REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS FEI XU Abstract. We define for each small category C a category algebra RC over a base ring R and study its representations. When C is an EI-category, we develop a theory of vertices and sources for RC-mod, which parameterizes the indecomposable RCmodules. As a main application, we use our theory to find formulas for computing higher (inverse) limits over C. Keywords. Category algebras, relative projectivity, vertices and sources, higher limits and cohomology ring. 1. Introduction 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. The product on the basis elements of RC is defined by composition and then it is linearly extended to all elements of RC. The category algebra is introduced in the first place as a tool to study representations of the category on which it is defined. Representations of categories arise when we consider any diagram of modules, or take inverse or direct limits. We have in mind the constructions of group modules using representations of the Tits building of a Chevalley group by Ronan and Smith [37], the homology decompositions of classifying spaces of groups using representations of the orbit categories and other categories by Dwyer [10], and the works of Broto-Levi-Oliver [6], Grodal [18], Jackowski-McClure-Oliver [26] and Symonds [40]. A representation of C over R is a (covariant) functor from the small category to the category of R-modules, i.e., R-mod. The representations of C over R form an abelian category (R-mod) C (the functor category), which has sufficiently many projectives and injectives. The starting point of our research is the following result of Mitchell. Proposition 1.1 (Mitchell [30]) For any small category C with finitely many objects, its representations can be identified with the unital RC-modules. Since we have the equivalence (R-mod) C RC-mod when Ob C is finite, we usually call any functor in (R-mod) C an RC-module. Particularly, the constant functor R, which sends every object to R and every morphisms to the identity, will be called the trivial RC-module, because it plays a similar role as the trivial group module does in group representations and cohomology. The categories we study in this article are In partial fulfillment of the requirements for the PhD degree at the University of Minnesota. 1

2 FEI XU usually finite. A small category C is said to be finite if its morphism set Mor(C) is finite, which implies Ob C is finite and RC is of finite R-rank with an identity 1 = x Ob C 1 x. When C is an EI-category in which every endomorphism is an isomorphism, Lück [28] has done some fundamental work on the representation theory of RC. For instance he classified the projective and simple RC-modules, and studied the restriction of projective modules to subalgebras. EI-categories are important to us because many interesting categories, such as posets, groups, orbit categories and fusion systems, are EI. Any EI-category C has a prominent property that there is a natural preorder on its set of objects: if x, y Ob C then x y if and only if Hom C (x, y). The preorder allows us to obtain a filtration for any non-zero RC-module M so that every factor concentrates on a single isomorphism class of objects of C (that is, as a functor any factor takes non-zero values only at a single isomorphism class of objects). The preorder also allows us to construct very useful full subcategories such as C x for every x, where Ob C x = {y Ob C Hom C (y, x) }. Similar constructions include C <x, C >x and C x for each and every x Ob C. Our main results will be proved under the assumption that the category C in question is finite EI. Let D be a subcategory of a small category C. Then there are two well-known functors: the restriction C D : RC-mod RD-mod and the induction C D : RD-mod RC-mod. We call an RC-module M D-projective (or projective relative to D) if M is isomorphic to a direct summand of M C D C D. When D C is a full subcategory, we ve got the following results. Theorem 1.2 Let D be a full subcategory of a finite EI-category C. Then (1) M C D RD-mod is indecomposable if M RC-mod is indecomposable and D-projective; (2) N C D RC-mod is indecomposable and D-projective if N RD-mod is indecomposable; (3) if M RC-mod is D-projective then M C D C D = M. The above theorem gives us a parametrization of RC-modules through the full subcategories of C. Let D C be a full subcategory. We define RC D -mod to be the full subcategory of RC-mod, consisting of all modules which are D-projective. Theorem 1.3 Let D be a full subcategory of a finite EI-category C. Then RD-mod is equivalent to RC D -mod. A subcategory D is called convex if the composite x α y β z is a morphism in Mor(D) then both α and β belong to Mor(D). Theorem 1.4 Let C be a finite EI-category and M an indecomposable RC-module. Then there exists a smallest full convex subcategory of C, relative to which M is projective.

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 3 The subcategory in Theorem 1.4 is called the vertex of M, denoted by V M. The restriction of M, M C V M, is an indecomposable RV M -module and is called the source for M. Our next two statements focus on the trivial module R because of its special role in the representation theory and computing higher limits. For any functor ι : D C and any x Ob C there exists an overcategory denoted by ι x (Mac Lane [31]), which can be used to define the left Kan extension lim of ι. When ι? ι : D C is the inclusion, the left Kan extension is isomorphic to the induction C D = RC RD, from where we can establish the following result. Proposition 1.5 Let ι : D C be an inclusion. The RC-module R is D-projective if and only if every ι x, x Ob C, is connected. We note that when C is a poset in Proposition 1.5, ι x = D x for any x Ob C. The above characterization, however, is not easy to use in practice to narrow down the possible choices of the vertex of R. Inspired by the work of Bouc [5], Puig [34] and Quillen [36], among others, we define an object x Ob C to be weakly essential if C <x is empty or has more than one component. Following Symonds, the full subcategory consisting of all such objects in Ob C is named Wess 0 (C). There is a larger subcategory, denoted by Wess(C), whose objects x satisfy the condition that C <x is not contractible. Proposition 1.6 Let C be a finite EI-category. Then Wess 0 (C) V R. In Section 4.6 we will elaborate on this point and prove that if (C, I) is a finite category with subobjects then C is EI, I is a poset and Wess 0 (I) completely determines V R. The category Wess(I) is as equally important as Wess 0 (I) in this situation. Using a result of Jackowski-S lomińska [27], we show for any finite category with subobjects (C, I) and any RC-module M, lim M = lim M C D C D, where D C is a full subcategory and shares the same object set with Wess(I). Let I be a collection of p-subgroups of a group G in the sense of Dwyer [10]. Then I can be made into a poset by the inclusions of subgroups. There is a transport category Tr I (G) associated with I, which has the same object set while the morphisms are the conjugations. The opposite of the transport category Tr op I (G), along with Iop, is a category with subobjects. A p-subgroup H G is called p-radical if N G (H)/H has no normal p-subgroup. Quillen [36] conjectured that in this case Wess(I op ) consists of the p-radical subgroups of G. We note that in this case Wess 0 (I) is the same as the H p of Grodal [18]. When we study an arbitrary representation M of a small category C, the R-module Hom RC (R, M) = lim M naturally comes into play. It is well-known that the values C of the derived functors of lim C = Hom RC (R, ) on any M RC-mod, lim M = C Ext RC(R, M), are isomorphic to the cohomology groups of C with coefficients in M, i.e, H (C, M). Let M, N be two RC-modules. We can consider more generally the Ext groups Ext i RC(M, N), i 0. When M = N, Ext RC(M, M) := i 0 Ext i RC(M, M)

