Morita Equivalence

Rabib IslamDepartment of Mathematics & Statistics

University of Ottawa

Summer 2015

Contents

1 Morita Equivalence 51.1 Category Theory: Basic Definitions and Theorems . . . . . . . . . . . . 51.2 Module Theory: Basic Definitions and Theorems . . . . . . . . . . . . . 81.3 Morita Equivalence: Definition and Examples . . . . . . . . . . . . . . . 141.4 Morita Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Bicategory of Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6 Morita Theory Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Dualities in Representation Theory 382.1 Group Representations: Connection to Modules . . . . . . . . . . . . . . 382.2 Schur-Weyl Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3 GLn −GLm and Skew GLn −GLm Duality . . . . . . . . . . . . . . . . . . 47

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2

Preface

These notes were written and compiled during the Summer 2015 term under thesupervision of Professor Alistair Savage. The corresponding research was funded bythe Work-Study Program of the University of Ottawa and the NSERC Discovery Grantof Professor Savage. This document is intended for a junior to senior undergraduateaudience interested in module theory and category theory.

Acknowledgements

The author would like to thank Professor Savage for his guidance and wisdom, as wellas for his correction of errors.

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Introduction

The theory of modules over rings dates back to the late 19th century, and they wereused to great success in an 1882 paper by Richard Dedekind and Anton Weber, [DW82],which was concerned with algebraic geometry. Rings, however, had not undergoneaxiomatic development until around 40 years later, at the hands of Emmy Noetherand Wolfgang Krull.

Categories, on the other hand, did not appear in the literature until 1945, whenSamuel Eilenberg and Saunders Mac Lane published their paper [EM45]. While theoriginal motivation for categories, functors, and natural transformations were certaintopics of group theory and topology, category theory has become an immensely usefuland prevalent tool in modern algebra, and is also studied today for its own sake.

A natural question concerning rings is: do the modules of a ring contain all theinformation about the structure of that ring? The answer to this question is “no.” Thisleads to a new sense of “equivalence” of rings which is coarser than isomorphism. Thatis, two rings are “equivalent” when their categories of modules are equivalent (in thecategory-theoretical sense). Japanese mathematician Kiiti Morita is known for havingstated and proved a theorem which gave an equivalent condition for this equivalencein an influential 1958 paper, [Mor58]. This theorem plays an important role in modernalgebra, and in his honour, this equivalence of module categories between two ringshas been dubbed “Morita equivalence.”

In this paper, we begin by reviewing the material necessary to define Morita equiva-lence, and we examine two classical examples of Morita equivalence. We then delveinto what is now called “Morita theory”, which is in regards to the key theorems ofMorita in [Mor58], the results leading up to them, and their immediate consequences.We continue on by introducing bicategories, with a focus on the bicategory of bi-modules. We then look at ways in which this bicategorical perspective can be usedin Morita theory, including the neat encapsulation such a perspective provides tothe concept of Morita equivalence. Starting in the second chapter, we introduce therepresentation theory of groups, another old, yet current area of mathematics, andwe examine how this connects with module theory. We then use this connection toidentify some examples of Morita equivalence in representation theory, particularly inthe case of representation-theoretic dualities of groups.

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1 Morita Equivalence

1.1 Category Theory: Basic Definitions and Theorems

We begin by stating many of the basic category-theoretic notions used throughout thispaper. Readers experienced in category theory can safely skip this section.

Definition 1.1.1 (Category). A category C consists of

• a collection Ob(C) of objects;

• for every pair of objects A,B ∈Ob(C), a set HomC(A,B) of morphisms between Aand B;

• for every three objects A,B,C ∈Ob(C) an associative binary operation

: HomC(B,C)×HomC(A,B)→HomC(A,C),

called composition; and

• for every object A ∈ Ob(C), a morphism idA ∈ HomC(A,A) called the identitymorphism onA such that for every morphism f ∈HomC(X,Y ) whereX,Y ∈Ob(C),we have

idY f = f and f idX = f .

Definition 1.1.2 (Functor). Given categories C and C′, a functor F from C to C′, denotedF : C → C′, consists of

• a function F : Ob(C)→Ob(C′); and

• for every two objects A,B ∈Ob(C), a function

F : HomC(A,B)→HomC′ (F(A),F(B)),

which must satisfy the following axioms:

• given an object A ∈ C, we have F(idA) = idF(A); and

• given objects A,B,C ∈ C and morphisms f : A → B and g : B → C, we haveF(g f ) = F(g) F(f ).

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Definition 1.1.3 (Natural transformation). Given categories C and C′, and functorsF : C → C′ and G : C → C′, a natural transformation α from F to G, denoted α : F→ G,is a family of maps

(αA : F(A)→ G(A))A∈Csuch that for A,A′ ∈ C and f : A→ A′, we have

G(f ) αA = αA′ F(f ).

It is also said that α is natural in A.If αA is an isomorphism for every A ∈ C, then α is called a natural isomorphism.

Definition 1.1.4 (Equivalence of categories). Given categories C and C′, a functorF : C → C′ is called an equivalence of categories if there is a functor F′ : C′ → C andnatural isomorphisms α : idC→ F′ F and α′ : idC′ → F F′.

Definition 1.1.5 (Product category). Given two categories C andD, the product categoryC ×D is defined as the category with

• objects (C,D) for C ∈Ob(C) and D ∈Ob(D);

• morphisms (f ,g) for f ∈ HomC(C,C′) and g ∈ HomD(D,D ′) for objects C,C′ ∈Ob(C) and D,D ′ ∈Ob(D);

• composition defined as pointwise composition of morphisms: (f ′, g ′) (f ,g) =(f ′ f ,g ′ g);

• and identity objects 1(C,D) = (1C ,1D) for C ∈Ob(C) and D ∈Ob(D).

Definition 1.1.6 (Bifunctor). A bifunctor F : A×B → C is a functor from the productcategory A×B to the category C.

Definition 1.1.7 (Opposite category). Given a category C, the opposite category Cop isdefined as the category with

• objects C for C ∈Ob(C);

• morphisms HomCop(C,C′) = HomC(C′,C);

• composition defined in reverse: g Cop f = f C g;

• and the same identity objects as in C.

Definition 1.1.8 (Contravariant functor). A contravariant functor F : C →D is a functorF : Cop→D. For contrast, functors which are not contravariant are sometimes calledcovariant.

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Example 1.1.9. Given a category C and A ∈Ob(C), the functor HomC(A,−) : C → Set isdefined by

C 7→HomC(A,C)HomC(A,−)(f ) : g 7→ f g

for f : C→ C′ and g : A→ C.There is also the contravariant functor HomC(−,A) : C → Set, defined by

C 7→HomC(C,A)HomC(−,A)(f ) : g 7→ g f

for f : C→ C′ and g : C′→ A.We also have the bifunctor HomC(−,−) : Cop ×C → Set, defined by

(A,B) 7→HomC(A,B)HomC(−,−)(f ,g) : h 7→ g h f

for f : A→ B, g : X→ Y , and h : B→ X.

Definition 1.1.10 (Adjoint functors). Let F : C →D and G : D→ C be functors. If thereis a natural isomorphism

τ : HomD(F−,−)→HomC(−,G−),

then (F,G) is called an adjoint pair, G is called the right adjoint of F, and F is calledthe left adjoint of G. Note that the condition that τ is natural means that τ is requiredto be natural in the elements (A,B) ∈Ob(C ×D).

Example 1.1.11. Every equivalence of categories forms an adjoint pair with its inverse.Thus, if F is an equivalence and G is its inverse, since G is also an equivalence, F isboth the left and right adjoint of G, and vice versa.

Definition 1.1.12 (Subcategory). Given categories C and D, D is a subcategory of C if

• Ob(D) is a subcollection of Ob(C);

• for all A,B ∈Ob(D), HomD(A,B) is a subcollection of HomC(A,B);

• for all A ∈Ob(D), idA ∈HomD(A,A);

• if f ,g are morphisms in D and the domain of g is the codomain of f , then g fis a morphism in D; and

• the composition law in D is equal to the composition law in C.

Definition 1.1.13 (Full subcategory). Given a category C, a subcategoryD of C is calleda full subcategory of C if, for every A,B ∈Ob(D),

HomD(A,B) = HomC(A,B).

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Definition 1.1.14 (Skeleton). Given a category C, a subcategory D of C is a skeleton ofC if each object of C is isomorphic to exactly one object of D.

Proposition 1.1.15 ([ML98, p. 93]). Every category C is equivalent to its skeletons.

Definition 1.1.16 (Full, faithful functor). Given two categories C and D, a functorF : C → D is called full when the map of F on morphisms is surjective, and is calledfaithful if the map of F on morphisms is injective. If F is both full and faithful, then itis called fully faithful.

Definition 1.1.17 (Essentially surjective functor). Given two categories C and D, afunctor F : C →D is said to be essentially surjective if each object X ∈Ob(D) is isomor-phic to F(A) for some object A ∈ C.

Theorem 1.1.18 ([ML98, p. 93]). Given two categories C and D and a functor F : C →D,the following are equivalent:

• F is an equivalence of categories; and

• F is fully faithful and essentially surjective.

1.2 Module Theory: Basic Definitions and Theorems

We recall some basic facts in module theory that are normally covered in an under-graduate course of algebra, as well as some facts that are more advanced.

Definition 1.2.1 (Ring). A ring R consists of

• a non-empty set R;

• a binary operation “+′′ called addition such that (R,+) is an abelian group withidentity 0 ∈ R; and

• a binary operation “·′′ called multiplication that is associative, distributive over“+′′, and has an identity 1 ∈ R.

Definition 1.2.2 (Left module). Given a ring R, a left R-module M consists of

• a non-empty set M;

• a binary operation “+” such that (M,+) is an abelian group with identity 0 ∈M;

• a map R×M→M called a left action such that for r, s ∈ R and x,y ∈M, we have

1. r(x+ y) = rx+ ry;

2. (r + s)x = rx+ sx;

3. (rs)x = r(sx); and

4. 1Rx = x.

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We may write RM to emphasize the left R-module structure.

Definition 1.2.3 (Right module). Given a ring R, a right R-module M consists of

• a non-empty set M;

• a binary operation “+” such that (M,+) is an abelian group with identity 0 ∈M;

• a map M ×R→M called a right action such that for r, s ∈ R and x,y ∈M, we have

1. (x+ y)r = xr + yr;

2. x(r + s) = xr + xs;

3. x(sr) = (xs)r; and

4. x1R = x.

We may write MR to emphasize the right R-module structure.

Example 1.2.4. A ring R is a left R-module and a right R-module, with left multipli-cation and right multiplication actions respectively. These modules are called the leftand right regular modules, respectively.

Example 1.2.5. Given a ring R, the n-fold direct product Rn is a left R-module anda right R-module with respect to left and right pointwise multiplication actionsrespectively.

Definition 1.2.6 (Module homomorphism). Given a ring R and left R-modules M,M ′,a map f : M→M ′ is called a module homomorphism if, for x,y ∈M and r ∈ R,

• f (x+ y) = f (x) + f (y); and

• f (rx) = rf (x).

Homomorphisms of right modules are defined similarly.

We denote the category of left modules over a ring R by RMod. Its objects aremodules and its morphisms are module homomorphisms.

Definition 1.2.7 (Bilinear map). Given vector spaces X, Y , and Z over a field K, abilinear map is a function B : X ×Y → Z such that for x ∈ X and y ∈ Y , the map

x 7→ B(x,y)

is a linear map X→ Z, and the map

y 7→ B(x,y)

is a linear map from Y to Z.

