will have onR.

Definition 11. A ringRislocal if it has a unique maximal ideal.

1AnR-algebra will be defined formally in Definition 16(i).

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It is well-known that a ringRis local if and only if its non-units form an ideal.

This equivalent definition is used in [2, Ch. 1].

Note that any factor of a local ring is in turn local.

Lemma 12. If Ris a local ring thenR/J is a local ring for any proper idealJ in R.

Proof. LetR be a local ring, and supposeJ is a proper ideal in R. Let mdenote the maximal ideal inR. Thenm/J is an ideal inR/J, and we claim that it is the unique maximal ideal.

SupposeY is a non-trivial, proper ideal inR/J. ThenY is of the form Y =X/J,

where X is an ideal inRsuch that

J (X (R.

Sincemis the maximal ideal inR then X ⊆m, hence

Y =X/J ⊆m/J.

The last condition on our ringRwill be that is is an artinian ring. This property is defined below along with the similar concept of noetherianness.

Definition 13.

(i) A left (right) artinian ringRis a ring such that any descending chain of left (right) ideals stabilizes. That is, given a descending chain

R=I0⊇I1⊇I2⊇...

of left (right) ideals inR, then there isN ∈Nsuch that In =Imfor all m, n≥N.

(ii) A left (right) noetherian ring R is a ring such that any ascending chain of left (right) ideals stabilizes. That is, given an ascending chain

0 =I0⊆I1⊆I2⊆...

of left (right) ideals inR, then there isN ∈Nsuch that In =Imfor all m, n≥N.

The following definition is analogous to Definition 13, but forR-modules.

Definition 14.

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(i) AnartinianR-moduleM is anR-moduleM such that any descending chain M =M₀⊇M₁⊇M₂⊇...

ofR-sumbodules ofM stabilizes. That is, there isN ∈Nsuch thatMn=Mm

for allm, n≥N.

(ii) AnoetherianR-moduleM is anR-moduleM such that any ascending chain 0 =M0⊆M1⊆M2⊆...

ofR-sumbodules ofM stabilizes. That is, there isN ∈Nsuch thatM_n=M_m for allm, n≥N.

The following lemma is useful in situations where it is desirable to derive artinianness or noetherianness for anR-module.

Lemma 15. Let

0 A f B g C 0

be an exact sequence ofR-modules. Then the two following statements are equivalent.

(i) B is an artinian (noetherian)R-module.

(ii) AandC are artinian (noetherian) R-modules.

Proof. We will prove the lemma for artinianR-modules. The proof in the case of noetherian R-modules is similar.

(i)⇒(ii) : SupposeB is an artinian R-module.

We first show thatCis an artinian R-module. Let

C=C0⊇C1⊇C2⊇... (2.5)

be a descending chain ofR-submodules of C. We need to show that (2.5) stabilizes. Note that (2.5) induces a descending chain

B=g⁻¹(C0)⊇g⁻¹(C1)⊇g⁻¹(C2)⊇... (2.6) ofR-submodules ofB. We claim that ifg⁻¹(Cn) =g⁻¹(Cm), thenCn=Cm. Assumeg⁻¹(Cn) =g⁻¹(Cm), and letc∈Cn. Sincegis ontoC, there isb∈B such that g(b) = c. Henceb ∈ g⁻¹(Cn), then by hypothesis b ∈ g⁻¹(Cm).

This means thatg(b)∈Cm. ThusCn⊆Cm. By symmetry, we conclude that C_n =C_m.

SinceBis an artinianR-module, we know that (2.6) stabilizes. That is, there isN ∈Nsuch thatg⁻¹(Cn) =g⁻¹(Cm) for allm,n≥N. Then by the above result,Cn=Cmfor allm,n≥N, thus (2.5) stabilizes.

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We now show thatAis an artinianR-module. Let

A=A0⊇A1⊇A2⊇... (2.7)

be a descending chain of R-submodules of A. We need to show that (2.7) stabilizes. Note that (2.7) induces a descending chain

B=f(A0)⊇f(A1)⊇f(A2)⊇... (2.8) of R-submodules of B. We claim that iff(A_n) = f(A_m), then A_n =A_m. Assume f(An) = f(Am), and let a ∈ An. Then f(a) ∈ f(An), thus by hypothesis, f(a) ∈f(Am). Then there is a⁰ ∈Am such that f(a⁰) =f(a).

