Fei Xu

A Mini Course on Category Algebras

November 9, 2011

Contents

1 Category algebras and their representations . . . . . . . . . . . . . . 3

1.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Category algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Representations of categories and Mitchell’s theorem . . . . . . . . 6 1.2.1 Representations of categories . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Mitchell’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 EI categories, projectives, injectives and simples . . . . . . . . . . . . 8 1.3.1 EI condition and its implications . . . . . . . . . . . . . . . . . . . 8 1.3.2 Some representation theory . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.3 Projectives and simples, first approach . . . . . . . . . . . . . . 12 1.3.4 Injective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Combinatorially constructed modules . . . . . . . . . . . . . . . . . . . . . 16

2 Basic tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Restriction on algebras and modules . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Kan extensions of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Overcategories and undercategories . . . . . . . . . . . . . . . . . 24 2.2.3 Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Classifications of projectives and simples for EI categories … 28 2.4 Dual modules and Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Transporter categories and their algebras . . . . . . . . . . . . . . . . . 33

3.1 Transporter categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 G-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.2 Transporter categories as Grothendieck constructions . 35

3.2 Local categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.1 Two diagrams of categories . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.2 Frobenius Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Transporter category algebras are skew group algebras . . . . . . 39 3.4 Transporter category algebras are Gorenstein . . . . . . . . . . . . . . 40 3.5 Projective, injective and geometric dimensions . . . . . . . . . . . . . 40 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Preface

Groups and partially ordered sets (posets in short) can be regarded as small categories. More precisely, a group can be described as a category with one object such that all morphisms are isomorphisms, while a poset can be considered as a category with several objects such that from one object to another there exists at most one morphism. There are well-established representation theories for finite groups and finite posets, via group algebras and incidence algebras. What we present here is a representation theory of small categories generalizing both of the aforementioned theories. In this generality, the category algebras of small categories are used to replace group algebras and incidence algebras.

The first chapter defines the category algebra of a small category. Examples are used to illustrate this construction. Then the representations of a small category are introduced and we shall prove that these representations are identified with the modules of the category algebra we have defined, under a mild assumption. Afterwards, various types of important modules of category algebras are discussed.

The second chapter introduces basic homological tools for studying category algebras and their modules. Roughly speaking, we want to know if there is a functor 𝔲 : D → C between two small categories how we may compare representations of D and C. This is achieved by establishing functors between the categories of representations of these two small categories. An application of these functors is the characterization of indecomposable projective, injective and simple representations of a class of finite categories.

The third chapter contains one of the most important motivating examples in the representation theory of category algebras. It shows us how we may build new categories with a group and a poset. These new categories are called transporter categories. They are perhaps the best finite categories after finite groups and finite posets that we understand very well. Their category algebras are Gorenstein, usually more complicated than the two extreme cases of group algebras and incidence algebras, but still with good behavior in many aspects. Although we shall not touch it, it is enlightening to indicate that this is exactly the place where representation theory of categories, local representation theory of groups and homotopy theory of classifying spaces meet.

These notes are written for a short course given at University of Science and Technology of China (USTC) in September 2011. I am grateful to Xiao-Wu Chen for his invitation to visit USTC and the opportunity to present these materials. Some of the materials were also used for a course at Universitat Autònoma de Barcelona during the 2010-2011 academic year.

Universitat Autònoma de Barcelona, Fei Xu October 2011 xu@mat.uab.cat

Conventions We assume the reader knows basic category theory and homological algebra. Throughout these notes, the base ring R is always a commutative ring with identity. A module will normally be a left module, unless otherwise specified. If S is a set, then RS stands for the free R-module generated by S.

Given a category C, we denote by Ob C and Mor C the classes of objects and morphisms in C. For any two objects x, y ∈ Ob C the morphisms from x to y is a set written as Hom_{C}(x, y). A morphism α ∈ Hom_{C}(x, y) is often pictured as x→^{α}y. If α ∈ Hom_{C}(x, y) and β ∈ Hom_{C}(y, z) are two composable morphisms in a category C, we write βα for the composite x→^{α}y→^{β}z in Hom_{C}(x, z).

All functors are covariant if not specified.

Chapter 1

Category algebras and their representations

Abstract The concept of a category algebra is introduced here. We establish Mitchell’s theorem which shows relationship between representations of a small category and modules over the corresponding category algebra. We are particularly interested in a special class of categories, called finite EI category, and their algebras because we can classify the projective, injective and simple modules.

1.1 Definition and examples

1.1.1 Category algebras

We introduce the main concept here. Roughly speaking constructing the category algebra is a natural way to linearize a category.

Recall that a category C is called small if Mor C is a set and it is finite if Mor C is a finite set.

Definition 1.1.1. Let C be a small category and R a commutative ring. The category algebra RC is a free R-module whose basis is the set of morphisms of C. We define a product on the basis elements of RC by

α ∗ β = \{ \begin{matrix}α ∘ β, & if α and β can be composed in C; \\ 0 , & otherwise \end{matrix}

and then extend this product linearly to all elements of RC. With this product, RC becomes an associative R-algebra.

If Ob C is finite, it is easy to see that ∑_{x∈Ob C} 1_{x} is the identity of RC where 1_{x} is the identity endomorphism of x. If a category C is finite, then RC is of finite R-rank and furthermore Ob C is forced to be finite which implies that RC has an identity.

Remark 1.1.2. Another way to linearize a category C over a ring R is to construct a new category C_{R} whose objects are the same as C while the morphisms between any two objects x, y ∈ C_{R} are Hom_{C_{R}}(x, y) := RHom_{C}(x, y). This category C_{R} is additive and one can obtain the category algebra RC by forgetting the category structure in an obvious way. It appeared in Peter Gabriel’s thesis “Des Catégories Abéliennes”.

The first example shall help the reader to get some feeling of the category algebras.

Example 1.1.3. Suppose C is the following finite category

[figure: x; 1_{x}; α; y; 1_{y}; g]

with g^{2} = 1_{y} and α = gα. (One may verify that the axioms for a category are satisfied by our specifications.)

Then the category algebra RC is of rank 4 with identity 1_{RC} = 1_{x} + 1_{y}. It contains two (group) algebras R{1_{x}} and R{1_{y}, g}, as well as something to “tie them together”.

We say C is connected if C as a (directed) graph is connected. Every category C can be written as the disjoint union of connected components C = ⊎_{i∈J}C_{i}, where each C_{i} is a connected full subcategory and J is an index set. As a consequence the category algebra RC becomes a direct product of ideals RC_{i}, i ∈ J. Thus in order to study the properties of RC it suffices to study the properties of each RC_{i}. For simplicity and some technical reasons we often make the connectedness assumption.

1.1.2 Further examples

Let us consider three motivating examples.

Example 1.1.4. Let G be a (discrete) group. Then the group can be regarded as a category with only one object •, whose morphisms are the elements of G. The group algebra RG is the same as the category algebra RG. A left RG-module M can be regarded as the representation of G given by a certain functor Φ : G → R-mod, sending • to M and Aut_{G}(•) = G into Aut_{R}(M). The trivial RG-module \underline{R} is exactly the trivial module of RG.

Note that in this case G ≅ G^{op} and thus RG ≅ RG^{op} and RG^{e} ≅ RG ⊗ RG ≅ R(G × G).

A quiver 𝔮 = (Γ_{0}, Γ_{1}) is a directed graph having Γ_{0} and Γ_{1} as the set of vertices and the set of arrows, respectively. The path algebra of 𝔮 can be thought as a category algebra as follows. Any directed graph 𝔊 = (Γ_{0}, Γ_{1}) may be used to generate a category C_{𝔊} on the same set Γ_{0} of objects, where the morphisms of this category are the strings of composable arrows of 𝔊. It is called the free category generated by 𝔊.

Example 1.1.5. The path algebra of a quiver 𝔮 is the category algebra RC_{𝔮} of the free category C_{𝔮}.

It is necessary to point out that, given a category, its category algebra is usually different from its path algebra, if we consider the category as a quiver at the same time. Nevertheless, there is a relationship between these two algebras, as we now explain.

Proposition 1.1.6. Let 𝔮 be a category. We may regard 𝔮 as a quiver and form the free category C_{𝔮} over 𝔮. There is a natural functor 𝔲 : C_{𝔮} → 𝔮, which extends to a surjective homomorphism, still denoted by 𝔲 : RC_{𝔮} → R𝔮, from the path algebra of 𝔮 to the category algebra of 𝔮, such that its kernel I is generated by {←^{α_{1}}←^{α_{2}} − ←^{α_{1}α_{2}}}, where α_{1} and α_{2} are arrows of 𝔮. This epimorphism induces a natural isomorphism of R-algebras RC_{𝔮}/I and R𝔮.

Proof. The functor 𝔲 is defined as follows. For each x ∈ Ob C_{𝔮}, ϕ(x) = x. For each α ∈ Mor C_{𝔮}, 𝔲(α) = the composite of the maps in the string α. It can be extended linearly to an epimorphism 𝔲 : RC_{𝔮} → R𝔮, having the kernel I. ⊓ ⊔

The last example of a category algebra is the incidence algebra of a locally finite poset (partially ordered set). A (closed) interval of P is a subposet [x, y] ⊂ P which consists of all objects z such that x ≤ z ≤ y for a given pair of objects x, y in P. The incidence algebra I(P, R) is an R-algebra of all functions f : Int(P) → R, where Int(P) is the set of intervals of P. The multiplication (also called convolution) if defined by

(fg)([x, y]) = ∑_{x≤z≤y} f([x, z])g([z, y]).

It has an identity δ such that

δ([x, y]) = \{ \begin{matrix}1, & if x = y; \\ 0 , & otherwise. \end{matrix}

On the other hand since whenever x ≤ y in a poset we can replace ≤ with an arrow x → y, a poset is always a category.

Example 1.1.7. Let P be a finite poset. Then the incidence algebra I(P, R) of P is isomorphic to the category algebra RP.

For convenience, we denote by n the following poset of (n + 1)-objects

0 < 1 < ⋯ < n − 1 < n.

1.2 Representations of categories and Mitchell’s theorem

1.2.1 Representations of categories

A fundamental property of the category algebra RC is that it provides a mechanism for investigating R-representations of C, which we define now.

Definition 1.2.1. An (R-)representation of a category C is a (covariant) functor M : C → R-mod.

When the base ring is understood, we often abbreviate an R-representation as a representation of C. All representations of C form a functor category (Rmod)^{C}, which is an abelian category with enough projectives and injectives and we can talk about subfunctors and quotient functors.

1.2.2 Mitchell’s theorem

The following theorem was established by B. Mitchell in his paper “Rings with several objects”.

Theorem 1.2.2 (Mitchell). For any small category C, there exist functors ι : (R-mod)^{C} → RC-mod and σ : RC-mod → (R-mod)^{C} such that

1. σ ∘ ι ≅ Id_{(R-mod)^{C}}; and

2. ι is fully faithful.

Moreover if Ob C is finite then ι ∘ σ ≅ Id_{RC-mod}. Thus if Ob C is finite, the R-representations of C can be identified with the unital RC-modules.

Proof. Assume F : C → R-mod is a representation of C. We construct a free R-module 𝔐_{F} = ⨁_{x∈Ob C} F (x). For any m ∈ F (x) and morphism α ∈ Mor C, we ask α · m = F (α)(m), if t(α) = x, or α · m = 0 if t(α) ≠ x. By extending this operation linearly we obtain an RC-module structure on 𝔐_{F}. This construction can be easily verified to define a functor ι : (R-mod)^{C} → RC-mod.

Conversely, if M is an RC-module, we may define a functor 𝔉_{M} by 𝔉_{M}(x) = 1_{x} · M. Since, if α ∈ Hom_{C}(x, y) and m ∈ 1_{x} · M, α · m = (1_{y} ∘ α) · m = 1_{y} · (α · m) ∈ 1_{y} · M, we see that 𝔉_{M} : C → R-mod is well defined. It induces a functor σ : RC-mod → (R-mod)^{C}.

