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. Representations of categories, category algebras, relative projectivity, vertices and sources, higher limits, cohomology ring.

1. Introduction

Let C be a small category and R a commutative ring with an identity. The category algebra RC is a free R-module whose basis is the set of all morphisms of C. The product on the basis elements of RC is defined by composition and then it is linearly extended to a product on all elements of RC. The category algebra is introduced in the first place as a tool to study representations and cohomology of the category on which it is defined. 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 starting point of our research is the following result of Mitchell.

Proposition 1.1 (Mitchell [29]) For any small category C with finitely many objects, the category of covariant (resp. contra-variant) functors from C to R-mod can be identified with the unital left (resp. right) RC-modules.

When the category in question is a group or a partially ordered set (poset), the category algebra is either a group algebra or an incidence algebra and our work doesn’t provide many new results for such a category algebra. It is the recent active investigations of categories associated with groups that motivates our research. We have in mind the constructions of group modules using representations of the Tits building of a Chevalley group by Ronan and Smith [36], the homology decompositions of classifying spaces of groups using representations of orbit categories and other categories by Dwyer [11], the theory of p-local finite groups of Broto-Levi-Oliver [8], as well as the work of Grodal [18], Jackowski-McClure-Oliver [25], Linckelmann [28] and Symonds [39]. The purpose of this paper is to provide a general framework for studying representations and cohomology of abstract small categories and to prove some general results which can be used to understand the representation and cohomology theory of the particular categories we mentioned above.

In partial fulfillment of the requirements for the PhD degree at the University of Minnesota.

To achieve the goal, we need to extract some fundamental properties from the prototypes we have got and use them as axioms to construct a class of abstract small categories, which should be general enough to contain all existing examples while on the other hand is specific enough for us to develop the representation and cohomology theory. This class of categories will be the class of finite EI-categories which we now define. A small category C is said to be finite if its morphisms form a finite set Mor C. It implies that the set of objects Ob C is finite and that RC is of finite R-rank with an identity 1 = ∑_{x∈Ob C} 1_{x}. A small category C is EI if every automorphism is an isomorphism. Any EI-category C has a prominent property that there exists 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 RC-module M: {0} = M_{0} ⊊ M_{1} ⊊ ⋯ ⊊ M_{n} = M so that every factor concentrates on a single isomorphism class of objects of C (that is, as a functor each factor takes non-zero values only at a single isomorphism class of objects). The preorder also allows us to construct very useful auxiliary 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.

Lück has classified the projective and simple RC-modules for EI-categories, and has studied the restriction of projective modules to certain subalgebras in [27]. The reader can find a short description of his key results in Section 3.1. In this paper we exploit further properties and other aspects of the representations of finite EI-categories, with the applications to computing higher limits in mind.

Let D be a subcategory of a small category C. Then RD is a subalgebra of RC. There are two naturally defined functors: the restriction ↓_{D}^{C} : RC-mod → RD-mod and the induction ↑_{D}^{C} : RD-mod → RC-mod. The induction is the usual tensor product RC ⊗_{RD} −, while the restriction is more sophisticatedly defined since RC is not a unital RD-module. The reader is advised to read from Definition 2.2.2 through Proposition 2.2.3 for more information. We call an RC-module M D-projective (or projective relative to D) if M is isomorphic to a direct summand of M ↓_{D}^{C} ↑_{D}^{C}. When D ⊂ C is a full subcategory, we have the following results.

Theorem 1.2 Let D be a full subcategory of a finite EI-category C. Then

(1) M ↓_{D}^{C} ∈ RD-mod is indecomposable if M ∈ RC-mod is indecomposable and D-projective;

(2) N ↑_{D}^{C} ∈ RC-mod is indecomposable and D-projective if N ∈ RD-mod is indecomposable;

(3) if M ∈ RC-mod is D-projective then M ↓_{D}^{C} ↑_{D}^{C} ≅ 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 of any two morphisms α, β ∈ Mor C being a morphism in Mor D implies 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 the smallest full convex subcategory of C, relative to which M is projective.

The subcategory in Theorem 1.4 is called the vertex of M, denoted by V_{M}. The restriction of M, M ↓_{V_{M}}^{C}, is an indecomposable RV_{M}-module by Theorem 1.2 and is called the source for M. Just as in group representation theory, there is a trivial module (or constant functor) \underline{R}, which sends every object to R and every morphism to the identity and plays an important role in this paper. For any functor ι : D → C and any x ∈ Ob C there exists an overcategory denoted by ι ↓_{x} (Mac Lane [30]), which can be used to define the left Kan extension \underrightarrow{lim}_{ι↓_{?}} − of ι. When ι : D ↪ C is the inclusion, the left Kan extension is isomorphic to the induction ↑_{D}^{C} = RC ⊗_{RD} − : RD-mod → RC-mod. Using (3) of Theorem 1.2, we can give the following characterization of the full subcategories D, relative to which \underline{R} is projective.

Proposition 1.5 Suppose C is a finite EI-category and D is a full subcategory. Let ι : D ↪ C be the inclusion. Then the RC-module \underline{R} is D-projective if and only if every ι ↓_{x}, x ∈ Ob C, is connected.

The above characterization, however, is not easy to use in practice to narrow down the possible choices of the vertex of \underline{R}. Inspired by the work of Puig [33], Thévenaz [40] and Symonds [39], we define an object x ∈ Ob C to be weakly essential if C_{<x} is empty or has more than one component. Following Symonds [39], 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), containing x ∈ Ob C such that C_{<x} is not contractible, see Quillen [34], [35], Bouc [6] and Symonds [39].

Proposition 1.6 Let C be a finite EI-category. Then Wess_{0}(C) ⊂ V\underline{R}.

