Morita Equivalence

Home , Artinian module, Morita equivalence

Rabib Islam Department of Mathematics & Statistics University of Ottawa

Summer 2015 Contents

1 Morita Equivalence5 1.1 Category Theory: Basic Definitions and Theorems...... 5 1.2 Module Theory: Basic Definitions and Theorems...... 8 1.3 Morita Equivalence: Definition and Examples...... 14 1.4 Morita Theory...... 18 1.5 Bicategory of Bimodules...... 24 1.6 Morita Theory Revisited...... 28

2 Dualities in Representation Theory 38 2.1 Group Representations: Connection to Modules...... 38 2.2 Schur-Weyl Duality...... 44 − − 2.3 GLn GLm and Skew GLn GLm Duality...... 47

2 Preface

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

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

3 Introduction

The theory of modules over rings dates back to the late 19th century, and they were used to great success in an 1882 paper by Richard Dedekind and Anton Weber, [DW82], which was concerned with algebraic geometry. Rings, however, had not undergone axiomatic development until around 40 years later, at the hands of Emmy Noether and Wolfgang Krull. Categories, on the other hand, did not appear in the literature until 1945, when Samuel Eilenberg and Saunders Mac Lane published their paper [EM45]. While the original motivation for categories, functors, and natural transformations were certain topics of group theory and topology, category theory has become an immensely useful and 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 the information about the structure of that ring? The answer to this question is “no.” This leads to a new sense of “equivalence” of rings which is coarser than isomorphism. That is, two rings are “equivalent” when their categories of modules are equivalent (in the category-theoretical sense). Japanese mathematician Kiiti Morita is known for having stated and proved a theorem which gave an equivalent condition for this equivalence in an inﬂuential 1958 paper, [Mor58]. This theorem plays an important role in modern algebra, and in his honour, this equivalence of module categories between two rings has been dubbed “Morita equivalence.” In this paper, we begin by reviewing the material necessary to deﬁne Morita equivalence, and we examine two classical examples of Morita equivalence. We then delve into what is now called “Morita theory”, which is in regards to the key theorems of Morita 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 bimodules. We then look at ways in which this bicategorical perspective can be used in Morita theory, including the neat encapsulation such a perspective provides to the concept of Morita equivalence. Starting in the second chapter, we introduce the representation theory of groups, another old, yet current area of mathematics, and we examine how this connects with module theory. We then use this connection to identify some examples of Morita equivalence in representation theory, particularly in the case of representation-theoretic dualities of groups.

4 1 Morita Equivalence

1.1 Category Theory: Basic Deﬁnitions and Theorems

We begin by stating many of the basic category-theoretic notions used throughout this paper. 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 Hom (A,B) of morphisms between A and B; C

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

◦: Hom (B,C) × Hom (A,B) → Hom (A,C), C C C called composition; and ∈ C ∈ • for every object A Ob( ), a morphism idA Hom (A,A) called the identity morphism on A such that for every morphism f ∈ Hom C(X,Y ) where X,Y ∈ Ob(C), we have C ◦ ◦ idY f = f and f idX = f .

Deﬁnition 1.1.2 (Functor). Given categories C and C0, a functor F from C to C0, denoted F : C → C0, consists of

• a function F : Ob(C) → Ob(C0); and • for every two objects A,B ∈ Ob(C), a function

F : Hom (A,B) → Hom (F(A),F(B)), C C0 which must satisfy the following axioms: ∈ C • given an object A , we have F(idA) = idF(A); and • given objects A,B,C ∈ C and morphisms f : A → B and g : B → C, we have F(g ◦ f ) = F(g) ◦ F(f ).

5 Deﬁnition 1.1.3 (Natural transformation). Given categories C and C0, and functors F : C → C0 and G : C → C0, a natural transformation α from F to G, denoted α : F → G, is a family of maps → (αA : F(A) G(A))A ∈C such that for A,A0 ∈ C and f : A → A0, we have

G(f ) ◦ α = α ◦ F(f ). A A0 It is also said that α is natural in A. ∈ C If αA is an isomorphism for every A , then α is called a natural isomorphism.

Definition 1.1.4 (Equivalence of categories). Given categories C and C0, a functor F : C → C0 is called an equivalence of categories if there is a functor F0 : C0 → C and natural isomorphisms α : id → F0 ◦ F and α0 : id → F ◦ F0. C C0 Definition 1.1.5 (Product category). Given two categories C and D, the product category C × D is defined as the category with

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

• morphisms (f ,g) for f ∈ Hom (C,C0) and g ∈ Hom (D,D0) for objects C,C0 ∈ C D Ob(C) and D,D0 ∈ Ob(D);

• composition deﬁned as pointwise composition of morphisms: (f 0,g0) ◦ (f ,g) = (f 0 ◦ f ,g0 ◦ g); ∈ C ∈ D • and identity objects 1(C,D) = (1C,1D) for C Ob( ) and D Ob( ). Deﬁnition 1.1.6 (Bifunctor). A bifunctor F : A × B → C is a functor from the product category A × B to the category C.

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

• objects C for C ∈ Ob(C);

• morphisms Hom op (C,C0) = Hom (C0,C); C C • composition deﬁned in reverse: g ◦ op f = f ◦ g; C C • and the same identity objects as in C.

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

6 Example 1.1.9. Given a category C and A ∈ Ob(C), the functor Hom (A,−): C → Set is deﬁned by C

C 7→ Hom (A,C) C Hom (A,−)(f ): g 7→ f ◦ g C for f : C → C0 and g : A → C. There is also the contravariant functor Hom (−,A): C → Set, defined by C C 7→ Hom (C,A) C Hom (−,A)(f ): g 7→ g ◦ f C for f : C → C0 and g : C0 → A. We also have the bifunctor Hom (−,−): Cop × C → Set, defined by C (A,B) 7→ Hom (A,B) C Hom (−,−)(f ,g): h 7→ g ◦ h ◦ f C 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 there is a natural isomorphism

τ : Hom (F−,−) → Hom (−,G−), D C then (F,G) is called an adjoint pair, G is called the right adjoint of F, and F is called the left adjoint of G. Note that the condition that τ is natural means that τ is required to 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 is both 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), Hom (A,B) is a subcollection of Hom (A,B); D C ∈ D ∈ • for all A Ob( ), idA Hom (A,A); D • if f ,g are morphisms in D and the domain of g is the codomain of f , then g ◦ f is 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 subcategory D of C is called a full subcategory of C if, for every A,B ∈ Ob(D),

Hom (A,B) = Hom (A,B). D C

7 Definition 1.1.14 (Skeleton). Given a category C, a subcategory D of C is a skeleton of C 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 functor F : C → D is called full when the map of F on morphisms is surjective, and is called faithful if the map of F on morphisms is injective. If F is both full and faithful, then it is called fully faithful. Definition 1.1.17 (Essentially surjective functor). Given two categories C and D, a functor F : C → D is said to be essentially surjective if each object X ∈ Ob(D) is isomorphic 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 undergraduate 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 “+00 called addition such that (R,+) is an abelian group with identity 0 ∈ R; and

• a binary operation “·00 called multiplication that is associative, distributive over “+00, 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

4.1 Rx = x.

8 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 multiplication and right multiplication actions respectively. These modules are called the left and right regular modules, respectively. Example 1.2.5. Given a ring R, the n-fold direct product Rn is a left R-module and a right R-module with respect to left and right pointwise multiplication actions respectively.

Deﬁnition 1.2.6 (Module homomorphism). Given a ring R and left R-modules M,M0, a map f : M → M0 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 are modules and its morphisms are module homomorphisms. Deﬁnition 1.2.7 (Bilinear map). Given vector spaces X, Y , and Z over a ﬁeld K, a bilinear 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 of Morita equivalence.