4 FEI XU possesses a ring structure with the multiplication given by the Yoneda splice. Theorem 1.7 Let D be a full subcategory of a finite EI-category C and M an RCmodule which is D-projective. If RC is a right flat RD-module then we have Ext RC(M, N) = Ext RD(M C D, N C D), for any RC-module N. When M = N, this is a ring isomorphism. We comment that if M = R, Theorem 1.7 becomes a result of Jackowski and S lomińska [27]. When it gets down to the earth and there is no way to reduce the size of C the only thing we can do is to use a projective resolution of M (or an injective resolution of N) to compute the Ext groups directly. In the end we investigate various projective resolutions of RC-modules, especially the minimal ones, and use them to establish other formulas for computing the Ext groups. Proposition 1.8 Let C be an EI-category and R a commutative ring. Then Ext RC(S x,v, S y,u ) = Ext RC y x (S x,v, S y,u ), where C y x = C x C y. Especially we have Ext RC(S x,v, S x,u ) = Ext R Aut(x)(V, U). We note that in some situations we can express the Ext groups Ext RC y x (S x,r, S y,r ) as cohomology groups of some space, and thus we generalize a result on posets due to Igusa and Zacharia [24]. Finally the projective dimension of an RC-module M tells us when the groups Ext RC(M, N) vanish for any N RC-mod. Using our knowledge about the minimal resolutions, we can show when RC has a finite global dimension. Theorem 1.9 Let C be a finite EI-category. Then RC has finite global dimension if and only if for all x Ob C Aut C (x) 1 R. This article is organized as follows. In Section 2 we define category algebras and their representations. In Section 3 we go over some basic homological properties of category algebras, where several reduction formulas on computing higher limits are presented. The focus of Section 4 is the representation theory of EI-categories. We ll give a description of the simple and the projective RC-modules, as well as the restriction of the projective modules. Our theory of vertices and sources will be introduced, and we ll establish reduction formulas for computing Ext groups. In Section 5 we describe the minimal projective resolutions for RC-modules and consider their applications. Acknowledgement I wish to thank my thesis advisor, Prof. Peter J. Webb, for leading me into this area, for numerous helpful conversations and comments and for much encouragement.

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 5 2. Category algebras: definitions and examples Throughout this paper, the base ring R is always a commutative ring with an identity. A module will be a finitely generated left module, if it is not otherwise specified. 2.1. Definition. Definition 2.1.1. Let C be a category and R a commutative ring. The category algebra RC is the free R-module whose basis is the set of morphisms of C. We define a product on the basis elements of RC by { f g, if f and g can be composed in C f g = 0, otherwise and then extend this product linearly to all elements of RC. With this product, RC becomes an associative R-algebra. A category C is said to be finite if Mor(C) is a finite set. Note that this implies Ob C is finite, and that RC is of finite R-rank. If Ob C is finite, it s easy to see that x Ob C 1 x is the identity of RC where 1 x is the identity of Aut C (x). We say C is connected if C as a (directed) graph is connected. Every category C is a 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 make the following assumption. Convention In this article, we assume C is connected. 2.2. Representations of categories and examples. We shall show that a fundamental property of a category algebra RC is that it provides a mechanism for discussing representations of C, in a sense which we now define. Definition 2.2.1. A representation of a category C over a commutative ring R is a (covariant) functor M : C R-mod. The functor category (R-mod) C is an abelian category with enough projectives and injectives so we can talk about subfunctors and quotient functors and do homological algebra on it. As we ve mentioned in the introduction to this paper, Mitchell [30] proved (R-mod) C RC-mod if Ob C is finite. Since our real intention is to study finite categories, for convenience we usually won t distinguish RC-mod and (R-mod) C if it doesn t cause any serious trouble. Throughout this article, we re going to use R to denote the constant functor or trivial module. For any category C, R : C R-mod, is defined by R(x) = R for all x Ob C and R(f) = Id for all f Mor C. Let G be a (discrete) group and R a commutative ring. Then the group can be regarded as a category Ĝ with only one object, whose morphisms are the elements of G. The group algebra RG is the same as the category algebra RĜ. A left RG-module