We may readily check that these two functors satisfy 1 and 2. ⊓ ⊔

Similar statements can be made between right RC-modules and contravariant functors from C to R-mod.

Remark 1.2.3. Throughout these notes we will be particularly interested in RC-modules lying in ι{(R-mod)^{C}}, because many important modules are indeed of this form. Thus we opt to focus on finite categories, although many results make sense for arbitrary small categories. An advantage we obtain is that there exists a well developed representation theory of finite-dimensional algebras. Consequently for practical reasons we will not distinguish between RC-mod and (R-mod)^{C}, and normally refer to an object in these categories as an RC-module.

From now on, we only consider finite category algebras.

Among all RC-modules, there is a distinguished one that we introduce below. It plays an important role in our theory.

Definition 1.2.4. For any category C, the constant functor or trivial module \underline{R} : C → R-mod, is defined by \underline{R}(x) = R for all x ∈ Ob C and \underline{R}(α) = Id_{R} for all α ∈ Mor C.

We provide several examples where both the module and functor aspects are described.

Example 1.2.5. Suppose C is finite. Then we have (R-mod)^{C} ≃ RC-mod by Theorem 1.2.2. For convenience, given a functor 𝔉 we say the RC-module ι(𝔉) corresponds to it, and similarly given an RC-module 𝔐 the functor 𝔉_{𝔐} corresponds to it, as in the proof of Theorem 1.2.2.

1. The constant functor \underline{R} corresponds to the module R Ob C, on which RC acts via α · x = y, if α ∈ Hom_{C}(x, y), or zero otherwise.

2. The regular module RC corresponds to a functor such that

RC(x) = 1_{x} · RC = RHom_{C}(−, x).

3. The opposite algebra (RC)^{op} is isomorphic to the algebra of the opposite category C^{op}, and thus a set of base elements are Mor C^{op}. Consider the enveloping category of C, namely C^{e} = C × C^{op}. Then we have

RC^{e} ≅ (RC)^{e} = (RC) ⊗_{R} (RC)^{op},

the enveloping algebra of RC. Consequently RC is an RC^{e}-module via (α, β^{op})·γ = αγβ, for any α, β, γ ∈ Mor C. As a functor RC : C^{e} → R-mod, we have

RC(y, x) = RHom_{C}(x, y)

if (x, y) ∈ Ob C^{e}.

Occasionally we will need to discuss right RC-modules. A dual version of Mitchell’s theorem says that the right RC-modules corresponds to the contravariant functors of C. There exists a natural functor (−)^{∧} = Hom_{R}(−, R) :

RC-mod → mod-RC. In case R is a field, ∧ induces an anti-isomorphism between RC-mod and mod-RC. We can give an explicit description of the dual module M^{∧} = Hom_{R}(M, R) (a right RC-module) of M ∈ RC-mod.

Lemma 1.2.6. Suppose M ∈mod-RC. Then its dual M^{∧} ∈ RC-mod has values M^{∧}(x) = M(x)^{∧} = Hom_{R}(M(x), R), and each α ∈ Mor C acts via M(α)^{∧} = Hom_{R}(M(α)(−), R).

Proof. As a free R-module, M^{∧} = ⨁_{x∈Ob C} M(x)^{∧}. Each α ∈ Hom_{C}(x, y) acts as M(α) : M(y) → M(x) and thus induces a map M(α)^{∧} : M(x)^{∧} → M(y)^{∧}. One can readily check that M^{∧}(x) is exactly M(x)^{∧} so we know the structure of M^{∧} as a right RC-module. ⊓ ⊔

The next statement follows from direct calculation.

Corollary 1.2.7. We have \underline{R}^{∧} = \underline{R}. Here the first \underline{R} is a left module and the second is a right module.

1.3 EI categories, projectives, injectives and simples

In this section, we investigate the representation theory of EI-categories. We always assume the base ring R is an algebraically closed field.

Some of the general theory of EI-categories is given by T. tom Dieck in his book “Transformation Groups”, much of which was due to W. Lück, see his book “Transformation Groups and Algebraic K-Theory”.

1.3.1 EI condition and its implications

Definition 1.3.1. An EI-category is a small category C in which all endomorphisms are isomorphisms.

One of the important features of EI-categories is described as follows. Given an EI-category C, there is a preorder defined on Ob C, that is, y ≤ x if and only if Hom_{C}(y, x) ≠ ∅. Let [y] be the isomorphism class of an object y ∈ Ob C. This preorder induces a partial order on the set Is C of isomorphism classes of Ob C (specified by [y] ≤ [x] if and only if Hom_{C}(y, x) ≠ ∅), which plays an important role in studying representations and cohomology of EIcategories. Because of the existence of an order for the isomorphism classes of objects in any EI-category, EI-categories are sometimes referred to as ordered categories by some algebraic topologists.

Definition 1.3.2. For any EI category C and any object x ∈ Ob C, we can define a full subcategory D_{≤x} ⊂ D consisting of all y ∈ Ob D such that [y] ≤ [x], or equivalently Hom_{C}(y, x) ≠ ∅. Similarly we can define other full subcategories of D: D_{<x}, D_{≥x} and D_{>x}.

An object in an EI category C is called maximal if C_{>x} = ∅, and is minimal if C_{<x} = ∅.

Definition 1.3.3. Let C be an EI category. A subcategory D of C is called an ideal if for any x ∈ Ob D, C_{≥x} ⊂ D.

A subcategory D of C is called an coideal if for any x ∈ Ob D, C_{≤x} ⊂ D.

By definition, ideals and coideals of an EI category are full subcategories. Naively speaking, an EI category consists of a bunch of groups and certain

non-isomorphisms connecting them. Note that usually a non-isomorphism does not induce a group homomorphism between the two automorphism groups of the source and target of the non-isomorphism. Nonetheless, the partial order on Is C allows us to filtrate an RC-module by modules of various group algebras inside RC.

Suppose M ∈ RC-mod and x ≅ y ∈ Ob C. We easily deduce that 1_{x}M ≅ 1_{y}M as R-modules. In the language of functors, we have M(x) ≅ M(y). If furthermore C is EI, then M(x) becomes an RAut_{C}(x)-module.

Definition 1.3.4. Suppose C is a small category. An RC-module M is said to be atomic if M as a functor satisfies the condition that M(y) ≠ 0 if and only if y ≅ x. Under the circumstance we say M is supported on C_{[x]}.

Let C be a finite EI-category and M ∈ RC-mod. Then for each maximal object x ∈ Ob C, one can construct an atomic submodule M_{x} ⊂ M such that M_{x}(y) = M(y) if y ≅ x or M_{x}(y) = 0 otherwise. Then we have a short example

0 → M_{x} → M → M/M_{x} → 0.

On the other hand, if x ∈ Ob C is minimal, one can introduce a submodule Mx̂ ⊂ M such that Mx̂(y) = M(y) if y ≅ x̸, and otherwise Mx̂(y) = 0. For each minimal x we have a short exact sequence

0 → Mx̂ → M → M/Mx̂ → 0

such that M/Mx̂ is atomic. In either case, we can repeat the process for Mx̂. Since C is finite, we will eventually obtain a filtration of M with every factor atomic. We can further refine this filtration until every factor is simple.

For each RC-module, even if C is not finite, the above analysis can visualize its structure, which will help us in future investigations. For example, every simple RC-module has to be atomic.

1.3.2 Some representation theory

Here we recall some standard facts in representation theory. Suppose A is a finite-dimensional algebra over a field k. An A-module M is indecomposable if M is not a direct sum of two non-trivial submodules. An element e ∈ A is an idempotent if e ≠ 0 and e^{2} = e. For example 1 is always an idempotent and if e ≠ 1 is an idempotent then so is 1 − e. Two idempotents e_{1} and e_{2} are orthogonal if e_{1}e_{2} = e_{2}e_{1} = 0. An idempotent e is primitive if one cannot write e = e_{1} + e_{2} such that both e_{1} and e_{2} are orthogonal idempotents.

Suppose e ∈ A is a non-identity idempotent. Then e and 1 − e are a pair of orthogonal idempotents with 1 = e + (1 − e). It results in a decomposition

A = Ae ⊕ A(1 − e).

The module Ae is projective and one can check that it is indecomposable (not a direct sum of two non-trivial modules) if and only if e is primitive.

Since A is finite-dimensional, 1 can be decomposed as a sum of finitely many pairwise orthogonal primitive idempotents (called a primitive decomposition)

1 = e_{1} + e_{2} + ⋯ + e_{n}.

Consequently the regular module A becomes a direct sum

A = Ae_{1} ⊕ Ae_{2} ⊕ ⋯ ⊕ Ae_{n}.

and we obtain a list of indecomposable projective A-modules, Ae_{i}. Up to isomorphism this is a complete list of indecomposable projective A-modules.

In fact if 1 = e′_{1} + e′_{2} + ⋯ + e′_{m} is another sum of primitive idempotents in A, then m = n and there exists an invertible element a ∈ A such that 1 = a1a^{−1} = ae′_{1}a^{−1}+ae′_{2}a^{−1}+⋯+ae′_{n}a^{−1} such that up to an reordering one has e_{i} = ae′_{i}a^{−1} for all possible i. Following from this fact, the automorphism of A, induced by a, maps each direct summand Ae′_{i} of A = Ae′_{1} ⊕ Ae′_{2} ⊕ ⋯ ⊕ Ae′_{n} isomorphically to Ae_{i}. Thus the number of idempotents in a primitive decomposition is equal to the number of indecomposable direct summand of A.

Example 1.3.5. Suppose C is a finite category and R is a commutative ring with identity. Then 1_{RC} is a sum of orthogonal idempotents ∑_{x∈Ob C} 1_{x}. Hence we have a decomposition of the regular module

RC = ⨁_{x∈Ob C} RC · 1_{x} = ⨁_{x∈Ob C} RHom_{C}(x, −),

and each summand RHom_{C}(x, −) is a projective RC-module. However since an idempotent 1_{x} may be decomposable in RC, the above decomposition can be refined, in a way depending on the choice of R. If this happens, the corresponding projective module RHom_{C}(x, −) is a direct sum of other projective modules. See Example 3.4.7.

In order to describe the simple A-modules, we have to introduce a few more terminologies. Let J(A) be the Jacobson radical of A. It is the intersection of all left maximal ideals of A. In particular it is nilpotent in the sense that there exists an integer n > 0 such that J(A)^{n} = 0. By contrast an element a ∈ A is nilpotent if a^{n} = 0 for some positive integer n. If an ideal satisfies the condition that every element is nilpotent, then this ideal is contained in the radical.

Example 1.3.6. If C is an EI-category, the any non-isomorphism, as an element in the category algebra RC is nilpotent. Since they form an ideal of RC, it implies that all non-isomorphisms are contained in J(RC). A important fact is that every element in J(A) is nilpotent, but the converse if not true (see Example 1.3.8 (4)).

For each A-module, the radical of M, RadM, is the intersection of all maximal submodules of M. For example the regular module has its radical RadA = J(A) because left ideal of A is exactly the same as a left submodule of A. In general RadM = J(A)M. It is easy to see that M/RadM has a trivial radical. Any A-module with trivial radical is called semi-simple. A semi-simple module is called simple if it is indecomposable. Equivalently an A-module is simple if it does not contain any non-trivial submodule. For example A/RadA is semi-simple. Every simple A-module occurs as a direct summand in this semi-simple module up to isomorphism.