Since f is injective, then a =a⁰, thus a ∈ Am. Then we have shown that An⊆Am, hence, by symmetry, An=Am.

Again, sinceB is an artinianR-module, we know that (2.8) stabilizes. That is, there is M ∈ N such that f(A_n) = f(A_m), and thus A_n = A_m, for all m,n≥M. Hence (2.7) stabilizes. ThenA andC are proven to be artinian R-modules.

(ii)⇒(i): SupposeAand Care artinian R-modules. Let

B=B₀⊇B₁⊇B₂⊇... (2.9)

be a descending chain of R-submodules of B. We need to show that (2.9) stabilizes. As in the first part of the proof, we consider the induced descending chains

C=g(B₀)⊇g(B₁)⊇g(B₂)⊇... (2.10) A=f⁻¹(B0)⊇f⁻¹(B0)⊇f⁻¹(B0)⊇ (2.11) ofR-submodules ofAandC, respectively. SinceAandC are artinian, then (2.10) and (2.11) stabilize; there is N ∈ N such that g(Bn) = g(Bm) and f⁻¹(Bn) =f⁻¹(Bm) for allm, n≥N. We claim that Bn =Bm for all m, n≥N.

Consider some fixedm, n≥N, and supposeBm6=Bn. Since theBi’s arise from a descending chain, it is clear that one of these submodules is contained in the other. Without loss of generality we assume Bn ⊆ Bm. Suppose b∈Bm\Bn. Then

g(b)∈g(Bm) =g(Bn), thus there isb⁰ ∈B_n such that

g(b⁰) =g(b).

Then

b−b⁰ ∈Ker(g) = Im(f), (2.12) Moreover,b⁰∈Bn⊆Bm, so

b−b⁰∈Bm. (2.13)

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By (2.12) there isa∈Asuch thatf(a) =b−b⁰, and by (2.13), a∈f⁻¹(B_m) =f⁻¹(B_n).

Thenf(a) =b−b⁰∈Bn, so

b= (b−b⁰) +b⁰∈Bn,

contradicting the assumption. We conclude thatBm=Bn for allm,n≥N, thus (2.9) stabilizes. This completes the proof;B is proven to be an artinian R-module.

A special case of an artinian R-module is an artin R-algebra. Consider the following definition.

Definition 16.

(i) AnR-algebraΛ is a ring which is also anR-module, such that the following holds: For allα,β,λ∈Λ and r,s∈R, then

(rα+sβ)λ=r(αλ) +s(βλ) and

α(rβ+sλ) =r(αβ) +s(αλ).

(ii) AnartinR-algebra is anR-algebra which is finitely generated asR-module.

Informally, the length of an S-module M (over some ring S) is obtained by considering all ways of writing descending chains of S-submodules ofM where all containments are proper, and taking the length of the longest such. For a more technical definition, see [2, Ch. 1].

It can be shown that a module over an artinR-algebra is finitely generated if and only if it has finite length, and this again occurs if and only if the module is noetherian. We shall implicitly use this result throughout this thesis, as we from now on let Λ denote a fixed artin R-algebra. Note that Λôp is then also an artin R-algebra, hence a result involving Λ and Λôp is generally still valid if these two rings are interchanged. Particularly, a functor (with no parameters from Λ or Λôp)

mod(Λ)→mod(Λ^op) is also a functor

mod(Λ^op)→mod(Λ).

Definition 17. LetS be a ring.

(i) For anyM ∈Mod(S), we letlS(M) denote the length ofM as S-module.

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(ii) We let mod(S) be the full subcategory of Mod(S) whose objects are the S- modules of finite length;

Ob(mod(S)) :={M ∈Ob(Mod(S))|lS(M)<∞}.

Note that mod(S) is also an abelian category, because the lengths of the kernel and the cokernel of a morphism in mod(S) must also be of finite length. Whenever we apply a hom functor in this thesis, it will be from one of the abelian categories which we have seen, hence, by Lemma 5, it will always be left exact.

2.2.1 Some useful results

In Lemma 19 we will see that the artin R-algebra Λ is also an artinian ring, but first we need to establish some relations between modules and homomorphisms of R and Λ in our current framework.