6 FEI XU M can be regarded as the representation of Ĝ given by a certain functor Φ : Ĝ R- mod, sending to Aut R (M). The trivial RĜ-module R is exactly the trivial module of RG. Given a quiver q. Then the category algebra of the free category C q generated by q is the same as the path algebra of the quiver q. Suppose Γ is a poset. Then it can be regarded as a category in a natural way. The incidence algebra of Γ (see [7]) is the same as the category algebra RΓ. 3. Basic homological properties The R-module Hom RC (R, M) naturally emerges when we study a representation M : C R-mod. In fact, the derived functors of Hom RC (R, ) (or more generally of Hom RC (M, )) and their values are of great importance to us. 3.1. Cohomology theory of small categories. In this section we introduce the cohomology theory of small categories and go over some basic properties. The cohomology theory of small categories has been discussed in various places in literature, and readers are referred to Baues-Wirsching [3], Generalov [15] and Oliver [33] for more details. One can also find in Gabriel-Zisman [14] and Hilton-Stammbach [22] the homology theory of small categories. Definition 3.1.1. Let C be a small category. The n-th cohomology group, H n (C, M), of C with coeffiecients in module M RC-mod is defined by H n (C, M) := Ext n RC(R, M). It was shown by Roos [39] and Gabriel-Zisman [14] that Ext RC(R, M) = lim M, C the higher inverse limits of M over C. For future reference, we review the construction of overcategories associated with a functor between two small categories (Mac Lane [31]). Let C and D be small categories equipped with a functor µ : D C. For each y Ob C, the overcategory µ y consists of objects (x, α), where x Ob D and α Hom C (µ(x), y). A morphism from (x, α) to (x, α ) in the overcategory is given by a morphism β Hom D (x, x ), which satisfies αµ(β) = α. For any µ : D C and any y Ob C, we may construct a simplicial complex {C (µ y )} from the simplicial set associated to the small category µ y, which can be regarded as the cellular chain complex on the space µ y. It s easy to see, for each integer n, C n (µ? ) is an RC-module. When D = C and µ = Id, we normally write Id(C) y as C y for any y Ob C. It is known that C n (C? ) is a projective RC-module and furthermore, we can form a projective resolution for R: C (C? ) R 0. Note that it is exact because the overcategories C y, y Ob C, are all contractible. Overcategories associated with the inclusion ι : D C of a subposet D into a poset C have been intensively studied by group representation and cohomology theorists, see Benson [5] and Grodal [18]. Under the circumstance, the overcategory ι y is isomorphic to D y for any y Ob C. For any M RC-mod Ext RC(M, M) has a ring structure with product given by the Yoneda splice, but it is the case where M = R that is of great interest to us.

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 7 Definition 3.1.2. We call the ring Ext RC(R, R) = i 0 Ext i RC(R, R) the cohomology ring of the category algebra RC. The product in this ring is defined by the Yoneda splice. Given a small category C, it is well-known that there exists a ring isomorphism Ext RC(R, R) = H ( C, R), where C stands for the topological realization of the nerve NC. 3.2. Basic properties and some reduction theorems. We prove some basic properties, many of which follow directly from simple reasoning. Definition 3.2.1. Suppose µ : C C is a (covariant) functor. We define Res µ : (Rmod) C (R-mod) C to be the restriction along µ. Given a functor M (R-mod) C, we have Res µ M = M µ (R-mod) C. Given a functor µ : C C, the restriction Res µ : (R-mod) C (R-mod) C has a counterpart, also denoted by Res µ, between the corresponding module categories: Res µ : RC-mod RC -mod. Since µ : C C extends linearly to a natural map of R-modules µ : RC RC, it s reasonable to ask if µ is always an algebraic homomorphism because if it is, the so-called change-of-base-ring or the representation-theoretic restriction, RC RC : RC-mod RC -mod, coincides with Res µ. The answer to the question is no, and here is a simple example. Let C be a category with two objects and only identity maps, and let C be a category with one object and the identity map along with the unique functor µ : C C. Then the map µ : RC RC is not an algebra homomorphism for the product of the two morphisms in C is zero while the product of their images is not. We show when µ will be an algebra homomorphism. Proposition 3.2.2. A functor µ : C C extends linearly to an algebra homomorphism µ : RC RC if and only if µ is injective on Ob C. When this happens, the induced functor followed by 1 C, 1 C RC RC : RC-mod RC -mod, is exactly Res µ. Proof. We know µ(βα) = µ(β)µ(α) for any pair of composable morphisms α, β in C. The injectivity of µ implies two morphisms α, β Mor(C ) are composable if and only if µ(α), µ(β) Mor(C) are composable. If µ is injective on Ob C, then we define a map µ : RC RC as the linear extension of functor µ, i.e., µ( i r iα i ) = i r i µ(α i ) for any r i R, α i Mor(C ). This µ is indeed an algebra homomorphism because our previous observation of µ implies µ(( j r jβ j )( i r iα i )) = µ( j r jβ j ) µ( i r iα i ) is always true. On the other hand if the linear extension µ : RC RC is an algebra homomorphism then we must have µ(0) = 0 and then µ(1 x ) µ(1 y ) = µ(1 x 1 y ) = 0 unless x = y. This suggests that µ is injective on Ob C. In Section 4, we will take C to be a full subcategory of C. Then the restriction Res ι : RC-mod RC -mod is determined by the algebra homomorphism ῑ : RC RC, hence by ι : C C. For this reason we don t distinguish Res ι and RC RC, and will write RC RC and Res ι as C C which is common in representation theory.