The next seven definitions are useful in understanding one of the first examples ofMorita equivalence.

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Definition 1.2.8 (Algebra). Given a fieldK, an (associative) algebra A overK is a vectorspace equipped with an associative, bilinear map A×A→ A called multiplication inA.

Definition 1.2.9 (Jacobson radical). Given an algebra A and an A-module M, theJacobson radical rad(M) of M is the intersection of all maximal A-submodules of M.

Definition 1.2.10 (Artinian module). Given a commutative ring R, an R-algebra A,and a A-moduleM, we say thatM is an artinian module if, whenever there is a sequence

M ⊇M1 ⊇M2 ⊇M3 ⊇ · · ·

of A-submodules of M, then there exists n0 ∈N such that Mn =Mn0for all n ≥ n0.

Definition 1.2.11 (Artinian algebra). Given a commutative ring R and an R-algebra A,A is said to be a left (right) artinian algebra if the left (right) regular module is artinian.

Definition 1.2.12 (Basic algebra). Given an artinian algebra A, if for every pair ofsimple submodules S1 and S2 of A/ rad(A) one has S1 S2 =⇒ S1 = S2, then A iscalled a basic algebra.

Definition 1.2.13 (Idempotent). Given a ring R, an element x ∈ R is called an idempo-tent of R if x · x = x.

Definition 1.2.14 (Orthogonal idempotent). Given a ring R, two idempotents e1 ande2 are called orthogonal idempotents if e1e2 = e2e1 = 0. An idempotent e , 0 is primitiveif whenever e = e1 + e2 with e1, e2 orthogonal idempotents, then either e1 = 0 or e2 = 0.

Definition 1.2.15 (Balanced map). Given a right R-moduleM, a left R-module N , andan abelian group G, a map f : M ×N → G is R-balanced if

• f (m1 +m2,n) = f (m1,n) + f (m2,n);

• f (m,n1 +n2) = f (m,n1) + f (m,n2); and

• for r ∈ R, f (mr,n) = f (m,rn).

Definition 1.2.16 (Tensor product of modules). Given a right R-module M, a leftR-module N , an abelian group T , and an R-balanced map τ : M×N → T , we call (T ,τ)a tensor product of M and N if for every abelian group G and every R-balanced mapf : M ×N → G, there is a unique group homomorphism g : T → G such that

g τ = f .

Proposition 1.2.17. Given a right R-moduleM and a left R-moduleN , if (T ,τ) and (T ′, τ ′)are tensor products of M and N , then T and T ′ are isomorphic abelian groups. In otherwords, tensor products are unique up to isomorphism.

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Proof. Since τ ′ is R-balanced and T is a tensor product, there is a unique grouphomomorphism f : T → T ′ such that

τ ′ = f τ.

Since τ is R-balanced and T ′ is a tensor product, there is a unique group homomor-phism g : T ′→ T such that

τ = g τ ′.

We thus obtainτ = g f τ and τ ′ = f g τ ′,

which shows that f is a group isomorphism. Therefore, T T ′.

Proposition 1.2.18. A tensor product exists for any right R-module M and left R-moduleN .

Proof. Denote by X the module with basis em,n : m ∈M,n ∈N . Let Y be the subgroupof X generated by the following elements:

em+m′ ,n − em,n − em′ ,n;em,n+n′ − em,n − em,n′ ;emr,n − em,rn,

where m,m′ ∈M,n,n′ ∈N,r ∈ R. Denote by ι : M ×N → X the canonical injection andπ : X→ X/Y the projection. Set α = π ι.

We show that α is R-balanced. However, this is equivalent to checking that theimages under α of the elements defining Y are 0, but this follows automatically.

Let G be an abelian group and let f : M ×N → G be an R-balanced map. Then, thereis a unique group homomorphism h : X→ G defined by h(em,n) = f (m,n), and it is suchthat h ι = f . We show that the generators of A are in kerh.

h(em+m′ ,n−em,n−em′ ,n) = h(em+m′ ,n)−h(em,n)−h(em′ ,n) = f (em+m′ ,n)− f (em,n)− f (em′ ,n) = 0

h(em,n+n′ − em,n − em,n′ ) = h(em,n+n′ )− h(em,n)− h(em,n′ ) = f (em,n+n′ )− f (em,n)− f (em,n′ ) = 0

h(emr,n − em,rn) = h(emr,n)− h(em,rn) = f (emr,n)− f (em,rn) = 0

This shows that the map p : X/Y → G defined by p(em,n) = f (m,n) is well-defined (andunique). Finally, we obtain that p α is the unique group homomorphism satisfyingp α = f . Thus, (X/Y ,p) is a tensor product for M and N .

Let (T ,τ) be a tensor product of a right R-module M and a left R-module N . Sincetensor products are unique up to isomorphism, we use the standard notation T =M⊗RN and τ(m,n) =m⊗n. By an abuse of notation, we callM⊗RN the tensor productof M and N .

Definition 1.2.19 (Free module). Given a ring R, a free left R-module is a left R-modulethat is isomorphic to the direct sum of some copies (possibly an infinite number) of R.

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Proposition 1.2.20. Given a ring R, every left R-module is a quotient of a free left R-module.

Proof. Let M be a module with generating set B. Now let F be a free left R-modulewith B as a basis. Then, we construct a module homomorphism f : F→M induced byf (b) = b for b ∈ B. That is, define f by

f

∑i

ribi

=

∑i

ribi

,for ri ∈ R and bi ∈ B. Then f is evidently a surjection, and by the first isomorphismtheorem, we have

F/ kerf M.

Definition 1.2.21 (Simple module). Given a ring R, a left R-module M is called simpleif M has no non-zero proper submodule.

Definition 1.2.22 (Semisimple module). Given a ring R, a left R-module M is calledsemisimple if M is the direct sum of simple modules.

Proposition 1.2.23. Given a ring R, every simple module of R is isomorphic to R/M forsome maximal ideal M of R, and for every maximal ideal M, R/M is isomorphic to a simplemodule of R.

Proof. Given a maximal idealM, we know thatR/M has no proper ideals. Furthermore,we know that every ideal of R/M corresponds to a proper submodule of R/M as anR-module. Thus R/M is a simple module.

Let N be a simple R-module. Given 0 , n ∈ N , the map r 7→ rn is a surjection, asRn is a non-zero submodule of N , and thus Rn contains N . Thus R/I N for someideal I of R. Now suppose I is not maximal. Then there is an ideal J of R such thatI ( J ( R. However this means that J corresponds to some nontrivial proper ideal ofR/I N , and so N has a nonzero proper submodule. This is a contradiction, so I mustbe maximal.

Definition 1.2.24 (Semisimple module category). Given a ring R, the category of left(right) R-module RMod is said to be semisimple if every module M ∈ Ob(RMod) issemisimple.

Lemma 1.2.25. Let R be a ring, and let M,N1, and N2 be (left) R-modules. Then

HomR(M,N1 ⊕N2) HomR(M,N1)⊕HomR(M,N2)

as abelian groups.

Proof. We first define the projection maps

π1 : N1 ⊕N2→N1

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andπ2 : N1 ⊕N2→N2

by π1(n1,0) = n1 and π2(n2,0) = n2. Then given f ∈ HomR(M,N1 ⊕ N2), the mapρ(f ) = (π1 f ,π2 f ) is a group isomorphism HomR(M,N1 ⊕N2)→ HomR(M,N1)⊕HomR(M,N2). Firstly, ρ is a homomorphism because

ρ(f + g) = (π1 (f + g),π2 (f + g))= (π1 f +π1 g,π2 f +π2 g)= (π1 f ,π2 f ) + (π1 g,π2 g)= ρ(f ) + ρ(g).

It is also the case that ρ has an inverse homomorphism. Given

(f ,g) ∈HomR(M,N1)⊕HomR(M,N2),

we define ρ−1(f ,g) = h where h(m) = (f (m), g(m)). Then

ρ(ρ−1(f ,g)) = ρ(h) = (π1 h,π2 h) = (f ,g).

Also, for f ∈HomR(M,N1 ⊕N2),

ρ−1(ρ(f )) = ρ−1(π1 f ,π2 f ) = f .

Thus ρ is a group isomorphism.

Lemma 1.2.26 ([HS97, Proposition 3.4]). Let R be a ring, and let M1,M2, and N be (left)R-modules. Then

HomR(M1 ⊕M2,N ) HomR(M1,N )⊕HomR(M2,N )

as abelian groups.

Proof. We first define the embedding maps

ι1 : M1→M1 ⊕M2

andι2 : M2→M1 ⊕M2

by ι1(m1) = (m1,0) and ι2(m2) = (0,m2). Then, given f ∈HomR(M1 ⊕M2,N ), the mapσ (f ) = (f ι1, f ι2) is a group isomorphism HomR(M1 ⊕M2,N )→ HomR(M1,N )⊕HomR(M2,N ). Firstly, σ is a homomorphism because

σ (f + g) = ((f + g) ι1, (f + g) ι2)= (f ι1 + g ι1, f ι2 + g ι2)= (f ι1, f ι2) + (g ι1, g ι2)= σ (f ) + σ (g).

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It is also the case that σ has an inverse homomorphism. Given

(f ,g) ∈HomR(M1,N )⊕HomR(M2,N ),

we define σ−1(f ,g) = h where h(m1,m2) = f (m1) + g(m2). Then

σ (σ−1(f ,g)) = σ (h) = (h ι1,h ι2) = (f ,g).

Also, for f ∈HomR(M1 ⊕M2,N ),

σ−1(σ (f )) = σ−1(f ι1, f ι2) = f .

Thus σ is a group isomorphism.

Definition 1.2.27 (Simple ring). A non-zero ring is called a simple ring if its only idealsare 0 and the ring itself.

We see from the definition of a simple ring and Proposition 1.2.23 that simple ringseach have a unique simple module up to isomorphism.

Definition 1.2.28 (Semisimple ring). A ring R is said to be left semisimple if R issemisimple as a left R-module. Similarly, R is right semisimple if R is semisimple as aright R-module.

Proposition 1.2.29 ([Lam01, Corollary 3.7]). A left semisimple ring is always rightsemisimple, and vice versa.

Due to this proposition, we will henceforth refer to left and right semisimple ringsas semisimple. It may be of interest to the reader that this proposition is in fact acorollary of a more general version of Lemma 2.1.15.

Theorem 1.2.30 (Jacobson density, [Isa09, p. 185]). Let U be a simple right R-moduleand write D = EndR(U ). Let γ be any D-linear operator on U and let X ⊆U be any finiteD-linear independent subset. Then there exists an element r ∈ R such that xr = γ(x) for allx ∈ X.

1.3 Morita Equivalence: Definition and Examples

We now present the definition of Morita equvialence and we follow up with twoclassical examples.

Definition 1.3.1 (Morita equivalence). Two rings R and S are called Morita equivalentif RMod and SMod are equivalent categories.

It is evident that if two rings are isomorphic, they are Morita equivalent. However,the converse in general is not true. We will see in Example 1.3.3 two rings that are notisomorphic, yet are Morita equivalent. However, the following partial converse holds.

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Theorem 1.3.2 ([Lam99, p. 494]). If two rings are Morita equivalent, then their centersare isomorphic. In particular, if the rings are commutative, then they are isomorphic.