9 Deﬁnition 1.2.8 (Algebra). Given a ﬁeld K, an (associative) algebra A over K is a vector space equipped with an associative, bilinear map A × A → A called multiplication in A.

Deﬁnition 1.2.9 (Jacobson radical). Given an algebra A and an A-module M, the Jacobson 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-module M, we say that M is an artinian module if, whenever there is a sequence ⊇ ⊇ ⊇ ⊇ ··· M M1 M2 M3 ∈ N ≥ of A-submodules of M, then there exists n0 such that Mn = Mn0 for 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 of ⇒ simple submodules S1 and S2 of A/ rad(A) one has S1 S2 = S1 = S2, then A is called a basic algebra.

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

Deﬁnition 1.2.14 (Orthogonal idempotent). Given a ring R, two idempotents e1 and e2 are called orthogonal idempotents if e1e2 = e2e1 = 0. An idempotent e , 0 is primitive if 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-module M, a left R-module N, and an 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 left R-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 map f : 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-module M and a left R-module N, if (T ,τ) and (T 0,τ0) are tensor products of M and N, then T and T 0 are isomorphic abelian groups. In other words, tensor products are unique up to isomorphism.

10 Proof. Since τ0 is R-balanced and T is a tensor product, there is a unique group homomorphism f : T → T 0 such that

τ0 = f ◦ τ.

Since τ is R-balanced and T 0 is a tensor product, there is a unique group homomorphism g : T 0 → T such that τ = g ◦ τ0. We thus obtain τ = g ◦ f ◦ τ and τ0 = f ◦ g ◦ τ0, which shows that f is a group isomorphism. Therefore, T T 0. Proposition 1.2.18. A tensor product exists for any right R-module M and left R-module N. { ∈ ∈ } Proof. Denote by X the module with basis em,n : m M,n N . Let Y be the subgroup of X generated by the following elements:

e − e − e ; m+m0,n m,n m0,n e − e − e ; m,n+n0 m,n m,n0 − emr,n em,rn, where m,m0 ∈ M,n,n0 ∈ N,r ∈ R. Denote by ι: M × N → X the canonical injection and π : X → X/Y the projection. Set α = π ◦ ι. We show that α is R-balanced. However, this is equivalent to checking that the images 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, there → is a unique group homomorphism h: X G defined by h(em,n) = f (m,n), and it is such that h ◦ ι = f . We show that the generators of A are in kerh. h(e −e −e ) = h(e )−h(e )−h(e ) = f (e )−f (e )−f (e ) = 0 m+m0,n m,n m0,n m+m0,n m,n m0,n m+m0,n m,n m0,n h(e − e − e ) = h(e ) − h(e ) − h(e ) = f (e ) − f (e ) − f (e ) = 0 m,n+n0 m,n m,n0 m,n+n0 m,n m,n0 m,n+n0 m,n m,n0 − − − 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 (and unique). Finally, we obtain that p ◦ α is the unique group homomorphism satisfying p ◦ 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. Since tensor products are unique up to isomorphism, we use the standard notation T = ⊗ ⊗ ⊗ M R N and τ(m,n) = m n. By an abuse of notation, we call M R N the tensor product of M and N. Definition 1.2.19 (Free module). Given a ring R, a free left R-module is a left R-module that is isomorphic to the direct sum of some copies (possibly an infinite number) of R.

11 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-module with B as a basis. Then, we construct a module homomorphism f : F → M induced by f (b) = b for b ∈ B. That is, define f by     X  X  f  r b  =  r b ,  i i  i i i i ∈ ∈ for ri R and bi B. Then f is evidently a surjection, and by the first isomorphism theorem, we have F/ kerf M. Definition 1.2.21 (Simple module). Given a ring R, a left R-module M is called simple if M has no non-zero proper submodule.

Deﬁnition 1.2.22 (Semisimple module). Given a ring R, a left R-module M is called semisimple 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 for some maximal ideal M of R, and for every maximal ideal M, R/M is isomorphic to a simple module of R.

Proof. Given a maximal ideal M, we know that R/M has no proper ideals. Furthermore, we know that every ideal of R/M corresponds to a proper submodule of R/M as an R-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, as Rn is a non-zero submodule of N, and thus Rn contains N. Thus R/I N for some ideal I of R. Now suppose I is not maximal. Then there is an ideal J of R such that I ( J ( R. However this means that J corresponds to some nontrivial proper ideal of R/I N, and so N has a nonzero proper submodule. This is a contradiction, so I must be 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) is semisimple.

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

12 and ⊕ → π2 : N1 N2 N2 ∈ ⊕ by π1(n1,0) = n1 and π2(n2,0) = n2. Then given f HomR(M,N1 N2), the map ◦ ◦ ⊕ → ⊕ ρ(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),

1 we deﬁne ρ− (f ,g) = h where h(m) = (f (m),g(m)). Then 1 ◦ ◦ ρ(ρ− (f ,g)) = ρ(h) = (π1 h,π2 h) = (f ,g). ∈ ⊕ Also, for f HomR(M,N1 N2), 1 1 ◦ ◦ ρ− (ρ(f )) = ρ− (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 and → ⊕ ι2 : M2 M1 M2 ∈ ⊕ by ι1(m1) = (m1,0) and ι2(m2) = (0,m2). Then, given f HomR(M1 M2,N), the map ◦ ◦ ⊕ → ⊕ σ(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).

13 It is also the case that σ has an inverse homomorphism. Given ∈ ⊕ (f ,g) HomR(M1,N) HomR(M2,N),

1 we deﬁne σ − (f ,g) = h where h(m1,m2) = f (m1) + g(m2). Then 1 ◦ ◦ σ(σ − (f ,g)) = σ(h) = (h ι1,h ι2) = (f ,g). ∈ ⊕ Also, for f HomR(M1 M2,N), 1 1 ◦ ◦ σ − (σ(f )) = σ − (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 ideals are 0 and the ring itself.

We see from the deﬁnition of a simple ring and Proposition 1.2.23 that simple rings each 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 is semisimple as a left R-module. Similarly, R is right semisimple if R is semisimple as a right R-module.

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

Due to this proposition, we will henceforth refer to left and right semisimple rings as semisimple. It may be of interest to the reader that this proposition is in fact a corollary 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-module ⊆ and write D = EndR(U). Let γ be any D-linear operator on U and let X U be any ﬁnite D-linear independent subset. Then there exists an element r ∈ R such that xr = γ(x) for all x ∈ X.

1.3 Morita Equivalence: Deﬁnition and Examples

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

Deﬁnition 1.3.1 (Morita equivalence). Two rings R and S are called Morita equivalent if 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 not isomorphic, yet are Morita equivalent. However, the following partial converse holds.

14 Theorem 1.3.2 ([Lam99, p. 494]). If two rings are Morita equivalent, then their centers are 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 categories → F : RMod SMod. We follow the proof of [Lam99, p. 470]. Let M,M0,M00 be left R-modules, and let f : M → M0 be a module homomorphism. → Define the map F : RMod SMod by M 7→ Mn | 7→ | F(f ):(m1,...,mn) (f (m1),...,f (mn)) , ∈ n { | ∀ ∈ } where m1,...,mn M and M = (m1,m2,...,mn) : i,mi M is the module with the action defined by matrix multiplication from the left. We see that F is a functor since, for a module homomorphism g : M0 → M00, we have ◦ ◦ ◦ | F(g f )(m1,...,mn) = (g f (m1),...,g f (mn)) | = F(g)(f (m1),...,f (mn)) ◦ | = F(g) F(f )(m1,...,mn) , and one can easily check that F(f ) is a module homomorphism. Let M,M0 be left S-modules, and let f : M → M0 be a module homomorphism. → Define the map G : SMod RMod by 7→ M E11M 7→ G(f ): E11m E11f (m), ∈ where m M and Eij is the matrix with 1 in the (i,j) position and 0 elsewhere. Letting ∈ rij 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 : M0 → M00 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,M0 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).