8 FEI XU Proposition 3.2.3. Let C and C be equivalent small categories. Then (1) (R-mod) C (R-mod) C, an equivalence which sends the constant functor to the constant functor. If both Ob C and Ob C are finite then RC and RC are Morita equivalent; and (2) the nerves NC and NC are homotopy equivalent. Proof. We prove the first assertion, and the second is proved in Baues and Wirsching [3]. We show the two functor categories (R-mod) C and (R-mod) C are equivalent. Then it implies the module categories RC-mod and RC -mod are equivalent, hence RC and RC are Morita equivalent. In fact if µ : C C and ν : C C are equivalences, we have Res µ Res ν = IdRC : (R-mod) C (R-mod) C because of the following diagram M(νµ(x)) = (Res µ Res ν M)(x) = (Id C M)(x) = M(x) M(νµ(α))=(Res µ Res ν M)(α) (Id C M)(α)=M(α) M(νµ(y)) = (Res µ Res ν M)(y) = (Id C M)(y) = M(y) where M (R-mod) C, α : x y Mor C and Id RC is the identity functor. Similarly we can show Res ν Res µ = IdRC : (R-mod) C (R-mod) C. Clearly the constant functor restricts to the constant functor always. It s well-known that if a functor µ : C C has a left adjoint ν : C C, then Res ν is also the left adjoint of Res µ, and hence Ext RC(Res ν M, N) = Ext RC (M, Res µ N) for any M RC -mod and N RC-mod, because both Res µ and Res ν are exact (see for example Jackowski-McClure-Oliver[26] II, Proposition 5.1). If µ is indeed an equivalence, we have a stronger result. Corollary 3.2.4. Let µ : C C be an equivalence of two small categories. Then we have Ext RC(M, N) = Ext RC (Res µm, Res µ N), for M, N RC-mod. In particular there is a ring isomorphism Ext RC(R, R) = Ext RC (R, R). Proof. The thing is, RC and RC are Morita equivalent by the functor Res µ, which takes R to R. In [3] Baues and Wirsching had a special case of the above result. Let µ : C C be an equivalence. They proved Ext RC(R, N) = Ext RC (R, Res µ N) based on a homotopy equivalence between the cochain complexes used to compute H (C, N) and H (C, Res µ N). We have seen that a certain pair of adjoint functors µ, ν between D and C gives Ext RC(R, N) = Ext RD(R, Res µ N) because of the existence of another pair of (induced) adjoint functors Res µ, Res ν between the corresponding functor categories. The

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 9 truth is, a single functor µ : D C is enough to induce a pair of adjoint functors on the level of functor categories. This time the left adjoint of Res µ, which is not necessarily induced by any functor from C to D, is the left Kan extension K : RD-mod RC-mod. Recall that the left Kan extension of a functor ι : D C is defined by K(M)(y) = lim ι y M π, where M RD-mod, y Ob C and π : ι y D is the projection (x, α) x. We summarize some useful properties of the left Kan extension in Lemma 3.2.5. The first part of Lemma 3.2.5 was observed by Dwyer-Kan [11] and Hollender-Vogt [23], and the second is new, though a special form of it has appeared in Symonds [40]. The possibility of constructing a projective resolution of R using the left Kan extension was discovered by Jackowski-S lomińska [27]. Lemma 3.2.5. Let µ : D C be a functor. Then (1) there are isomorphisms of complexes of projective RC-modules K(C (D? )) = C (µ? ). If µ : D C is injective on Ob D, then the above complexes are isomorphic to RC RD C (D? ); (2) if every µ y is non-empty and connected, then K(R) = R. Thus if for every y Ob C the overcategory µ y is R-acyclic (i.e. all reduced homology groups of its classifying space with coefficients in R vanish), we obtain a projective resolution K(C (D? )) = C (µ? ) K(R) = R 0 of the trivial RC-module R, which, evaluated at y, is isomorphic to the reduced simplicial chain complex of µ y. One can deduce from the above lemma that lim n Res D µm = Ext n RD(R, Res µ M) = Hn(Hom RC (C (µ? ), M)). When every µ y, y Ob C, is R-acyclic, C (µ? ) becomes a projective RC-resolution of R, and thus the right hand side of the above isomorphism becomes higher limit over C. Corollary 3.2.6. (Jackowski-S lomińska [27]) Let µ : D C satisfy the condition that every µ y, y Ob C is R-acyclic. Then lim M = lim Res C D ι M for any RC-module M. Note that if all the overcategories µ y, y Ob C are contractible, then D C by Quillen s Theorem A. When we turn to our representation-theoretic settings, the left adjoint of the restriction Res µ : RC-mod RD-mod is normally written as the induction C D = RC RD, provided RC is a right RD-module. This means that K = C D as two functors, and that the conditions in the above corollary may be replaced by others using C D. More precisely, the key to establish Corollary 3.2.6 is to ensure P D C D R C D 0 is a projective resolution, and so we can require the subcategory D C to satisfy the condition that RC is a right flat RD-module (i.e.

10 FEI XU C D is exact) and R C D = R. This reminds us of the Eckmann-Shapiro Lemma in the cohomology theory of algebras which compares Ext groups after a base ring change, and the theory of vertices and sources in group representation theory, which studies induced modules (see Benson [4] and [5]). 4. EI-categories, relative projectivity, vertices and sources In this section, we investigate the representation theory of EI-categories and its applications to cohomology theory, especially to the computation of higher limits. We always assume the base ring R is a field or a complete discrete valuation ring, in order to have the unique decomposition property for every RC-module. When R is a field of characteristic p > 0, we denote it by F p (instead of F q for q = p n etc), and require it to be large enough (e.g. algebraically closed etc.) if necessary. Definition An EI-category is a small category C in which all endomorphisms are isomorphisms. Some of the general theory of EI-categories is given by tom Dieck in [9] (I.11), much of which was due to Lück, see also [28]. 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 EI-categories. 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 authors, see Oliver [33] and Jackowski and S lomińska [27]. For any EI-category C, any subcategory D 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 D (y, x). Similarly we can define other full subcategories of D: D <x, D x and D >x. Convention In the rest of this article, we re going to assume that C is finite (i.e. Mor(C) is finite). Then RC becomes an R-algebra of finite rank and 1 RC is a sum of primitive orthogonal idempotents. When we consider a full subcategory D of an EI-category C, we suppose D has the following property: if x Ob D, then [x] Ob D, where [x] is the isomorphism class of x in C. Let C be an EI-category and D C a full subcategory. The above condition on D is a natural requirement, which won t change the nature of any questions to be considered here and does protect us from some unnecessary non-essential technical troubles. The thing is that, if we are to investigate an RC-module M, then (as a functor) M has to take an isomorphic value on every object of an isomorphism class of objects in C. For any full subcategory D C, it is possible to find another full