In Section 3.6 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\underline{R}. The other category Wess(I) is as equally important as Wess_{0}(I) in this situation, see Propositions 3.5.6 and 3.5.7.

Since RC-mod is an abelian category with enough projectives and injectives, we can consider the groups Ext_{RC}^{∗}(M, N) for two arbitrary RC-modules. When M = N, Ext_{RC}^{∗}(M, M) := ⊕_{i≥0} Ext_{RC}^{i}(M, M) possesses a ring structure with the multiplication given by the Yoneda splice. The following theorem is proved by using Theorem 1.2 and the Eckmann-Shapiro Lemma (see Benson [4]).

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 ↓_{D}^{C}, N ↓_{D}^{C}),

for any RC-module N. When M = N, this is a ring isomorphism.

It is well-known that \underleftarrow{lim}_{C}^{∗} N ≅ Ext_{RC}^{∗}(\underline{R}, N) for any RC-module N. In this case, Theorem 1.7 becomes a result of Jackowski and Słomińska [26]. The representation theory of an EI-category C enables us to describe projective resolutions of RC-modules, and hence leads us to some other interesting results. In the following proposition, S_{x,V} and S_{y,U} are two arbitrary simple RC-modules. The structure of simple RC-modules and their projective covers was studied by Lück [27] and is described in Theorem 3.1.2 and the paragraph after it.

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_{x}^{y}}^{∗}(S_{x,V} , S_{y,U}),

where C_{x}^{y} = C_{≥x} ∩ C_{≤y}. Especially we have Ext_{RC}^{∗}(S_{x,V} , S_{x,U}) ≅ Ext_{R Aut_{C}(x)}^{∗}(V, U).

Using our knowledge about the minimal resolutions, we can also show when RC has 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 paper is organized as follows. In Section 2 we define category algebras and their representations and go over some basic homological properties of category algebras. The central piece of the paper is Section 3 on the representation theory of EI-categories. We will give a description of the simple and projective RC-modules, as well as the restriction of projective modules to subalgebras. Our theory of vertices and sources will be developed, and we will apply it to establish reduction formulas for computing Ext groups. In Section 4 we describe the minimal projective resolutions for RC-modules and consider their applications.

Acknowledgement I would like to thank my thesis advisor, Prof. Peter Webb, for leading me into this area, for numerous helpful conversations and comments and for much encouragement.

I wish to thank the referee for his or her useful comments which help remove a couple of errors and make this paper better presented.

2. Category algebras and basic properties

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 = \{ \begin{matrix}f ∘ g, & if f and g 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.

A category C is said to be finite if all its morphisms form a finite set Mor(C). Note that this implies C has finitely many objects (i.e. 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 sum 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 paper, we assume C is connected.

2.2. Representations of categories. 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 [29] proved the category of covariant functors from C to R-mod is equivalent to the category of left RC-modules, i.e. (R-mod)^{C} ≃ RC-mod, if Ob C is finite. The functors which establish the equivalences are described as follows. If F ∈ (R-mod)^{C} is a covariant functor, then we define an RC-module M_{F} to be M_{F} = ⊕_{x∈Ob C}M(x). Conversely if M ∈ RC-mod, we can define a functor F_{M} such that F_{M}(x) = 1_{x} · M. 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 paper, we’re going to use \underline{R} to denote the constant functor or trivial module. For any category C, \underline{R} : C → R-mod, is defined by \underline{R}(x) = R for all x ∈ Ob C and \underline{R}(f) = Id for all f ∈ Mor C.

Any group G 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, and a left RG-module M is a representation of Ĝ in an obvious way. The trivial RĜ-module \underline{R} is exactly the trivial module of RG. As further examples of category algebras we observe that when 𝔮 is a quiver, the category algebra of the free category C_{𝔮} generated by 𝔮 (see Mac Lane [30]) is the same as the path algebra of the quiver 𝔮, and that when Γ is a poset the incidence algebra of Γ (see [9]) is the same as the category algebra RΓ.

Definition 2.2.2. Suppose μ : D → C is a (covariant) functor. We define Res_{μ} : (Rmod)^{C} → (R-mod)^{D} to be the restriction along μ. Given a functor M ∈ (R-mod)^{C}, we have Res_{μ} M = M ∘ μ ∈ (R-mod)^{D}.

Given a functor μ : D → C, the restriction Res_{μ} : (R-mod)^{C} → (R-mod)^{D} has a counterpart, also denoted by Res_{μ}, between the corresponding module categories: Res_{μ} : RC-mod → RD-mod. In fact, if M = ⊕_{x∈Ob C}M(x) is an RC-module, then Res_{μ} M = ⊕_{y∈Ob D}M(μ(y)) ∈ RD-mod. On the other hand μ : D → C extends linearly to a natural map of R-modules μ̄ : RD → RC, which is not necessarily an algebra homomorphism, and here is a simple example. Let D 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 : D → C. Then the map μ̄ : RD → RC is not an algebra homomorphism for the product of the two morphisms in D is zero while the product of their images is not. When μ̄ is an algebra homomorphism, it induces the representation-theoretic restriction ↓_{RD}^{RC}: RC-mod → RD-mod.

Proposition 2.2.3. A functor μ : D → C extends linearly to an algebra homomorphism μ̄ : RD → RC if and only if μ is injective on Ob D. When this happens, the induced functor followed by 1_{RD}, 1_{RD}· ↓_{RD}^{RC}: RC-mod → RD-mod, is exactly Res_{μ}.