The quotient A/RadA is itself an algebra with identity ¯1, the image of 1 ∈ A. A pairwise-orthogonal primitive decomposition 1 = e_{1} + e_{2} + ⋯ + e_{n} gives rise to ¯1 ∈ A/RadA, ¯1 = ē_{1} + ē_{2} + ⋯ + ē_{n}, which is again a sum of pairwise-orthogonal primitive idempotents in A/RadA. This simple observation actually establishes a one-to-one correspondence between the sets of isomorphism classes of indecomposable projective A-modules and of simple A-modules.

Proposition 1.3.7. Every indecomposable projective A-module, up to isomorphism, is of the form Ae, for some primitive idempotent e ∈ A. Moreover, Ae/Rad(Ae) is a simple A-module and every simple A-module arises in this way.

Moreover the number of idempotents in a primitive decomposition of 1 ∈ A, the number of indecomposable summands of A and the number of indecomposable summands of A/Rad(A) equal to each other.

There exists a large collection of good references on representation theory of associative algebras. However for those who do not plan to go over the whole theory, just bear in mind the basic constructions and important facts that we record here. Then through upcoming examples one can see how they work.

1.3.3 Projectives and simples, first approach

We shall use the recalled materials in last section to characterize projective and simple modules of certain finite category algebras. The following examples mean to tell the reader how these module look like. Then in Chapter 2, after developing enough tools, we can complete the picture by providing a full description of indecomposable projective and simple modules of a finite EI category algebra.

Example 1.3.8. 1. Let G = {g | g^{2} = 1_{•}} be the cyclic group of order 2, regarded as a category with one object. If k = ℂ is the field of complex numbers, the identity 1_{•} can be written as \frac{1_{•}+g}{2} + \frac{1_{•}−g}{2} , a decomposition into a sum of orthogonal primitive idempotents. The regular module is a direct sum of two one dimensional modules ℂG ≅ ℂ(1_{•} + g) ⊕ ℂ(1_{•} − g). Thus both ℂ(1_{•} + g) and ℂ(1_{•} − g) are projective. They are simple as well because they cannot have non-trivial submodules. It means ℂG is semisimple with trivial radical. The module ℂ(1_{•} +g) is the trivial module and ℂ(1_{•} − g) is called the sign representation. However when k is a field of characteristic 2, 1_{•} is primitive. Hence kG is indecomposable. The regular module has exactly one non-trivial submodule k(1_{•} + g), which has to be the radical Rad(kG). Then kG/Rad(kG) is one-dimensional and is simple. It is the only simple kG-module, the trivial module.

2. The poset 1 = 0 → 1 is a category with two objects 0 and 1. For any field k, the identity 1_{k1} = 1_{0} + 1_{1} in the category algebra k1. The two identity morphisms 1_{0} and 1_{1} are primitive orthogonal idempotents so k1 = k{1_{0}, α} ⊕ k1_{1}. The first indecomposable summand has exactly one non-trivial submodule k{α}, the radical Rad(k{1_{0}, α}) of k{1_{0}, α}. Then it gives rise to a one-dimensional simple module S_{0}. As a functor, S_{0}(0) = k and S_{0}(1) = 0. The second summand is of dimension one so it is already simple. If we denote it by S_{1}. As a functor S_{1}(0) = 0 and S_{1}(1) = k.

3. Now we examine the category C of Example 1.1.3 that is neither a group nor a poset.

[figure: x; 1_{x}; α; y; 1_{y}; g]

with g^{2} = 1_{y} and α = gα. We always have 1_{kC} = 1_{x}+1_{y} so kC = k{1_{x}, α}⊕ k{1_{y}, g}. Similar to 2, the first summand, named P_{x}, is indecomposable and has radical k{α}. The quotient of P_{x} by its radical is a one-dimensional simple module S_{x} (analogues to S_{0} as above). According to 1, the second direct summand is decomposable if k = ℂ. Whence we have ℂG ≅ ℂ(1_{•} + g) ⊕ ℂ(1_{•} − g). It means when ℂC has three indecomposable projective modules and the same number of simple modules.

The category algebra ℂC

Indecomposable projective modules Simples module
P_{x,1} = ℂ{1_{x}, α}  S_{x,1} = ℂ{1_{x}}
P_{y,1} = ℂ{1_{y} + g}  S_{y,1} = ℂ{1_{y} + g}
P_{y,−1} = ℂ{1_{y} − g}  S_{y,−1} = ℂ{1_{y} − g}

When k is of characteristic 2, k{1_{y}, g} is indecomposable. Whence kC only has two indecomposable projective and simple modules.

The category algebra kC, chark = 2

Indecomposable projective modules Simple modules
P_{x,1} = k{1_{x}, α}  S_{x,1} = k{1_{x}}
P_{y,1} = k{1_{y}, g}  S_{y,1} = P_{y,1}/k{1_{y} + g}

4. A useful example to bear in mind is the following category (a groupoid) D that is equivalent to •

\xymatrix{x \ar[r]^{α} & y. \ar[l]^{α^{−1}}}

One can write RD = RD · 1_{x} ⊕ RD · 1_{y} = R{1_{x}, α} ⊕ R{1_{y}, α^{−1}}. Observe that we have an isomorphism (−) ∘ α^{−1} : R{1_{x}, α} → R{1_{y}, α^{−1}} between RD-modules. When R = k is a field, every non-zero element of a1_{x} + bα ∈ kD · 1_{x} generates the whole module. Hence kD · 1_{x} has no non-trivial submodule and thus is simple. Note that it corresponds to the (simple) functor S_{x,1} which takes values S_{x,1}(x) = k1_{x} and S_{x,1}(y) = kα. Similarly kD · 1_{y} is also a simple module, corresponding to the (simple) functor S_{y,1} given by S_{y,1}(x) = kα^{−1} and S_{y,1}(y) = k1_{y}, which is isomorphic to S_{x,1}. The algebra kD is semi-simple but contains nilpotent elements α and α^{−1}. (Note that α and α^{−1} generate k{1_{x}, α} and k{1_{y}, α^{−1}}, respectively, both containing non-nilpotent elements.) The algebra kD is Morita equivalent to k•.

5. Finally let us consider the following category E

[figure: x; α; y; β; 1_{y}; e]

such that αβ = e, βe = β, eα = α and e^{2} = e. Note that α and β are not invertible in E. Hence E is not a groupoid and the two objects are not isomorphic in E. However the following part of E forms a category

[figure: x; α; y; β; e]

that is isomorphic to D. Consequently the category algebra RE is isomorphic to R{1_{x}, α, β, e}×R{1_{y}} ≅ RD×R•. Note that this is not isomorphic to R(D × {1_{y}}) = RD ⊗_{R} R{1_{y}}. This category is a simple example of inverse categories, see [15]. The category algebra of an inverse category is canonically isomorphic to a direct product of groupoid algebras.

Sometimes we may want to use injective modules, so we finish this section with several remarks concerning the injectives. In general injective modules behave better than the projectives in the sense that for any ring A and any Amodule M, there exists a minimal injective A-module I_{M} such that M admits an injection into I_{M}. This module is called the injective hull of M. There are module categories with enough injective but not with enough projectives. However if A is a finite-dimensional algebra, then for any M ∈ A-mod there exists a minimal projective module P_{M} which admits a surjection onto M. It is called the projective cover of M. For instance, in the tables of Example 1.3.8 (3), each row consists of a simple module as well as its projective cover. When we do representation theory we often prefer working with projective modules because simple modules come from their quotients.

As we pointed out earlier one has an anti-isomorphism (−)^{∧} = Hom_{k}(−, k) from A-mod to mod-A. Suppose P is a projective A-module, then P^{∧} is an injective right A-module. The anti-isomorphism provides a bijection between projective left (resp. right) A-modules and injective right (resp. left) A-modules. Thus knowing all projectives leads to getting all injectives. An important case is the group algebra of a finite group G, or more generally a finite-dimensional cocommutative Hopf algebra. They are self-injective, which means the regular module is an injective module. In this case, the sets of projective and injective modules coincide.

In the end we mention an important concept in algebra.

Definition 1.3.9. Let A and B be two rings. Then A is Morita equivalent to B if A-mod is equivalent to B-mod.

Two Morita equivalent rings have the same representation theory. We will prove that a category equivalence induces a Morita equivalence between category algebras. Thus in Example 1.1.3 (4) RC is Morita equivalent to R = R•.

1.3.4 Injective modules

Here we assume the base ring R is a field.

In Lemma 1.2.6 and the paragraph precedes it we mentioned how to use duality to obtain injective modules. In summary, given a finite EI category algebra RC and an indecomposable projective RC-module P, its R-dual P^{∧} becomes an indecomposable injective right module. Moreover all indecomposable injective right RC-modules are constructed in this way (up to isomorphism). In a similar fashion to produce all indecomposable injective RCmodules, we can first find all indecomposable projective right RC-modules and then take their duals. In this section we shall use incidence algebras to illustrate the process.

Example 1.3.10. Let us consider the following poset P : x → y. Its category algebra RP has two indecomposable projective left modules, pictured as

P_{x} : R → R, P_{y} : 0 → R

and two indecomposable projective right modules

yP : R ← R, _{x}P : R ← 0.

By duality, we immediately obtain two indecomposable injective left RPmodules

I_{y} =_{y} P^{∧} : R → R, I_{x} =_{x} P^{∧} : R → 0.

We can continue to show that RP has finite injective dimension because RP ≅ P_{x} ⊕ P_{y} and we have injective resolutions

0 → P_{x} → I_{y} → 0

and

0 → P_{y} → I_{y} → I_{x} → 0.

We say inj.dim_{RP}RP = 1. Similarly one finds inj.dimRP_{RP} = 1.

Proposition 1.3.11. Suppose P is a finite poset and M is an RP-module. Then M has both finite projective and injective dimension.

Proof. In fact, one can prove that all left RP-modules and all right RPmodules are of finite projective dimension. We leave the reader to add the details. ⊓ ⊔

Incidence algebras and groups algebras are examples of Gorenstein algebras.

Definition 1.3.12. Let A be a finite-dimensional algebra. Then it is Gorenstein if inj.dim_{A}A and inj.dimA_{A} are finite.

Self-injective algebras, such as Hopf algebras and symmetric algebras, are always Gorenstein. The converse is not correct because of Example 1.3.10. We shall come back to Gorenstein algebras in Chapter 3.

1.4 Combinatorially constructed modules

Here we briefly discuss how combinatorial constructions attain modules.

Definition 1.4.1. Let C be a small category. For every n ≥ 0, we define N_{n}C to be the set of all n-chains of morphisms in C, that is, the set of all symbols of the form

x_{0}→^{α_{1}}x_{1}→^{α_{2}} ⋯ → x_{n−1}→^{α_{n}}x_{n}.

These sets are part of a combinatorial construction, a simplicial set named the nerve of C, written as NC (there are many references on simplicial sets and we do not plan to go into the direction).

We easily see that N_{0}C = Ob C and N_{1}C = Mor C. Since the regular module RC = RN_{1}C, we would like to say a few words about the other RN_{n}C.

Lemma 1.4.2. The free R-module RN_{0}C carries a natural RC-module structure such that RN_{0}C ≅ \underline{R}.

Proof. This is Example 1.2.5 (1). ⊓ ⊔

For each n ≥ 1, we can introduce an RC-module structure on RN_{n}C by

α · (x_{0}→^{α_{1}}x_{1}→^{α_{2}} ⋯ → x_{n−1}→^{α_{n}}x_{n}) = x_{0}→^{α_{1}}x_{1}→^{α_{2}} ⋯ → x_{n−1} →^{αα_{n}}y

if α ∈ Hom_{C}(x_{n}, y), or zero otherwise. One may readily check that in this way RN_{1}C is exactly the regular module RC. These combinatorially constructed modules play a significant role in the representation and (co)homology theories of groups and small categories (which we will not go over here), partially due to the fact that they form a complex of RC-modules

ℂ_{∗}(C, R) : ⋯ → RN_{n}C → RN_{n−1}C → ⋯ → RN_{0}C → 0.