Because of the above theorem, Morita equivalence is interesting solely in the situa-tion of noncommutative rings.

Example 1.3.3. Let R be a ring and S be the ring of n× n matrices with entries in R.We show that R and S are Morita equivalent by finding an equivalence of categoriesF : RMod→ SMod. We follow the proof of [Lam99, p. 470].

Let M,M ′,M ′′ be left R-modules, and let f : M→M ′ be a module homomorphism.Define the map F : RMod→ SMod by

M 7→Mn

F(f ) : (m1, . . . ,mn)ᵀ 7→ (f (m1), . . . , f (mn))ᵀ,

where m1, . . . ,mn ∈M and Mn = (m1,m2, . . . ,mn)ᵀ : ∀i,mi ∈M is the module with theaction defined by matrix multiplication from the left. We see that F is a functor since,for a module homomorphism g : M ′→M ′′, we have

F(g f )(m1, . . . ,mn) = (g f (m1), . . . , g f (mn))ᵀ

= F(g)(f (m1), . . . , f (mn))ᵀ

= F(g) F(f )(m1, . . . ,mn)ᵀ,

and one can easily check that F(f ) is a module homomorphism.Let M,M ′ be left S-modules, and let f : M → M ′ be a module homomorphism.

Define the map G : SMod→ RMod by

M 7→ E11M

G(f ) : E11m 7→ E11f (m),

where m ∈M and Eij is the matrix with 1 in the (i, j) position and 0 elsewhere. Lettingrij be the matrix with r ∈ R in the (i, j) position, we see that

rE11M = r11E11M = E11r11M ⊆ E11M,

so E11M is a left R-module with respect to multiplication.If g : M ′→M ′′ is a module homomorphism, then

G(g f )(E11m) = E11g f (m)= G(g)(E11f (m))= G(g) G(f )(E11m),

and one can easily show G(f ) is a module homomorphism, so G is a functor.Let M,M ′ be left R-modules. We want to show that G F is naturally isomorphic to

idRMod. Let βM : M→ (m,0, . . . ,0) : m ∈M = G F(M) be the map defined by

βM(m) = (m,0 . . . ,0).

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Then βM is an isomorphism and for f : M→M ′, we have

(G F)(f ) βM(m) = (G F)(f )(m,0, . . . ,0)ᵀ

= (G F)(f )E11(m,m, . . . ,m)ᵀ

= E11(f (m), f (m), . . . , f (m))ᵀ

= (f (m),0, . . . ,0)ᵀ

= βM ′ idRMod(f )(m),

so β : idRMod→ G F is a natural isomorphism.

Let M be a left S-module. To show that F is an equivalence of categories, wemust also show that F G id

SMod. We do this by finding a natural isomorphismαM : M→ (E11M)n = F(G(M)). Define the map αM by

αM(m) = (E11m,E12m,. . . ,E1nm)ᵀ,

where m ∈M. Notice that αM(m+m′) for m′ ∈M, so to show that αM is a module ho-momorphism, we must demonstrate that, given r ∈ R, we have αM(rEijm) = rEijαM(m),because rEij : r ∈ R,1 ≤ i ≤ n,1 ≤ j ≤ n forms a spanning set for S. The left-hand sideevaluates to

αM(rEijm) = (0, . . . ,0, rE1jm︸︷︷︸j-th position

,0, . . . ,0)ᵀ,

and the right-hand side evaluates to

rEijα(m) = rEij(E11m,. . . ,E1nm)ᵀ = (0, . . . ,0, rE1jm︸︷︷︸j-th position

,0, . . . ,0)ᵀ,

so the equality is satisfied.To show that αM is an isomorphism, we show that it is injective and surjective. First,

let αM(m) = 0 for some m ∈M. Then

(E11m,. . . ,E1nm)ᵀ = (0, . . . ,0)ᵀ =⇒ E1jm = 0 for all j

=⇒ 0 =∑j

Ej1E1jm =∑j

Ejjm = Inm =m,

so αM is injective. Now, let p1, . . . ,pn ∈ E11M; one can then pick m1, . . . ,mn ∈M suchthat pi = E11mi for 1 ≤ i ≤ n. Then,

αM(pi) = (E11pi , . . . ,E1npi)ᵀ

= (E11E11mi ,E12E11mi , . . . ,E1nE11mi)ᵀ

= (pi ,0, . . . ,0)ᵀ,

so(0, . . . ,0, pi︸︷︷︸

i-th position

,0, . . . ,0)ᵀ = Ei1(pi ,0, . . . ,0)ᵀ = Ei1αM(pi) = αM(Ei1pi),

16

and finally,

αM(E11p1 +E21p2 + · · ·+En1pn) = αM(E11p1) + · · ·+αM(En1pn) = (p1, . . . ,pn)ᵀ;

this shows us that αM is surjective, and thus an isomorphism.Let M,M ′ be two left S-modules, with m ∈ M, and let f : M → M ′ be a module

homomorphism. Then

(F G)(f ) αM(m) = (F G)(f )(E11m,. . . ,E1nm)= (F G)(f )(E11E11m,E11E12m,. . . ,E11E1nm)= (E11f (E11m), . . . ,E11f (E1nm))= (E11E11f (m), . . . ,E11E1nf (m))= (E11f (m), . . . ,E1nf (m))= αM ′ id

SMod(f )(m),

so α is a natural isomorphism idSMod→ F G.

We finally see that F : RMod→ SMod is an equivalence of categories, so R and S areMorita equivalent.

For every field K and finite-dimensional K-algebra A, there is an associated basicalgebra (see Definition 1.2.12) defined in the following way: let Ae be a minimalprogenerator of the full subcategory of AMod consisting of finite-dimensional leftA-modules, with e = e2 ∈ A. Then Ab = eAe is called the basic algebra associated to A.One can choose e in a particular way. We can decompose the identity

1A =s1∑j=1

e1j + · · ·+sn∑j=1

enj =n∑i=1

sj∑j=1

eij

into the sum of pairwise primitive orthogonal idempotents eij (see Definition 1.2.14)such that

Aeij Aeik for j,k ∈ 1, . . . , si, Aeij Aei′k for i , i′.

Then we may let e =∑nn=1 ei1. In fact, every decomposition of A into a finite number of

left ideals induces a partition 1A = e1 + · · ·+er , as is shown by the following proposition,to which we present the proof due to an error in the book.

Proposition 1.3.4 ([Coh03, Proposition 4.3.1]). Let R be any ring. Any decomposition ofR as a direct sum of a finite number of left ideals

R = a1 ⊕ · · · ⊕ ar (1.1)

corresponds to a decomposition of 1 as a sum of pairwise orthogonal idempotents

1 = e1 + · · ·+ er , (1.2)

where ai = Rei , and for any idempotent e, Re is indecomposable if and only if e is primitive.

17

Proof. Given (1.1), we write 1 as∑i ei where ei ∈ ai . Then ek =

∑i ekei ; since the

sum (1.1) is direct, we see that ekei = 0 of i , k and e2k = ek, so (1.2) follows.

Conversely, given (1.2), put ai = Rei ; then any x ∈ R can be written as x =∑i xei ,

where xei ∈ ai , hence R =∑i ai and this sum is direct, for if

∑i xiei = 0, then right

multiplication by ek gives xkek = 0.For any idempotent e, Re is a direct summand of R: R = Re+R(1− e); now the above

correspondence shows that any direct decomposition of Re corresponds to writing eas a sum of two orthogonal idempotents. It follows that Re is indecomposable if andonly if e is primitive.

Thus, if A decomposes into

A = Ae11 ⊕Ae12 ⊕ · · · ⊕Aensn ,

and e is the idempotent given in Ab = eAe, then Ae is obtained from A by removing allthe duplicate indecomposable submodules, as these duplicates are killed by e.

Ae = (Ae11 ⊕Ae12 ⊕ · · · ⊕Aensn)e = Ae11 ⊕Ae21 ⊕ · · · ⊕Aen1.

The ideas discussed in the above discussion are key to the proof of the followingtheorem.

Theorem 1.3.5 ([SY11, Theorem 6.16]). Every finite-dimensional K-algebra A is Moritaequivalent to its basic algebra Ab.

1.4 Morita Theory

We now seek to explore key theorems with regards to Morita equivalence, and theirconsequences, that Kiiti Morita found and proved in [Mor58]. Our exposition isheavily influenced by [Lam99].

Definition 1.4.1 (Trace ideal). The trace ideal of a left (right) R-module M is denotedtr(M) and is defined by

tr(M)B∑

f ∈HomRMod(M,R)

f (M).

Definition 1.4.2 (Generator). For a ring R, a right R-module P is called a generatorfor ModR if HomR(P ,−) : ModR→ Ab is a faithful functor. In otherwords, if for M,N ∈Ob(ModR) there is a morphism f : M→ R, then HomR(P ,f ) is nonzero.

The following theorem makes clear the many interesting properties of generators.

Theorem 1.4.3 ([Lam99, Theorem 18.8]). For any right R-module P , the following areequivalent:

1. P is a generator;

18

2. tr(P ) = R;

3. R is a direct summand of a finite direct sum⊕

i P ;

4. R is a direct summand of a direct sum⊕

i P ; and

5. every M ∈Ob(ModR) is an surjective image of some direct sum⊕

i P .

Definition 1.4.4 (Projective module). A left R-module P is said to be projective if, forany surjective homomorphism of left R-modules g : B→ C, and any R-homomorphismh : P → C, there exists an R-homomorphism h′ : P → B such that h = g h′.

Definition 1.4.5 (Progenerator). Given a ring R, a left R-module P is a progenerator ifP a is finitely generated, projective generator.

Definition 1.4.6 (Full idempotent). Given a ring R, an idempotent x ∈ R of R is calleda full idempotent of R if RxR = R. That is, the two-sided ideal generated by x is thewhole ring R.

Example 1.4.7. An example of a full idempotent e ∈ Mn(R) is e = Ekk for 1 ≤ k ≤ n.This is because EikEkkEkj = Eij for 1 ≤ i ≤ n and 1 ≤ j ≤ n, and Eij | 1 ≤ i ≤ n,1 ≤ j ≤ nis a basis for Mn(R).

Definition 1.4.8 (Bimodule). Given two rings R and S, an (R,S)-bimodule M is anabelian group such that,

• M is a left R-module and a right S-module; and

• for r ∈ R,s ∈ S,m ∈M, we have (rm)s = r(ms).

We may write RMS to emphasize the (R,S)-bimodule structure.

Example 1.4.9. A ring R is itself an (R,R)-bimodule with respect to the ring multipli-cation.

Example 1.4.10. Given a ring R, the ring Mm×n(R) of m×n matrices with entries fromR is an (Mm(R),Mn(R))-bimodule with respect to standard matrix multiplication.

Example 1.4.11. Given a commutative ring R, a left R-module M is also a right R-module. Let r ∈ R and m ∈M. One defines the right action by mr = rm; because R isa commutative ring, the axioms for a right module are satisfied. The left and rightaction also commute, so M is an (R,R)-bimodule.

Definition 1.4.12 (Bimodule homomorphism). Given two (R,S)-bimodules M andN , a map f : M → N is a bimodule homomorphism if it is a homomorphism of leftR-modules and a homomorphism of right S-modules.

We see in the next two propositions that tensor products interact in a useful waywith bimodules.

19

Proposition 1.4.13. If M is an (R,S)-bimodule and N is a left S-module, then M ⊗S N isa left R-module.