15 → Then βM is an isomorphism and for f : M M0, we have ◦ ◦ ◦ | (G F)(f ) βM(m) = (G F)(f )(m,0,...,0) ◦ | = (G F)(f )E11(m,m,...,m) | = E11(f (m),f (m),...,f (m)) | = (f (m),0,...,0) = β ◦ id (f )(m), M0 RMod → ◦ 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, we must also show that F ◦ G id Mod. We do this by finding a natural isomorphism → n S αM : M (E11M) = F(G(M)). Define the map αM by | αM(m) = (E11m,E12m,...,E1nm) , ∈ ∈ where m M. Notice that αM(m + m0) for m0 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 side evaluates to | αM(rEijm) = (0,...,0,rE1jm,0,...,0) , |{z} j-th position and the right-hand side evaluates to | | rEijα(m) = rEij(E11m,...,E1nm) = (0,...,0,rE1jm,0,...,0) , |{z} j-th position 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) = (0,...,0) = E1jm = 0 for all j X X ⇒ = 0 = Ej1E1jm = Ejjm = Inm = m, j j ∈ ∈ so αM is injective. Now, let p1,...,pn E11M; one can then pick m1,...,mn M such ≤ ≤ that pi = E11mi for 1 i n. Then, | αM(pi) = (E11pi,...,E1npi) | = (E11E11mi,E12E11mi,...,E1nE11mi) | = (pi,0,...,0) , so | | (0,...,0, pi ,0,...,0) = Ei1(pi,0,...,0) = Ei1αM(pi) = αM(Ei1pi), |{z} i-th position

16 and finally, ··· ··· | αM(E11p1 + E21p2 + + En1pn) = αM(E11p1) + + αM(En1pn) = (p1,...,pn) ; this shows us that αM is surjective, and thus an isomorphism. Let M,M0 be two left S-modules, with m ∈ M, and let f : M → M0 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)) = α ◦ id (f )(m), M0 S Mod so α is a natural isomorphism id Mod → F ◦ G. →S We finally see that F : RMod SMod is an equivalence of categories, so R and S are Morita equivalent. For every field K and finite-dimensional K-algebra A, there is an associated basic algebra (see Definition 1.2.12) defined in the following way: let Ae be a minimal progenerator of the full subcategory of AMod consisting of finite-dimensional left A-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

s Xs1 Xsn Xn Xj ··· 1A = e1j + + enj = eij j=1 j=1 i=1 j=1 into the sum of pairwise primitive orthogonal idempotents eij (see Definition 1.2.14) such that Ae Ae for j,k ∈ {1,...,s }, Ae Ae for i i . ij ik i ij i0k , 0 Pn Then we may let e = n=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 of R 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 P ∈ P Proof. Given (1.1), we write 1 as i ei where ei ai. Then ek = i ekei; since the 2 sum (1.1) is direct, we see that ekei = 0 of i , k and ek = ek, so (1.2) follows. ∈ P Conversely, given (1.2), put ai = Rei; then any x R can be written as x = i xei, ∈ P P 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 e as a sum of two orthogonal idempotents. It follows that Re is indecomposable if and only 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 all the 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 following theorem.

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

1.4 Morita Theory

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

Definition 1.4.1 (Trace ideal). The trace ideal of a left (right) R-module M is denoted tr(M) and is defined by X tr(M) B f (M). f Hom Mod(M,R) ∈ R Definition 1.4.2 (Generator). For a ring R, a right R-module P is called a generator − → ∈ for 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 are equivalent:

1. P is a generator;

18 2. tr(P ) = R; L 3. R is a direct summand of a finite direct sum i P ; L 4. R is a direct summand of a direct sum i P ; and L ∈ Mod 5. every M Ob( R) 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, for any surjective homomorphism of left R-modules g : B → C, and any R-homomorphism h: P → C, there exists an R-homomorphism h0 : P → B such that h = g ◦ h0. Definition 1.4.5 (Progenerator). Given a ring R, a left R-module P is a progenerator if P a is finitely generated, projective generator.

Deﬁnition 1.4.6 (Full idempotent). Given a ring R, an idempotent x ∈ R of R is called a full idempotent of R if RxR = R. That is, the two-sided ideal generated by x is the whole 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 n is a basis for Mn(R). Deﬁnition 1.4.8 (Bimodule). Given two rings R and S, an (R,S)-bimodule M is an abelian 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 multiplication. × Example 1.4.10. Given a ring R, the ring Mm n(R) of m n matrices with entries from × R 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 is a commutative ring, the axioms for a right module are satisﬁed. The left and right action also commute, so M is an (R,R)-bimodule.

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

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

19 ⊗ Proposition 1.4.13. If M is an (R,S)-bimodule and N is a left S-module, then M S N is a left R-module. ⊗ P ⊗ Proof. Every element of M R N can be written as a finite sum i mi ni, so we define the left action to be   X  X r  m ⊗ n  = (rm ) ⊗ n .  i i i i i i We verify that this is an action. X X ⊗ ⊗ (r + r0) mi ni = (r + r0)mi ni i i X ⊗ = (rmi + r0mi) ni i X X ⊗ ⊗ = rmi ni + r0mi ni i i X X ⊗ ⊗ (rr0) mi ni = (rr0mi) ni i i X ⊗ = r(r0mi) ni i X ⊗ = r r0mi ni i ⊗ This shows that M R N 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 S N is an (R,T )-bimodule. P ⊗ P ⊗ Proof. Similar to the previous proof we see that ( i mi ni)t = i mi (nit) defines a right action. It remains to show that the left and right action commute.    X   X r  m ⊗ n t = r m ⊗ n t  i i  i i i i X ⊗ = rmi nit i X ⊗ = r mi nit i    X  = r m ⊗ n t  i i i ⊗ We 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 endomorphism ring, denoted EndR(M) is the ring of all left (right) module homomorphisms of M into itself, with addition being pointwise addition, and multiplication being composition of homomorphisms.

Definition 1.4.16 (Faithfully balanced). Given rings R and S, an (R,S)-bimodule → M is called faithfully balanced if the ring homomorphisms λM : R End(MS) and → ∈ ∈ ∈ ρM : 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

ps B s(p).

Note that this rule necessitates that composition is written in reverse. We verify that this is a right action: for s,s0 ∈ S and p,p0 ∈ P ,

(p + p0)s = s(p + p0) = s(p) + s(p0); p(s + s0) = (s + s0)(p) = s(p) + s0(p); p(ss0) = p(s ◦ s0) = ((p)s)s0 = (ps)s0

p1S = p.

Thus P is a right S-module. The left and right actions on P commute because all elements 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 written ∈ on 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 module homomorphism. Also, given the elements q ∈ Q, p ∈ P , and s ∈ S, Q is a left S-module with respect to the action p(sq) = (ps)q (“PSQ-associativity”). Again, this is evidently a left action on Q as ps ∈ P and q is a module homomorphism. The left and right actions on 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 p0,q0,r0,s0 will denote elements in P,Q,R, and S, respectively. We now perform some calculations. Note that pq ∈ R by the application of q to p (as noted above, Q is acting on the right of P ) and qp ∈ S, which is deﬁned as

(p0)(qp) = (p0q)p (“PQP -associativity00).