The next lemma is considered basic knowledge, and will occasionally be applied without reference. Note that for any ringS, then the identity map

1_S :S→S

in Hom_S(S, S) can be regarded as multiplication by the identity element ofSitself, so we will use the same notation 1_S for the identity element ofS.

Lemma 18. The following statements are true:

(i) Mod(Λ)⊆Mod(R),

(ii) HomΛ(A, B)⊆HomR(A, B)for allA,B ∈Mod(Λ), (iii) mod(Λ)⊆mod(R).

Proof.

(i) Let M ∈ Mod(Λ). Then M is an R-module under the following binary operation:

R×M →M (r, m)7→(r·1_Λ)

| {z }

∈Λ

m. (2.14)

(ii) LetA,B∈Mod(Λ), and supposef ∈HomΛ(A, B). Then fora1,a2∈Aand λ∈Λ,

f(λa1+a2) =λf(a1) +f(a2).

Thus for allr∈R, we see that f(ra1+a2)^(2.14)= f((r1Λ)

| {z }

∈Λ

a1+a2) = (r1Λ)f(a1) +f(a2)^(2.14)= rf(a1) +f(a2).

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(iii) IfM ∈mod(Λ), then for somen∈Nthere exists a Λ-module epimorphism g: Λⁿ→A.

By (ii),g is also anR-module epimorphism.

Recall that Λ is an artin algebra, that is, Λ is finitely generated as an R- module. Then for somem∈Nthere exists anR-module epimorphism

h:R^m→Λ, and thus there is anR-module epimorphism hⁿ:R^mn→Λⁿ.

By composinggwithhⁿ we get anR-module homomorphism ghⁿ :R^mn→A,

thusAis finitely generated as anR-module.

Note that sinceR is a commutative ring, Lemma 18(i) implies that any Λ^op- module is an R-Λ-bimodule. With Lemma 15 and Lemma 18(i) at hand, we are ready to show that Λ is an artinian ring.

Lemma 19. The artinR-algebraΛ is an artinian ring.

Proof. Note that an R-submodule of R is the same as an ideal in R, so since R is an artinian ring then R is also an artinian R-module. We will now proceed as follows.

I) We first show thatRⁿ is an artinian R-module.

II) We use I) to show that Λ is an artinianR-module.

III) Finally we show that II) implies that Λ is an artinian ring.

I) We show that Rⁿ is an artinianR-module by induction onn. Suppose the statement holds forn=k−1. Consider the R-module epimorphism

gk:R^k→R^k−1 {ri}^k_i=17→ {ri}^k_i=2. We see that Ker(gk) =R, thus

0 R R^k R^k−1 0

gk

is an exact sequence of R-modules. By Lemma 15, since R and R^k−1 are artinianR-modules, then so isR^k.

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II) Since Λ is a finitely generatedR-algebra, we know that Λ can be written as Λ =Rλ1+Rλ2+...+Rλn (2.15) for some finite subset{λ_i}ⁿ_i=1⊆Λ. We claim that the maps

fλ_i :R→Λ r7→rλi.

fori∈ {1, ..., n}areR-module homomorphisms. Fors,r₁,r₂∈Rthen fλ_i(sr1+r2) := (sr1+r2)λi=sr1λi+r2λi=sfλ_i(r1) +fλ_i(r2).

Then

[f_λ₁, f_λ₂, ..., f_λ_n] :Rⁿ →Λ

is also an R-module homomorphism, and by (2.15) it is onto Λ. Again, by Lemma 15, then Λ is an artinianR-module.

III) Let

Λ =I0⊇I1⊇I2⊇...

be a descending chain of ideals in Λ. That is,Ii∈Mod(Λ) for alli≥0. Then by Lemma 18(i), allIi’s areR-modules, and since Λ is an artinianR-module, the chain must stabilize.

We often wish to show that some functorF goes from mod(S) to mod(S⁰) for ringsS andS⁰. Then in addition to assigningS-module structure to the codomain ofF, we need to show that the resultingS-module is of finite length. For this, the following two lemmas are useful.

Lemma 20. Let A, B ∈ mod(S) for a ring S, and suppose f ∈ HomS(A, B).

ThenCok(f)∈mod(S).

Proof. For somen∈N, there is an S-epimorphism

Sⁿ B 0.

Then by composing with the canonical projection fromB onto Cok(f), we get an S-epimorphism Rⁿ Cok(f) 0.