Proof. Every element of M ⊗RN can be written as a finite sum∑imi ⊗ni , so we define

the left action to be

r

∑i

mi ⊗ni

=∑i

(rmi)⊗ni .

We verify that this is an action.

(r + r ′)∑i

mi ⊗ni =∑i

(r + r ′)mi ⊗ni

=∑i

(rmi + r ′mi)⊗ni

=∑i

rmi ⊗ni +∑i

r ′mi ⊗ni

(rr ′)∑i

mi ⊗ni =∑i

(rr ′mi)⊗ni

=∑i

r(r ′mi)⊗ni

= r∑i

r ′mi ⊗ni

This shows that M ⊗RN is a left R-module.

Proposition 1.4.14. If M is an (R,S)-bimodule and N is an (S,T )-bimodule, then M⊗SNis an (R,T )-bimodule.

Proof. Similar to the previous proof we see that (∑imi ⊗ni)t =

∑imi ⊗ (nit) defines a

right action. It remains to show that the left and right action commute.

r

∑i

mi ⊗ni

t = r

∑i

mi ⊗nit

=∑i

rmi ⊗nit

= r∑i

mi ⊗nit

=

r∑i

mi ⊗ni

tWe conclude that M ⊗S N is an (R,T )-bimodule.

20

Definition 1.4.15 (Endomorphism ring). Given a left (right) R-module M, the endo-morphism ring, denoted EndR(M) is the ring of all left (right) module homomorphismsof M into itself, with addition being pointwise addition, and multiplication beingcomposition of homomorphisms.

Definition 1.4.16 (Faithfully balanced). Given rings R and S, an (R,S)-bimoduleM is called faithfully balanced if the ring homomorphisms λM : R → End(MS) andρM : S→ End(RM) defined for r ∈ R,s ∈ S,m ∈M by

λM(r)(m) = r ·m and ρM(s)(m) =m · s

are ring isomorphisms.

We follow closely the exposition of [Lam99, p. 485-490] up until Proposition 1.4.18.Let R be a ring and P be a left R-module. Let S = EndR(P ) be the endomorphism

ring of P acting on the right of P by evaluation. That is, for p ∈ P and s ∈ S, define

psB s(p).

Note that this rule necessitates that composition is written in reverse. We verify thatthis is a right action: for s, s′ ∈ S and p,p′ ∈ P ,

(p+ p′)s = s(p+ p′) = s(p) + s(p′);p(s+ s′) = (s+ s′)(p) = s(p) + s′(p);p(ss′) = p(s s′) = ((p)s)s′ = (ps)s′

p1S = p.

Thus P is a right S-module. The left and right actions on P commute because allelements of S are module homomorphisms. Thus P is an (R,S)-bimodule.

Let Q = HomR(P ,R) be the set of left R-module homomorphisms P → R writtenon the right of P in the sense of EndR(P ) above. Then, given the elements q ∈ Q,p ∈ P , and r ∈ R, Q is a right R-module with respect to the action p(qr) = (pq)r (“PQR-associativity”). This is evidently a right action on Q as pq ∈ R and q is a modulehomomorphism. Also, given the elements q ∈Q, p ∈ P , and s ∈ S, Q is a left S-modulewith respect to the action p(sq) = (ps)q (“P SQ-associativity”). Again, this is evidently aleft action on Q as ps ∈ P and q is a module homomorphism. The left and right actionson Q commute: given p ∈ P , q ∈Q, s ∈ S, and r ∈ R,

(p(sq))r = ((ps)q)r = (ps)(qr) = p(s(qr)).

Thus, Q is an (S,R)-bimodule. Henceforth, p,q, r, s and p′,q′, r ′, s′ will denote elementsin P ,Q,R, and S, respectively.

We now perform some calculations. Note that pq ∈ R by the application of q to p (asnoted above, Q is acting on the right of P ) and qp ∈ S, which is defined as

(p′)(qp) = (p′q)p (“PQP -associativity′′).

21

We verify that qp is an R-module homomorphism. Letting p′′ ∈ P ,

(p′ + p′′)(qp) = ((p′ + p′′)q)p = (p′q+ p′′q)p = p′qp+ p′′qp;rp′(qp) = (rp)qp = r(p)qp.

We now show that(qp)q′ = q(pq′) (“QPQ-associativity′′).

This is checked by showing that the left and right sides are equal functions on P :

p′((qp)q′) = (p′(qp))q′ (“P SQ-associativity′′)= ((p′q)p)q′ (“PQP -associativity′′)= (p′q)(pq′) (“RPQ-associativity′′ : q is a left R-module homomorphism)= p′(q(pq′)) (“PQR-associativity′′).

Lemma 1.4.17 ([Lam99, Lemma 18.15]). In the above notation:

1. (p,q) 7→ pq defines an (R,R)-bimodule homomorphism α : P ⊗S Q→ R;

2. (q,p) 7→ qp defines an (S,S)-bimodule homomorphism β : Q⊗R P → S.

Proof. We see that α is well defined due to P SQ-associativity. Further, α is an (R,R)-bimodule homomorphism due to RPQ-associativity and PQR-associativity. Similarly,β is well defined due to QRP -associativity, and is a bimodule homomorphism due toSQP -associativity and QPS-associativity.

In the above context, the six-tuple (R,P ,Q,S;α,β) is called the Morita Context associ-ated to P . This Morita Context is fixed for the next two propositions and the followingcorollary.

Proposition 1.4.18 ([Lam99, Proposition 18.17]). 1. The left R-module module P isa generator if and only if α is surjective.

2. Assume P is a generator. Then

a) α is an isomorphism of (R,R)-bimodules;

b) Q HomS(P ,S) as (S,R)-bimodules;

c) P HomS(Q,S) as (R,S)-bimodules; and

d) R EndS(PS) EndS(SQ) as rings.

Proposition 1.4.19 ([Lam99, Proposition 18.19]). 1. The left R-module P is finitelygenerated and projective if and only if β is surjective.

2. Assume P is finitely generated projective. Then

a) β is an isomorphism of (S,S)-bimodules;

b) Q HomR(P ,R) as (R,S)-bimodules;

22

c) P HomR(S,R) as (S,R)-bimodules; and

d) S EndR(RP ) EndR(QR) as rings.

Corollary 1.4.20 ([Lam99, Corollary 18.21]). If RP is a progenerator (i.e. a finitelygenerated projective generator), then RPS and SQR are faithfully balanced bimodules.

The next three theorems are those of [Mor58], and they help us see exactly whentwo rings are Morita equivalent.

Theorem 1.4.21 ([Lam99, Theorem 18.24] “Morita I”). Let RP be a progenerator, and(R,P ,Q,S;α,β) the associated Morita context. Then

1. −⊗S Q : ModR→ ModS and −⊗R P : ModR→ ModS are mutually inverse categoryequivalences.

2. P ⊗S − : SMod→ RMod and Q⊗R − : RMod→ SMod are mutually inverse categoryequivalences.

Theorem 1.4.22 ([Lam99, Theorem 18.26] “Morita II”). Let R and S be two rings, and

F : RMod→ SMod, G : SMod→ RMod

be mutually inverse category equivalences. Let Q = F(RR) and P = G(SS). Then P is an(R,S)-bimodule and Q is an (S,R)-bimodule such that F Q⊗R − and G = P ⊗S −.

Definition 1.4.23 ((S,R)-progenerator). Given two rings R and S, an (S,R)-bimoduleP is called an (S,R)-progenerator if SPR is faithfully balanced and PR is a progenerator.

Theorem 1.4.24 ([Lam99, Theorem 18.28] “Morita III”). Given two rings R and S, theisomorphism classes of category equivalences SMod→ RMod are in bijective correspondencewith the isomorphism classes of (S,R)-progenerators. Composition of category equivalencescorresponds to tensor products of such progenerators.

The most significant consequences of Morita’s theorems are the following.

Proposition 1.4.25 ([Lam99, Proposition 18.32]). If RMod and SMod are equivalent,then ModR and ModS are equivalent.

Proposition 1.4.26 ([Lam99, Proposition 18.33]). Given rings R and S, the following areequivalent:

1. R is Morita equivalent to S.

2. R End(SP ) for some progenerator P of SMod.

3. R eMn(S)e for some full idempotent e in a matrix ring Mn(S).

We give some justification here for Theorem 1.3.2.

23

Proposition 1.4.27 ([Lam99, Lemma 18.41]). Let M be a faithfully balanced (R,S)-bimodule. It follows that Z(R) Z(S), and R and S are isomorphic to the endomorphismring of bimodule homomorphisms of M.

Corollary 1.4.28 ([Lam99, Corollary 18.42]). If R and S are Morita equivalent rings,then Z(R) Z(S) as rings.

Proof. Since R and S are Morita equivalent, by Proposition 1.4.26, we know thatS End(RP ) for RP a progenerator. Thus the Morita Context (R,P ,Q,S;α,β) associatedwith P exists. We know by Corollary 1.4.20 that RPS is a faithfully balanced (R,S)-bimodule. By applying Proposition 1.4.27 to P , we see that Z(R) Z(S).

1.5 Bicategory of Bimodules

Here, we introduce a new tool with which to look at the relationship between rings,bimodules between them, and the homomorphisms between these bimodules: thebicategory. We begin discussing the idea of a 2-category, as this is helps us in dealingwith bicategories.

Definition 1.5.1 (2-category). A 2-category C consists of

• a collection Ob(C) of objects or 0-cells;

• for every pair of objects A,B ∈ Ob(C), a category HomC(A,B) whose objectsf ,g, . . . : A→ B are called 1-morphisms or 1-cells, whose morphisms α,β, . . . arecalled 2-morphisms or 2-cells, and whose composition is denoted 1 and calledvertical composition;

A B

f

g

h

α

βA B

f

h

β1α

• for every three objects A,B,C ∈Ob(C) an associative, unital functor

0 : HomC(B,C)×HomC(A,B)→HomC(A,C),

called horizontal composition with identities idA and ididA .

A B C

f

g

f ′

g ′

α α′ A C

f ′f

g ′g

α′0α

24

Definition 1.5.2 (2-functor). Given 2-categories C and D, a 2-functor F from C to D isa consists of:

• a function Ob(C)→Ob(D);

• given two objects A,B ∈Ob(C), a function

F : HomC(A,B)→HomC′ (F(A),F(B));

• given two 1-morphisms f ,g ∈HomC(A,B), a function

HomHomC(A,B)(f ,g)→HomHomD(F(A),F(B))(F(f ),F(g))

which much satisfy the following:

• given an object A ∈Ob(C), we have F(idA) = idF(A);

• given objects A,B,C ∈ Ob(C) and morphisms f : A→ B and g : B→ C, we haveF(g f ) = F(g) F(f );

• given objects A,B ∈Ob(C) and a morphism f : A→ B, we have F(idf ) = idF(f );

• given 1-morphisms f ,g,h ∈HomC(X,Y ), and 2-morphisms α : f → g and β : g→h,

F(β 1 α) = F(β) 1 F(α);

and

• given 1-morphisms f ,g ∈ HomC(X,Y ), f ′, g ′ ∈ HomC(Y ,Z) and 2-morphismsα : f → g,α′ : f ′→ g ′,

F(α′ 0 α) = F(α′) 0 F(α).

Example 1.5.3 (2-category of small categories). There exists a 2-category Cat whoseobjects are small categories, whose 1-morphisms are functors, and whose 2-morphismsare natural transformations.