(The differentials come from the face maps of the simplicial set NC.) The homology of the above complex is called the homology of C with coefficients in R. They are of fundamental importance in homology representations of groups as well as algebraic topology, for example, see for example [4, Chapter 7] and [10].

Basic tools

Abstract Let 𝔲 : D → C be a functor between two small categories. We show how it induces functors between RD-mod and RC-mod. In many interesting situations, these functors are very well understood so they are used to do concrete computations. Examples are given to illustrate our techniques.

2.1 Restriction on algebras and modules

We prove some basic properties, many of which follow directly from simple reasoning.

Definition 2.1.1. Suppose 𝔲 : D → C is a (covariant) functor. We define Res_{𝔲} : RC-mod → RD-mod to be the restriction along 𝔲. Given a module M ∈ RC-mod, we have Res_{𝔲}M = M ∘ 𝔲 ∈ RD-mod.

Lemma 2.1.2. Let 𝔲 : D → C be a functor. Then Res_{𝔲}\underline{R} = \underline{R}.

Since a functor 𝔲 : D → C also extends linearly to a natural map of R-modules 𝔲̄ : RD → RC, it is reasonable to ask if 𝔲̄ is always an algebraic homomorphism because if it is, the so-called change-of-base-ring or the representation-theoretic restriction, ↓_{RD}^{RC}: RC-mod → RD-mod, should coincide with Res_{𝔲}. The answer to the question is no, and here is a simple example. Let D = 1 and C = •. There is a unique functor pt : D → C. The induced map p̄t : RD → RC is not an algebra homomorphism since the product of the two isomorphisms in D is zero while the product of their images is not.

Proposition 2.1.3. A functor 𝔲 : D → C extends linearly to an algebra homomorphism 𝔲̄ : RD → RC if and only if 𝔲 is injective on Ob D. When this happens, the induced functor followed by 1_{RD}, 1_{RD}∘ ↓_{RD}^{RC}: RC-mod → RDmod, is exactly Res_{𝔲}.

Proof. We know 𝔲(βα) = 𝔲(β)𝔲(α) for any pair of composable morphisms α, β in D. The injectivity of 𝔲 implies two morphisms α, β ∈ Mor(D) are composable if and only if 𝔲(α), 𝔲(β) ∈ Mor(C) are composable.

If 𝔲 is injective on Ob D, then we define a map 𝔲̄ : RD → RC as the linear extension of functor 𝔲, i.e., 𝔲̄(∑_{i} r_{i}α_{i}) = ∑_{i} r_{i}𝔲̄(α_{i}) for any r_{i} ∈ R, α_{i} ∈ Mor(D). This 𝔲̄ is indeed an algebra homomorphism because our previous observation of 𝔲 implies 𝔲̄((∑_{j} r_{j}β_{j})(∑_{i} r_{i}α_{i})) = 𝔲̄(∑_{j} r_{j}β_{j})𝔲̄(∑_{i} r_{i}α_{i}) is always true.

On the other hand if the linear extension 𝔲̄ : RD → RC is an algebra homomorphism then we must have 𝔲̄(0) = 0 and then 𝔲̄(1_{x})𝔲̄(1_{y}) = 𝔲̄(1_{x} · 1_{y}) = 0 unless x = y. This suggests that 𝔲 is injective on Ob C′.

When 𝔲̄ : RD → RC is an algebra homomorphism, we show 1_{RD}∘ ↓_{RD}^{RC}= Res_{𝔲} : RC-mod → RD-mod. Let M be an RC-module and γ ∈ Mor(D). Then it corresponds to some 𝔉_{M} ∈ (R-mod)^{C}. The RD-modules Res_{𝔲}M and 1_{RD} · (M ↓_{RD}^{RC}) are isomorphic because Res_{𝔲}M = 𝔐_{Res_{𝔲}(𝔉_{M})}, γ · M = 𝔲(γ)M for any γ ∈ Mor D and 1_{RD} kills the elements of M ↓_{RD}^{RC} that are not supported on any objects of D. ⊓ ⊔

Proposition 2.1.4. Let D and C be equivalent small categories. Then (Rmod)^{C} ≃ (R-mod)^{D}, an equivalence which sends the constant functor to the constant functor. If both Ob C and Ob D are finite then RC and RD are Morita equivalent.

Proof. We show that the two functor categories (R-mod)^{C} and (R-mod)^{D} are equivalent. Then it implies the module categories RC-mod and RD-mod are equivalent, hence RC and RD are Morita equivalent. In fact if 𝔲 : D → C and 𝔳 : C → D are equivalences, we have Res_{𝔲}Res_{𝔳} ≅ Id_{RD} := Id_{RD-mod} : (Rmod)^{D} → (R-mod)^{D} because of the following diagram

\xymatrix{M(𝔳𝔲(x)) = (Res_{𝔲}Res_{𝔳}M)(x) \ar[r]^{≅} \ar[d]_{M(𝔳𝔲(α))=(Res_{𝔲}Res_{𝔳}M)(α)} & (Id_{RD}M)(x) = M(x) \ar[d]^{(Id_{RD}M)(α)=M(α)} \\ M(𝔳𝔲(y)) = (Res_{𝔲}Res_{𝔳}M)(y) \ar[r]_{≅} & (Id_{RD}M)(y) = M(y)}

where M ∈ (R-mod)^{D}, α : x → y ∈ Mor D and Id_{RD} is the identity functor. Similarly we can show Res_{𝔳}Res_{𝔲} ≅ Id_{RC} : (R-mod)^{C} → (R-mod)^{C}. Clearly the constant functor restricts to the constant functor always. ⊓ ⊔

Remark 2.1.5. From Example 1.3.8 (5) we learned that the existence of a Morita equivalence RC ≃ RD induced by a functor 𝔲 : D → C does not guarantee that 𝔲 is an equivalence.

2.2 Kan extensions of modules

In most cases a functor 𝔲 : D → C does not come with an adjoint. Nevertheless the restriction Res_{𝔲} : RD-mod → RC-mod does have two adjoints. These two functors are the subjects of this section. They are called Kan extensions, generalizing limits, and are classical tools in homological algebra. Since their computations are not easy, it may be a good idea to try to understand upcoming constructions for the two extreme cases of finite groups and finite posets, in a first reading.

2.2.1 Limits

In these notes, direct and inverse limits of functors are key concepts so we recall their definitions here. For those who are familiar with the materials, you can just skip this section or quickly browse over the examples and quoted results that we need later on.

Suppose M ∈ Ob(T^{C}) is an object in a functor category. Then it can be identified with a commutative diagram of objects {M(x) | x ∈ Ob C} and morphisms {M(α) | α ∈ Mor C} in T . For convenience and future reference, we call this the diagram of M.

Definition 2.2.1. Let C be a small category and T an arbitrary category. Consider the functor category T^{C}. Suppose M ∈ Ob(T^{C}). Then it has an inverse limit, denoted by \underleftarrow{lim}_{C}M ∈ Ob T , if for each x ∈ Ob C there exists a morphism θ_{x} : \underleftarrow{lim}_{C}M → M(x) such that, by adding the object \underleftarrow{lim}_{C}M and the morphisms {θ_{x} | x ∈ Ob C} to the diagram of M, we obtain an enlarged commutative diagram, and this object \underleftarrow{lim}_{C}M is universal in the sense that if t is another object in Ob T that enjoys the same properties as \underleftarrow{lim}_{C}M then we must have a unique morphism Θ_{t} : t → \underleftarrow{lim}_{C}M making the whole diagram commutative as pictured as follows

[figure: t; θ′_{y}; θ′_{x}; Θ_{t}; \underleftarrow{lim}_{C}M; θ_{x}; θ_{y}; M(x); M(α); M(y); .]

Dually the functor M has a direct limit in T , denoted by \underrightarrow{lim}_{C}M ∈ Ob T , if for each x ∈ Ob C there exists a morphism τ_{x} : M(x) → \underleftarrow{lim}_{C}M such that, by adding the object \underrightarrow{lim}_{C}M and the morphisms \{τ_{x} | x ∈ Ob C\} to the diagram of M, we obtain an enlarged commutative diagram, and this object \underrightarrow{lim}_{C}M is universal in the sense that if t is another object in Ob T that enjoys the same properties as \underrightarrow{lim}_{C}M then we must have a unique morphism Ξ_{t} : \underrightarrow{lim}_{C}M → t making the whole diagram commutative as pictured as follows

[figure: M(x); M(α); τ_{x}; τ′_{x}; M(y); τ_{y}; τ′_{y}; \underrightarrow{lim}_{C}M; Ξ_{t}; t]

It follows directly from the definition that, when a limit exists, it is unique up to isomorphism. If either the index category or the functor itself is structurally simple, we may explicitly compute the limits.

Example 2.2.2. 1. Let C = ℤ and T = 𝔖𝔢𝔱. Then a functor M : ℤ → 𝔖𝔢𝔱 is represented by a chain of set maps

⋯ → M(−1) → M(0) → M(1) → M(2) → · · · .

In particular if M_{0} is given by the following diagram in which each map is an inclusion

⋯ ⊂ M_{0}(−1) ⊂ M_{0}(0) ⊂ M_{0}(1) ⊂ M_{0}(2) ⊂ ⋯,

then \underleftarrow{lim}_{ℤ}M_{0} = ⋂_{i∈ℤ} M_{0}(i) and \underrightarrow{lim}_{ℤ}M_{0} = ⋃_{i∈ℤ} M_{0}(i).

2. If C = ℤ and T = V ect_{k}, the category of finite-dimensional k-vector spaces. Suppose N is represented by the following

⋯ → 0 ⊊ N(1) ⊊ N(2) ⊊ N(3) ⊊ · · · .

Then \underrightarrow{lim}_{ℤ}N does not exist in V ect_{k}, the category of finite-dimensional k-vector spaces, but \underleftarrow{lim}_{ℤ}N = \{0\}.

3. Let T be a category and I a set considered as a discrete category. Then a functor M : I → T is simply an I-indexed set of objects in T . The the coproduct of these objects is defined as ⨁_{I} x_{i} = \underrightarrow{lim}_{I}M and the product of these objects is ∏_{I} x_{i} = \underleftarrow{lim}_{I}M.

4. Let C be the following poset

\xymatrix{ & x \ar[d]^{α} \\ y \ar[r]_{β} & z}

and T = Vect_{k}, the category of all k-vector spaces. Each functor M ∈ Ob T^{C} is represented by

\xymatrix{ & M(x) \ar[d]^{M(α)} \\ M(y) \ar[r]_{M(β)} & M(z).}

Then \underleftarrow{lim}_{C}M is called the pullback of the latter diagram and \underrightarrow{lim}_{C}M = M(z). Dually if D is the following poset

\xymatrix{a \ar[r]^{f} \ar[d]_{g} & b \\ c}

and N ∈ Ob T^{D} then \underrightarrow{lim}_{D}N is called the pushout of the diagram

\xymatrix{N(a) \ar[d]_{N(g)} \ar[r]^{N(f)} & N(b) \\ N(c)}

and \underleftarrow{lim}_{D}N = N(a).

The definition of limits can be rewritten by using a simple, yet very important, construction. It shall pave the way to define Kan extensions.

Definition 2.2.3. There is a constant functor

𝔎 : T → T^{C}

such that, for any t ∈ Ob T , 𝔎(t) is defined by 𝔎(t)(x) = t and 𝔎(α) = 1_{t} for any x ∈ Ob C and α ∈ Mor C.

In the literature, the functor 𝔎 : T → T^{C} is often named the “diagonal functor”. Since the terminology is used for another purpose in our theory, we shall stick with our notion which seems to be more appropriate.