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

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

On the other hand if the linear extension μ̄ : RD → RC is an algebra homomorphism then we must have μ̄(0) = 0 and then μ̄(1_{x})μ̄(1_{y}) = μ̄(1_{x} ·1_{y}) = 0 unless x = y. This implies that μ is injective on Ob D. □

In Section 3, we will take D to be a full subcategory of C and μ = ι, the inclusion. Then the restriction Res_{ι} : RC-mod → RD-mod is determined by the algebra homomorphism ῑ: RD → RC, hence by ι : D → C. For this reason we don’t distinguish Res_{ι} and ↓_{RD}^{RC}, and will write ↓_{RD}^{RC} and Res_{ι} as ↓_{D}^{C} which is common in representation theory.

Proposition 2.2.4. Let C and D be equivalent small categories. Then

(1) (R-mod)^{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; and

(2) the nerves NC and ND are homotopy equivalent.

Proof. We prove the first assertion. The second is well-known and a proof of it can be found in Baues and Wirsching [3].

We show 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 μ : D → C and ν : C → D are equivalences, we have Res_{μ} Res_{ν} ≅ Id_{RD} : (R-mod)^{D} → (R-mod)^{D} because of the following diagram

\xymatrix{M(νμ(x)) = (Res_{μ} Res_{ν} M)(x) \ar[r]^{≅} \ar[d]_{M(νμ(α))=(Res_{μ} Res_{ν} M)(α)} & (Id_{C′} M)(x) = M(x) \ar[d]^{(Id_{C′} M)(α)=M(α)} \\ M(νμ(y)) = (Res_{μ} Res_{ν} M)(y) \ar[r]_{≅} & (Id_{C′} 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_{ν} Res_{μ} ≅ Id_{RC} : (R-mod)^{C} → (R-mod)^{C}. Clearly the constant functor restricts to the constant functor always. □

2.3. Basic homological properties. The category RC-mod is abelian and has enough projective and injectives so we can consider the Ext groups Ext_{RC}^{∗}(M, N) for M, N ∈ RC-mod. 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 = \underline{R} that is of great interest to us.

Definition 2.3.1. We call the ring Ext_{RC}^{∗}(\underline{R}, \underline{R}) = ⊕_{i≥0} Ext_{RC}^{i}(\underline{R}, \underline{R}) the cohomology ring of the category algebra RC. The product in this ring is defined by the Yoneda splice.

There exists a ring isomorphism

Ext_{RC}^{∗}(\underline{R}, \underline{R}) ≅ H^{∗}(|C|, R),

where |C| stands for the topological realization of the nerve NC.

Direct computation of cohomology groups is in general very difficult, and so people have been searching for reduction formulas. It’s well-known that if a functor μ : D → C has a left adjoint ν : C → D, then Res_{ν} is also the left adjoint of Res_{μ}, and hence Ext_{RC}^{∗}(Res_{ν} M, N) ≅ Ext_{RD}^{∗}(M, Res_{μ} N) for any M ∈ RD-mod and N ∈ RC-mod, because both Res_{μ} and Res_{ν} are exact (see for example Jackowski-McClure-Oliver [25] II, Proposition 5.1). If μ is indeed an equivalence, we have the following result.

Lemma 2.3.2. Let μ : D → C be an equivalence of two small categories. Then we have

Ext_{RC}^{∗}(M, N) ≅ Ext_{RD}^{∗}(Res_{μ}M, Res_{μ}N),

for M, N ∈ RC-mod. In particular there is a ring isomorphism

Ext_{RC}^{∗}(\underline{R}, \underline{R}) ≅ Ext_{RD}^{∗}(\underline{R}, \underline{R}).

Proof. The thing is, RC and RD are Morita equivalent by the functor Res_{μ}, which takes \underline{R} to \underline{R}. □

The groups Ext_{RC}^{∗}(\underline{R}, M) are very useful to us, since they can be used to recover the cohomology theory of small categories that has been discussed in various places in the literature, see Baues-Wirsching [3], Generalov [16] and Oliver [32].

Definition 2.3.3. 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_{RC}^{n}(\underline{R}, M).

It was shown by Roos [38] and Gabriel-Zisman [15] that

Ext_{RC}^{∗}(\underline{R}, M) ≅ \underleftarrow{lim}_{C}^{∗}M,

the higher inverse limits of M over C. The computation of higher limits occupies an important place in group cohomology theory so we record some relevant results below.

Let C and D be small categories equipped with a functor μ : D → C. For each y ∈ Ob C, the overcategory μ ↓_{y} (comma category in Mac Lane [30]) 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′. We can define a functor μ ↓_{?}: C → sCat (the category of small categories), and thus a functor C_{∗}(μ ↓_{?}) : C → RC-Cplx (the category of complexes of RC-modules) through the simplicial complexes given by the overcategories. When D = C and μ = Id, we normally write Id(C) ↓_{y} as C ↓_{y} for any y ∈ Ob C. It is well-known that {C_{∗}(C ↓_{?})} is a projective resolution (the bar resolution) of \underline{R} ∈ RC-mod, see for instance Grodal [18].

The restriction Res_{μ} : RC-mod → RD-mod, induced by any functor μ : D → C, always has a left adjoint, called the left Kan extension K : RD-mod → RC-mod and defined by

K(M)(y) = \underrightarrow{lim}_{μ↓_{y}}M ∘ π,

where M ∈ RD-mod, y ∈ Ob C and π : μ ↓_{y}→ D is the projection (x, α) ↦ x. Let P → \underline{R} → 0 be a projective resolution of the RD-module \underline{R}. Then

\underleftarrow{lim}_{D}^{n}Res_{μ}M ≅ Ext_{RD}^{n}(\underline{R}, Res_{μ}M) ≅ H_{n}(Hom_{RC}(K(P), M)).