Example 1.5.4 (2-category of topological spaces). There exists a 2-category Top whoseobjects are topological spaces, whose 1-morphisms are continuous maps, and whose2-morphisms are homotopy classes.

Definition 1.5.5 (Bicategory). A bicategory C consists of

• a collection Ob(C) of objects or 0-cells;

• for every pair of objects A,B ∈ Ob(C), a category HomC(A,B) whose objectsf ,g, . . . : A→ B are called 1-morphisms or 1-cells, whose morphisms α,β, . . . arecalled 2-morphisms or 2-cells, and whose composition is denoted 1 and calledvertical composition;

25

• for every three objects A,B,C ∈Ob(C), a functor

0 : HomC(B,C)×HomC(A,B)→HomC(A,C),

called horizontal composition, such that for every five objects A,B,C,D,E andevery four 1-morphisms f : A → B,g : B → C,h : C → D,k : D → E, there are2-isomorphisms

αf ,g,h : h (g f )→ (h g) fλf : idBf → f

ρf : f idA→ f

such that the following diagrams commute.

((kh)g)f ((kh)g)f

(kh)(gf ) k((hg)f )

k(h(gf ))

α

α01

α

α10α

(g idB)f g(idB f )

gf

α

ρ01

10λ

Example 1.5.6 (Bicategory of bimodules). There exists a bicategory Bim whose ob-jects are rings, whose 1-morphisms from R to S are (S,R)-bimodules, and whose2-morphisms are bimodule homomorphisms. The composition

HomBim(S,T )×HomBim(R,S)→HomBim(R,T )

is the tensor product over S.This is presented as a bicategory and not a 2-category because, for an (R,S)-bimodule

M, an (S,T )-bimodule N , and a (T ,V )-bimodule P , there is an isomorphism

M ⊗S (N ⊗T P ) (M ⊗S N )⊗T P ,

however this does not constitute an equality. An isomorphism can be constructedusing the universal property of tensor products.

26

Theorem 1.5.7 ([Hon, Theorem 3.2]). Every bicategory is equivalent to a 2-category.

Remark 1.5.8. In light Theorem 1.5.7, one can treat Bim as a 2-category by pickingand using a 2-category equivalent to Bim.

For any given bimodule, there is a tensor product functor. Specifically, letting R,Sbe rings, and M an (R,S)-bimodule, we can define M ⊗S − : SMod→ RMod for any leftS-module N and any left module homomorphism f defined on N by

N 7→M ⊗S N

M ⊗S (f ) :∑i

mi ⊗ni 7→∑i

mi ⊗ f (ni),

and this is a functor because, for any additional left module homomorphism g definedon N ,

M ⊗S (g f )

∑i

mi ⊗ni

=∑i

mi ⊗ ((g f )(ni))

= (M ⊗S g)

∑i

mi ⊗ f (ni)

= (M ⊗S g)(M ⊗S f )

∑i

mi ⊗ni

.Similarly, every bimodule homomorphism gives rise to a natural transformationbetween a pair of the aforementioned tensor product functors. In particular, for (R,S)-bimodules M and N , and a bimodule homomorphism f : M → N , we get a naturaltransformation α : M ⊗S − → N ⊗S − defined, for any left S-module A, ai ∈ A, andmi ∈M, by

αA

∑i

mi ⊗ ai

=∑i

f (mi)⊗ ai .

This is natural because, for any left module homomorphism g : A → B for left S-modules A,B, we have

N ⊗S (g) αA

∑i

mi ⊗ ai

= (N ⊗S (g))

∑i

f (mi)⊗ ai

=

∑i

f (mi)⊗ g(ai)

= αB

∑i

mi ⊗ g(ai)

= αB (M ⊗S (g))

∑i

mi ⊗ ai

.

27

This gives a 2-functor F : Bim→ Cat defined by

R 7→ RMod,

RMS 7→ RMS ⊗S −, and(RMS → RNS) 7→ (M ⊗S −→N ⊗S −).

To see that F is a 2-functor, we first note that, for two composable bimodules RMS andSNT ,

F(TMS SNR) = F(TMS ⊗S SNR)= TMS ⊗S SNR ⊗R −= (TMS ⊗S −) (SNR ⊗R −)= F(TMS) F(SNR).

Next, for two vertically composable bimodule homomorphisms f : RMS → RNS andg : RNS → RPS , and for a left S-module A,

F(g 1 f )A

∑i

mi ⊗ ai

=∑i

(g 1 f )(mi)⊗ ai

= F(g)A

∑i

f (mi)⊗ ai

= (F(g)A 1 F(f )A)

∑i

mi ⊗ ai

.Finally, for two horizontally composable bimodule homomorphisms f : SNT → SN

′T

and g : RMS → RM′S , and left T -module A,

F(g 0 f )A

∑i

mi ⊗ni ⊗ ai

=

∑i

g(mi)⊗ f (ni)⊗ ai

= (F(g) 0 F(f ))

∑i

mi ⊗ni ⊗ ai

.1.6 Morita Theory Revisited

We now look for ways of applying knowledge of bicategories, and in particular Bim,towards an understanding of Morita theory. We begin by addressing how Bim nicelydeals with the concept of Morita equivalence.

Definition 1.6.1 (Isomorphism of objects). Given a category C and two objects A,B ∈Ob(C), an isomorphism f : A→ B is a morphism which is both right-invertible andleft-invertible.

28

Definition 1.6.2 (Equivalence of objects). Given a bicategory C, two objects A,B ∈Ob(C) are called equivalent if there exist morphisms f : A → B and g : B → A suchthat f g idB and gf idA. The maps f and g satisfying this property are calledequivalences.

Theorem 1.6.3. The rings R and S are Morita equivalent if and only if R and S areequivalent in Bim.

Proof. By Corollary 1.6.18, we know that if R and S are Morita equivalent, then thereare 1-morphisms SQR and RPS in Bim such that when horizontally composed,

RPS ⊗S SQR RRR 1RMod and SQR ⊗R RPS SSS 1

SMod,

showing that RPS is an isomorphism S→ R in Bim. By again applying Corollary 1.6.18,the converse holds by inspection.

Now, we move towards giving a proof of Corollary 1.6.18 using a bicategoricalperspective. First, we must understand exact sequences so that we can deal withTheorem 1.6.17.

Definition 1.6.4 (Exact sequence). A sequence of left R-modules and module homo-morphisms

· · · →Mn+1fn+1−−−→Mn

fn−−→Mn−1→ ·· ·

is called an exact sequence if imfn+1 = kerfn for all n.

Definition 1.6.5 (Short exact sequence). A sequence of the form

0→ Af−→ B

g−→ C→ 0,

where 0 is the trivial module, is called a short exact sequence

Definition 1.6.6 (Left exact functor). A functor T : RMod→ Ab is called left exact ifexactness of

0→ Af−→ B

g−→ C

implies exactness of

0→ T (A)T (f )−−−−→ T (B)

T (g)−−−−→ T (C).

Definition 1.6.7 (Right exact functor). A functor T : RMod→ Ab is called right exact ifexactness of

Af−→ B

g−→ C→ 0

implies exactness of

T (A)T (f )−−−−→ T (B)

T (g)−−−−→ T (C)→ 0.

29

Lemma 1.6.8 ([Rot09, Theorem 2.63]). Given a right R-module M, the functor

M ⊗R − : RMod→ Ab

is right exact.

It is also useful to know how to deal with direct sums in module categories. This iswhere the notions of preadditive categories and additive functors can be used.

Definition 1.6.9 (Preadditive category). A category C is a preadditive category if,

• for every A,B ∈Ob(C), there is a binary operation

+A,B : HomC(A,B)×HomC(A,B)→HomC(A,B),

referred to simply as +;

• for every A,B ∈Ob(C), (HomC(A,B),+) is an abelian group with identity 0A,B;

• for every A,B,C ∈ Ob(C), f ,g ∈ HomC(A,B), and h ∈ HomC(B,C), we have h(f +g) = hf + hg;

• and for every A,B,C ∈ Ob(C), f ,g ∈ HomC(A,B), and h ∈ HomC(C,A), we have(f + g)h = f h+ gh.

Definition 1.6.10 (Biproduct). Given a preadditive category C, a biproduct for objectsA,B ∈Ob(C) is a diagram

A C Bi1

p1

p2

i2

whose morphisms p1,p2, i1, i2 satisfy

p1i1 = 1A, p2i2 = 1B, i1p1 + i2p2 = 1C .

Example 1.6.11. Given a ring R, the category of left R-modules RMod is a preadditivecategory: addition of module homomorphisms is pointwise addition and the finitedirect sum of modules is a biproduct.

Definition 1.6.12 (Additive functor). Given preadditive categories C and D, a functorF : C → D is an additive functor if for every A,B ∈ Ob(C), the map HomC(A,B) →HomD(FA,FB) is an abelian group homomorphism.

Proposition 1.6.13 ([Wei94, Theorem 2.6.1]). If F : C →D and G : D→ C are additivefunctors that form an adjoint pair (F,G), then F is right exact and G is left exact.

Lemma 1.6.14 ([Rot09, p. 74]). Given a right R-module M and a left R-module N , thefunctors

M ⊗R − : RMod→ Ab and −⊗RN : ModR→ Ab

are additive functors.

30

Proposition 1.6.15 ([ML98, p. 197]). Given preadditive categories C and D where C hasall binary biproducts, then a functor T : C →D commutes with biproducts if and only if Tis an additive functor.

According to [Mey, p. 1], Theorem 1.6.17 was simultaneously found by Eilenberg,Gabriel, and Watts. We follow Watts’s proof. But first, we need a lemma.

Lemma 1.6.16. For a left R-module M, M RRR ⊗RM as left R-modules.

Proof. Consider the map f : RRR ⊗RM → M defined by f (∑i ri ⊗mi) =

∑i rimi for

ri ∈ R and mi ∈M. As

f

a∑i=0

ri ⊗mi +b∑

i=a+1

ri ⊗mi

= f

a+b∑i=0

ri ⊗mi

=a+b∑i=0

rimi

=a∑i=0

rimi +b∑

i=a+1

rimi

= f

a∑i=0

ri ⊗mi

+ f

b∑i=0

ri ⊗mi

,and

f

r∑i

ri ⊗mi

= f

∑i

rri ⊗mi

=

∑i

rrimi

= r∑i

rimi

= rf

∑i

ri ⊗mi

,we see that f is a module homomorphism. It is evident that f is surjective as, for anym ∈M, f (1R ⊗m) =m. Finally, f is injective, as

f

∑i

ri ⊗mi

= 0 =⇒∑i

rimi = 0

=⇒∑i

1⊗ rimi = 0

=⇒∑i

ri ⊗mi = 0.

Together, this shows that f is an isomorphism of left R-modules.

31

Theorem 1.6.17 ([Wat60, Theorem 1]). Let T : RMod → SMod be a right-exact func-tor that commutes with direct sums. Then there is an (S,R)-bimodule Q and a naturalequivalence of functors ψ : Q⊗R −→ T .