Now we present alternative characterizations of limits. The definition of an inverse limit can be rephrased as saying that there exists \underleftarrow{lim}_{C}M ∈ Ob C with a natural transformation Γ : 𝔎(\underleftarrow{lim}_{C}M) → M which is universal in the sense that if there is another object t, along with a natural transformation Υ : 𝔎(t) → M, then there exists a unique morphism θ_{t} : t → \underleftarrow{lim}_{C}M hence a natural transformation 𝔎(θ_{t}) : 𝔎(t) → 𝔎(\underleftarrow{lim}_{C}M) such that Γ ∘ 𝔎(θ_{t}) = Υ.

\xymatrix{𝔎(t) \ar[dr]^{Υ} \ar@{.>}[dd]_{𝔎(θ_{t})} \\  & M \\ 𝔎(\underleftarrow{lim}_{C}M) \ar[ur]_{Γ}}

Especially we obtain a morphism

Ω : Hom_{T ^{C}}(𝔎(t), M) → Hom_{T} (t, \underleftarrow{lim}_{C}M),

given by Ω(Υ) = θ_{t}. It is an isomorphism because each t → \underleftarrow{lim}_{C}M extends to a functor 𝔎(t) → M. Thus if every M ∈ Ob(T^{C}) has a limit in T , then \underleftarrow{lim} : T^{C} → T is the right adjoint of 𝔎 and Ω becomes the corresponding adjunct. We characterize the direct limit \underrightarrow{lim}_{C} in a similar way but we leave it to the reader. When T is “large” enough with respect to C, we can introduce the limits in an economical way.

Proposition 2.2.4. If 𝔎 has a right adjoint ℜ : T^{C} → T , then for each M ∈ Ob(T^{C}), ℜ(M) is the inverse limit of M. If 𝔎 has a left adjoint 𝔏 : T^{C} → T , then for each M ∈ Ob(T^{C}), 𝔏(M) is the direct limit of M.

For future reference, we recall the following concepts.

Definition 2.2.5. A category T is called complete if for any small category C and any functor M ∈ T^{C} the inverse limit \underleftarrow{lim}_{C}M exists. A category T is called cocomplete if for any small category C and any functor M ∈ T^{C} the direct limit \underrightarrow{lim}_{C}M exists.

A category T is called finitely complete (respectively finitely cocomplete) if for any finite category C and any functor M ∈ T^{C} the inverse limit \underleftarrow{lim}_{C}M (respectively the direct limit \underrightarrow{lim}_{C}M) exists.

The categories 𝔖𝔢𝔱 and R-Mod are both complete and cocomplete.

A coproduct is a direct limit and a product in an inverse limit. In abelian categories, the existence of coproducts, respectively product, is equivalent to the cocompleteness, respectively completeness, condition on a category T .

Proposition 2.2.6. Let T be an abelian category. Then

1. it is complete (or finitely complete) if and only if all products (or all finite products) exist; and

2. it is cocomplete (or finitely cocomplete) if and only if all coproducts (or all finite coproducts) exist.

Proof. We only prove (1). Since a product is an inverse limit, we just have to demonstrate the opposite direction. Suppose C is a small (or finite) category and M : C → T is a functor. Let

ℂ^{0}(C; M) = {f : Ob C → ∏_{x∈Ob C} M(x) | f(x) ∈ M(x)}

and

ℂ^{1}(C; M) = {f : Mor C → ∏_{α∈Mor C} M(y) | f(x→^{α}y) ∈ M(y)}.

We define δ : ℂ^{0}(C; M) → ℂ^{1}(C; M) by

δ(f)(x→^{α}y) = f(y) − M(α)[f(x)].

Then we can verify that the kernel of δ is \underleftarrow{lim}_{C}M. ⊓ ⊔

For example the category of finitely generated R-modules, R-mod, is both finitely complete and finitely cocomplete. Completeness and cocompleteness pass to functor categories. We record this and another result that shall be useful in understanding Kan extensions.

Proposition 2.2.7. Let C be a small category. If T is complete (resp. cocomplete), then T^{C} is complete (resp. cocomplete).

The above result tells us that 𝔖𝔢𝔱^{C} and (R-Mod)^{C} are both complete and cocomplete if C is small. Furthermore (R-mod)^{C} is finitely complete and cocomplete if C is finite.

We record two results on preserving limits. The first is about change-of-target categories.

Proposition 2.2.8. Let 𝔉 : T → T′ be the left adjoint of 𝔊 : T′ → T .

1. Suppose M : C → T has a direct limit. Then 𝔉(\underrightarrow{lim}_{C}M) ≅ \underrightarrow{lim}_{C}𝔉 ∘ M.

2. Suppose N : D → T′ has an inverse limit. Then 𝔊(\underleftarrow{lim}_{D}N) ≅ \underleftarrow{lim}_{D}𝔊 ∘ N.

The second concerns change-of-source categories. We begin with an observation. Suppose 𝔲 : D → C is a functor. Then for any M ∈ Ob(T^{C}), there are canonical morphisms \underleftarrow{lim}_{C}M → \underleftarrow{lim}_{D}Res_{𝔲}M and \underrightarrow{lim}_{D}Res_{𝔲}M → \underrightarrow{lim}_{C}M, coming from the universal property, for example, of inverse limits,

[figure: \underleftarrow{lim}_{C}M; α_{y}; α_{x}; Θ; \underleftarrow{lim}_{D}Res_{𝔲}M; ϕ_{x}; ϕ_{y}; Res_{𝔲}M(a) = M(𝔲(a)); Res_{𝔲}M(ϕ)=M(𝔲(ϕ)); Res_{𝔲}M(b) = M(𝔲(b)); ,]

in which ϕ : a → b is a morphism in D. The map between direct limits can be established similarly.

Proposition 2.2.9. Suppose 𝔲 : D → C has a left adjoint 𝔳 : C → D. Then for any T , Res_{𝔲} has a right adjoint Res_{𝔳}. In particular, if 𝔲 and 𝔳 are category equivalences, Res_{𝔲} and Res_{𝔳} are equivalences.

If T is complete and cocomplete, then for any M ∈ Ob(T^{C}), \underrightarrow{lim}_{C}M ≅ \underrightarrow{lim}_{D}Res_{𝔲}M, and for any N ∈ Ob(T )^{D}, \underleftarrow{lim}_{D}N ≅ \underleftarrow{lim}_{C}Res_{𝔳}N.

2.2.2 Overcategories and undercategories

In order to introduce the adjoints of a restriction, we have to provide some important categorical constructions. These categorical constructions are of great importance in both homological algebra and algebraic topology.

Definition 2.2.10. Let 𝔲 : D → C be a functor between (small) categories and x ∈ Ob C. The category over x, 𝔲/x, consists of objects {(a, α) | a ∈ Ob D, α ∈ Hom_{C}(𝔲(a), x)}. For any two objects (a, α), (b, β), a morphism from (a, α) to (b, β) is a morphism μ ∈ Hom_{D}(a, b) making the following diagram commutative

\xymatrix{𝔲(a) \ar[dr]^{α} \ar[dd]_{𝔲(μ)} \\  & x, \\ 𝔲(b) \ar[ur]_{β}}

The category under x, written as x\𝔲, is defined in a dual fashion. It consists of objects {(α, a) | a ∈ Ob D, α ∈ Hom_{C}(x, 𝔲(a))}. For any two objects (α, a), (β, b), a morphism from (α, a) to (β, b) is a morphism μ ∈ Hom_{D}(a, b) making the following diagram commutative

\xymatrix{ & 𝔲(a) \ar[dd]^{𝔲(μ)} \\ x \ar[ur]^{α} \ar[dr]_{β} \\  & 𝔲(b)}

The categories defined above are customarily called overcategories and undercategories, associated with 𝔲 : D → C. We will see later on that Id_{C}/x and x\Id_{C}, for any x ∈ Ob C, are already very interesting.

Remark 2.2.11. 1. From definition, an object in the overcategory 𝔲/x, (a, α), can be pictured as 𝔲(a)→^{α}x, and consequently a morphism μ : (a, α) →

(b, β) can be equivalently interpreted as a sequence 𝔲(a) →^{𝔲(μ)}𝔲(b)→^{β}x. This kind of rewritings will be useful for us when dealing with chains of morphisms in 𝔲/x. Similar reinterpretation can be made for objects and morphisms in undercategories too but we leave it to the reader.

2. There is a canonical functor 𝔓_{x} : 𝔲/x → D (resp. 𝔓^{x} : x\𝔲 → D given on objects as projection to the first (resp. the second) component and on morphisms as the identity. For simplicity, we shall denote such functors just as 𝔓.

3. If γ : x → y is a morphism in C, then it naturally induces a functor γ_{∗} : 𝔲/x → 𝔲/y and a functor γ^{∗} : y\𝔲 → x\𝔲.

Example 2.2.12. 1. Let pt : C → • be the canonical functor. Then pt/• ≅

•\pt ≅ C.

2. Let G be a group and H a subgroup. Then the inclusion functor 𝔦_{H} : H ↪ G gives exactly one overcategory 𝔦_{H}/•. By direct calculation, the objects are {(•, g)|g ∈ G}, and biject with the elements of G. There is a morphism from one object (•, g_{1}) to another (•, g_{2}) if there exists a h ∈ H such that g_{1} = g_{2}h. Obviously h = g_{2}^{−1}g_{1}. Thus there is at most one morphism from an object to another. Since h is invertible, there exists a morphism between two objects if and only if their are isomorphic in 𝔦_{H}/•. In other words, two objects (•, g_{1}) to another (•, g_{2}) are isomorphic if and only if g_{1}H = g_{2}H. Because the category 𝔦_{H}/• consists of [G : H] many groupoids, each of which is equivalent to the trivial category, 𝔦_{H}/• is equivalent to the discrete set G/H of left cosets (regarded as a category). Similarly the undercategory •\𝔦_{H} has objects {(g, •)|g ∈ G}. There is a morphism from (g_{1}, •) to (g_{2}, •) if and only if there exists a (unique) h ∈ H such that hg_{1} = g_{2} or equivalently h = g_{2}g_{1}^{−1}. The undercategory is equivalent to H\G, the set of right cosets. We have an isomorphism 𝔦_{H}/• → •\𝔦_{H} given by

(•, g) ↦ (g^{−1}, •) and (•, g_{1}) →^{g_{2}^{−1}g_{1}}(•, g_{2}) ↦ (g_{1}^{−1}, •) →^{g_{2}^{−1}g_{1}}(g_{2}^{−1}, •).

When G = H, 𝔦_{G}/• = Id_{G}/• ≅ •\Id_{G} = •\𝔦_{G} is the Cayley graph of G.

3. Suppose 𝔲 : D → C is a functor between two posets. Then for any x ∈ Ob C, 𝔲/x is isomorphic to the subposet of D consisting of objects {a ∈ Ob D | Hom_{C}(𝔲(a), x) ≠ ∅}, while x\𝔲 is isomorphic to the subposet of D consisting of objects {b ∈ Ob D | Hom_{C}(x, 𝔲(b)) ≠ ∅}.

The following observation follows directly from definitions and will be useful to us. For any functor 𝔲 : D → C one can define a (covariant) opposite functor 𝔲^{op} : D^{op} → C^{op} such that 𝔲^{op}(x) = x and 𝔲^{op}(α^{op}) = 𝔲(α)^{op}. Be aware that it is different from the dual functor given before Definition 1.1.9.

Lemma 2.2.13. Suppose 𝔲 : D → C is a functor. Consider its opposite functor 𝔲^{op} : D^{op} → C^{op}. Then for any x ∈ Ob C = Ob C^{op} we have (𝔲/x)^{op} ≅ x\𝔲^{op} and (x\𝔲)^{op} ≅ 𝔲^{op}/x.

2.2.3 Kan extensions

In this section, we assume T to be a complete and cocomplete abelian category. The reader should bear in mind in order to see that Kan extensions generalize direct and inverse limits. Indeed Kan extensions are a bunch of limits, carefully assembled together.