21 We verify that qp is an R-module homomorphism. Letting p00 ∈ P ,

(p0 + p00)(qp) = ((p0 + p00)q)p = (p0q + p00q)p = p0qp + p00qp; rp0(qp) = (rp)qp = r(p)qp.

We now show that (qp)q0 = q(pq0) (“QPQ-associativity00). This is checked by showing that the left and right sides are equal functions on P :

p0((qp)q0) = (p0(qp))q0 (“PSQ-associativity00) = ((p0q)p)q0 (“PQP -associativity00) = (p0q)(pq0) (“RP Q-associativity00 : q is a left R-module homomorphism) = p0(q(pq0)) (“PQR-associativity00).

Lemma 1.4.17 ([Lam99, Lemma 18.15]). In the above notation: 7→ ⊗ → 1. (p,q) pq defines an (R,R)-bimodule homomorphism α : P S Q R; 7→ ⊗ → 2. (q,p) qp defines an (S,S)-bimodule homomorphism β : Q R P S. Proof. We see that α is well defined due to PSQ-associativity. Further, α is an (R,R)- bimodule homomorphism due to RP Q-associativity and PQR-associativity. Similarly, β is well defined due to QRP -associativity, and is a bimodule homomorphism due to SQP -associativity and QPS-associativity. In the above context, the six-tuple (R,P ,Q,S;α,β) is called the Morita Context associated to P . This Morita Context is fixed for the next two propositions and the following corollary.

Proposition 1.4.18 ([Lam99, Proposition 18.17]). 1. The left R-module module P is a 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 ﬁnitely generated 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 ﬁnitely generated projective generator), then RPS and SQR are faithfully balanced bimodules. The next three theorems are those of [Mor58], and they help us see exactly when two 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 category equivalences. ⊗ − → ⊗ − → 2. P S : SMod RMod and Q R : RMod SMod are mutually inverse category equivalences.

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)-bimodule P 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, the → isomorphism classes of category equivalences SMod RMod are in bijective correspondence with the isomorphism classes of (S,R)-progenerators. Composition of category equivalences corresponds 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 are equivalent:

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 endomorphism ring 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 that S End(RP ) for RP a progenerator. Thus the Morita Context (R,P ,Q,S;α,β) associated with 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: the bicategory. We begin discussing the idea of a 2-category, as this is helps us in dealing with 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 Hom (A,B) whose objects f ,g,... : A → B are called 1-morphisms or 1-cells, whose morphismsC α,β,... are ◦ called 2-morphisms or 2-cells, and whose composition is denoted 1 and called vertical composition; f f

α g β α A B A ◦1 B β

h h

• for every three objects A,B,C ∈ Ob(C) an associative, unital functor ◦ × → 0 : Hom (B,C) Hom (A,B) Hom (A,C), C C C

called horizontal composition with identities idA and ididA .

f 0 f f f 0 ◦

α α α0 0α A B 0 C A ◦ C g g 0 g g 0◦

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

• a function Ob(C) → Ob(D);

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

F : Hom (A,B) → Hom (F(A),F(B)); C C0 • given two 1-morphisms f ,g ∈ Hom (A,B), a function C → HomHom (A,B)(f ,g) HomHom (F(A),F(B))(F(f ),F(g)) C D which much satisfy the following: ∈ C • given an object A Ob( ), we have F(idA) = idF(A); • given objects A,B,C ∈ Ob(C) and morphisms f : A → B and g : B → C, we have F(g ◦ f ) = F(g) ◦ F(f ); ∈ C → • given objects A,B Ob( ) and a morphism f : A B, we have F(idf ) = idF(f ); • given 1-morphisms f ,g,h ∈ Hom (X,Y ), and 2-morphisms α : f → g and β : g → h, C ◦ ◦ F(β 1 α) = F(β) 1 F(α); and

• given 1-morphisms f ,g ∈ Hom (X,Y ),f 0,g0 ∈ Hom (Y,Z) and 2-morphisms C C α : f → g,α0 : f 0 → g0, ◦ ◦ F(α0 0 α) = F(α0) 0 F(α).

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

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

Deﬁnition 1.5.5 (Bicategory). A bicategory C consists of

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

25 • for every three objects A,B,C ∈ Ob(C), a functor ◦ × → 0 : Hom (B,C) Hom (A,B) Hom (A,C), C C C called horizontal composition, such that for every ﬁve objects A,B,C,D,E and every four 1-morphisms f : A → B,g : B → C,h: C → D,k : D → E, there are 2-isomorphisms ◦ ◦ → ◦ ◦ αf ,g,h : h (g f ) (h g) f ◦ → λf : idB f f ◦ → ρf : f idA f

such that the following diagrams commute.

α 1 ((kh)g)f ◦0 ((kh)g)f

α α

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

α 1 α ◦0 k(h(gf ))

α (g idB)f g(idB f ) ρ 1 ◦0 1 λ ◦0 gf

Example 1.5.6 (Bicategory of bimodules). There exists a bicategory Bim whose objects are rings, whose 1-morphisms from R to S are (S,R)-bimodules, and whose 2-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 constructed using 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 picking and using a 2-category equivalent to Bim. For any given bimodule, there is a tensor product functor. Specifically, letting R,S ⊗ − → be rings, and M an (R,S)-bimodule, we can define M S : SMod RMod for any left S-module N and any left module homomorphism f defined on N by 7→ ⊗ N M S N X X ⊗ ⊗ 7→ ⊗ M S (f ): mi ni mi f (ni), i i and this is a functor because, for any additional left module homomorphism g defined on N,   X  X M ⊗ (g ◦ f ) m ⊗ n  = m ⊗ ((g ◦ f )(n )) S  i i i i i i   X  = (M ⊗ g) m ⊗ f (n ) S  i i  i   X  = (M ⊗ g)(M ⊗ f ) m ⊗ n . S S  i i i Similarly, every bimodule homomorphism gives rise to a natural transformation between 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 natural ⊗ − → ⊗ − ∈ transformation α : M S N S defined, for any left S-module A, ai A, and m ∈ M, by i   X  X α  m ⊗ a  = f (m ) ⊗ a . A  i i i i i i This is natural because, for any left module homomorphism g : A → B for left S- modules A,B, we have     X  X  N ⊗ (g) ◦ α  m ⊗ a  = (N ⊗ (g)) f (m ) ⊗ a  S A  i i S  i i i i X ⊗ = f (mi) g(ai) i   X  = α  m ⊗ g(a ) B  i i  i   X  = α ◦ (M ⊗ (g)) m ⊗ a . B S  i i i

27 This gives a 2-functor F : Bim → Cat deﬁned by 7→ R RMod, 7→ ⊗ − RMS RMS S , and → 7→ ⊗ − → ⊗ − (RMS RNS) (M S N S ).

To see that F is a 2-functor, we ﬁrst note that, for two composable bimodules RMS and SNT , ◦ ⊗ F(T MS SNR) = F(T MS SSNR) ⊗ ⊗ − = T MS SSNR R ⊗ − ◦ ⊗ − = (T MS S ) (SNR R ) ◦ = F(T MS) F(SNR). → Next, for two vertically composable bimodule homomorphisms f : RMS RNS and → g : RNS RPS, and for a left S-module A,   X  X F(g ◦ f )  m ⊗ a  = (g ◦ f )(m ) ⊗ a 1 A  i i 1 i i i i   X  = F(g)  f (m ) ⊗ a  A  i i i   X  = (F(g) ◦ F(f ) ) m ⊗ a . A 1 A  i i i → Finally, for two horizontally composable bimodule homomorphisms f : SNT SNT0 → and g : RMS RMS0 , and left T -module A,     X  X  F(g ◦ f )  m ⊗ n ⊗ a  =  g(m ) ⊗ f (n ) ⊗ a  0 A  i i i  i i i i i   X  = (F(g) ◦ F(f )) m ⊗ n ⊗ a . 0  i i i i

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 nicely deals 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 and left-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 such that f g idB and gf idA. The maps f and g satisfying this property are called equivalences.