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 11 subcategory E such that D E C and E D. The construction of E is easy. We just let Ob E consist of all objects in C which are isomorphic to some object(s) in D and then the structure of E becomes clear. Since the representation and cohomology theories of E are the same as those of D, we can study the connections between RC-mod and RE-mod, in order to understand those between RC-mod and RD-mod. 4.1. Projective modules and simple modules. Now we start describing the projective and simple modules for an EI-category. This part of the work is due to Lück [28], as is described by tom Dieck in [9]. The base ring R is assumed to be a field or a complete discrete valuation ring. Proposition 4.1.1. (Lück [28]) Any projective RC-module is isomorphic to a direct sum of indecomposable projective modules of the form RC e, where e R Aut C (x) is a primitive idempotent, for some x Ob C. Since each indecomposable projective module is a direct summand of some RC 1 x = R Hom C (x, ), x Ob C, and all the non-isomorphisms in Hom C (x, ) span a submodule that is contained in the radical of RC 1 x, RC 1 x is the projective cover of a semi-simple module, which is non-zero only on the isomorphism class [x]. Theorem 4.1.2. (Lück [28]) Let C be an EI-category. For each object x Ob C and simple R Aut C (x)-module V there is a simple RC-module M such that [x] Is C is exactly the set of objects on which M is non-zero, and M(x) = V. On the other hand, if M is a simple RC-module, then there exists a unique isomorphism class of objects [x] Is C on which M is non-zero, and furthermore each M(x) is a simple R Aut C (x)- module. These two processes are inverse to each other. Thus the isomorphism classes of the simple RC-modules biject with the pairs ([x], V ), where x Ob C and V is a simple R Aut C (x) module, taken up to isomorphism. We denote a simple RC-module by S x,v, if it comes from a simple R Aut(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 R Aut(x) e is the projective cover of the simple R Aut(x)-module V, then RC e is the projective cover of S x,v. The simple modules are atomic in the sense we now define. Definition 4.1.3. A functor M : C R-mod is called atomic, concentrated on an isomorphism class of objects [x] Ob C if M(y)(= 1 y M) 0 if and only if y = x. For convenience, we just say M is concentrated on x, instead of [x]. We will call an RC-module M atomic if the corresponding functor is. With the description of indecomposable projectives, we can show when the trivial module R is projective. This generalizes a result of Symonds [40]. Proposition 4.1.4. Let C be a finite EI-category. Then 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(x) has a single orbit on Hom(x, y), and Aut(x) is invertible in R.

12 FEI XU Proof. If R is projective then R = P x,v for certain indecomposable projective modules P x,v. The only V which can arise are V = R, and R must be projective as an R Aut(x)-module, forcing Aut(x) to be invertible in R for the x which appear in the direct sum, as in the first proof. Since P y,r (z) = 0 unless y z, P x,r must appear as a summand for each isomorphism class of minimal x. Now P x,r (z) = Hom(x, z) Aut(x) R = R n, where n is the number of orbits of Aut(x) on Hom(x, z). For P x,r (z) to be a summand of R we must have n = 1. Finally, {minimal x} P x,r at an object z is R t where t = number of isomorphism classes of minimal [x] with x z, so each component has a unique minimal x. The other direction is easy. The conditions imply that R = {minimal x} P x,r and this is projective. 4.2. Restrictions of projective modules. We assume R is a field or a complete discrete valuation ring. Let ι : D C be the inclusion of a full subcategory D. We provide equivalent conditions on D to the conditions that the restriction Res ι = C D : RC-mod RD-mod preserves (left or right) projectives. Note that if C D preserves left projectives then it preserves any projective resolutions of RC-modules, and that if C D preserves right projectives then RC becomes a right projective RD-module, and hence C D = RC RD is exact (and preserves projectives of course). The main theorem 4.2.5 is a special form of tom Dieck [9] Proposition 11.39, see also Xu [45]. Definition 4.2.1. Let D be a subcategory of C. Then we say a non-isomorphism f Mor(C) is irreducible (resp. co-irreducible) with respect to D if there s no way to write f = f 1 f 2 where f 1, f 2 are two non-isomorphisms and f 1 (resp. f 2 ) Mor(D). Given x Ob C\ Ob D and y Ob D the subset of Hom C (x, y) of all non-isomorphisms which are irreducible with respect to D is denoted by Irr D (x, y). Similarly given x Ob D and y Ob C\ Ob D we define coirr D (x, y) to be the subset of Hom C (x, y) consisting of non-isomorphisms that are co-irreducible with respect to D. We will mainly describe the action of C D on left RC-modules (i.e. covariant functors), and the action of C D on the right RC-modules (i.e. contra-variant functors) can be easily deduced in a dual fashion. Definition 4.2.2. Let x Ob D and z Ob D. Suppose f Hom C (x, z) and f admits two factorizations f = f 1 f 2 = f 1 f 2 with f 1, f 1 Mor(D) and f 2, f 2 irreducible with respect to D. Then we say f is good if for every such pair of factorizations f 2, f 2 have isomorphic targets. We say D is good with respect to x Ob C\ Ob D if for all z Ob D with Hom C (x, z), all f Hom C (x, z) are good. We say D is good if D is good with respect to all x Ob C\ Ob D. The goodness condition is surprisingly useful in describing the restriction of representable functors.