Proof. Given M ∈Ob(RMod) and m ∈M, define φm : R→M by φm(r) = rm. We definea function ψM0 : TR ×M → TM by ψM0 (r ,m) = (Tφm)(r) for r ∈ TR and m ∈M. Notethat T is additive by Proposition 1.6.15. Further, ψR0 : TR ×R→ TR defines a rightaction, because for r ∈ TR, r1, r2 ∈ R,

ψR0 (ψR0 (r , r1), r2) = ψR0 ((Tφr1)(r), r2)

= (Tφr2)((Tφr1)(r))

= T (φr2 φr1)(r) by functoriality of T

= T (φr1r2)(r)

= ψR0 (r , r1r2);

ψR0 (r , r1 + r2) = (Tφr1+r2)(r)

= T (φr1 +φr2)(r)

= (Tφr1)(r) + (Tφr2)(r) by additivity of T

= ψR0 (r , r1) +ψR0 (r , r2);

for r1, r2 ∈ TR,r ∈ R,

ψR0 (r1 + r2, r) = T (φr)(r1 + r2)= T (φr)(r1) + T (φr)(r2) since Tφr is a module homomorphism

= ψR0 (r1, r) +ψR0 (r2, r);

and for 1R ∈ R, r ∈ TR,

ψR0 (r ,1R) = (Tφ1R)(r) = (T 1R)(r) = 1TR(r) = r .

Finally, the left and right actions on TR commute: for s ∈ S, r ∈ R, r ∈ TR,

sψR0 (r , r) = s(Tφr)(r) = (Tφr)(sr) = ψR0 (sr, r).

Thus TR is an (S,R)-bimodule. TR will henceforth be denoted by Q.Note that ψM0 , for M ∈ Ob(RMod) is R-balanced because it is defined by an action.

Thus, ψM0 descends to an R-homomorphism ψM : Q ⊗RM → TM, and the family(ψM)M∈Ob(RMod) is a natural transformation ψ : Q ⊗R − → T . For any left R-modulehomomorphism f : M→M ′,

T (f ) ψM(x⊗m) = T (f )(Tφm(x))= T (f φm)(x)= T (φf (m))(x)

= ψM′(x⊗ f (m))

= ψM′Q⊗R (f )(x⊗m).

32

Further, ψR is a natural isomorphism, as C ⊗R R = TR⊗R R TR by Lemma 1.6.16.T commutes with direct sums, and so does Q ⊗R − by Lemma 1.6.14, so ψF is anisomorphism whenever F is a free left R-module, as F is the direct sum of copies of R.

Let M ∈ Ob(RMod). By Proposition 1.2.20, we can obtain a free left R-module Fadmitting a short exact sequence:

0→ L→ F→M→ 0

Seeing that T and Q ⊗R − are right exact (the latter by Lemma 1.6.8), the followingdiagram of exact rows commutes.

Q⊗R L Q⊗R F Q⊗RM 0

T L T F TA 0

α1

ψL

β1

ψF ψM

α2 β2

Note that ψL and ψM are surjective because they are right actions on T L and TM,respectively. To show that ψM is injective, we “chase” the diagram.

Let a ∈ kerψM . Since β1 is surjective, there is b ∈ Q ⊗R F such that β1(b) = a. Letc = ψF(b) ∈ T F. Then

β2(c) = β2(ψF(b)) = ψM(β1(b)) = ψM(a) = 0.

Since c ∈ kerβ2, we know that c ∈ imα2, so we can pick d ∈ T L such that α2(d) = c.Since ψL is surjective, we can pick e ∈Q⊗R L such that ψL(e) = d. Since ψF is injective,b is the unique preimage of c through ψF . Since the diagram commutes, we know thatα1(e) = b. Finally, by exactness,

a = β1(b) = β1(α1(e)) = 0.

Thus, ψM is injective, and hence an isomorphism.

We can now use our knowledge of Bim to prove the following corollary.

Corollary 1.6.18 ([Mey, Corollary 1.1]). The rings R and S are Morita equivalent if andonly if there is an (R,S)-bimodule P and an (S,R)-bimodule Q such that RPS ⊗S SQR RRRand SQR ⊗R RPS SSS as bimodules.

Proof. (⇐= ) Using the 2-functor F defined previously, we have that

RPS ⊗S SQR ⊗R − = F(RPS ⊗S SQR) F(RRR) = RRR ⊗R −,

so RPS ⊗S SQR⊗R− 1RMod by Lemma 1.6.16. Similarly, SQR⊗R RPS ⊗S − 1

SMod. ThusRPS ⊗S − is an equivalence of categories SMod→ RMod.

( =⇒ ) Let T : RMod → SMod be an equivalence of categories. It is well-knownthat equivalences are additive, and so T is additive. Since T is additive and theleft adjoint to its inverse, we see that T commutes with direct sums and is right

33

exact by Proposition 1.6.13, so T satisfies the conditions of Theorem 1.6.17. Thusthere is an (S,R)-bimodule Q such that Q ⊗R − is equivalent to T . Similarly, for theinverse equivalence U : SMod→ RMod, there is an (R,S)-bimodule P such that P ⊗S −is equivalent to U . Thus

F(RRR) = RRR ⊗R − 1RMod RPS ⊗S SQR ⊗R − = F(RPS ⊗S SQR),

so RRR RPS ⊗S SQR, and similarly SSS SQR ⊗R RPS .

We now give an example of how Corollary 1.6.18 can be applied to determining aMorita equivalence.

Example 1.6.19. Let R be a ring and consider its n×n matrix ring Mn(R). Then the setof n× 1 matrices Mn×1(R) is a left Mn(R)-module with respect to matrix multiplicationand a right R-module with respect to scalar multiplication, and both actions commute,so Mn×1(R) is a (Mn(R),R)-bimodule. Similarly, M1×n(R) is an (R,Mn(R))-bimodule.

We check that f : M1×n(R)×Mn×1(R)→ R defined by

f ((r1, . . . , rn), (s1, . . . , sn)ᵀ) =n∑i=1

risi

is Mn(R)-balanced and g : Mn×1(R)×M1×n(R)→Mn(R) defined by

g((r1, . . . , rn)ᵀ, (s1, . . . , sn)) = (risj)

is R-balanced. However, it is evident that both of these maps are given by matrixmultiplication, and since matrix multiplication is associative and distributive, we havethat f and g are Mn(R)-balanced and R-balanced, respectively. Thus f and g bothdescend to well-defined bimodule homomorphisms f : M1×n(R)⊗Mn(R) Mn×1→ RRRand g : Mn×1 ⊗RM1×n(R)→Mn(R) Mn(R)Mn(R) defined by

f ((r1, . . . , rn)⊗ (s1, . . . , sn)ᵀ) =n∑i=1

risi

andg((r1, . . . , rn)ᵀ ⊗ (s1, . . . , sn)) = (risj).

We check that f and g are bimodule isomorphisms. First, f is surjective: for anyr ∈ R,

f ((r,1, . . . ,1)⊗ (1, . . . ,1)ᵀ) = r.

Note that every simple tensor (r1, . . . , rn)⊗ (s1, . . . , sn)ᵀ can be rewritten in a particular

34

way:

(r1, . . . , rn)⊗ (s1 . . . sn)ᵀ = (r1, . . . , rn)⊗ s1 0 ··· 0............

sn 0 0 0

(1,0, . . . ,0)ᵀ

= (r1, . . . , rn)

s1 0 ··· 0............

sn 0 0 0

⊗ (1,0, . . . ,0)ᵀ

=

∑i

risi ,0, . . . ,0

⊗ (1,0 . . . ,0)ᵀ

The map is thus also injective:

f ((r1, . . . , rn)⊗ (s1, . . . , sn)ᵀ) = 0 =⇒n∑i=1

risi = 0

=⇒

∑i

risi ,0, . . . ,0

⊗ (1,0, . . . ,0)ᵀ = 0

=⇒ (r1, . . . , rn)⊗ (s1, . . . , sn)ᵀ = 0.

Since f is R-balanced, it is a bimodule homomorphism, and hence a bimodule isomor-phism. Now, g is surjective:

g((0, . . . ,0, 1︸︷︷︸i-th position

,0, . . . ,0)ᵀ ⊗ (0, . . . ,0, 1︸︷︷︸j-th position

,0, . . . ,0)) = Eij ,

and Eij | 1 ≤ i ≤ n,1 ≤ j ≤ n forms a basis for Mn(R). Note that every simple tensor(r1, . . . , rn)⊗ (s1, . . . , sn)ᵀ can be rewritten in a particular way:

(r1, . . . , rn)ᵀ ⊗ (s1, . . . , sn) = ((1,0, . . . ,0)ᵀr1 + · · ·+ (0, . . . ,0,1)ᵀrn)⊗ (s1, . . . , sn)= (1,0, . . . ,0)ᵀr1 ⊗ (s1, . . . , sn) + · · ·+ (0, . . . ,0,1)ᵀrn ⊗ (s1, . . . , sn)= (1,0, . . . ,0)ᵀ ⊗ (r1s1, . . . , r1sn) + · · ·+ (0, . . . ,0,1)ᵀ ⊗ (rns1, . . . , rnsn).

We thus also see that g is injective:

g((r1, . . . , rn)ᵀ ⊗ (s1, . . . , sn)) = 0 =⇒∀i,∀j, risj = 0

=⇒ (1,0, . . . ,0)ᵀ ⊗ (r1s1, . . . , r1sn)+ · · ·+ (0, . . . ,0,1)ᵀ ⊗ (rns1, . . . , rnsn) = 0

=⇒ (r1 . . . , rn)ᵀ ⊗ (s1, . . . , sn) = 0.

Since g is Mn(R)-balanced, it is a bimodule homomorphism, and hence a bimoduleisomorphism.

By Corollary 1.6.18, we see that R and Mn(R) are Morita equivalent, as we sawearlier in Example 1.3.3.

35

In the following two lemmas and one proposition, we attempt to understand Defini-tion 1.4.16 from a bicategorical perspective. That is, we explore what it means for abimodule to be faithfully balanced in Bim.

Lemma 1.6.20. The map f : R→ EndR(RR) defined by f (r) = φr , where φr is left multi-plication by r, is a ring isomorphism.

Proof. We first show that all the endomorphisms of RR are left multiplications byelements of R. Let ζ ∈ EndR(RR). Then, for s ∈ R,

ζ(s) = ζ(1s) = ζ(1)s = φζ(1)(s).

Now we show that every left multiplication by an element of R is a right R-modulehomomorphism. Let r, r ′ ∈ R and m ∈ RR. Then

φr(mr′) = rmr ′ = φr(m)r ′.

Also, for m′ ∈ RR

φr(m+m′) = r(m+m′) = rm+ rm′ = φr(m) +φr(m′).

Finally, we show that f is a ring homomorphism. For r, r ′ ∈ R and m ∈ RR,

f (r + r ′)(m) = φr+r ′ (m) = (r + r ′)m = rm+ r ′m = φr(m) +φr ′ (m) = f (r)(m) + f (r ′)(m);

f (rr ′)(m) = φrr ′ (m) = rr ′m = φr(φr ′ (m)) = (φr φr ′ )(m) = f (r)f (r ′)(m);

andf (1)(m) = φ1(m) = 1m =m.

Since f is a ring homomorphism, every module endomorphism of RR is a left multipli-cation by some element of R, and every left multiplication by some element of R is amodule endomorphism, we see that f is a ring isomorphism.

Lemma 1.6.21. The map f : Rop → EndR(RR) defined by f (r) = ψr , where ψr is rightmultiplication by r, is a ring isomorphism.

Proposition 1.6.22. Given rings R and S, if RMS is a faithfully balanced (R,S)-bimodule,then the maps f : End(

ZRR)→ End(

ZMS) and g : End(SopS

opZ

)→ End(RMZ) defined for

r ∈ R, s ∈ S, and m ∈M by

f (φr)(m) = r ·m and g(ψs)(m) =m · s

are ring isomorphisms, where φr is left multiplication by r and ψs is right multiplication bys.