When P = C_{∗}(D ↓_{?}), it was known to Dwyer-Kan [12] and Hollender-Vogt [22] that K(C_{∗}(D ↓_{?})) ≅ C_{∗}(μ ↓_{?}), while the latter will become a projective resolution of the RC-module \underline{R} if it’s exact. A category E is called R-acyclic if its reduced homology groups H̃_{∗}(|E|, R) vanish.

Proposition 2.3.4. (Jackowski-Słomińska [26] 5.4) Let μ : D → C satisfy the condition that every μ ↓_{y}, y ∈ Ob C, is R-acyclic. Then \underleftarrow{lim}_{C}^{∗} M ≅ \underleftarrow{lim}_{D}^{∗} Res_{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 [34]. When we turn to our representation-theoretic settings and assume RC is a right (non-unital) RD-module, Res_{μ} becomes the usual restriction ↓_{D}^{C} whose left adjoint is the induction ↑_{D}^{C} = RC ⊗_{RD} −. Under the circumstances, both Lemma 2.3.2 and Proposition 2.3.4 can be rewritten using ↑_{D}^{C} and ↓_{D}^{C}.

3. 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 can be found in the book of tom Dieck [10] (I.11), much of which was due to Lück [27]. One of the important features of EIcategories 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 [32] and Jackowski-Słomińska [26]. 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 second condition in the convention (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 first reason 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, and the second is that for any full subcategory D ⊂ C, there always exists a natural full subcategory E such that D ⊂ E ⊂ C, E ≃ D and E meets our convention.

3.1. Projective modules and simple modules. Now we start describing the projective and simple modules for an EI-category. The base ring R is assumed to be a field or a complete discrete valuation ring.

Proposition 3.1.1. (Lück [27]) 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 3.1.2. (Lück [27]) 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 corresponds to a pair ([x], V) where V is a simple R Aut(x)-module, and 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. More precisely, 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 3.1.3. 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 \underline{R} is projective. This generalizes Lemma 2.5 of Symonds [39].

Proposition 3.1.4. 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(x) has a single orbit on Hom(x, y), and | Aut(x)| is invertible in R.

Proof. If \underline{R} is projective then \underline{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 \underline{R} = ⨁_{\{minimal x\}} P_{x,R} and this is projective. □

For certain subcategories D ⊂ C, the restriction ↓_{D}^{C} : RC-mod → RD-mod may preserve projective modules.

Definition 3.1.5. 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 ↓_{D}^{C} preserves projectives (resp. right projectives).

Lemma 3.1.6. If D is an ideal in Ob C, then ↓_{D}^{C} preserves left projective modules. If D is a co-ideal in Ob C, then ↓_{D}^{C} preserves right projective modules.

Proof. We only prove the first assertion by computing R Hom_{C}(x, −) ↓_{D}^{C} explicitly. If x ∈ Ob D, then R Hom_{C}(x, −) ↓_{D}^{C} = R Hom_{D}(x, −). If x ∉ Ob D, then by definition R Hom_{C}(x, −) ↓_{D}^{C} = 0. Hence R Hom_{C}(x, −), and consequently RC, are projective RD-modules. In fact RC ↓_{D}^{C} = RD. □

For a complete description of the restrictions of projective modules, one can consult tom Dieck [10] I.11.

3.2. Relative projectivity. Suppose A is an R-subalgebra of an R-algebra B. Let M be a B-module. Then there is a natural epimorphism ϵ : B ⊗_{A} M → M given by the multiplication ϵ(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.

Let ι : D ↪ C be a subcategory and M ∈ RD-mod. Then M ↑_{D}^{C} ∈ RC-mod evaluated at any y ∈ Ob C equals

M ↑_{D}^{C}(y) ≅ K(M)(y) = \underrightarrow{lim}_{ι↓_{y}}M ≅ ∑_{x≤y}RHom_{C}(x, y)⊗_{RD}M(x),

where K is the left Kan extension described in the paragraphs preceding Proposition 2.3.4.

Definition 3.2.1. Let M be an RC-module. If the RC-module epimorphism

ϵ = ϵ_{M} : M ↓_{D}^{C} ↑_{D}^{C} = RC ⊗_{RD} M → M

is split, then we say M is projective relative to D, or relatively D-projective.

We have some equivalent descriptions of the relative projectivity of an RC-module M.

Proposition 3.2.2. Let D ⊂ C be a subcategory and M an RC-module. Then the following statements are equivalent:

(1) The canonical surjective map ϵ : M ↓_{D}^{C} ↑_{D}^{C} → M splits;

(2) M is a direct summand of M ↓_{D}^{C} ↑_{D}^{C};

(3) M is a direct summand of N ↑_{D}^{C}, 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. Since the proof for category algebras is similar to that for group algebras, we omit it and refer the reader to Xu [44] for details. □

Suppose M and N are two RC-modules. Then we write M|N if M is isomorphic to a direct summand of N.

Proposition 3.2.3. Let C be a category. Then

(1) if E ⊂ D are subcategories of C and M is relatively E-projective then M is relatively D-projective;

(2) if E ⊂ D are subcategories of C, N is an RD-module which is relatively Eprojective, and M is a direct summand of N ↑_{D}^{C}, then M is relatively Eprojective.

Proof. Since M is a direct summand of M ↓_{E}^{C}↑_{E}^{C} which can be written as (M ↓_{E}^{C}↑_{E}^{D}) ↑_{D}^{C}, we have M|N ↑_{D}^{C} for an RD-module. So M is relatively D-projective as stated in part 1.

From N|N ↓_{E}^{D}↑_{E}^{D}, we get

M | N ↑_{D}^{C}| (N ↓_{E}^{D}↑_{E}^{D}) ↑_{D}^{C}= (N ↓_{E}^{D}) ↑_{E}^{C} .