Theorem 2.2.14. Let 𝔲 : D → C be a functor between small categories. Then the restriction Res_{𝔲} : T^{C} → T^{D} admits a left adjoint LK_{𝔲}, called the left Kan extension along 𝔲, as well as a right adjoint RK_{𝔲}, called the right Kan extension along 𝔲.

Given M ∈ Ob(T^{D}) the left and right Kan extensions along 𝔲 are given by

LK_{𝔲}M = \underrightarrow{lim}_{𝔲/−}Res_{𝔓}M and RK_{𝔲}M = \underleftarrow{lim}_{−\𝔲}Res_{𝔓}M,

where 𝔓 is the functor in Remark 2.2.11 (2).

The proof is quite formal and can be found in many books on homological algebras.

Since not every functor 𝔲 : D → C has an adjoint, we have to look for other functors for comparing RD-mod and RC-mod. Clearly the restriction Res_{𝔲} possesses two adjoint functors, the left and right Kan extensions LK_{𝔲} and RK_{𝔲}. Here will be our first attempt to study Kan extensions of modules. We note that, because Res_{𝔲} is exact, LK_{𝔲} preserves projectives while RK_{𝔲} preserves injectives.

Proposition 2.2.15. Let G be a finite group, H a subgroup and 𝔦 : H ↪ G the inclusion functor. Then LK_{𝔦} ≅ RK_{𝔦} ≅↑_{H}^{G} ≅⇑_{H}^{G}.

Proof. Based on our computations in Example 2.2.12 (2), as well as Proposition 2.2.9, if M ∈ RH-mod,

\underrightarrow{lim}_{𝔦_{H}/•}M ≅ \underrightarrow{lim}_{G/H}M ≅ ⊕_{g∈G/H}gH ⊗_{k} M.

One can see that G permutes these direct summands and the limit \underrightarrow{lim}_{𝔦_{H}/•}M is isomorphic to RG ⊗_{RH} M. ⊓ ⊔

Suppose M ∈ RD-mod and 𝔲 : D → C is a functor. Then we obtain an RC-module LK_{𝔲}M. As a functor, for each x ∈ Ob C, (LK_{𝔲}M)(x) is given by a direct limit, and if α : x → y is a morphism, it induces a map (LK_{𝔲}M)(x) → (LK_{𝔲}M)(y) by the universal property of \underrightarrow{lim}. This specifies the RC-action on LK_{𝔲}M.

Suppose P is a poset and Q is a subposet. We call Q an ideal of P if for a pair of object x ∈ Ob Q and y ∈ Ob P such that x ≤ y in P then y belongs to Q. Before we start the first calculation we mention that if D is a full subcategory of a small category C then any functor over D can be naively considered as a functor over C by asking its value to be zero on any objects that do not belong to D. In other words, any RD-module is naturally an RC-module under the assumption.

Proposition 2.2.16. Suppose P is a poset and Q is an ideal together with the inclusion 𝔦 : Q ↪ P. Then for any M ∈ RQ-mod, the RP-module LK_{𝔦}M is exactly M, regarded as an RP-module.

Proof. One can find that (LK_{𝔦}M)(x) ≅ \underrightarrow{lim}_{Q_{≤x}}M, where Q_{≤x} is a subposet of Q consisting of objects smaller or equal to x. By assumption this poset is either empty or has a terminal object x. Consequently (LK_{𝔦}M)(x) is either zero if x ∉ Ob Q or M(x) otherwise. Hence we are done.

Alternatively we can compute RP ⊗_{RQ} M. For any x ∈ Ob P, we have

1_{x} · (RP ⊗_{RQ} M) = ∑_{y≤x} RHom_{P}(y, x) ⊗ M = ∑_{y≤x} RHom_{P}(y, x) ⊗ 1_{y} · M.

This is not zero if and only if 1_{y} · M ≠ 0 which means y ∈ Ob Q. By assumption, it forces x belongs to Q. Thus 1_{x} · (RP ⊗_{RQ} M) ≠ 0 if and only if x ∈ Ob Q. When this happens, 1_{x} · (RP ⊗_{RQ} M) = 1_{x} · M. Thus we obtain the same result. ⊓ ⊔

The concept of an idea can be more generally defined for EI-categories.

Proposition 2.2.17. Suppose 𝔲 : D → C satisfies the condition that for every x ∈ Ob C 𝔲/x is connected. Then LK_{𝔲}\underline{R} = \underline{R}.

Proof. When 𝔲/x is connected for an x ∈ Ob C, we have \underrightarrow{lim}_{𝔲/x}\underline{R} = R. Then by the universal property of direct limit we see any α : x → y must induce the identity morphism on R. Thus LK_{𝔲}\underline{R} = \underline{R}. ⊓ ⊔

Definition 2.2.18. Let C be a small category and x an object. Then we define a full subcategory C_{x} which consists of one object x and all of its endomorphisms. We also define C_{[x]} to be the full subcategory of C containing all objects that are isomorphic to x.

Proposition 2.2.19. Let C be a small category and x ∈ Ob C. Consider the inclusion ι : C_{x} ↪ C. Then LK_{ι}[REnd_{C}(x)] ≅ RHom_{C}(x, −).

Proof. For every y ∈ Ob C, the overcategory ι/y has objects {(x, α)} where α runs over the set Hom_{C}(x, y). A morphism (x, α) → (x, β) is an endomorphism g ∈ End_{C}(x) such that α = βg. Hence we can fit RHom_{C}(x, y) into the commutative diagram

\xymatrix{[REnd_{C}(x)](x, α) \ar[rr]^{g∘−} \ar[dr]_{α∘−} &  & [REnd_{C}(x)](x, β) \ar[dl]^{β∘−} \\  & RHom_{C}(x, y) & ,}

in which [REnd_{C}(x)](x, α) = [REnd_{C}(x)](x, β) = REnd_{C}(x), and α ∘ −, β ∘ − are compositions. If M is another R-module that can replace RHom_{C}(x, y) and make a new commutative diagram, then, by examining the images of End_{C}(x), one can easily establish a canonical map RHom_{C}(x, y) → M such that the whole diagram is commutative. This means that RHom_{C}(x, y) ≅ \underrightarrow{lim}_{ι/y}[REnd_{C}(x)] and thus LK_{ι}[REnd_{C}(x)] ≅ RHom_{C}(x, −). ⊓ ⊔

2.3 Classifications of projectives and simples for EI categories

Now we can start describing the projective and simple modules of an EI category algebra. This part of the work is due to Lück [17], as is described by tom Dieck in [6]. The base ring R is assumed to be an algebraically closed field.

Let C be a small category and x ∈ Ob C an object. Suppose P_{x} is a projective RC_{x}-module (or in other words a projective REnd_{C}(x)-module). Then its left Kan extension LK_{ι}P_{x}, along ι : C_{x} ↪ C, is a projective kC-module. Especially LK_{ι}[REnd_{C}(x)] ≅ RHom_{C}(x, −) by Proposition 3.2.11.

Let us assume furthermore C is a finite EI category. Then End_{C}(x) = Aut_{C}(x) for every x ∈ Ob C. From Example 3.4.4 we already learned that RC decomposes into a direct sum ⊕_{x∈Ob C}RC · 1_{x}. Now we try to analyze the indecomposable direct summands.

Suppose Is C is the full subcategory of C, consisting of all objects and all isomorphisms. Then R Is C is a subalgebra of RC.

Lemma 2.3.1. If 1_{RC} = ∑_{i=1}^{n} e_{i} is a primitive decomposition of 1_{RC} in R Is C, then it is also a primitive decomposition of 1_{RC} in RC.

Proof. Given any primitive decomposition of 1_{RC} in RC, the number of idempotents in this decomposition is equal to the number of indecomposable direct summands of the regular module of RC, which is equal to the number of indecomposable direct summands of RC/Rad(RC) by Proposition 3.4.5. Let us take the decomposition 1_{RC} = ∑_{i=1}^{n} e_{i}. We need to show it is primitive. To this end, we prove n equals the number of indecomposable direct summands in RC/Rad(RC).

Since all non-isomorphisms generate an ideal I of RC, which is contained in Rad(RC) and which induces an algebra isomorphism RC/I ≅ R Is C, from the isomorphism RC/Rad(RC) ≅ (RC/I)/(Rad(RC)/I), we know the two sides have the same numbers of indecomposable direct summands. From definition one can check that Rad(RC)/I is the radical of RC/I. Then by Proposition 3.4.5, applied to both RC and RC/I ≅ R Is C, we see the primitive decompositions of 1_{RC} in both RC and R Is C must have the same number of idempotents. Hence we are done. ⊓ ⊔

The category Is C is a disjoint union of groupoids, each of which comes from an isomorphism class of some object. Recall that, for each object x ∈ Ob C, we denote by [x] the set of objects isomorphic to x, and C_{[x]} the full subcategory consisting of these objects.

Lemma 2.3.2. 1. If x ≅ y are two isomorphic objects, and f_{y} ∈ Hom_{C}(x, y) is an isomorphism, then the assignment α ↦ α · f_{y} for each α ∈ RC · e defines an isomorphism of RC-modules RC · 1_{y} → RC · 1_{x}.

2. If 1_{x} = ∑_{i=1}^{n} e_{i} is a primitive decomposition in RAut_{C}(x), then 1_{y} = ∑_{i=1}^{n} f_{y}e_{i}f_{y}^{−1} is a primitive decomposition in RAut_{C}(y). Furthermore if we fix for each y ≅ x an isomorphism f_{y} ∈ Hom_{C}(x, y), then

∑_{y∈Ob C_{[x]}} ∑_{i=1}^{n} f_{y}e_{i}f_{y}^{−1}

is a primitive decomposition of the identity 1_{RC_{[x]}} in the groupoid algebra RC_{[x]}.

Proof. The isomorphism is straightforward to prove. Now if 1_{x} = ∑_{i=1}^{n} e_{i} is a primitive decomposition in RAut_{C}(x), certainly 1_{y} = ∑_{i=1}^{n} f_{y}e_{i}f_{y}^{−1} is a decomposition in RAut_{C}(y). It has to be primitive, because if it were not, then a primitive decomposition would be a sum of more than n idempotents in RAut_{C}(y). However f_{y}^{−1}(−)f_{y} maps such a primitive decomposition of 1_{y} to a decomposition of 1_{x}, which contradicts with the assumption that 1_{x} = ∑_{i=1}^{n} e_{i} is a primitive decomposition. ⊓ ⊔

The reader can compare the above statements with Example 3.4.7 (4).

Corollary 2.3.3. Let C be a finite EI category. One can write

1_{RC} = ∑_{x∈Ob C} ∑_{j=1}^{n_{x}} e_{xj},

where n_{x} is a positive integer for each x ∈ Ob C and 1_{x} = ∑_{j=1}^{n_{x}} e_{xj}. As a consequence, RC = ⊕_{x∈Ob C} ⊕_{j=1}^{n_{x}} RC · e_{xj} for some primitive pairwise orthogonal idempotents e_{xj} ∈ RAut_{C}(x), x ∈ Ob C.

Any projective RC-module is isomorphic to a direct sum of indecomposable projective modules of the form RC · e, where e ∈ RAut_{C}(x) is a primitive idempotent, for some x ∈ Ob C.

Given a primitive orthogonal decomposition 1_{RC} = ∑_{i} e_{i} such that each e_{i} belongs to some group algebra RAut_{C}(x), each summand of RC ≅ ⨁_{e_{i}} RC ·e_{i} is indeed a left Kan extension

LK_{ι}[RAut_{C}(x)e_{i}] = {LK_{ι}[RAut_{C}(x)]}e_{i} ≅ RHom_{C}(x, −)e_{i} = RCe_{i}

because LK_{ι} commutes with direct sums. Here ι : C_{x} ↪ C is the inclusion. In each RCe_{i}, its radical contains ∑_{z≅x̸} RHom_{C}(x, z)e_{i} which are linear combinations of non-isomorphisms in Hom_{C}(x, −).