Proof. By the proofs of Lemma 1.6.20 and Lemma 1.6.21, it is clear that f and gare well-defined. Also, End(

ZRR) = End(RR), End(

ZMS) = End(MS), End(SopS

opZ

) =End(SopSop), and End(RMZ

) = End(RM). Note that f is the map from Lemma 1.6.20composed with λM from Definition 1.4.16, both of which are ring isomorphisms, andg is the map from Lemma 1.6.21 composed with ρM from Definition 1.4.16, both ofwhich are also ring isomorphisms. Thus f and g are both ring isomorphisms.

36

We now quote two propositions from Rachel Brouwer who studied the use of Bim inMorita theory.

Proposition 1.6.23 ([Bro03, Proposition 2.4.1]). Given rings R, and S, let RMS be an(R,S)-bimodule. Then RMS is an invertible morphism in Bim if and only if

• MS is a progenerator for ModS ; and

• R EndSop(MS).

Proposition 1.6.24 ([Bro03, Subsection 2.4]). Assuming one of Theorem 1.6.3 and Theo-rem 1.4.24, one can prove the other using Proposition 1.6.23.

37

2 Dualities in Representation Theory

2.1 Group Representations: Connection to Modules

Here, we introduce group representations, and we show that they are just a certaintype of module. We use this fact to study group representations using module theory.

Definition 2.1.1 (Group representation). Given a group G and a field K, a (linear)representation of G (over K) is a group homomorphism

G→GL(V ),

where V is a finite-dimensional vector space over K with dim(V ) ≥ 1.

Definition 2.1.2 (Group algebra). Given a group G and a field K, the group algebraK[G] of G over K is the algebra with the elements of G as a basis and with multiplica-tion on the basis elements given by group multiplication.

Note that K[G] has the multiplicative identity 1G, meaning that K[G] can be seenas a ring. Note also that the elements of K[G] for any K and G are of the form∑g∈G agg. Multiplication on arbitrary elements of K[G] is given by the bilinearity of

the multiplication on the basis elements:∑g∈G

agg

∑h∈G

bhh

=∑k∈G

∑gh=k

agbh

k.Definition 2.1.3 (Endomorphism ring). Given a vector space V , the endomorphismring of V , denoted End(V ), is the set of all linear endomorphisms of G where additionin End(V ) is pointwise addition, and multiplication in End(V ) is composition of linearmaps.

Proposition 2.1.4 ([Web14, Proposition 1.2]). Given a group G and a field K, everyK[G]-module provides a representation of G over K, and every representation of G over Kinduces a K[G]-module action on V .

Proof. Let ρ : G→GL(V ) be a representation on some K-vector space V . Then∑g∈G

agg

v =∑g∈G

agρ(g)(v)

38

defines a K[G]-module action on V . We verify this claim. First, distributivity over thevector addition is satisfied:∑g∈G

agg

(v + v′) =∑g∈G

agρ(g)(v + v′)

=∑g∈G

ag(ρ(g)(v) + ρ(g)(v′)) distributivity of matrix multiplication

=∑g∈G

agρ(g)(v) + agρ(g)(v)

=∑g∈G

agρ(g)(v) +∑g∈G

agρ(g)(v′)

=

∑g∈G

agg

v +

∑g∈G

agg

v′.Distributivity over the group algebra addition is satisfied:∑

g∈Gagg +

∑g∈G

bgg

v =

∑g∈G

(ag + bg)g

v=

∑g∈G

(ag + bg)ρ(g)v

=∑g∈G

agρ(g)v + bgρ(g)v

=∑g∈G

agρ(g)v +∑g∈G

bgρ(g)v

=

∑g∈G

agg

v +

∑g∈g

bgg

v.

39

The module action commutes with the group algebra multiplication:∑g∈G

agg

∑h∈G

bhh

v =

∑k∈G

∑gh=k

agbh

kv

=∑k∈G

∑gh=k

agbh

ρ(k)v

=∑gh∈G

∑g∈G

ag

∑h∈H

bh

ρ(g)ρ(h)v

=∑g∈G

agg

∑h∈G

bhρ(h)v

=

∑g∈G

agg

∑h∈G

bhh

v .

The identity preserves elements:

1Gv = ρ(1G)v = v.

Thus, every representation induces a module action as specified above.Now suppose that V is a K[G]-module. The module action induces a ring homomor-

phism σ : K[G]→ End(V ) defined by

σ

∑g∈G

agg

(v) =

∑g∈G

agg

v.Note that σ descends to a map σ : G→ GL(V ). First, for any g ∈ G, ρ(g) has an inverse:

ρ(g)ρ(g−1) = ρ(gg−1) = ρ(1G) = 1End(V ).

Thus, the range of σ is at most GL(V ). Also, σ is evidently a group homomorphism.Thus σ is a representation on G.

Definition 2.1.5 (Irreducible representation). Given a group G and a field K, theirreducible representations of G over K are those representations whose correspondingK[G]-modules are simple.

Henceforth, given a group G and a field K, the representation ρ : G→ GL(V ) of Gover K will refer to the corresponding K[G]-module V given by the proof of Proposi-tion 2.1.4.

Definition 2.1.6 (Direct sum of representations). Given a group G, a field K, and tworepresentations V1 and V2, the direct sum V1⊕V2 is a representation of G whose actionis given by

g(v1,v2) = (gv1, gv2).

40

Definition 2.1.7 (Tensor product of representations). Given a group G, a field K, andtwo representations V1 and V2, the tensor product V1 ⊗V2 is a representation whoseaction is given by

g(v1 ⊗ v2) = (gv1 ⊗ gv2).

Denote by Sn the symmetric group on n symbols.

Theorem 2.1.8 (Maschke, [Lan02, Theorem 1.2]). Let G be a finite group of order n,and let K be a field whose characteristic does not divide n. Then all K[G]-modules aresemisimple.

Corollary 2.1.9. The complex representations of Sn are decomposable into direct sums ofirreducible representations.

Definition 2.1.10. A Young diagram is a collection of boxes in left-justified rows withthe number of boxes in each row less than or equal to the number of boxes in theprevious row.

Given a positive integer n, a partition of n is notated here as a weakly decreasing,eventually zero sequence (λ1,λ2, . . . ,0, . . .) of positive integers such that

∑∞i=1λi = n.

Given a Young diagram with n boxes and k rows, and letting κi be the number ofboxes in the i-th row, the sequence (κ1, . . . ,κk ,0, . . .) is a partition of n. Conversely, apartition (λ1, . . . ,λk ,0, . . .) of n gives a Young diagram whose i-th row has λi boxes.

Example 2.1.11. The partition (7,6,4,3,0, . . .) corresponds to the following Youngdiagram.

Proposition 2.1.12 ([Ful97, Proposition 7.2.1]). The Young diagrams with n boxes are inbijection with the irreducible complex representations of Sn.

Definition 2.1.13 (Polynomial representation). A representation V of GL(V ) over Cis called a polynomial representation if, for all g ∈GL(V ), the entries of the matrix givenby the action of g on V are polynomial in the matrix entries of g.

Proposition 2.1.14 ([Ful97, Theorem 8.2.2]). The Young diagrams with at most n rowsare in bijection with the irreducible polynomial representations of GLn(C).

It is a well-known fact that the general linear group GLn(C) is an example of whatis called a reductive group. A key characteristic of reductive groups is that theirrepresentations over fields of characteristic 0 are decomposable into direct sums ofirreducible representations. See [Bor91, p. 273] for some discussion on the latterstatement.

41

Lemma 2.1.15 (Schur, [Coh03, Lemma 6.3.2]). Let K be an algebraically closed field, A aK-algebra, and U,V any two simple A-modules, finite-dimensional over K. Then

HomA(U,V )

K if U V

0 otherwise.

where the first isomorphism is of algebras if U = V .

The following proposition is key in the proof of Proposition 2.2.2.

Proposition 2.1.16. Let G and H be two groups such thatC[G]Mod and

C[H]Mod aresemisimple module categories with the same number of isomorphism classes of simpleobjects. Then C[G] and C[H] are Morita equivalent.

Proof. Since categories are equivalent to their skeletons by Proposition 1.1.15, itsuffices to show that a skeleton C of

C[G]Mod is equivalent to a skeleton D ofC[H]Mod.

SinceC[G]Mod and

C[H]Mod are semisimple module categories, every object in eithercategory is decomposable into a direct sum of simple objects. Let V1,V2, . . . ,Vn be thesimple objects of C, and let W1,W2, . . . ,Wn be the simple objects of D. Let F : CMod→DMod be a functor whose map on objects is defined by

F

n⊕i=1

V ⊕nii

=n⊕i+1

W ⊕nii

for ni ∈N. It is seen immediately that F is surjective.Let

ιa,b : Va→n⊕i=1

V ⊕nii

for 1 ≤ a ≤ n and 1 ≤ b ≤ ni be the canonical injection and let

πa,b :⊕i=1

V⊕mji → Va

for 1 ≤ a ≤ n and 1 ≤ b ≤mj be the canonical projection. Then for every map

f ∈HomC[G]

n⊕i=1

V ⊕nii ,n⊕j=1

V⊕mjj

and for every 1 ≤ i ≤ n,1 ≤ j ≤ n,1 ≤ a ≤ ni , and 1 ≤ b ≤mj , we get a collection of maps

ηi,j,a,b(f ) = πj,b f ιi,a ∈HomC[G](Vi ,Vj).

It is noteworthy that η is compatible with composition: for

g ∈HomC[G]

n⊕j=1

V⊕mjj ,

n⊕k=1

V⊕`kk

42

and 1 ≤ c ≤ `k,

ηi,k,a,c(g f ) = πk,c g f ιi,a =∑j,b

πk,c g ιj,b πj,b f ιi,a =∑j,b

ηj,k,b,c(g)ηi,j,a,b(f ).

Further, we know by Lemma 1.2.25 and Lemma 1.2.26 that

f 7→n⊕

i,j=1

ni⊕a

mj⊕b

ηi,j,a,b(f )

is an abelian group isomorphism.By Lemma 2.1.15, we know that ηi,j,a,b(f ) = 0 when i , j and that there is an

isomorphism of algebras φi : EndC[G](Vi) → End

C[H](Wi) for 1 ≤ i ≤ n. By againapplying Lemma 1.2.25 and Lemma 1.2.26, we find an isomorphism

HomC[G]

n⊕i=1

V ⊕nii ,n⊕j=1

V⊕mjj

→HomC[G]

n⊕i=1

W ⊕nii ,n⊕j=1

W⊕mjj

,and we set the map on morphisms of F to be this map:

F(f ) =n∑i=1

ni∑j=1

mi∑k=1

φi(ηi,i,j,k(f ))

This map is compatible with composition as well:

F(g f ) =n∑i=1

ni∑j=1

mi∑k=1

φi(ηi,i,j,k(g f ))

=n∑i=1

ni∑j=1

mi∑k=1

φi(ηi,i,j,k(g)) φi(ηi,i,j,k(f ))

=

n∑i=1

ni∑j=1

mi∑k=1

φi(ηi,i,j,k(g))

n∑i=1

ni∑j=1

mi∑k=1

φi(ηi,i,j,k(f ))

= F(g) F(f ).