If a Λ-module is finitely generated asR-module, is is also finitely generated as Λ-module:

Lemma 21. IfM ∈Mod(Λ)∩mod(R), thenM ∈mod(Λ).

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Proof. Recall that

rm:= (r1Λ)m defines theR-module structure onM. Suppose

{m₁, m₂, ..., m_n} ⊆M

generatesM asR-module. Then anym∈M can be written as m=

n

X

i=1

rimi=

n

X

i=1

(ri1Λ)

| {z }

∈Λ

mi⊆

n

X

i=1

Λmi,

implying that M is finitely generated as Λ-module.

In addition to being abelian groups, the homomorphism sets (in our current situation) admit miscellaneous module structures depending on their respective source and target objects. Here we will give a few examples. It might seem redun- dant to include both (i) and (ii) of the following lemma, but we find it useful in order to get a better grasp on how these structures interact.

Lemma 22.

(i) LetM ∈Mod(Λ), and letN ∈Mod(R). ThenHomR(M, N)is aΛ^op-module with multiplication

HomR(M, N)×Λ→HomR(M, N) defined by

(hλ)(m) :=h(λm) (2.16)

for allm∈M.

(ii) LetM ∈Mod(Λ^op), and letN ∈Mod(R). ThenHom_R(M, N)is aΛ-module with multiplication

Λ×Hom_R(M, N)→Hom_R(M, N) defined by

(λh)(m) :=h(mλ) (2.17)

for allm∈M.

(iii) LetM,N ∈Mod(Λ). ThenHomΛ(M, N)is anR-module with multiplication R×HomΛ(M, N)→HomΛ(M, N)

defined by

(rh)(m) :=h(rm) =r(h(m)) (2.18) for allm∈M.

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(iv) LetM,N∈Mod(R). ThenHom_R(M, N)is anR-module with multiplication R×HomR(M, N)→HomR(M, N)

defined by

(rh)(m) :=h(rm) =r(h(m)) (2.19) for allm∈M.

(v) LetM,N ∈Mod(Λ). ThenHom_Λ(M, N)is anR-submodule ofHom_R(M, N).

Proof.

(i) We first show that hλ∈ HomR(M, I) for any h∈HomR(M, I) andλ∈ Λ.

Letr∈Randm1,m2∈M. Then

(hλ)(rm₁+m₂) =h(λ(rm₁+m₂))

=h(λ(rm1) +λm2)

=h(r(λm₁) +λm₂)

=rh(λm1) +h(λm2)

=r(hλ)(m1) + (hλ)(m2).

We now show that

h(λ₁λ₂) = (hλ₁)λ₂

for allh∈HomR(M, I) andλ1,λ2∈Λ. For anym∈M, then (h(λ₁λ₂))(m) =h((λ₁λ₂)m)

=h(λ1(λ2m))

= (hλ₁)(λ₂m)

= ((hλ1)λ2)(m).

We leave it up to the reader to check that the distributive laws hold for this action of Λ on HomR(M, I).

(ii) Similar to (i). Besides, if M ∈Mod(Λ^op) then we can interpret M as a left Λ^op-module as well as a right Λ-module, and apply (i). Then if * denotes the multiplication

Λ^op×M →M andh∈HomR(M, N),m∈M andλ∈Λ, we have

(λh)(m) = (h∗λ)(m)⁽ⁱ⁾=h(λ∗m) =h(mλ).

(iii) Let r∈R andh∈Hom_Λ(M, N). By Lemma 18(ii) thenh∈Hom_R(M, N), so

h(r·m) =r·h(m)

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for allm∈M. For anys∈Randm∈M, then

(rh)(sm) =r(h(sm)) =r(sh(m)) =s(rh(m)) =s(rh)(m),

so rh ∈ HomR(M, N). It is evident that 1R·h = h. For associativity of multiplication of HomR(M, N) with R, we recall that R is a commutative ring, and we leave it up to the reader to verify that the following distributive laws are satisfied:

– (r₁+r₂)h=r₁h+r₂hfor allr₁,r₂∈R andh∈Hom_R(M, N), – r(h1+h2) =rh1+rh2for allr∈R andh1,h2∈HomR(M, N), – rh∈Hom(M, N) for all r∈Randh∈HomR(M, N).

(iv) Similar to (iii).