We continue to characterize the simple RC-modules. Directly from the EI condition we have seen that a simple module S has to be atomic. It matches with our description of indecomposable projective modules. The quotient of RCe_{i}, for a primitive idempotent e_{i} ∈ RAut_{C}(x), by its radical is an atomic module supported on C_{[x]}. Moreover for each y ≅ x, S(y) must be a simple RAut_{C}(y)-module. In fact all simple RC-modules are exactly those simple modules of R Is C, which are obtained from simple modules of automorphism group algebras RAut_{C}(x).

Theorem 2.3.4 (Lück). Let C be a finite EI-category. The isomorphism classes of the simple RC-modules biject with the pairs ([x], V), where x ∈ Ob C and V is a simple RAut_{C}(x)-module, taken up to isomorphism.

Proof. First of all, we already know that all simple RC-modules are atomic. Thus simple RC-modules are exactly those simple RC_{[x]}-modules, with x running over Ob C.

Secondly for a fixed x, RC_{[x]}-mod is equivalent to RAut_{C}(y), for any y ≅ x, through restrictions induced by the equivalences C_{y} → C_{[x]} and C_{[x]} → C_{y}. Hence simple RC_{[x]}-modules biject with simple RC_{y}-modules, for any y ≅ x. Since C_{y} is the group Aut_{C}(y), we have proved the assertion. ⊓ ⊔

Because of the above theorem, it is natural to denote a simple RC-module by S_{x,V}, if it comes from a simple RAut_{C}(x)-module V, for some x ∈ Ob C. For consistency, we use P_{x,V} for the projective cover of S_{x,V}, whose structure is determined by its value at the object x. If RAut_{C}(x) · e is the projective cover of the simple RAut_{C}(x)-module V, then RC · e is the projective cover of S_{x,V}. The reader may revisit Example 1.3.8 to get better understanding of our results and notations in this section.

With the description of indecomposable projectives, we can show when the trivial module \underline{R} is projective.

Proposition 2.3.5. Let C be a finite EI-category. Then \underline{R} is projective if and only if each connected component of C has a unique isomorphism class of minimal objects [x], with the properties that for all y in the same connected component as x, Aut_{C}(x) acts transitively on Hom(x, y), and |Aut_{C}(x)| is invertible in R.

Proof. Without loss of generality, we may assume C is connected.

If \underline{R} is projective then \underline{R} ≅ ⨁ P_{y,W} for certain indecomposable projective modules P_{y,W}. Since \underline{R} is indecomposable and takes constant value at all objects, we must have that \underline{R} ≅ P_{x,V} for some x ∈ Ob C, x is minimal, and all minimal objects are isomorphic. Moreover because P_{x,V} (x) = R, the projective cover of the simple kAut_{C}(x)-module V, we get V = R and R must be projective as an RAut_{C}(x)-module. Thus \underline{R} is projective if and only if \underline{R} ≅ P_{x,R}, all minimal objects are isomorphic to x and |Aut_{C}(x)|^{−1} ∈ R.

Now, assume all minimal objects are isomorphic to x and |Aut_{C}(x)|^{−1} ∈ R. By Proposition 3.2.11 RHom_{C}(x, −) ≅ LK_{ι}[RAut_{C}(x)]. Then \underline{R} ≅ P_{x,R} can be explicitly constructed, using the idempotent e = \frac{1}{|Aut_{C}(x)|} ∑_{g∈Aut_{C}(x)} g, as

\underline{R} ≅ P_{x,R} ≅ LK_{ι}[RAut_{C}(x)e] = \{LK_{ι}[RAut_{C}(x)]\}e = RHom_{C}(x, −)e

because LK_{ι} commutes with direct sum. It is equivalent to saying that at each y ∈ Ob C, R ≅ RHom_{C}(x, y)e. This happens if and only if Aut_{C}(x) to act transitively on Hom_{C}(x, y). ⊓ ⊔

2.4 Dual modules and Kan extensions

In group representations, the two Kan extensions are isomorphic. Many important properties of group representations rely on this fact. Unfortunately it is not the case for category representations. To obtain a balanced understanding of the homological properties of category algebras, we describe some compatibility between the two Kan extensions. Here we assume the base ring is a field R = k.

Let M ∈ mod-kC be a right module. Its dual M^{∧} = Hom_{k}(M, k) becomes a left kC-module. The functor (−)^{∧} = Hom_{k}(−, k) is an anti-isomorphism between the two module categories. Suppose G is a group and H is a subgroup. In group representation theory, there are two well-known isomorphisms (M ↓_{H}^{G})^{∧} ≅ M^{∧} ↓_{H}^{G}, if M ∈ kG-mod, and (N ↑_{H}^{G})^{∧} ≅ N^{∧} ↑_{H}^{G}, if N ∈ kH-mod. We shall extend these to category representations.

Lemma 2.4.1. Let τ : D → C be a functor between finite categories. Then we have Res_{τ} M^{∧} ≅ (Res_{τ} M)^{∧}, for any M ∈mod-kC.

Proof. The isomorphism follows from Lemma 1.2.6. ⊓ ⊔

Lemma 2.4.2. With the same assumption as above, we have RK_{τ} N^{∧} ≅ (LK_{τ} N)^{∧}, for any N ∈ mod-kD.

Proof.

RK_{τ} N^{∧} ≅ Hom_{kC-mod}(kC, RK_{τ} N^{∧})
≅ Hom_{kD-mod}(Res_{τ} kC, N^{∧})
≅ Hom_{mod-kD}(N, (Res_{τ} kC)^{∧})
≅ Hom_{mod-kD}(N, Res_{τ} kC^{∧})
≅ Hom_{mod-kC}(LK_{τ} N, kC^{∧})
≅ Hom_{kC-mod}(kC, (LK_{τ} N)^{∧})
≅ (LK_{τ} N)^{∧}.

⊓ ⊔

The last isomorphism can be immediately combined with computations from the preceding section to get the right Kan extensions of certain modules.

Since this duality exchanges projective and injective modules, whenever the base ring is a field or a complete local ring if we can characterize the projective modules then the structures of injective modules are understood through duality.

Transporter categories and their algebras

Abstract We study a special class of finite EI categories, namely the transporter categories. They originate from certain local categories of finite groups, and can be used to compare representations and cohomology of groups and of other local categories. In fact a group and all its subgroups are some sort of transporter categories. We pay attention to the transporter categories and try to demonstrate their similarities with and differences from groups, in terms of homological properties. We shall apply tools developed in the preceding two chapters.

Conventions In this chapter we will compare modules of a group algebra with modules of various category algebras of certain categories constructed over the given group. To avoid confusions, we use symbols like M, N for group modules while 𝔐, 𝔑 for modules of category algebras.

3.1 Transporter categories

Transporter categories are among the simplest local categories considered in group representation theory and homotopy theory of classifying spaces.

3.1.1 G-categories

Definition 3.1.1. Consider G as a category with one object •. A G-category is a functor F : G → ℭ𝔞𝔱.

In other words, a G-category is a category C, equipped with a group homomorphism G = Aut_{G}(•) → Aut_{ℭ𝔞𝔱}(C). The simplest example is the trivial category (i.e. a point) with trivial action by G. Recall that a set is regarded as a poset with trivial relations, and a poset is regarded as a category.

Example 3.1.2. Suppose H ⊂ G is a subgroup. Consider the discrete set of left cosets G/H = {gH | g ∈ G}. Then G acts on it by left multiplication, permuting these cosets.

Example 3.1.3. Suppose H ⊂ G is a subgroup. Then the discrete set of conjugacy class ^{G}H = {^{g}H | g ∈ G} is also a G-category with G acting by conjugations.

The above two examples are examples of G-sets, a special kind of G-posets.

Definition 3.1.4. A collection of subgroups of G is a set of subgroups of G, closed under conjugations by elements in G.

Example 3.1.5. A collection of subgroups of G is naturally a poset with inclusions as relations. Hence any collection of subgroups is a G-poset.

There are various interesting collections of subgroups of G.

Example 3.1.6. 1. The collection of all p-subgroups of G is denoted by S_{p}^{e}.

2. The collection of all non-identity p-subgroups of G is denoted by S_{p}. Below is a concrete example of the poset of all non-trivial 2-subgroups of Σ_{4}, S_{2}(Σ_{4}):

\xymatrix{ &  &  & ⋯ \ar[d] & ⋯ \ar[d] \\  &  & D_{8} & D_{8} & D_{8} \\  & C_{2} × C_{2} \ar[ur] & C_{4} \ar[u] & V \ar[ul] \ar[u] \ar[ur] \\ C_{2} \ar[ur] & C_{2} \ar[u] & C_{2} \ar[ul] \ar[u] \ar[ur] & C_{2} \ar[u] & C_{2} \ar[ul]}

Here D_{8} is the dihedral group of order 8, V is a Kleine four group and those C_{?} are cyclic groups with orders specified in the subscripts. Due to its size, we do not record the full poset. Neglecting the two D_{8}’s on the right, the remaining poset is S_{2}(Σ_{4})_{≤D_{8}} (the D_{8} in the subscript is the leftmost of the three). To obtain the full poset, one just needs to copy the subposet S_{2}(Σ_{4})_{<D_{8}} and paste it at where the dots appear. Then remove the repeating subgroups because V is a subgroup of all three D_{8}.

3. The collection of all non-identity elementary p-subgroups of G is denoted by A_{p}. It is a G-subposet of S_{p}. There are other G-subposets of S_{p} that are quite useful, see for instance [4, Chapter 6].

3.1.2 Transporter categories as Grothendieck constructions

Various G-posets of subgroups of G demonstrate certain local structural information of G. However it is not complete. For example given H ∈ G and the discrete set ^{G}H, any two non-identical objects are conjugate in G while this relationship is not seen in ^{G}H. Thus we want a category that presents all intrinsic connections among any chosen collection of subgroups of G.

Definition 3.1.7. Let G be a group and P a G-poset. We define the transporter category on P to be a Grothendieck construction G ∝ P := Gr_{G}P. More precisely, G ∝ P has the same objects as P, that is, Ob(G ∝ P) = Ob P. For x, y ∈ Ob(G ∝ P), a morphism from x to y is a pair (g, gx ≤ y) for some g ∈ G.

In the literature the transporter categories are mostly considered as auxiliary constructions before passing to various quotient categories of them. Here we want to stress on the perhaps unique property, among various categories constructed from a group, that transporter categories admit natural functors to the group itself. It singles out this particular type of categories and is the starting point of this chapter. Here in order to emphasize the similarities and connections between transporter categories and subgroups, we follow a definition which is well known to algebraic topologists. The symbol G ∝ P is not standard and is used because this particular Grothendieck construction resembles a semidirect product, yet is different.

This neat but seemingly abstract definition can be easily seen to give the usual transporter categories. For example, when P is the poset of non-trivial p-subgroups, we get G ∝ P = Tr_{p}(G), the p-transporter category of G. The advantage of taking our approach is shown by the upcoming examples, where each subgroup of G is identified as a transporter category, up to a category equivalence.

From Definition 3.1.9 one can easily see that there is a natural embedding ι_{P} : P ↪ G ∝ P via (x ≤ y) ↦ (e, x ≤ y). On the other hand, the transporter category admits a natural functor π_{P} : G ∝ P → G, given by x ↦ • and (g, gx ≤ y) ↦ g. Thus we always have a sequence of functors

P↪^{ι_{P}}G ∝ P→^{π_{P}}G

such that π_{P} ∘ ι_{P}(P) is the trivial subgroup or subcategory of G. For convenience, in the rest of this chapter we often neglect the subscript P and write ι = ι_{P}, π = π_{P}.

Example 3.1.8. 1. If G acts trivially on P, then G ∝ P = G × P.