Note that the above composition of sums makes sense because these sums are coordinate-wise, and the composition is strictly across these coordinates; in other words, thecomposition is zero when the coordinates do not match.

Since the map of F on morphisms is an isomorphism, we see that F is fully faithful,as well as surjective as mentioned above. Thus, by Theorem 1.1.18, F is an equivalenceof categories.

43

2.2 Schur-Weyl Duality

In this section, we treat the classical Schur-Weyl duality theorem (Theorem 2.2.1) andwe show that it exhibits a Morita equivalence.

The notation now defined will be carried onto the statement of the next theorem.Let V denote a complex vector space of dimension n. Let V ⊗r denote the r-fold tensorproduct of V . We endow V ⊗r with the following C[GL(V )]-action: for g ∈GL(V ),

g(v1 ⊗ v2 ⊗ · · · ⊗ vr) = g(v1)⊗ g(v2)⊗ · · · ⊗ g(vr).

We also give V ⊗r the following C[Sr]-action: for σ ∈ Sr ,

σ (v1 ⊗ v2 ⊗ · · · ⊗ vr) = vσ−1(1) ⊗ vσ−1(2) ⊗ · · · ⊗ vσ−1(r).

It is evident that the C[GL(V )]-action and C[Sr]-action commute with one another,making V ⊗r a (C[GL(V )],C[Sr])-bimodule. For a given Young diagramD, letUD be theirreducible polynomial representation of GL(V ) corresponding to the Young diagramD, and let WD be the irreducible complex representation of Sr on V ⊗r correspondingto the Young diagram D.

Theorem 2.2.1 (Schur-Weyl duality, [How95, Section 2.4.3]). Under the joint action ofGL(V ) and Sr , the space V ⊗r decomposes into a direct sum

V ⊗r ⊕D

UD ⊗WD ,

where D runs over Young diagrams with r boxes and at most n rows.

Proposition 2.2.2. Given the algebra homomorphisms

C[GL(V )]α−→ End(V ⊗r)

β←−C[Sr]

induced by the module actions of C[GL(V )] and C[Sr] on V ⊗r , we have that imαMod isMorita equivalent to imβMod.

Proof. Let f ∈⊕

D End(UD). Then f =⊕

D fD for maps fD ∈ End(UD), and we canmap

ω : f 7→⊕D

(fD ⊗ idWD).

The map ω is injective: suppose g ∈⊕

D End(UD), g =⊕

D gD and that fD ⊗ idWD=

gD ⊗ idWDfor all D. Then for x ∈UD , y ∈WD nonzero,

fD ⊗ idWD= gD ⊗ idWD

=⇒ fD(x)⊗ y = gD(x)⊗ y=⇒ (fD(x)− gD(x))⊗ y = 0.

44

However, if fD(x)− gD(x) is nonzero, then we can find a basis of UD such that fD(x)−gD(x) is a basis vector. Similarly, y can be considered a basis vector of WD . However,this would allow (fD(x)− gD(x))⊗ y to be a basis vector for UD ⊗WD , and basis vectorscannot be 0. This is a contradiction. Thus fD(x)− gD(x) = 0 for all D, which impliesf = g.

We see thatΩ = ω(f ) : f ∈

⊕D

End(UD)

must consist entirely of maps of⊕

D End(UD ⊗WD) that commute with the C[Sr]-action on

⊕UD ⊗WD . That is, Ω commutes with imβ.

As left C[GL(V )]-modules,

UD ⊗WD (UD ⊗Cw1)⊕ (UD ⊗Cw2)⊕ · · · ⊕ (UD ⊗CwdimWD) U⊕dimWD

D ,

where w1,w2, . . . ,wdimWD is a basis for WD . Thus,⊕

D

UD ⊗WD ⊕D

U⊕dimWDD .

Therefore, we have the isomorphism

End

⊕D

UD ⊗WD

End

⊕D

U⊕dimWDD

.Pulling out the direct sums with Lemma 1.2.25 and Lemma 1.2.26, we obtain

End

⊕D

U⊕dimWDD

⊕D,D ′

Hom(UD ,UD ′ )⊕(dimWD )(dimWD′ ).

We know that the UD are simple modules, however, and by Lemma 2.1.15, we see thatHom(UD ,UD ′ ) 0 when D ,D ′. Thus,⊕

D,D ′Hom(UD ,UD ′ )

⊕(dimWD )(dimWD′ ) ⊕D

Hom(UD ,UD)⊕(dimWD )(dimWD )

⊕D

Hom(U⊕dimWDD ,U⊕dimWD

D ′

)

⊕D

End(U⊕dimWDD

).

We wish to show that every map in imα is of the form⊕

D(fD ⊗ idWD). Let (FD)D be

a tuple of endomorphisms FD ∈ End(U⊕dimWDD

) End(UD ⊗WD). Then FD(u ⊗w) =∑

i,j aiui ⊗wj for any u ∈U , w ∈W .Note that every WD is a right C[Sr]-module. Theorem 1.2.30 tells us that for any of

the WD , and for any γ ∈ End(WD), and for any basis X, we can pick s ∈C[Sr] such that

45

ws = γ(w) for w ∈ X. Without loss of generality, let w = w1. Let γ be the projectiononto Cw1 =Cw, and let s be the element of C[Sr] which corresponds to γ . Then

FD (idUD ⊗s)(u ⊗w) = FD(u ⊗w1) = FD(u ⊗w) =∑i

aiui ⊗wi ,

and

(idUD ⊗s) FD(u ⊗w) = (idUD ⊗s)(∑i

aiui ⊗wi) = a1u1 ⊗w1 = a1u1 ⊗w.

Since FD must commute with the action of C[Sr] on WD , for any w ∈WD , it happensthat FD(u ⊗w) = a1u1 ⊗w, and we obtain that, as an endomorphism of UD ⊗WD , wewill have

FD(u ⊗w) = u′ ⊗wfor any w ∈ WD , and for any u ∈ UD . Hence FD = fD ⊗ idWD

for some fD ∈ End(U ).

Thus ω surjects onto End(U⊕dimWDD

). This makes

ω :⊕D

End(UD)→⊕D

End(U⊕dimWDD

)an isomorphism of algebras.

Now, since imα ⊕

D End(U⊕dimWDD

), we see that by the isomorphism ω,

imα ⊕D

End(UD),

and by similar reasoning,imβ

⊕D

End(WD).

We know that End(UD) for eachD is a simple ring as End(UD) MdimUD (C),MdimUD (C)is Morita equivalent to C by Example 1.3.3, C is a simple ring as C is a field, and byProposition 1.2.23, C has one simple module. The simple modules of

⊕D End(UD)

are in correspondence with the maximal ideals of⊕

D End(UD) by Proposition 1.2.23which are in turn enumerated by the Young diagrams D as the maximal ideals are ofthe form

End(UD1)⊕ · · · ⊕End(UDi−1

)⊕ 0 ⊕End(UDi+1)⊕ · · · ⊕End(UDd )

where d is the number of Young diagrams. The same reasoning applies to show thatthe simple modules of

⊕D End(WD) are enumerated by the Young diagrams D. Thus

imαMod and imβMod are semisimple module categories with the same number ofisomorphism classes of simple modules. Finally, by Proposition 2.1.16, we see thatsince imαMod and imβMod have the same number of isomorphism classes of simplemodules, imα and imβ must be Morita equivalent.

It is of note that imβ in the above proposition is called the Schur algebra SC

(n,r),and has been the subject of extensive study.

46

2.3 GLn −GLm and Skew GLn −GLm Duality

We now discuss two other dualities found in [How95] in which one finds Moritaequivalence.

Definition 2.3.1 (Tensor algebra). Given a vector space V , we call the associativealgebra T(V ) =

⊕∞i=0V

⊗i with associative product

(v1 ⊗ · · · ⊗ vk)(w1 ⊗ · · · ⊗wm) = v1 ⊗ · · · ⊗ vk ⊗w1 ⊗ · · · ⊗wm

the tensor algebra on V .

Definition 2.3.2 (Symmetric algebra). Given a vector space V , let I be the two-sidedideal of T(V ) generated by x⊗ y − y ⊗x for all x,y ∈ V . Then we call S(V ) = T(V )/I thesymmetric algebra on V .

Definition 2.3.3 (Exterior algebra). Given a vector space V , let I be the two-sidedideal of T(V ) generated by x⊗ y + y ⊗ x for all x,y ∈ V . Then we call Λ(V ) = T(V )/Ithe exterior algebra on V .

Note that given a group algebraK[G] and a (left)K[G]-module V , the tensor algebraT(V ) has a natural left K[G]-action:

g(v1 ⊗ · · · ⊗ vn) = (gv1)⊗ · · · ⊗ (gvn)

for all g ∈ G, which is extended by linearity. Further, the ideal (x⊗ y − y ⊗ x) ⊂ T(V ) isclosed under the R-action on the T(V ).

r(x⊗ y − y ⊗ x) = r(x⊗ y)− r(y ⊗ x) = (rx)⊗ (ry)− (ry)⊗ (rx).

Thus, the R-action on T(V ) descends to an R-action on the symmetric algebra S(V ). Ina similar fashion, the R-action on T(V ) descends to an R-action on the exterior algebraΛ(V ).

The notation developed here carries on into the next two theorems. Let U and Vbe two complex vector spaces. We give U ⊗ V the following C[GL(U )]-action: forg ∈GL(U ),

g(u ⊗ v) = g(u)⊗ v.

We also give U ⊗V a right C[GL(V )]-action: for h ∈GL(V ),

(u ⊗ v)h = u ⊗ h(v).

TheC[GL(U )]- andC[GL(V )]-actions commute, makingU⊗V a (C[GL(U )],C[GL(V )])-module. We can thus treat S(U ⊗V ) and Λ(U ⊗V ) as (C[GL(U )],C[GL(V )])-modules.Let UD and V D be the irreducible polynomial representations of GL(U ) and GL(V ),respectively, corresponding to the Young diagrams D.

47

Theorem 2.3.4 ((GLn,GLm)-duality, [How95, 15-16]). As a (C[GL(U )],C[GL(V )])-module,S(U ⊗V ) decomposes into a direct sum

S(U ⊗V ) ⊕D

UD ⊗V D ,

where D runs over all Young diagrams with at most mindimU,dimV rows.

Corollary 2.3.5. Given the algebra homomorphisms

C[GL(U )]α−→ End(S(U ⊗V ))

β←−C[GL(V )]

induced by the module actions of C[GL(V )] and C[GL(V )] on S(U ⊗ V ), we have thatimαMod is equivalent to imβMod.

Proof. This is justified in the same way as Proposition 2.2.2.

Given a Young diagram D, its transpose Dᵀ is the Young diagram whose columnsare the rows of D (and thus whose rows are the columns of D).

Theorem 2.3.6 (Skew (GLn,GLm)-duality, [How95, 50]). As a (C[GL(U )],C[GL(V )])-module, Λ(U ⊗V ) decomposes into a direct sum

Λ(U ⊗V ) ⊕D

UD ⊗V Dᵀ ,

where D runs over all Young diagrams with at most n rows and with rows of length at mostm.

Corollary 2.3.7. Given the algebra homomorphisms

C[GL(U )]α−→ End(Λ(U ⊗V ))

β←− C[GL(V )]

induced by the module actions of C[GL(V )] and C[GL(V )] on Λ(U ⊗ V ), we have thatimαMod is equivalent to imβMod.

Proof. This is justified in the same way as Proposition 2.2.2.

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