(v) We have from (iii) and (iv) that Hom_Λ(M, N) and Hom_R(M, N) are both R-modules with the same multiplicative structure, and by Lemma 18(ii), Hom_Λ(M, N)⊆Hom_R(M, N).

We have seen that given an abelian category C and an object X ∈ C, then Hom_C(X,−) and Hom_C(−, X) are functors

C →Ab.

In light of Lemma 22, it is interesting to ask whether HomC(X,−) and Hom_C(−, X) can be regarded as functors to module categories if we let their domainCbe some module category. The answer is yes, there are multiple examples of where this occurs. Unfortunately it is too tedious to go through all of them, but demonstrations for a few cases will be carried out in the proofs of Proposition 29 and Proposition 40.

There are a series of result which are valid for finitely generated modules. It is therefore of significant whether a hom set with some module structure is finitely generated as such.

Lemma 23. LetA,B∈mod(R). Then (i) HomR(A, B)∈mod(R).

(ii) HomΛ(A, B)∈mod(R).

Proof.

(i) For somen∈N, there is anR-module homomorphism fromRⁿontoA, giving rise to the following exact sequence ofR-modules:

Rⁿ A 0

(2.20)

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We apply Hom_R(−, B) to (2.20). It follows from Lemma 22(iv) that the following exact sequence is ofR-modules:

0 HomR(A, B) HomR(Rⁿ, B)

(2.21) LetS=R,X_i=Rfori∈ {1, ..., n}, andY =B. Then sinceRis commutative, by Lemma 8 there is an isomorphism ofR-modules

HomR(R, B)ⁿ→HomR(Rⁿ, B).

This result together with Lemma 9(ii) now implies that HomR(Rⁿ, B)'Bⁿ as R-modules, and sinceB is finitely generated as anR-module then so is Bⁿ. Thus Bⁿ is noetherian, and by applying Lemma 15 to (2.21) we see that Hom(A, B) is a noetherian R-module. By [1, Proposition 10.9, Ch. 3], all submodules of noetherianR-modules are finitely generated asR-modules, thus Hom_R(A, B)⊆mod(R).

(ii) By Lemma 22(v), HomΛ(A, B) is anR-submodule of HomR(A, B), thus since HomR(A, B) is a noetherianR-module, then HomΛ(A, B) is a finitely gener- atedR-module.

The previous result has the following convenient consequence:

Proposition 24. LetX ∈mod(Λ). Then (i) EndΛ(X)is an artin R-algebra.

(ii) IfI∈End_Λ(X) is an ideal, thenEnd_Λ(X)/I is an artin algebra.

Proof.

(i) Lemma 18 implies thatX ∈mod(R), and then, by Lemma 23(ii), EndΛ(X) is a finitely generatedR-module. Moreover, composition of morphisms defines anR-algebra structure on EndΛ(X); it is easy to see that the map

EndΛ(X)×EndΛ(X)→EndΛ(X) (f, g)7→f◦g isR-bilinear. Thus EndΛ(X) is an artinR-algebra.

(ii) Since End_Λ(X) is an artin algebra, then by definition, for somen∈Nthere is anR-module epimorphism

Rⁿ →End_Λ(X).

By composing thisR-epimorphism with the canonical projection from EndΛ(X) onto EndΛ(X)/I, we see that EndΛ(X)/I is also an artinR-algebra.

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In Chapter 3 we will fix an element X ∈mod(Λ), and then we will construct some isomorphisms with the usage of diagrams in mod(R) obtained from applying functors – studied in this section – to objects in mod(Λ), mod(Λôp) and mod(R), respectively. As it turns out, the R-modules of greatest interest (namely the do- mains and codomains of the mentioned isomorphisms) are also endoved with either Γ-module or Γôp-module structure for a factor Γ of EndΛ(X), and, more impor- tantly, these structures are preserved by the isomorphisms. That is, we will be constructing isomorphisms of Γ-modules and of Γôp-modules. The artinianness of Γ and Γôp will then be a great advantage, since any result which is derived for Λ in this section is evidently also valid for Γ and Γôp. ²

flytte? Moreover, we will see in Section 2.10 that with certain conditions on X ∈mod(Λ), then Γ (and similarly Γ^op) is a local ring whose factor modulo its radical is a simple Γ-module, a property which will be of great importance for our work in Section 3.2.2.