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 13 Lemma 4.2.3. Let [x] Ob D =. The full subcategory D is good with respect to x if and only if RHom C (x, ) C D= [y] Ψ x RHom D (y, ) RIrr D (x, y), where Ψ x is the set of all isomorphism classes [y] Ob D x such that Irr D (x, y) is non-empty. The above lemma implies that R Hom C (x, ) C D is projective if and only if every R Hom D (y, ) R Irr D (x, y) is projective. Suppose R Hom D (y, ) R Irr D (x, y) is a projective RD-module. Then its evaluation at y, i.e. R Irr D (x, y), must be a projective R Aut D (y)-module. However, the converse is not true in general. A condition on D, which is stronger than the goodness, is formulated by tom Dieck ([9], 11.38(iii)) and helps us resolve this problem. (TD) Let x Ob C\ Ob D, y Ob D and f Hom C (x, y). If f admits two factorizations f 1 f 2 = f 1 f 2 with f 2, f 2 irreducible with respect to D, then there is an isomorphism g Mor(D) such that f 2 = gf 2 and gf 1 = f 1. The following lemma is actually extracted from tom Dieck s Proposition 11.39 [9]. Lemma 4.2.4. Suppose D is a full subcategory of C and x Ob C\ Ob D with [x] Ob D =. If all morphisms in Hom C (x, z), z Ob D satisfy condition (TD), then the RD-module R Hom D (y, ) R Irr D (x, y) is projective if and only if the R Aut(y)- module R Irr D (x, y) is projective. These key lemmas can be used to establish the following theorem. Theorem 4.2.5. (tom Dieck [9]) Suppose R is a field or a complete discrete valuation ring. Let D C be a full subcategory and [x] Ob D =. Then R Hom C (x, ) C D is projective if and only if the following conditions are satisfied: (1) for each y Ob D, every f Hom C (x, y) satisfies the condition (TD); and (2) for each y Ob D, R Irr D (x, y) is a projective R Aut(y)-module. We leave the formulation of a dual version of the above theorem about the restriction of right representable functors for the interested readers. Now we show there exists D C for which RC becomes a (left or right) projective RD-module. Definition 4.2.6. Suppose D C is a (full) subcategory. We say D is an ideal in C if for any x Ob D we have C x D. Similarly, we say D is a co-ideal in C if for any x Ob D we have C x D. Let D C be a full subcategory. Then we can form a full subcategory of C, named C\D, which consists of all objects not belonging to D. From the definitions it s easy to verify that if D is an ideal (resp. a co-ideal) then C\D is a co-ideal (resp. an ideal). Note that if D C is an ideal (resp. a co-ideal) then RD becomes a right ideal (resp. a left ideal) in RC. If a full subcategory D forms an ideal (resp. a co-ideal) in C, then C D preserves projectives (resp. right projectives).

14 FEI XU Lemma 4.2.7. If D is an ideal in Ob C, then C D preserves left projective modules. If D is a co-ideal in Ob C, then C D preserves right projective modules. Proof. We only prove the first assertion by computing R Hom C (x, ) C D explicitly. If x Ob D, then R Hom C (x, ) C D = R Hom D(x, ). If x Ob D, then by definition R Hom C (x, ) C D = 0. Hence R Hom C(x, ), and consequently RC, are projective RD-modules. In fact RC C D = RD. 4.3. Relative projectivity. In this section, we give basics of the theory of relative projectivity, which provides us a general framework to study the induced modules. Suppose A is an R-subalgebra of an R-algebra B. Let M be a B-module. Then there is a natural epimorphism ɛ : B A M M given by the multiplication ɛ(b m) = bm. We shall only consider the case of a category algebra RC with a subalgebra RD, for some subcategory D of C. For consistency, we assume C is finite EI though in some definitions and results of this section the condition is not necessary. Definition 4.3.1. Let M be an RC-module. If the RC-module epimorphism ɛ = ɛ M : M C D C D = RC RD M M is split, then we say M is projective relative to D, or simply D-projective. Let s list some basic but relevant results. First of all we have some equivalent descriptions of the relative projectivity of an RC-module M. Proposition 4.3.2. Let D C be a subcategory and M an RC-module. Then the following statements are equivalent: (1) The canonical surjective map ɛ : M C D C D M splits; (2) M is a direct summand of M C D C D ; (3) M is a direct summand of N C D, where N is an RD-module. (4) if 0 M M M 0 is an exact sequence of RC-modules which splits as an exact sequence of RD-modules, then it splits as an exact sequence of RC-modules. Proof. The statements 1, 2 and 4 are proved to be equivalent in the context of Artin algebras, see for instance [2], Section VI Proposition 3.6. When C is a finite group, statement 3 is well-known to be equivalent to the others, and the proof for category algebras is similar to that for group algebras. One can find a proof of this proposition in Xu [45]. Suppose M and N are two RC-modules. Then we write M N if M is isomorphic to a direct summand of N. Proposition 4.3.3. Let C be a category. Then (1) if E D are subcategories of C and M is E-projective then M is D-projective; (2) if E D are subcategories of C, N is an RD-module which is E-projective, and M is a direct summand of N C D, then M is E-projective; (3) if an RC-module M is D-projective, and it s projective as RD-module, then M is a projective RC-module.