It means M is relatively E-projective, which completes the proof for part 2. □

We need the following terminology before introducing our next two results.

Definition 3.2.4. 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 ≅ x̸ and Hom(y, x) non-empty. Similarly we can define M-maximal objects.

For example, the \underline{R}-minimal objects are the minimal objects of C, and the \underline{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 3.2.5. Let M be an RC-module and D ⊂ C a subcategory.

(1) If M is relatively D-projective: M ↓_{D}^{C} ↑_{D}^{C} ≅ M ⊕ M′, then Ob D contains all M-minimal objects;

(2) If M is relatively D-projective, then M(x) is relatively 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 ↓_{D}^{C} ↑_{D}^{C}(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 ↓_{D}^{C} ↑_{D}^{C} which is a contradiction. Hence Ob D contains all M-minimal objects.

In order to prove (2) we just evaluate the relation M | M ↓_{D}^{C}↑_{D}^{C} 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 3.2.6. Let M be an indecomposable RC-module that is relatively 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 ↓_{D}^{C} ↑_{D}^{C} = ⊕_{i∈I}M ↓_{D_{i}}^{C}↑_{D_{i}}^{C} and M|M ↓_{D}^{C} ↑_{D}^{C} 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. □

Using (2) of Lemma 3.2.5, we may reveal some partial information about the structure of D, relative to which M is projective. As an example if \underline{R} is relatively Dprojective, then for any minimal object x ∈ Ob C, \underline{R}(x) = R is relatively Aut_{D}(x)- projective as 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.

3.3. Vertices and sources. If D ⊂ C is a full subcategory and M ∈ RC-mod is relatively D-projective, we show M ↓_{D}^{C} ↑_{D}^{C} ≅ M (without extra summands). Based on this fact, the RC-modules can be parameterized using the set of full subcategories of C. We will establish a theory of vertices and sources for indecomposable modules, which functions in a similar way as its counterpart in group representation theory.

Proposition 3.3.1. If M is relatively D-projective for a full subcategory D ⊂ C, then M is generated by its values on D, that is, M ↓_{D}^{C} ↑_{D}^{C} ≅ M.

Proof. Suppose M ↓_{D}^{C} ↑_{D}^{C} = 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 ↓_{D}^{C} ↑_{D}^{C}(y) = M′(y) ⊕ M′′(y). We claim M ↓_{D}^{C} ↑_{D}^{C}(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 relatively D-projective (D full), then the natural surjection ϵ : M ↓_{D}^{C} ↑_{D}^{C} → M is an isomorphism. Let y ∈ Ob C. From ϵ_{y} : M ↓_{D}^{C} ↑_{D}^{C}(y)→^{≅}M(y) we get

ϵ_{y}(M ↓_{D}^{C} ↑_{D}^{C}(y)) = ϵ_{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 relatively 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 relatively D-projective. We can consider the category x ⇉ y with two non-isomorphisms and two trivial isomorphisms. The trivial module \underline{R} is projective relative to the whole category, not \{x\}–the full subcategory with one object x, although both non-isomorphisms send \underline{R}(x) = R isomorphically to \underline{R}(y) = R.

Theorem 3.3.2. Let D ⊂ C be a (connected) full subcategory and N an indecomposable RD-module. Then the RC-module N ↑_{D}^{C} is indecomposable and moreover is relatively D-projective.

Proof. Suppose N ↑_{D}^{C}= N_{1} ⊕ N_{2}, where N_{1}, N_{2} are both non-zero. Then N = N ↑_{D}^{C}↓_{D}^{C}= N_{1} ↓_{D}^{C} ⊕N_{2} ↓_{D}^{C}, and since N is indecomposable we must have N_{1} ↓_{D}^{C}= N and N_{2} ↓_{D}^{C}= 0 (or the other way around). Now that N ↑_{D}^{C} is generated by its values on D implies N_{2} = 0. Hence N ↑_{D}^{C} is indecomposable, and its relative D-projectivity follows from Definition 3.2.1. □

One can compare the above theorem with Green’s indecomposability theorem in group representation theory (see for instance Alperin [1] or Benson [4]).

Definition 3.3.3. 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 ↑_{{[x]}}^{C}. Then Theorem 3.3.2 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 3.3.4. Let M be an indecomposable RC-module which is relatively Dprojective for a (connected) full subcategory D ⊂ C. Then M ↓_{D}^{C} is indecomposable.

Proof. Suppose M ↓_{D}^{C}= M_{1}⊕⋯⊕M_{n} is a direct sum of indecomposable RD-modules. Then M ≅ M ↓_{D}^{C}↑_{D}^{C}= M_{1} ↑_{D}^{C} ⊕ ⋯ ⊕ M_{n} ↑_{D}^{C}. Since M is indecomposable, we have M|M_{i} ↑_{D}^{C} for some index i. This implies M(x)|M_{i} ↑_{D}^{C} (x) = M_{i}(x) for all x ∈ Ob D, hence M ↓_{D}^{C}= M_{i} is indecomposable. □

Theorems 3.3.2 and 3.3.4 also give us an equivalence of two module categories (a Green correspondence).

Definition 3.3.5. Let D ⊂ C be a connected full subcategory. We define RC_{D}-mod to be the full subcategory of RC-mod consisting of all relatively D-projective RC-modules.

For the sake of simplicity, we write Hom_{RC_{D}}(M, N) for the set of morphisms between two modules M, N ∈ RC_{D}-mod.

Proposition 3.3.6. The functor ↓_{D}^{C} : RC_{D}-mod → RD-mod is an equivalence with ↑_{D}^{C} as its inverse.