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

Proof. By Corollary 1.6.18, we know that if R and S are Morita equivalent, then there are 1-morphisms SQR and RPS in Bim such that when horizontally composed, ⊗ ⊗ RPS SSQR RRR 1RMod and SQR RRPS SSS 1S Mod, → 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 bicategorical perspective. First, we must understand exact sequences so that we can deal with Theorem 1.6.17.

Deﬁnition 1.6.4 (Exact sequence). A sequence of left R-modules and module homomorphisms ··· → −−−→fn+1 −−→fn → ··· Mn+1 Mn 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

f g 0 → A −→ B −→ 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 if exactness of f g 0 → A −→ B −→ C implies exactness of T (f ) T (g) 0 → T (A) −−−−→ T (B) −−−−→ T (C). → Deﬁnition 1.6.7 (Right exact functor). A functor T : RMod Ab is called right exact if exactness of f g A −→ B −→ C → 0 implies exactness of T (f ) T (g) T (A) −−−−→ T (B) −−−−→ 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 is where 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 : Hom (A,B) Hom (A,B) Hom (A,B), C C C referred to simply as +; ∈ C • for every A,B Ob( ), (Hom (A,B),+) is an abelian group with identity 0A,B; C • for every A,B,C ∈ Ob(C), f ,g ∈ Hom (A,B), and h ∈ Hom (B,C), we have h(f + g) = hf + hg; C C

• and for every A,B,C ∈ Ob(C), f ,g ∈ Hom (A,B), and h ∈ Hom (C,A), we have (f + g)h = f h + gh. C C Deﬁnition 1.6.10 (Biproduct). Given a preadditive category C, a biproduct for objects A,B ∈ Ob(C) is a diagram p 1 i2 A C B i1 p2 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 preadditive category: addition of module homomorphisms is pointwise addition and the ﬁnite direct sum of modules is a biproduct. Deﬁnition 1.6.12 (Additive functor). Given preadditive categories C and D, a functor F : C → D is an additive functor if for every A,B ∈ Ob(C), the map Hom (A,B) → Hom (FA,FB) is an abelian group homomorphism. C D Proposition 1.6.13 ([Wei94, Theorem 2.6.1]). If F : C → D and G : D → C are additive functors 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, the functors ⊗ − → − ⊗ → 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 has all binary biproducts, then a functor T : C → D commutes with biproducts if and only if T is 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 R M as left R-modules. ⊗ → P ⊗ P Proof. Consider the map f : RRR R M M deﬁned by f ( i ri mi) = i rimi for ∈ ∈ ri R and mi M. As  a b   a+b  X X  X  f  r ⊗ m + r ⊗ m  = f  r ⊗ m   i i i i  i i i=0 i=a+1 i=0 Xa+b = rimi i=0 Xa Xb = rimi + rimi i=0 i=a+1  a   b  X  X  = f  r ⊗ m  + f  r ⊗ m ,  i i  i i i=0 i=0 and      X  X  f r r ⊗ m  = f  rr ⊗ m   i i  i i i i X = rrimi i X = r rimi i   X  = rf  r ⊗ m ,  i i i we see that f is a module homomorphism. It is evident that f is surjective as, for any ∈ ⊗ m M, f (1R m) = m. Finally, f is injective, as   X  X f  r ⊗ m  = 0 =⇒ r m = 0  i i i i i i X ⇒ ⊗ = 1 rimi = 0 i X ⇒ ⊗ = ri mi = 0. i 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 functor that commutes with direct sums. Then there is an (S,R)-bimodule Q and a natural ⊗ − → equivalence of functors ψ : Q R T . ∈ ∈ → Proof. Given M Ob(RMod) and m M, define φm : R M by φm(r) = rm. We define M × → M ∈ ∈ a function ψ0 : TR M TM by ψ0 (r,m¯ ) = (T φm)(r¯) for r¯ TR and m M. Note R × → that T is additive by Proposition 1.6.15. Further, ψ0 : TR R TR defines a right ∈ ∈ action, because for r¯ TR, r1,r2 R, R R R ψ0 (ψ0 (r,r¯ 1),r2) = ψ0 ((T φr1 )(r¯),r2)

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

= T (φr1r2 )(r¯) R = ψ0 (r,r¯ 1r2);

R ψ0 (r,r¯ 1 + r2) = (T φr1+r2 )(r¯)

= T (φr1 + φr2 )(r¯)

= (T φr1 )(r¯) + (T φr2 )(r¯) by additivity of T R R = ψ0 (r,r¯ 1) + ψ0 (r,r¯ 2); ∈ ∈ for r¯1,r¯2 T R,r R, R ψ0 (r¯1 + r¯2,r) = T (φr)(r¯1 + r¯2) = T (φr)(r¯1) + T (φr)(r¯2) since T φr is a module homomorphism R R = ψ0 (r¯1,r) + ψ0 (r¯2,r); ∈ ∈ and for 1R R,r¯ TR, R ψ0 (r,¯ 1R) = (T φ1R )(r¯) = (T 1R)(r¯) = 1TR(r¯) = r.¯ Finally, the left and right actions on TR commute: for s ∈ S, r ∈ R, r¯ ∈ TR, R R sψ0 (r,r¯ ) = s(T φr)(r¯) = (T φr)(sr¯) = ψ0 (sr,r¯ ). Thus TR is an (S,R)-bimodule. TR will henceforth be denoted by Q. M ∈ Note that ψ0 , for M Ob(RMod) is R-balanced because it is deﬁned by an action. M M ⊗ → Thus, ψ0 descends to an R-homomorphism ψ : Q R M TM, and the family M ⊗ − → (ψ )M Ob( Mod) is a natural transformation ψ : Q R T . For any left R-module ∈ R homomorphism f : M → M0, ◦ M ⊗ T (f ) ψ (x m) = T (f )(T φm(x)) ◦ = T (f φm)(x) = T (φf (m))(x) = ψM0 (x ⊗ f (m))

M0 ◦ ⊗ ⊗ = ψ Q R (f )(x m).

32 R ⊗ ⊗ Further, ψ is a natural isomorphism, as C R R = TR R R TR by Lemma 1.6.16. ⊗ − F T commutes with direct sums, and so does Q R by Lemma 1.6.14, so ψ is an isomorphism 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 F admitting 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 following diagram of exact rows commutes.