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 15 Proof. Since M is a direct summand of M C E C E which can be written as (M C E D E ) C D, we have M N C D for an RD-module. So M is D-projective as stated in part 1. From N N D E D E, we get M N C D (N D E D E ) C D= (N D E ) C E. It means M is E-projective, which completes the proof for part 2. As for part 3, it is so because M C D must be a direct summand of RDn for some positive integer n. Then M C D C D is a direct summand of RDn C D = RC n, which means M is a direct summand of a free RC-module RC n. We need the following terminology before introducing our next two results. Definition 4.3.4. Let C be a (finite) EI-category. For each RC-module M, we define the M-minimal objects to be those x Ob C which satisfy the condition that M(y) = 0 if y = x and Hom(y, x) non-empty. Similarly we can define M-maximal objects. For example, the R-minimal objects are the minimal objects of C, and the R- maximal objects are the maximal objects of C. The S x,v -minimal and S x,v -maximal objects are the same: y [x]. We explain what s special about these M-minimal objects. Lemma 4.3.5. Let M be an RC-module and D C a subcategory. (1) If M is D-projective: M C D C D = M M, then Ob D contains all M-minimal objects; (2) If M is D-projective, then M(x) is Aut D (x)-projective as an R Aut C (x)-module for any M-minimal object x. Proof. If z is M-minimal and z Ob D, then M C D C D (z) = y Ob D R Hom C(y, z) RD M(y) = 0, since z is M-minimal. Therefore M cannot be a direct summand of M C D C D which is a contradiction. Hence Ob D contains all M-minimal objects. In order to prove (2) we just evaluate the relation M M C D C D at x, and then the result follows. We comment that M(x) is not necessarily an indecomposable R Aut C (x)-module even if M is indecomposable. Corollary 4.3.6. Let M be an indecomposable RD-module that is D-projective for a subcategory D C. Then D has a unique connected component relative to which M is projective. Proof. If D is a disjoint union of several connected components {D i } i I, then from M C D C D = i IM C D i C D i and M M C D C D we know M is projective relative to some D i. Such a D i has to be unique because it contains all the M-minimal objects by the preceding lemma. The M-minimal objects will be used in the next section to describe the vertex of an indecomposable M. Using (2) of Lemma 4.3.5, we may reveal some partial information about the structure of D, relative to which M is projective. As an example if R is D- projective, then for any minimal object x Ob C, R(x) = R is Aut D (x)-projective as

16 FEI XU an R Aut C (x)-module. When R = F p for some prime p dividing the order of Aut C (x), Aut D (x) has to contain a Sylow p-subgroup of Aut C (x), by a standard result from the theory of vertices and sources for group algebras, see for instance Alperin [1] and Webb [44]. 4.4. Vertices and sources. If D C is a full subcategory and M RC-mod is D-projective, we can show M C D C D = M (without extra summands). Based on this fact, the RC-modules can be parameterized using the set of full subcategories of C. Another important fact is that if M is indecomposable and is both D-projective and E-projective for D and E two ideals of C, M must be D E-projective. This fact enables us to establish a theory of vertices and sources, which functions in a similar way as its counterpart in group representation theory. The base ring R is a field or a complete discrete valuation ring, as usual. Proposition 4.4.1. If M is D-projective for a full subcategory D C, then M is generated by its values on D, that is, M C D C D = M. Proof. Suppose M C D C D = M M for some RC-module M, M with M = M and M (x) = 0 for all x Ob D. Let s take y Ob D and consider M C D C D (y) = M (y) M (y). We claim M C D C D (y) = y>x Ob D R Hom RC(x, y) RD M(x) equals M (y). In fact M (x) = 1 D M(x) for all x Ob D, and given any α Hom(x, y), α M (x) M (y), which means α M(x) M (y). When x and α run over all possible choices, we get exactly y>x Ob D R Hom RC(x, y) RD M(x) M (y) which is indeed an equality since the converse direction inclusion is certainly true. Thus the statement is correct. If M is D-projective (D full), then the natural surjection ɛ : M C D C D M is an isomorphism. Let y Ob C. From ɛ y : M C D C D (y) M(y) = we get ɛ y (M C D C D (y)) = ɛ y( x Ob D y R Hom C (x, y) RD M(x)) = x Ob D y R Hom C (x, y) M(x) = M(y). This explains why any D-projective RC-module M is generated by its values on objects in D. However, x Ob D y R Hom C (x, y) M(x) = M(y) for any y Ob C\ Ob D doesn t guarantee M is D-projective. We can consider the category x y with two non-isomorphisms and two trivial isomorphisms. Then the trivial module R is projective relative to the whole category, not {x} although both non-isomorphisms send R(x) = R isomorphically to R(y) = R. Theorem 4.4.2. Let D C be a (connected) full subcategory and N an indecomposable RD-module. Then the RC-module N C D is indecomposable and moreover is D-projective. Proof. Suppose N C D = N 1 N 2, where N 1, N 2 are both non-zero. Then N = N C D C D = N 1 C D N 2 C D, and since N is indecomposable we must have N 1 C D = N and N 2 C D = 0 (or the other way around). Now that N C D is generated by its values

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 17 on D implies N 2 = 0. Hence inductively we can show N C D is indecomposable, and its D-projectivity follows from Definition 4.3.1. One can compare this theorem with the Green s indecomposability theorem in group representation theory (see for instance Alperin [1] or Benson [4]). Definition 4.4.3. Let x be an object of an EI-category C. Then we use {x} to denote the full subcategory of C with a single object x. We use {[x]} to denote the full subcategory of C consisting of all objects which are isomorphic to x. Given an x, one can choose the full subcategory {[x]} and use an indecomposable R{[x]}-module N to generate an RC-module N C {[x]}. Then Theorem 4.4.2 asserts that such an induced module is indecomposable. This implies that RC is not of finite representation type if, for some x Ob C, R Aut C (x) is not. Theorem 4.4.4. Let M be an indecomposable RC-module which is D-projective for a (connected) full subcategory D C. Then M C D is indecomposable. Proof. Suppose M C D = M 1 M n is a direct sum of indecomposable RD-modules. Then M = M C D C D = M 1 C D M n C D. Since M is indecomposable, we have M M i C D for some index i. This implies M(x) M i C D (x) = M i(x) for all x Ob D, hence M C D = M i is indecomposable. Theorems 4.4.2 and 4.4.4 also give us an equivalence of two module categories (a Green correspondence). Definition 4.4.5. Let D C be a connected full subcategory. We define RC D -mod to be the full subcategory of RC-mod consisting of all D-projective RC-modules. For the sake of simplicity, we write Hom RCD (M, N) for the set of morphisms between two modules M, N RC D -mod. Proposition 4.4.6. The functor C D : RC D-mod RD-mod is an equivalence with C D as its inverse. Proof. From the previous results we see the two functors are well-defined on objects, while C D C D = Id CD and C D C D = Id D. Actions of the induction and the restriction on the morphisms are very clear. Furthermore on morphisms we have the following isomorphisms and Hom RD (M C D, N C D) = Hom RCD (M C D C D, N) = Hom RCD (M, N), Hom RCD (M C D, N C D) = Hom RD (M, N C D C D) = Hom RD (M, N). So C D C D and C D C D are also identities on morphisms, because both M and N are generated by their values on D. Note that the second isomorphism doesn t depend on the relative projectivity of the modules, since we always have Hom RC (M C D, N C D ) = Hom RD (M, N), as long as D C is a full subcategory.