Proof. From the previous results we see the two functors are well-defined on objects, while ↓_{D}^{C} ↑_{D}^{C} ≅ Id_{C_{D}} and ↑_{D}^{C} ↓_{D}^{C} ≅ Id_{D}. Actions of the induction and the restriction on the morphisms are very clear. Furthermore on morphisms we have the following isomorphisms

Hom_{RD}(M ↓_{D}^{C}, N ↓_{D}^{C}) ≅ Hom_{RC_{D}}(M ↓_{D}^{C} ↑_{D}^{C}, N) ≅ Hom_{RC_{D}}(M, N),

and

Hom_{RC_{D}}(M ↑_{D}^{C}, N ↑_{D}^{C}) ≅ Hom_{RD}(M, N ↑_{D}^{C} ↓_{D}^{C}) ≅ Hom_{RD}(M, N).

So ↓_{D}^{C} ↑_{D}^{C} and ↑_{D}^{C} ↓_{D}^{C} are also identities on morphisms, because both M and N are generated by their values on D. □

Now we’re ready to develop the theory of vertices and sources for category algebras. The following result will be used as a stepping stone to define the vertex of an indecomposable module.

Proposition 3.3.7. Let D and E be two ideals of C. Suppose M is an 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 ↓_{E}^{C} ≅ 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 3.3.8. Suppose D and E are two ideals of C. Let M be an RC-module, which is both relatively D-projective and E-projective. Then M is also relatively D∩Eprojective. Thus for any indecomposable RC-module M, there exists the 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 proposition. □

Obviously, Ṽ_{M} has to be connected, because if Ṽ_{M} = D_{1} ∪D_{2}, then M ↓C ; Ṽ_{M}= M ↓_{D_{1}}^{C} ⊕M ↓_{D_{2}}^{C}, and M must be projective relative to one of its connected components, which contradicts with the minimality of Ṽ_{M}.

Before defining the vertex of an indecomposable module, we introduce some auxiliary notation.

Definition 3.3.9. For any RC-module M we define the full subcategory of C, C_{M} to be a category whose object set is

Ob C_{M} = {y | [y] ≥ [x], some x with M(x) ≠ 0}.

Similarly we define C^{M} to be the full subcategory whose object set is

Ob C^{M} = {y | [y] ≤ [x], some x with M(x) ≠ 0}.

In other words, C_{M} consists of all objects above M-minimal objects and C^{M} consists of all objects below M-maximal objects. 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, In particular, we have C_{S_{x,V}} = C_{≥x}, C^{S_{x,V}} = C_{≤x} and C\underline{R} = C\underline{R} = C.

Definition 3.3.10. The full subcategory V_{M} = Ṽ_{M} ∩ C_{M} is called the vertex of M.

We provide two alternative descriptions of the vertex of M.

Definition 3.3.11. 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 3.3.12. 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 relatively 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 { \atop D ∩ E} ⊂ Ṽ_{M} in C, relative to which M is projective. But then by definition we have Ṽ_{M} = { \atop D ∩ E}, which implies D = Ṽ_{M} ∩ C_{M} = { \atop D ∩ E} ∩ C_{M} = D ∩ E.

2 ⇒ 3 : Let E be a full convex subcategory for which M is relatively 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 Ṽ_{M}, and this results in an inclusion D ⊂ V_{M}, which can be shown to be an equality by extend D to an ideal in Ṽ_{M}. □

Proposition 3.3.13. 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 ↑_{D}^{C} is relatively 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 ↓_{D}^{C} is an indecomposable RD-module whose vertex is V_{M}.

Proof. The first statement holds because of Theorem 3.3.2, and by Proposition 3.2.3 (2) we know V_{M} ⊂ V_{N}. After we prove the second part, we can show V_{M} is exactly V_{N}, if V_{N} is a convex subcategory of C.

The second statement is true because of Theorem 3.3.4. Suppose M ↓_{D}^{C} has the vertex E′ ⊂ D. Since M ≅ (M ↓_{D}^{C}) ↓_{V_{M}}^{D} ↑_{V_{M}}^{C} ≅ (M ↓_{V_{M}}^{C} ↑_{V_{M}}^{D}) ↑_{D}^{C}, we obtain M ↓_{D}^{C} ≅ M ↓_{V_{M}}^{C} ↑_{V_{M}}^{D} = (M ↓_{D}^{C}) ↓_{V_{M}}^{D} ↑_{V_{M}}^{D}. Hence M ↓_{D}^{C} is relatively V_{M}-projective and E′ ⊂ V_{M}. But by the first part, M ↓_{D}^{C} ↑_{D}^{C} has a vertex that is contained in E′. Since M ↓_{D}^{C} ↑_{D}^{C} ≅ M, we must have V_{M} ⊂ E′.

Now we go back to finish proving part 1. Let V_{M} be the vertex of M = N ↑_{D}^{C}, from part 2 and M ↓_{D}^{C} = N ↑_{D}^{C} ↓_{D}^{C} we get desired equality V_{M} = V_{N} because M ↓_{D}^{C} still has vertex V_{M} by part 2, while M ↓_{D}^{C} = (N ↑_{D}^{C}) ↓_{D}^{C} = N has vertex V_{N}. □

Remark 3.3.14. In the first part of Proposition 3.3.13, if V_{N} is not a convex subcategory in C, then it’s not necessarily true that V_{M} = V_{N}, where M = N ↑_{D}^{C}. One can check Example 3.4.8 in the next section, where we have a pair of categories D ⊂ C. If we choose the RD-module N = \underline{R}, then V_{N} = D, which is not convex in C. The induced module M = N ↑_{D}^{C} = \underline{R} ↑_{D}^{C} is isomorphic to the trivial RC-module \underline{R}, which is relatively V_{N}-projective and whose vertex is shown to be V_{M} = C.