2.3 The dual

In this section we will study an important exact hom functor, obtained from a special injectiveR-module. Consider the following definition.

Definition 25. LetS be a ring.

(i) Let M, N ∈Mod(S), and suppose M ⊆N. We say that N is an essential extension ofM if

X∩N 6= 0 for all submodulesX ⊆M.

(ii) An injective envelopeI of N ∈Mod(S) is an injective S-moduleI together with a monomorphismιofS-modules

ι:M →I, whereI is an essential extension of Im(ι).

For a ringS it can be shown that there exists an injective envelope, unique up to isomorphism in Mod(S), of anyS-module. Since R is a commutative ring, its localness indicates the existence of a unique maximal ideal, from now denoted by m. We letI be the injective envelope of the simpleR-module given byK:=R/m.

Definition 26. Thedual is the hom functor D:= HomR(−, I).

2Our only assumption on Λ is that it is an artinR-algebra.

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Note that the above definition is a little imprecise since we have not specified the domain and codomain of the dual D. This is because we wish to vary these categories. It turns out that the dualD has an interesting property when regarded as a functor between certain module categories, namely that of being a duality.

Definition 27. LetCand Dbe categories. A contravariant functor F :C → D

is called aduality if there exists a functor G:D → C such that

GF '

nat.1C

and

F G '

nat.1D.

We shall see that forF :=D and for appropriate choices for C andD, thenD is a duality withG=D. In order to prove this we will need the following lemma.

Lemma 28. There is an isomprphism ofR-modules Hom_K(K, I)'K.

Proof. Letν denote the inclusion ofK into its injective envelopeI, and consider the exact sequence ofR-modules given by

0 K ν I µ X 0,

where µ is the cokernel of ν. It can be shown that HomK(K,−) is a (left exact) functor from Mod(R) to Mod(R), yielding the following exact sequence of R-modules.

0 Hom_R(K, K)(ν◦ −)KHom_R(K, I)(µ◦ −)KHom_R(K, X).

We claim that (µ◦ −)I = 0. Letf ∈HomR(K, I). If f = 0, then (µ◦ −)I(f) =µf is obviously 0. Assumef 6= 0. Then since

Ker(f)⊆K

and K is a simple R-module, then Ker(f) = 0, that is, f is injective. Since Im(f)⊆Iis a nonzero submodule, andI is an essential extension of Im(ν), then

Im(f)∩Im(ν)6= 0.

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Furthermore, Im(f) and Im(ν) are both simple since they are isomorphic to K, and since Im(f)∩Im(ν) is a nonzero submodule of them both, then

Im(f) = Im(f)∩Im(ν) = Im(ν).

Then for anyx∈K, we have

f(x) =ν(x⁰) for some x⁰∈K, hence

µf(x) =µν(x⁰) = 0.

Thus the composition

µf = 0

for anyf ∈HomR(K, I), that is, (µ◦ −)I = 0, implying that (ν◦ −)I : HomR(K, K)→HomR(K, I) is an isomorphism ofR-modules.

We now show that

HomR(K, K)'K

as R-modules. Consider the canonicalR-module epimorphism

R K 0.

We apply the left exact functor HomR(−, K), and get the exact sequence

0 HomR(K, K) HomR(R, K)

ofR-modules. By Lemma 9(ii), then

HomR(R, K)'K.

Since K is simple and Hom_R(K, K) 6= 0, then the image of the inclusion of Hom_R(K, K) into Hom_R(R, K) must be all of Hom_R(R, K), hence

Hom_R(K, K)'K as claimed. This completes the proof.

We now have all the results required in order to prove the following convenient results for the dual D.

Proposition 29.

(i) We can regardD as an exact, contravariant functor (a) D: mod(R)→mod(R),

(b) D: mod(Λ)→mod(Λ^op).

(ii) For anyM ∈mod(R)then

l_R(DM) =l_R(M).

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(iii) The functorD is a duality in either of these three cases.

Proof. We know that D is a contravariant, left exact functor to the category of abelian groups in either of these three cases, and because I is injective then D is exact. What needs to be proven, is that D is also a functor to the claimed codomains.