⊗ α1 ⊗ β1 ⊗ Q R L Q R F Q R M 0

ψL ψF ψM α β TL 2 TF 2 TA 0

Note that ψL and ψM are surjective because they are right actions on TL and TM, respectively. To show that ψM is injective, we “chase” the diagram. ∈ M ∈ ⊗ Let a kerψ . Since β1 is surjective, there is b Q R F such that β1(b) = a. Let c = ψF(b) ∈ TF. Then

F M M β2(c) = β2(ψ (b)) = ψ (β1(b)) = ψ (a) = 0. ∈ ∈ ∈ Since c kerβ2, we know that c imα2, so we can pick d TL such that α2(d) = c. L ∈ ⊗ L F Since ψ is surjective, we can pick e Q R L such that ψ (e) = d. Since ψ is injective, b is the unique preimage of c through ψF. Since the diagram commutes, we know that α1(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 and ⊗ only if there is an (R,S)-bimodule P and an (S,R)-bimodule Q such that RPS SSQR RRR ⊗ and SQR RRPS SSS as bimodules. Proof. ( ⇐= ) Using the 2-functor F deﬁned previously, we have that ⊗ ⊗ − ⊗ ⊗ − RPS SSQR R = F(RPS SSQR) F(RRR) = RRR R , ⊗ ⊗ − ⊗ ⊗ − so RPS SSQR R 1 Mod by Lemma 1.6.16. Similarly, SQR RRPS S 1 Mod. Thus ⊗ − R → S RPS S is an equivalence of categories SMod RMod. ⇒ → ( = ) Let T : RMod SMod be an equivalence of categories. It is well-known that equivalences are additive, and so T is additive. Since T is additive and the left 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. Thus ⊗ − there is an (S,R)-bimodule Q such that Q R is equivalent to T . Similarly, for the → ⊗ − inverse 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 SSQR R = F(RPS SSQR), ⊗ ⊗ so RRR RPS SSQR, and similarly SSS SQR RRPS. We now give an example of how Corollary 1.6.18 can be applied to determining a Morita equivalence. × Example 1.6.19. Let R be a ring and consider its n n matrix ring Mn(R). Then the set × of n 1 matrices Mn 1(R) is a left Mn(R)-module with respect to matrix multiplication and 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 × × n | X f ((r1,...,rn),(s1,...,sn) ) = risi i=1 × → is Mn(R)-balanced and g : Mn 1(R) M1 n(R) Mn(R) defined by × × | g((r1,...,rn) ,(s1,...,sn)) = (risj) is R-balanced. However, it is evident that both of these maps are given by matrix multiplication, and since matrix multiplication is associative and distributive, we have that f and g are Mn(R)-balanced and R-balanced, respectively. Thus f and g both ˜ ⊗ → descend to well-defined bimodule homomorphisms f : M1 n(R) M (R) Mn 1 RRR ⊗ → × n × and g˜ : Mn 1 R M1 n(R) M (R) Mn(R)M (R) defined by × × n n Xn ˜ ⊗ | f ((r1,...,rn) (s1,...,sn) ) = risi i=1 and | ⊗ g˜((r1,...,rn) (s1,...,sn)) = (risj). We check that f˜ and g˜ are bimodule isomorphisms. First, f˜ is surjective: for any r ∈ R, | f˜((r,1,...,1) ⊗ (1,...,1) ) = r. ⊗ | Note that every simple tensor (r1,...,rn) (s1,...,sn) can be rewritten in a particular

34 way:

 s1 0 0  |  ···  | ( ) ⊗ ( ) = ( ) ⊗  . . . . (1 0 0) r1,...,rn s1 ...sn r1,...,rn  . . . .  , ,..., sn 0 0 0  s1 0 0   ···  | = ( ) . . . .  ⊗ (1 0 0) r1,...,rn  . . . .  , ,..., s 0 0 0  n  X  | =  r s ,0,...,0 ⊗ (1,0...,0)  i i  i The map is thus also injective:

Xn ˜ ⊗ | ⇒ f ((r1,...,rn) (s1,...,sn) ) = 0 = risi = 0 i=1   X  | =⇒  r s ,0,...,0 ⊗ (1,0,...,0) = 0  i i  i ⇒ ⊗ | = (r1,...,rn) (s1,...,sn) = 0.

Since f˜ is R-balanced, it is a bimodule homomorphism, and hence a bimodule isomorphism. Now, g˜ is surjective: | ⊗ g˜((0,...,0, 1 ,0,...,0) (0,...,0, 1 ,0,...,0)) = Eij, |{z} |{z} i-th position j-th position { | ≤ ≤ ≤ ≤ } and Eij 1 i n,1 j n forms a basis for Mn(R). Note that every simple tensor ⊗ | (r1,...,rn) (s1,...,sn) can be rewritten in a particular way: | ⊗ | ··· | ⊗ (r1,...,rn) (s1,...,sn) = ((1,0,...,0) r1 + + (0,...,0,1) rn) (s1,...,sn) | ⊗ ··· | ⊗ = (1,0,...,0) r1 (s1,...,sn) + + (0,...,0,1) rn (s1,...,sn) | ⊗ ··· | ⊗ = (1,0,...,0) (r1s1,...,r1sn) + + (0,...,0,1) (rns1,...,rnsn). We thus also see that g˜ is injective: | ⊗ ⇒ ∀ ∀ g˜((r1,...,rn) (s1,...,sn)) = 0 = i, j,risj = 0 ⇒ | ⊗ = (1,0,...,0) (r1s1,...,r1sn) ··· | ⊗ + + (0,...,0,1) (rns1,...,rnsn) = 0 ⇒ | ⊗ = (r1 ...,rn) (s1,...,sn) = 0.

Since g˜ is Mn(R)-balanced, it is a bimodule homomorphism, and hence a bimodule isomorphism. By Corollary 1.6.18, we see that R and Mn(R) are Morita equivalent, as we saw earlier 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 a bimodule 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 multiplication by r, is a ring isomorphism.

Proof. We ﬁrst show that all the endomorphisms of RR are left multiplications by ∈ ∈ elements 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-module ∈ ∈ homomorphism. Let r,r0 R and m RR. Then

φr(mr0) = rmr0 = φr(m)r0. ∈ Also, for m0 RR

φr(m + m0) = r(m + m0) = rm + rm0 = φr(m) + φr(m0). ∈ ∈ Finally, we show that f is a ring homomorphism. For r,r0 R and m RR, f (r + r )(m) = φ (m) = (r + r )m = rm + r m = φ (m) + φ (m) = f (r)(m) + f (r )(m); 0 r+r0 0 0 r r0 0 f (rr )(m) = φ (m) = rr m = φ (φ (m)) = (φ ◦ φ )(m) = f (r)f (r )(m); 0 rr0 0 r r0 r r0 0 and f (1)(m) = φ1(m) = 1m = m.

Since f is a ring homomorphism, every module endomorphism of RR is a left multiplication by some element of R, and every left multiplication by some element of R is a module endomorphism, we see that f is a ring isomorphism. op → Lemma 1.6.21. The map f : R EndR(RR) deﬁned by f (r) = ψr, where ψr is right multiplication by r, is a ring isomorphism.

Proposition 1.6.22. Given rings R and S, if RMS is a faithfully balanced (R,S)-bimodule, → op → then the maps f : End(ZRR) End(ZMS) and g : End(Sop SZ ) 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 by s. Proof. By the proofs of Lemma 1.6.20 and Lemma 1.6.21, it is clear that f and g op are well-defined. Also, End(ZRR) = End(RR), End(ZMS) = End(MS), End(Sop SZ ) = op End(Sop S ), and End(RMZ) = End(RM). Note that f is the map from Lemma 1.6.20 composed with λM from Definition 1.4.16, both of which are ring isomorphisms, and g is the map from Lemma 1.6.21 composed with ρM from Definition 1.4.16, both of which 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 in Morita 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 certain type 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 algebra K[G] of G over K is the algebra with the elements of G as a basis and with multiplication on the basis elements given by group multiplication. K K Note that [G] has the multiplicative identity 1G, meaning that [G] can be seen as a ring. Note also that the elements of K[G] for any K and G are of the form P K g G agg. Multiplication on arbitrary elements of [G] is given by the bilinearity of the∈ multiplication on the basis elements:      X  X X X        agg bhh =  agbhk.      g G h G k G gh=k ∈ ∈ ∈ Definition 2.1.3 (Endomorphism ring). Given a vector space V , the endomorphism ring of V , denoted End(V ), is the set of all linear endomorphisms of G where addition in End(V ) is pointwise addition, and multiplication in End(V ) is composition of linear maps.