Since M ↓_{V_{M}}^{C} ↑_{V_{M}}^{C} ≅ M and M ↓_{V_{M}}^{C} is indecomposable, M is determined up to isomorphism by the indecomposable RV_{M}-module M ↓_{V_{M}}^{C}.

Definition 3.3.15. Suppose M is an indecomposable RC-module with the vertex V_{M}. Then M ↓_{V_{M}} is called the source for M.

Recall that we denote by \{[x]\} the full subcategory of C (equivalent to { \atop Aut_{C}(x)}) consisting of objects in the isomorphism class [x].

Proposition 3.3.16. The vertex of the indecomposable projective module P_{x,V} is {[x]}. The source for P_{x,V} is P_{V} = P_{x,V} ↓_{{[x]}}^{C}, the projective cover of V as an R{[x]}- module.

Proof. Let P_{x,V} = RC · e_{x,V} for some primitive idempotent e_{x,V} ∈ R Aut(x). Then we can easily check that P_{x,V} ↓_{C_{≤x}}^{C}↑_{C_{≤x}}^{C}= RC ⊗_{RC_{≤x}} R Aut(x) · e_{x,V} = RCe_{x,V} ⊗_{RC_{≤x}} 1_{x} ≅ P_{x,V}. Since C_{P_{x,V}} = C_{≥x}, by definition the vertex of P_{x,V} is C_{≤x} ∩ C_{≥x} = {[x]}. □

Definition 3.3.17. A morphism α ∈ Hom_{C}(x, y) is irreducible if it’s not a composite of two non-isomorphisms. The subset of Hom_{C}(x, y), x, y ∈ Ob C, consisting of irreducible morphisms is denoted by Irr_{C}(x, y).

Note that our irreducible morphisms are different from the irreducible morphisms in the representation theory of Artin algebras [2].

Proposition 3.3.18. Let M be an indecomposable atomic module concentrated on [x] ⊂ Ob C. Let D be the full subcategory of C_{≥x} whose object set consists of [x] and those y ≅ x̸ which satisfy the condition that Irr_{C}(x, y) ≠ ∅. Then M is relatively D-projective, and V_{M} is the convex hull of D. The source for M is itself (but regarded as an RV_{M}-module).

Proof. It’s easy to verify that M ↓_{D}^{C} ↑_{D}^{C} ≅ M and there is no proper full subcategory of D having the same property, because if y ∈ Ob C_{≥x}, Irr_{C}(x, y) ≠ ∅ and y ∉ Ob D then 0 ≠ R Irr_{C}(x, y) ⊗_{RD} M(x) ⊂ R Hom(x, y) ⊗_{RD} M(x) ⊂ M ↓_{D}^{C}↑_{D}^{C} (y), which contradicts with the fact M ↓_{D}^{C}↑_{D}^{C} (y) ≅ M(y) = 0.

Since D ⊂ V_{M}, by Proposition 3.3.12 (3) V_{M} is exactly the smallest full convex subcategory containing D. □

The last example considers the vertex and source for an indecomposable module other than the indecomposable projective or atomic modules, and also discusses the representation type of RC.

Example 3.3.19. Given a category C

[figure: x; 1_{x}; f; i_{1}; i_{2}; y; 1_{y}; g] ,

with i_{1}f = i_{1}, i_{2}f = i_{2}, gi_{1} = i_{2} and gi_{2} = i_{1}. Let R = 𝔽_{2} be a field of characteristic 2. We consider the indecomposable module M such that M(x) = 𝔽_{2} and M(y) = 𝔽_{2}⊕𝔽_{2}. The maps i_{1}, i_{2} send M(x) = 𝔽_{2} to the first and the second component, respectively, of M(y) = 𝔽_{2} ⊕ 𝔽_{2}, and g interchanges the two entries of 𝔽_{2} ⊕ 𝔽_{2} (it’s easy to verify these define a functor M : C → R-mod, which is neither projective nor simple). The module M has vertex V_{M} = \{x\} ≅ { \atop Aut_{C}(x)} and source M(x) = 𝔽_{2}, since 𝔽_{2} ↑_{D}^{C}(y) = (i_{1} ⊗ 𝔽_{2}) ⊕ (i_{2} ⊗ 𝔽_{2}) as 𝔽_{2} Aut(y)-module.

We show 𝔽_{2}C has infinite representation type by constructing infinitely many nonisomorphic modules whose vertices are C. Define for any n ∈ N an (indecomposable) 𝔽_{2}C-module M_{n} such that M_{n}(x) = [𝔽_{2} Aut_{C}(x)]^{n} and M_{n}(y) = 𝔽_{2}i_{1} + 𝔽_{2}i_{2}. It is certainly not relatively {y}-projective, and is not {x}-projective when n > 1 because [RC ⊗_{R Aut_{C}(x)} M_{n}(x)](y) has dimension 2n. If n ≠ m are both bigger than 1 then M_{n} ≅ M̸_{m} since M_{n}(x) ≅ M̸_{m}(x).

Finally we get to the applications of our theory. The following statement is actually an Eckmann-Shapiro type lemma.

Lemma 3.3.20. Let M be an RC-module which is relatively D-projective for a full subcategory of C. If RC is a right flat RD-module, then

Ext_{RC}^{∗}(M, N) ≅ Ext_{RD}^{∗}(M ↓_{D}^{C}, N ↓_{D}^{C}).

In particular we get Ext_{RC}^{∗}(M, N) ≅ Ext_{RC_{M}}^{∗} (M ↓_{C_{M}}^{C} , N ↓_{C_{M}}^{C}). If M = \underline{R} then we have

\underleftarrow{lim}_{C}^{∗}N ≅ \underleftarrow{lim}_{D}^{∗}N ↓_{D}^{C} .