(i) (a) Let M ∈mod(R). Then by Lemma 23(i), Hom_R(M, I)∈mod(R). We must check that, for anyM,M⁰∈mod(R) andh∈Hom_R(M, M⁰), then Dh∈Hom_R(DM⁰, DM). Supposef⁰∈DM⁰ andr∈R. Then

Dh(rf⁰)(m) = (− ◦h)I(rf⁰)(m)

= (rf⁰h)(m⁰)

(2.19)

= r(f⁰h)(m⁰)

=r(− ◦h)I(f⁰)(m)

=rDh(f⁰)(m) for allm∈M, hence

Dh(rf⁰) =rDh(f⁰).

We leave it up to the reader to check that

Dh(f₁⁰+f₂⁰) =Dh(f₁⁰) +Dh(f₂⁰) (2.22) for allf₁⁰,f₂⁰ ∈DM⁰.

(b) Suppose M ∈ mod(Λ). In Lemma 22(i) we saw that HomR(M, I) ∈ Mod(Λ^op). Also, HomR(M, I) is finitely generated as R-module by Lemma 23(i), hence by Lemma 21 it follows that Hom_R(M, I)∈mod(Λ^op).

We must also show that D takes Λ-module homomorphisms to Λ^op- module homomorphisms. LetM,M⁰∈mod(Λ) andh∈HomΛ(M, M⁰).

Then forf⁰ ∈DM⁰ andλ∈Λ, we have

Dh(f⁰λ)(m) = (− ◦h)_I(f⁰λ)(m)

= (f⁰λ)(h(m)

| {z }

∈M⁰

)

=f⁰(λh(m))

=f⁰(h(λm))

= (− ◦h)_I(f⁰)(λm)

= Dh(f⁰)

| {z }

∈HomR(M,I)

(λm)

= (Dh(f⁰)λ)(m) for allm∈M, hence

Dh(f⁰λ) =Dh(f⁰)λ.

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By Lemma 18 then 2.22 also holds forf₁⁰, f₂⁰ ∈ DM⁰ in this case, and Dhis thus a Λ^op-module homomorphism.

(ii) LetM ∈mod(R). We will show that

lR(DM) =lR(M) (2.23)

by induction onl_R(M).

SupposelR(M) = 1. Then M is a simple R-module, and since R is a local ring, thenR has a unique maximal ideal. This implies thatR has only one simple module, thus

M 'K asR-modules. Then by Lemma (28), we see that

HomR(M, I)'M asR-modules. It follows that

lR(DM) =lR(HomR(M, I)) =lR(M) = 1.

Now suppose (2.23) holds for allR-modulesM of lengthn−1 for somen∈N, and letM⁰ be an R-module of length n. Then M⁰ has a submodule M of lengthn−1, and the cokernel of the inclusion of M into M⁰ is simple and thus isomorphic toK. Hence

0 M M⁰ K 0

is an exact sequence R-modules. By applying D, which was shown in to (i)(a) to be an exact functor mod(R)→mod(R), we get the following exact sequence ofR-modules:

0 DK DM⁰ DM 0.

Then by [2, Proposition 1.3, Ch. 1], we have that l_R(DM⁰) =l_R(DK)

| {z }

=1

+l_R(DM)

| {z }

=n−1

=n.

Thus (2.23) holds whenlR(M) =nfor alln∈N. (iii) (a) We claim thatα: 1_mod(R)→D² defined by

α_M :M →Hom_R(Hom_R(M, I), I) m7→[f 7→f(m)]

forM ∈mod(R) is a natural transformation of functors. Given a fixed M ∈ mod(R), it is clear that αM(m) ∈HomR(HomR(M, I), I) for all

(37)

m∈M: Supposef₁, f₂∈Hom_R(M, I) andr∈R, then α_M(m)(rf₁+f₂) = (rf₁+f₂)(m)

=rf1(m) +f2(m)

=rαM(f)(f1) +αM(m)(f2).

Moreover, form1,m2∈M andr∈R, then αM(rm1+m2)(f) =f(rm1+m2)

=rf(m₁) +f(m₂)

=rαM(m)(f1) +αM(m2)(f)

= (rα_M(m₁) +α_M(m₂))(f) for allf ∈HomR(M, I), thus

αM(rm1+m2)(f) =rαM(m1) +αM(m2), andαM is anR-module homomorphism.