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

Proof. Let ρ : G → GL(V ) be a representation on some K-vector space V . Then   X  X    aggv = agρ(g)(v)   g G g G ∈ ∈

38 defines a K[G]-module action on V . We verify this claim. First, distributivity over the vector addition is satisfied:   X  X    agg(v + v0) = agρ(g)(v + v0)   g G g G ∈ X∈ = ag(ρ(g)(v) + ρ(g)(v0)) distributivity of matrix multiplication g G X∈ = agρ(g)(v) + agρ(g)(v) g G X∈ X = agρ(g)(v) + agρ(g)(v0) g G g G ∈   ∈  X  X      =  aggv +  aggv0.     g G g G ∈ ∈ Distributivity over the group algebra addition is satisfied:     X X  X       agg + bggv =  (ag + bg)gv     g G g G g G ∈ ∈ X∈ = (ag + bg)ρ(g)v g G X∈ = agρ(g)v + bgρ(g)v g G X∈ X = agρ(g)v + bgρ(g)v g G g G ∈   ∈  X  X      =  aggv +  bggv.     g G g g ∈ ∈

39 The module action commutes with the group algebra multiplication:        X  X  X X           agg bhhv =   agbhkv        g G h G k G gh=k ∈ ∈ ∈   X X    =  agbhρ(k)v   k G gh=k ∈    X X  X    =  ag bhρ(g)ρ(h)v    gh G g G h H ∈ ∈ ∈  X X  = a g  b ρ(h)v g  h  g G h G ∈ ∈   X X   = a g  b hv. g  h   g G h G ∈ ∈ 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 homomorphism σ : K[G] → End(V ) defined by     X  X      σ  agg(v) =  aggv.     g G g G ∈ ∈ Note that σ descends to a map σ˜ : G → GL(V ). First, for any g ∈ G, ρ(g) has an inverse: 1 1 ρ(g)ρ(g− ) = ρ(gg− ) = 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, the irreducible representations of G over K are those representations whose corresponding K[G]-modules are simple. Henceforth, given a group G and a field K, the representation ρ : G → GL(V ) of G over 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 two ⊕ representations V1 and V2, the direct sum V1 V2 is a representation of G whose action is given by g(v1,v2) = (gv1,gv2).

40 Deﬁnition 2.1.7 (Tensor product of representations). Given a group G, a ﬁeld K, and ⊗ two representations V1 and V2, the tensor product V1 V2 is a representation whose action 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 are semisimple.

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

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

Given a positive integer n, a partition of n is notated here as a weakly decreasing, P eventually zero sequence (λ1,λ2,...,0,...) of positive integers such that i∞=1 λi = n. Given a Young diagram with n boxes and k rows, and letting κi be the number of boxes in the i-th row, the sequence (κ1,...,κk,0,...) is a partition of n. Conversely, a partition (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 Young diagram.

Proposition 2.1.12 ([Ful97, Proposition 7.2.1]). The Young diagrams with n boxes are in bijection with the irreducible complex representations of Sn. Deﬁnition 2.1.13 (Polynomial representation). A representation V of GL(V ) over C is called a polynomial representation if, for all g ∈ GL(V ), the entries of the matrix given by 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 rows C are in bijection with the irreducible polynomial representations of GLn( ). C It is a well-known fact that the general linear group GLn( ) is an example of what is called a reductive group. A key characteristic of reductive groups is that their representations over ﬁelds of characteristic 0 are decomposable into direct sums of irreducible representations. See [Bor91, p. 273] for some discussion on the latter statement.

41 Lemma 2.1.15 (Schur, [Coh03, Lemma 6.3.2]). Let K be an algebraically closed field, A a K-algebra, and U,V any two simple A-modules, finite-dimensional over K. Then  K  if U V HomA(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 that C[G]Mod and C[H]Mod are semisimple module categories with the same number of isomorphism classes of simple objects. Then C[G] and C[H] are Morita equivalent. Proof. Since categories are equivalent to their skeletons by Proposition 1.1.15, it C D suﬃces to show that a skeleton of C[G]Mod is equivalent to a skeleton of C[H]Mod. Since C[G]Mod and C[H]Mod are semisimple module categories, every object in either category is decomposable into a direct sum of simple objects. Let V1,V2,...,Vn be the C D → simple objects of , and let W1,W2,...,Wn be the simple objects of . Let F : Mod Mod be a functor whose map on objects is deﬁned by C D   Mn Mn  ni  ni F  V ⊕  = W ⊕  i  i i=1 i+1 ∈ N for ni . It is seen immediately that F is surjective. Let Mn → ni ιa,b : Va Vi⊕ i=1 ≤ ≤ ≤ ≤ for 1 a n and 1 b ni be the canonical injection and let M m ⊕ j → πa,b : Vi Va i=1 ≤ ≤ ≤ ≤ for 1 a n and 1 b mj be the canonical projection. Then for every map

 n n  M M m  ∈  ni j  f HomC[G]  V ⊕ , V ⊕   i j  i=1 j=1 ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ 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

 n n  M m M  ∈  j `k  g HomC[G]  V ⊕ , V ⊕   j k  j=1 k=1

42 ≤ ≤ and 1 c `k, X X ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ηi,k,a,c(g f ) = πk,c g f ιi,a = πk,c g ιj,b πj,b f ιi,a = ηj,k,b,c(g) ηi,j,a,b(f ). j,b j,b

Further, we know by Lemma 1.2.25 and Lemma 1.2.26 that

m Mn Mni Mj 7→ f ηi,j,a,b(f ) i,j=1 a b 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) EndC[H](Wi) for 1 i n. By again applying Lemma 1.2.25 and Lemma 1.2.26, we ﬁnd an isomorphism

 n n   n n  M M m  M M m   ni j  →  ni j  HomC[G]  V ⊕ , V ⊕  HomC[G]  W ⊕ , W ⊕ ,  i j   i j  i=1 j=1 i=1 j=1 and we set the map on morphisms of F to be this map:

Xn Xni Xmi F(f ) = φi(ηi,i,j,k(f )) i=1 j=1 k=1

This map is compatible with composition as well:

Xn Xni Xmi ◦ ◦ F(g f ) = φi(ηi,i,j,k(g f )) i=1 j=1 k=1 Xn Xni Xmi ◦ = φi(ηi,i,j,k(g)) φi(ηi,i,j,k(f )) i=1 j=1 k=1     Xn Xni Xmi  Xn Xni Xmi    ◦   =  φi(ηi,i,j,k(g))  φi(ηi,i,j,k(f ))     i=1 j=1 k=1 i=1 j=1 k=1 = 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, the composition 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 equivalence of categories.

43 2.2 Schur-Weyl Duality

In this section, we treat the classical Schur-Weyl duality theorem (Theorem 2.2.1) and we show that it exhibits a Morita equivalence. The notation now deﬁned will be carried onto the statement of the next theorem. r Let V denote a complex vector space of dimension n. Let V ⊗ denote the r-fold tensor r product of V . We endow V ⊗ with the following C[GL(V )]-action: for g ∈ GL(V ), ⊗ ⊗ ··· ⊗ ⊗ ⊗ ··· ⊗ g(v1 v2 vr) = g(v1) g(v2) g(vr). r C ∈ We also give V ⊗ the following [Sr]-action: for σ Sr,

σ(v ⊗ v ⊗ ··· ⊗ v ) = v 1 ⊗ v 1 ⊗ ··· ⊗ v 1 . 1 2 r σ − (1) σ − (2) σ − (r) C C It is evident that the [GL(V )]-action and [Sr]-action commute with one another, r C C D making V ⊗ a ( [GL(V )], [Sr])-bimodule. For a given Young diagram D, let U be the irreducible polynomial representation of GL(V ) corresponding to the Young diagram D r D, and let W be the irreducible complex representation of Sr on V ⊗ corresponding to the Young diagram D.