18 FEI XU Now we re ready to develop a theory of vertices and sources for category algebras. The next proposition and its corollary suggest that we should restrict our attention to the subcategories which are ideals in C, in order to give a proper definition of the vertex of an indecomposable module. The following result will be used as a stepping stone to define the vertex of an indecomposable module. Proposition 4.4.7. Let D and E be two ideals of C. Suppose M is an indecomposable RC-module. Then M D C E C = M D E C. Proof. We need to consider the structure of RE RC RD M. Since Ob E forms an ideal in Ob C, we get RC = RE as an RE-module. The only terms in this direct sum on which D is non-zero in the action from the right are the ones where x is in Ob D. Regarded as a right RD-module, RC can be identified with RD = x Ob D R Hom C(x, ). So as an RE-RD-bimodule, RC = x Ob(D E) R Hom C(x, ). Thus RC RE RC RD M = RC RE { x Ob(D E) RHom C(x, )} RD M = RC RE { x Ob(D E) RHom C(x, )} R(D E) M = RC RE RE R(D E) M = RC R(D E) M. We note that the above argument does not work for an arbitrary pair of full subcategories relative to which M is projective. Corollary 4.4.8. Suppose D and E are two ideals of C. Let M be an indecomposable RC-module, which is both D-projective and E-projective. Then M is also D E- projective. Thus for any indecomposable RC-module M, there exists a smallest ideal Ṽ M in C, relative to which M is projective. Proof. We just need to check that D E forms an ideal in C, and then the results follow from the above lemma. Obviously, ṼM has to be connected, because if ṼM = D 1 D 2, then M C g VM = M C D 1 M C D 2, and M must be projective relative to one of its connected components, which contradicts with the minimality of ṼM. At this point, one who is familiar with the definition of the vertex for an indecomposable group module might want to define the category ṼM as the vertex of M. However, there is a possibility to throw away some objects x ṼM for which M(x) = 0, and obtain a category smaller than ṼM for M, relative to which M is still projective. Definition 4.4.9. For any RC-module M we define the full subcategory of C, C M to be a category consisting of all y Ob C with Hom C (x, y) for some M-minimal object x. Similarly we define C M to be the full subcategory containing all z Ob C with Hom(z, x) for some M-maximal object x. In particular, we have C Sx,V = C x and C S x,v = C x.

REPRESENTATIONS OF CATEGORIES AND THEIR APPLICATIONS 19 In fact, C M is a co-ideal in C generated by the M-minimal objects, and C M is an ideal in C generated by the M-maximal objects. In particular, we have C R = C R = C. Definition 4.4.10. The full subcategory V M = ṼM C M is called the vertex of M. Note that V M is connected but is not necessarily an ideal in C, though it s an ideal in C M. Next we want to get an alternative description of the vertex of M, free of the use of C M. Definition 4.4.11. Let D C be a subcategory. Then D is said to be convex if whenever there is a sequence of morphisms x α y β z in C with x, z Ob D, then both α and β are in Mor(D). Ideals and co-ideals in C are full convex subcategories. Let M be an indecomposable RC-module. Then its vertex V M is convex. Note that in general a convex subcategory D doesn t have to be full. Since intersection of two convex subcategories is still convex, it s natural to define the convex hull of a subcategory D of C as the smallest convex subcategory containing D. This terminology will be used in the next two sections. Proposition 4.4.12. Let M be an indecomposable RC-module and D a full (connected) subcategory of C. Then the following statements are equivalent: (1) D is the vertex of M; (2) D is the smallest ideal in C M, relative to which M is projective; (3) D is the smallest full convex subcategory of C, relative to which M is projective. Proof. 1 2 : If D = V M, then by definition D is full convex and M is D-projective. Suppose E is an ideal in C M, relative to which M is projective. We claim D E. In fact, D is an ideal in C M, and so is D E. We can naturally extend D E to an ideal D E ṼM in C, relative to which M is projective. But then by definition we have Ṽ M = D E, which implies D = ṼM C M = D E C M = D E. 2 3 : Let E be a full convex subcategory for which M is E-projective. Then E contains all M-minimal objects, and thus E C M must be an ideal in C M. Since as an RC M -module M is projective relative to E C M, we have D E. 3 1 : Let E be an ideal in C, relative to which M is projective. Then E C M is a full convex subcategory in C, which means D E C M. We can take E to be ṼM, and this results in an inclusion D V M, which can be shown to be an equality by extend D to an ideal in ṼM. Proposition 4.4.13. Let D be a connected full subcategory of C and N an indecomposable RD-module with vertex V N D. Then the indecomposable RC-module M = N C D is V N-projective. If V N is a (connected and full) convex subcategory of C, then V M = V N. If M is an indecomposable RC-module whose vertex is V M, and D is a connected full subcategory containing V M, then M C D is an indecomposable RD-module whose vertex is V M.