2. Let G be a finite group and H a subgroup. We consider the set of left cosets G/H which can be regarded as a G-poset: G acts via left multiplication.

The transporter category G ∝ (G/H) is a connected groupoid whose skeleton is isomorphic to H. In this way one can recover all subgroups of G, up to category equivalences. Consequently we have k(G ∝ G/H) ≃ kH as well as a homotopy equivalence B(G ∝ G/H) ≃ BH.

3. From Example 3.1.6 (2) we build a concrete transporter category Σ_{4} ∝ S_{2}(Σ_{4}) :

[figure: ⋯; ⋯; D_{8}; 8; 8; D_{8}; 8; 8; D_{8}; 8; C_{2} × C_{2}; 8; 8; C_{4}; 8; 8; V; 24; 24; 24; 24; C_{2}; 4; 4; 8; C_{2}; 4; 4; 8; C_{2}; 8; 8; 8; 24; 8; C_{2}; 8; 24; 8; 8; C_{2}; 8; 24; 8]

Here the numbers are the numbers of morphisms from one object to another. Note that again this is part of the whole category. However, it contains a skeleton so we know what is missing.

Remark 3.1.9. One can check directly that if Hom_{G∝P}(x, y) ≠ ∅ then both Aut_{G∝P}(x) and Aut_{G∝P}(y) act freely on Hom_{G∝P}(x, y).

Another notable structural fact is that G ∝ P is a category with subobjects, which means each morphism can be uniquely factorized as an isomorphism followed by a morphism in P (regarded as a subcategory of G ∝ P). In fact we have (g, gx ≤ y) = (e, gx ≤ y) ∘ (g, gx = gx) as shown in the diagram

\xymatrix{ & y \\ x \ar[ur]^{(g,gx≤y)} \ar[rr]_{(g,gx=gx)} &  & gx. \ar[ul]_{(e,gx≤y)}}

3.2 Local categories

In this section, we illustrate the role of transporter categories in group representations. Then we continue to give several results on representations of transporter categories, which further demonstrate close connections between representations of groups and transporter categories. This section will end with two applications, of representation theory, to cohomology of transporter category algebras.

Our concept of a transporter category is quite general because each subgroup H of G can be recovered as a transporter category G ∝ (G/H), for the G-poset G/H, up to a category equivalence. Thus it makes sense if we deem transporter categories as generalized subgroups for a fixed finite group.

Definition 3.2.1. Any quotient category C of G ∝ P is called a local category of G. When P consists of p-subgroups of G, for a prime p | |G|, we also call such a quotient category a p-local category.

A local category is connected with the group by the following diagram

\xymatrix{ & G ∝ P \ar[dl]_{π} \ar[dr]^{ρ} \\ G &  & C}

Transporter categories were implicitly considered by Mark Ronan and Steve Smith [4, Chapter 7] in the 1980s for constructing group modules, and later on played a key role in Bill Dwyer’s work [10] on homology decomposition of classifying spaces. Dwyer used this diagram to establish connections among various homotopy colimits (e.g. classifying spaces), while Ronan and Smith constructed kG-modules via representations of G ∝ P (using the language of G-presheaves on P).

Example 3.2.2. 1. The p-transporter category Tr_{p}(G) = G ∝ S_{p} has all nonidentity subgroups as its objects. For any p-subgroups P, Q, the morphism set is often written as Hom_{G}(P, Q) = {^{g}P ⊂ Q | g ∈ G}. In particular Aut_{G}(P) = N_{G}(P), the normalizer.

2. The p-fusion system F_{p}(G) is a quotient category of G ∝ S_{p}, given by Hom_{F_{p}(G)}(P, Q) = Hom_{G}(P, Q)/C_{G}(P).

3. The p-orbit category O_{p}(G) is a quotient category of G ∝ P, given by Hom_{O_{p}(G)}(P, Q) = Q\Hom_{G}(P, Q).

3.2.1 Two diagrams of categories

We have seen that given a G-poset P there exists a diagram of categories and functors

\xymatrix{ & G ∝ P \ar[dl]_{π} \ar[dr]^{ρ} \\ G &  & C}

for any given local category C. Diagrams similar to this have been (implicitly) considered by Ronan-Smith and Dwyer. Here we pursue a direction that is closely related to the work of Ronan and Smith.

In practice one often finds that the functor G ∝ P → C is part of an extension (or an opposite extension) sequence of categories. (Among examples are various orbit categories, Brauer categories and Puig categories.) It means that for such a quotient category C there exists a category K which is a disjoint union of subgroups of Aut_{G∝P}(x), x running over Ob P = Ob(G ∝ P), such that we can add K into the picture

\xymatrix{K \ar[dr] \\  & G ∝ P \ar[dl]_{π} \ar[dr]^{ρ} \\ G &  & C}

and moreover K ↪ G ∝ P ↠ C is a sequence of category extension. If the extension sequence is well understood then it will helps us to find extra connections between k(G ∝ P)-mod and kC-mod. However due to the scope of these notes, we shall not discuss extensions of categories.

Before passing to the next section we make a definition.

Definition 3.2.3. For any M ∈ kG-mod, sometimes we denote by κ_{M} the restriction Res_{π}M and call it a constant value module.

3.2.2 Frobenius Reciprocity

Applying classical tools in homological algebra, particularly the Kan extensions, we obtain a Frobenius Reciprocity between kG-mod and kC-mod, where kC is the (k-)category algebra of C. It implies, to some extent, comparing kGmod with kC-mod provides an extended context for local representation theory. This observation illustrates a possible way to understand group representations and cohomology via those of suitable categories. From the preceding diagram we obtain a diagram of module categories

\xymatrix{ & k(G ∝ P)-mod \ar[dl]_{LK_{π},RK_{π}} \ar[dr]^{LK_{ρ},RK_{ρ}} \\ kG-mod \ar[ur]_{_{Res_{π}}} &  & kC-mod. \ar[ul]^{_{Res_{ρ}}}}

The adjunctions between the restrictions and Kan extensions have the following consequences.

Proposition 3.2.4 (Frobenius Reciprocity). Suppose P is a G-poset and C is a quotient category of G ∝ P as in the preceding diagrams. Let M, N ∈ kG-mod and 𝔐, 𝔑 ∈ kC-mod. Then

1. Hom_{kG}(M, RK_{π}Res_{ρ}𝔑) ≅ Hom_{kC}(LK_{ρ}Res_{π}M, 𝔑);

2. Hom_{kG}(LK_{π}Res_{ρ}𝔐, N) ≅ Hom_{kC}(𝔐, RK_{ρ}Res_{π}N).

By direct calculations, these particular Kan extensions in Proposition 6.2.1 are simplified:

• LK_{π} ≅ \underrightarrow{lim}_{P}, RK_{π} ≅ \underleftarrow{lim}_{P} (compute the only overcategory and undercategory and show that both are equivalent to P. The calculation is long and quite tedious so we skip.);

• LK_{ρ} ≅↑_{k(G∝P)}^{kC} (the induction), RK_{ρ} ≅⇑_{k(G∝P)}^{kC} (the co-induction), since ρ induces an algebra homomorphism k(G ∝ P) → kC when C is a quotient category.

Remark 3.2.5. For any 𝔑 ∈ kC-mod, we shall write the k(G ∝ P)-module Res_{ρ}𝔑 as 𝔑 because they share the same underlying vector space. Recall that κ_{M} = Res_{π}M. Then the Frobenius Reciprocity can be rewritten as

(i’) Hom_{kG}(M, \underleftarrow{lim}_{P}𝔑) ≅ Hom_{kC}(κ_{M} ↑_{k(G∝P)}^{kC}, 𝔑); (ii’) Hom_{kG}(\underrightarrow{lim}_{P}𝔐, N) ≅ Hom_{kC}(𝔐, κ_{N} ⇑_{k(G∝P)}^{kC}).

When P = G/H for some subgroup H, we have natural isomorphisms \underleftarrow{lim}_{G/H} ≅ \underrightarrow{lim}_{G/H} ≅↑_{H}^{G}. Then the above isomorphisms certainly become the

usual adjunctions between ↑_{H}^{G} and ↓_{H}^{G} (the usual Frobenius Reciprocity) with C = G ∝ (G/H) and ρ = Id, in light of the Morita equivalence between kC and kH.

Remark 3.2.6. Our Frobenius reciprocity is different from a similar result of Ronan-Smith, see [4, 7.2.4], where they (implicitly) had a diagram of the same shape. However their C = G ∝ Q, not necessarily a quotient of G ∝ P, is another transporter category and ρ is induced by a G-map P → Q. This prohibits us from considering various quotients of transporter categories. Moreover since a G-map P → Q usually does not induce an algebra homomorphism from k(G ∝ P) to k(G ∝ Q), their Kan extensions cannot be interpreted as induction and coinduction.

3.3 Transporter category algebras are skew group algebras

Definition 3.3.1. Let A be a k-algebra and G a group. It is a G-algebra if there exists a group homomorphism G → Aut(A).

Definition 3.3.2. Let A be a G-algebra. The skew group algebra A[G] is a vector space A ⊗_{k} kG with multiplications determined by

(a_{1} ⊗ g_{1}) ∗ (a_{2} ⊗ g_{2}) = a_{1}(g_{1} · a_{2}) ⊗ g_{1}g_{2}

where a_{i} ∈ A and g_{i} ∈ G. The symbol g_{1} ·a_{2} represents the image of a_{2} under the action of g_{1}.

Proposition 3.3.3. k (G ∝ P) ≅ kP[G].

Proof. The isomorphism is given on base elements by (g, gx ≤ y) ↦ (gx ≤ y) ⊗ g and conversely (x ≤ y) ⊗ g ↦ (g, g(g^{−1}x) ≤ y). ⊓ ⊔

3.4 Transporter category algebras are Gorenstein

We introduced Gorenstein algebras in Definition 1.3.12. Typical examples of Gorenstein algebras are self-injective algebras (e.g. finite-dimension Hopf algebras including group algebras) and algebras of finite globle dimension.

Theorem 3.4.1 (Auslander-Reiten [2]). A skew group algebra A[G] is Gorenstein if and only if A is. In fact inj.dim_{A}A = inj.dim_{A[G]}A[G].

Proposition 3.4.2. Let P be a finite poset. Then kP is always Gorenstein.

Proof. This has been discussed in Chapter 1. ⊓ ⊔

Indeed, kP is of finite global dimension.

Corollary 3.4.3. k (G ∝ P) is Gorenstein.

Thus the algebra k (G ∝ P) has good homological properties. For extensive discussions on homological properties of Gorenstein algebras, read papers by Auslander-Reiten [2] and Buchweitz [5].

3.5 Projective, injective and geometric dimensions

Let M be a k (G ∝ P)-module. Since k (G ∝ P) is Gorenstein, M is of finite projective dimension if and only if it is of finite injective dimension. For instance, the regular module k (G ∝ P) is projective (projective dimension zero) and it also has finite injective dimension (usually strictly larger than zero).

Proposition 3.5.1. If M ∈ k (G ∝ P)-mod has finite injective dimension, then inj.dimM ≤ dim P.

In most cases, the geometric dimension dim P is strictly larger then the injective dimensions of all modules with finite injective dimension. For example, if P is the poset x → y → z, then all kP-modules are of finite injective dimension satisfying inj.dimM ≤ 1. One can try to compute the injective dimensions of the following complete list of (up to isomorphism) indecomposable kP-modules:

three simple modules

S_{x} : k → 0 → 0, S_{y} : 0 → k → 0, S_{z} : 0 → 0 → k;

their projective covers

P_{x} : k → k → k, P_{y} : 0 → k → k, P_{z} = S_{z};

and their injective hulls

I_{x} = S_{x}, I_{y} : k → k → 0, I_{z} = P_{x}.

These 6 modules, up to isomorphism, exhaust all indecomposable kP-modules. They have injective dimensions zero or one, smaller than dim P = 2.

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