Theorem 2.2.1 (Schur-Weyl duality, [How95, Section 2.4.3]). Under the joint action of r GL(V ) and Sr, the space V ⊗ decomposes into a direct sum M r D D V ⊗ U ⊗ W , D where D runs over Young diagrams with r boxes and at most n rows.

Proposition 2.2.2. Given the algebra homomorphisms

C −→α r ←−β C [GL(V )] End(V ⊗ ) [Sr] C C r induced by the module actions of [GL(V )] and [Sr] on V ⊗ , we have that imαMod is Morita equivalent to imβMod. ∈ L L ∈ Proof. Let f D End(UD). Then f = D fD for maps fD End(UD), and we can map M 7→ ⊗ ω : f (fD idWD ). D L L The map is injective: suppose ∈ End( ), = and that ⊗ id = ω g D UD g D gD fD WD ⊗ ∈ ∈ gD idWD for 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 vectors − cannot be 0. This is a contradiction. Thus fD(x) gD(x) = 0 for all D, which implies f = g. We see that M { ∈ } Ω = ω(f ): f End(UD) D L ⊗ C must consist entirely of maps of D End(UD WD) that commute with the [Sr]- L ⊗ action on UD WD. That is, Ω commutes with imβ. As left C[GL(V )]-modules,

dimW ⊗ ⊗ C ⊕ ⊗ C ⊕ ··· ⊕ ⊗ C ⊕ D UD WD (UD w1) (UD w2) (UD wdimWD ) UD , { } where w1,w2,...,wdimWD is a basis for WD. Thus, M M ⊗ dimWD UD WD UD⊕ . D D Therefore, we have the isomorphism     M M    dimWD  End U ⊗ W  End U ⊕ .  D D  D  D D Pulling out the direct sums with Lemma 1.2.25 and Lemma 1.2.26, we obtain   M M  dimWD  (dimW )(dimW ) End U ⊕  Hom(U ,U )⊕ D D0 .  D  D D0 D D,D0

We know that the UD are simple modules, however, and by Lemma 2.1.15, we see that Hom(U ,U ) 0 when D D . Thus, D D0 , 0 M (dimW )(dimW ) M (dimW )(dimW ) Hom(U ,U ) D D0 Hom(U ,U ) D D D D0 ⊕ D D ⊕ D,D0 D M dimW dimW Hom U ⊕ D ,U ⊕ D D D0 D M dimWD End UD⊕ . D L ⊗ We wish to show that every map in imα is of the form (fD idW ). Let (FD) be D D D ∈ dimWD ⊗ ⊗ a tuple of endomorphisms FD End UD⊕ End(UD WD). Then FD(u w) = P ⊗ ∈ ∈ i,j aiui wj for any u U, w W . C Note that every WD is a right [Sr]-module. Theorem 1.2.30 tells us that for any of ∈ ∈ C the WD, and for any γ End(WD), and for any basis X, we can pick s [Sr] such that

45 ∈ ws = γ(w) for w X. Without loss of generality, let w = w1. Let γ be the projection C C C onto w1 = w, and let s be the element of [Sr] which corresponds to γ. Then X ◦ ⊗ ⊗ ⊗ ⊗ ⊗ FD (idUD s)(u w) = FD(u w1) = FD(u w) = aiui wi, i and X ⊗ ◦ ⊗ ⊗ ⊗ ⊗ ⊗ (idUD s) FD(u w) = (idUD s)( aiui wi) = a1u1 w1 = a1u1 w. i C ∈ Since FD must commute with the action of [Sr] on WD, for any w WD, it happens ⊗ ⊗ ⊗ that FD(u w) = a1u1 w, and we obtain that, as an endomorphism of UD WD, we will have ⊗ ⊗ FD(u w) = u0 w ∈ ∈ ⊗ ∈ for any w WD, and for any u UD. Hence FD = fD idWD for some fD End(U). dimWD Thus ω surjects onto End UD⊕ . This makes M M → dimWD ω : End(UD) End UD⊕ D D an isomorphism of algebras. L dimW ⊕ D Now, since imα D End UD , we see that by the isomorphism ω, M imα End(UD), D and by similar reasoning, M imβ End(WD). D C C We know that End(UD) for each D is a simple ring as End(UD) MdimUD ( ), MdimUD ( ) is Morita equivalent to C by Example 1.3.3, C is a simple ring as C is a ﬁeld, and by L Proposition 1.2.23, C has one simple module. The simple modules of End(U ) L D D are in correspondence with the maximal ideals of D End(UD) by Proposition 1.2.23 which are in turn enumerated by the Young diagrams D as the maximal ideals are of the 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 that L the 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 of isomorphism classes of simple modules. Finally, by Proposition 2.1.16, we see that since imαMod and imβMod have the same number of isomorphism classes of simple modules, 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 Morita equivalence.

Deﬁnition 2.3.1 (Tensor algebra). Given a vector space V , we call the associative L ∞ i algebra T(V ) = i=0 V ⊗ 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-sided ideal of T(V ) generated by x ⊗ y − y ⊗ x for all x,y ∈ V . Then we call S(V ) = T(V )/I the symmetric algebra on V .

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

Note that given a group algebra K[G] and a (left) K[G]-module V , the tensor algebra T(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 ) is closed 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 ). In a similar fashion, the R-action on T(V ) descends to an R-action on the exterior algebra Λ(V ). The notation developed here carries on into the next two theorems. Let U and V be two complex vector spaces. We give U ⊗ V the following C[GL(U)]-action: for g ∈ 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).

The C[GL(U)]- and C[GL(V )]-actions commute, making U ⊗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 U D and V D be the irreducible polynomial representations of GL(U) and GL(V ), respectively, corresponding to the Young diagrams D.

47 C C Theorem 2.3.4 ((GLn,GLm)-duality, [How95, 15-16]). As a ( [GL(U)], [GL(V )])-module, S(U ⊗ V ) decomposes into a direct sum M S(U ⊗ V ) U D ⊗ V D, D where D runs over all Young diagrams with at most min{dimU,dimV } rows.

Corollary 2.3.5. Given the algebra homomorphisms

α β C[GL(U)] −→ End(S(U ⊗ V )) ←− C[GL(V )] induced by the module actions of C[GL(V )] and C[GL(V )] on S(U ⊗ V ), we have that imαMod is equivalent to imβMod. Proof. This is justiﬁed in the same way as Proposition 2.2.2. Given a Young diagram D, its transpose D| is the Young diagram whose columns are the rows of D (and thus whose rows are the columns of D). C C Theorem 2.3.6 (Skew (GLn,GLm)-duality, [How95, 50]). As a ( [GL(U)], [GL(V )])- module, Λ(U ⊗ V ) decomposes into a direct sum

M | Λ(U ⊗ V ) U D ⊗ V D , D where D runs over all Young diagrams with at most n rows and with rows of length at most m.

Corollary 2.3.7. Given the algebra homomorphisms

α β C[GL(U)] −→ End(Λ(U ⊗ V )) ←− C[GL(V )] induced by the module actions of C[GL(V )] and C[GL(V )] on Λ(U ⊗ V ), we have that imαMod is equivalent to imβMod. Proof. This is justiﬁed in the same way as Proposition 2.2.2.

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