We now show thatαis a natural transformation. LetM,M⁰∈mod(R), and leth∈HomR(M, M⁰). Then

D²(h) = HomR(−, I)(HomR(−, I)(h))

= Hom_R(−, I)((− ◦h)_I)

= (− ◦(− ◦h)I)I. We must show that

M HomR(HomR(M, I), I)

M⁰ Hom_R(Hom_R(M⁰, I), I) α_M

αM⁰

h (− ◦(− ◦h)I)I

is a commutative diagram. Supposem∈M. Consider Diagram 2.24.

m [f 7→f(m)]

h(m) [f⁰7→f⁰h(m)]

?

(2.24) We must show that

(− ◦(− ◦h)I)I([f 7→f(m)]) = [f⁰7→f⁰h(m)].

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Forf⁰ ∈Hom_R(Hom_R(M⁰, I), I), then

(− ◦(− ◦h)I)I([f 7→f(m)])(f⁰) = [f 7→f(m)]◦(− ◦h)I(f⁰)

= [f 7→f(m)](f⁰h)

=f⁰h(m).

Thusαis a natural transformation.

Finally, we show thatαM is an isomorphism ofR-modules for anyM ∈ mod(R).

We begin by demonstrating the injectiveness ofα_M. LetM ∈mod(R), and supposem∈M is a nonzero element. We must show thatα_M(m)6=

0, that is, that

αM(m)(f) =f(m)6= 0 for somef ∈HomR(M, I). Consider the map

fˆ:Rm→I rm7→r+m.

Note thatr+m∈R/m=K ⊆I. We show that ˆf is well-defined: If rm=r⁰m, then (r−r⁰)m= 0,and sincemis nonzero this means that r−r⁰ is a non-unit inR. Recall thatm is the ideal inR generated by all the non-units. Hencer−r⁰ ∈m, implying that r+m=r⁰+m. It is easy to see that ˆf is an R-module homomorphism. Moreover, since Rm⊆M is a submodule, then the inclusion

ι:Rm→M

is an R-module monomorphism. Thus by the lifting property of an injective module, there exists anR-module homomorphismf :M →I such that

f ι= ˆf :

Rm M

I ι fˆ ∃f

Then

f(m) =f ι(m) = ˆf(m) = ˆf(1Rm) = 1R+m6= 0,

andαM is shown to be injective. By (ii) thenlR(M) =lR(D²M), hence by [2, Proposition 1.4, Ch. 1],αM is an isomorphism ofR-modules.

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(b) LetM ∈mod(Λ), and consider theR-module isomorphism α_M :M →Hom_R(Hom_R(M, I), I)

used in the proof of (iii)(a). We claim thatα_M is also a homomorphism of Λ-modules.

Letm∈M andλ∈Λ. Then forf ∈Hom_R(M, I), α_M(λm)(f) =f(λm) = (f λ)(m)

because of the Λ^op-module structure on Hom_R(M, I) of Lemma 22(i).

Moreover, the Λ-module structure on Hom_R(Hom_R(M, I)

| {z }

∈mod(Λ^op)

, I) (Lemma 22(ii)) now implies that

(λαM(m))(f) =αM(m)(f λ) = (f λ)(m) forf ∈HomR(M, I). Hence

αM(λm) =λαM(m).

Also, αM(m1+m2) is obviously equal to αM(m1) +αM(m2) for all m1, m2 ∈ M, thus αM is a Λ-module homomorphism. Since αM was shown to be bijective in the proof of (iii)(a), it follows that αM is an isomorphism of Λ-modules. It was also shown in (iii)(a) that αM is natural inM. This completes the proof.

(c) Similar to (iii)(b).

Most of the investigation of Chapter 3 will be carried out without any more assumptions on R than the ones presented in Section 2.2, ³ but in the very last section we will add the condition thatRbe a field. The motivation for making this restriction is that the finitely generatedR-modules then become finite dimensional R-vector spaces, and by choosing an R-basis of an R-vector space V we have a method for obtaining a set of elements of the dual spaceDV – which even turns out to form an R-basis of DV. This procedure will be explained in this section.

Recall that Kdenotes the field R/m, wheremis the maximal ideal inR.

Lemma 30. In the case that R is a field, then R = K and the functor D of Definition 27 is given by

D= HomK(−, K), and it is a functor

mod(K)→mod(K).

3Ris a commutative, local and artinian ring.

Computing Almost Split Sequences: An algorithm for computing almost split sequences of finitely generated modules over a finite dimensional algebra