Proof. The proof is almost the same as the classic proof of the Eckmann-Shapiro Lemma Ext_{RC}^{∗}(M′ ↑_{D}^{C}, N) ≅ Ext_{RD}^{∗}(M′, N ↓_{D}^{C}), see Benson [5]. One just has to replace M′ by M ↓_{D}^{C} afterwards. □

We conclude the applications of vertices and sources with an isomorphism of cohomology rings. General theory for the correspondence between extensions of modules over a finite-dimensional algebra A and the groups Ext_{A}^{∗}(−, −) can be found in Benson [4].

Proposition 3.3.21. Let D ⊂ C be a full subcategory, relative to which M is projective. Suppose R is a field or a complete discrete valuation ring and RC is a right flat RD-module. Then there is a ring isomorphism between Ext_{RC}^{∗}(M, M) and Ext_{RD}^{∗}(M ↓_{D}^{C}, M ↓_{D}^{C}).

Proof. Let 0 → M ↓_{D}^{C} → N_{n−1} → ⋯ → N_{0} → M ↓_{D}^{C} → 0 represent an element of Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}) for some positive integer n. Then it gives rise to an element of Ext_{RC}^{n}(M, M) vis induction 0 → M → N_{n−1} ↑_{D}^{C} → ⋯ → N_{0} ↑_{D}^{C} → M → 0, since M ↓_{D}^{C} ↑_{D}^{C} ≅ M and RC is a right flat RD-module. But this element of Ext_{RC}^{n}(M, M) restricts back to the given element of Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}). Hence the composite of these two maps Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}) → Ext_{RC}^{n}(M, M) → Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}) is the identity, which implies the first map Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}) → Ext_{RC}^{n}(M, M) is injective. Since we know these two Ext groups are isomorphic (Lemma 3.3.20), this map has to be bijective. Now it’s easy to check that this map respects the Yoneda splice, and thus defines a ring isomorphism. □

3.4. Structure of the vertex of the trivial module. Because of the special interests in the trivial module \underline{R}, we try to obtain a precise description of the vertex V\underline{R}. For the definition of left Kan extension, the reader is referred to the paragraphs preceding Proposition 2.3.4.

Proposition 3.4.1. Let ι : D ↪ C be the inclusion of a full subcategory. Then \underline{R} is relatively D-projective if and only if ι ↓_{y} is connected and non-empty for each y ∈ Ob C.

Proof. Let \underline{R} = \underline{R} ↓_{D}^{C} be the trivial RD-module. Then \underline{R} ↑_{D}^{C}(y) ≅ K(\underline{R})(y) = \underrightarrow{lim}_{ι↓_{y}} \underline{R} equals a direct sum of R over the connected components of ι ↓_{y}. Hence the statement follows. □

The proposition is not true for subcategories which are not full in C. A simple example will be a group G with a proper Sylow-p subgroup P, both of which are regraded as categories with a single object ∗. When R = 𝔽_{p}, \underline{R} is relatively P̂- projective, while ι ↓_{∗} has [G : P] connected components which is not connected. Now we turn to an alternative characterization of the category V\underline{R}.

Definition 3.4.2. An object x ∈ Ob C is weakly essential if the full subcategory C_{<x} is empty or has more than one component. The full subcategory of C consisting of all weakly essential objects is named Wess_{0}(C). There is a larger full subcategory Wess(C) ⊃ Wess_{0}(C) containing objects x ∈ Ob C so that the full subcategory C_{<x} is not contractible.

When C is a certain subgroup poset of a group, Quillen [34], [35] and Bouc [5] considered Wess(C), and Puig [33] (see also Thévenaz [40]) introduced the so-called essential objects of C which we don’t need and are contained in Wess_{0}(C). It is Symonds [39] who generalized Wess(C) to arbitrary posets, and defined Wess_{0}(C) for posets. Obviously the minimal objects of any category C are weakly essential, contained in both Wess_{0}(C) and V\underline{R}.

Lemma 3.4.3. Let D be a connected full subcategory of C. Suppose \underline{R} is relatively D-projective. Then for any y ∉ Ob D, D_{<y} is non-empty and connected.

Proof. Obviously D_{<y} is non-empty by Proposition 3.4.1. If D_{<y} were disconnected, then we prove

\underline{R} ↓_{D}^{C}↑_{D}^{C} (y) = ∑_{x∈Ob D} R Hom(x, y) ⊗ \underline{R}(x)

is a direct sum of at least two non-zero summands. Hence a contradiction since \underline{R} ↓_{D}^{C}↑_{D}^{C} (y) = R.

If D_{<y} were disconnected, then \underline{R} ↓_{D}^{C}↑_{D}^{C} (y) = ∑_{x∈Ob D} R Hom(x, y)⊗\underline{R}(x) contains two elements α ⊗ 1 and β ⊗ 1, where α ∈ Hom(x_{1}, y) and β ∈ Hom(x_{2}, y) for x_{1}, x_{2} from different components of D_{<y}. Let’s assume x_{1}, x_{2} minimal. Now, since \underline{R} ↓_{D}^{C}↑_{D}^{C} (y) = R has rank 1, we have rα ⊗_{RD} 1 = β ⊗_{RD} 1 for some r ∈ R. But it means that rαγ ⊗_{RD} 1_{x_{2}} = β ⊗_{RD} 1_{x_{2}} (or rα ⊗_{RD} 1_{x_{1}} = βγ ⊗_{RD} 1_{x_{1}}) for some γ ∈ R Hom(x_{2}, x_{1}) (or in R Hom(x_{1}, x_{2})), which implies Hom(x_{2}, x_{1}) (or Hom(x_{1}, x_{2})) is non-empty. So x_{1} and x_{2} belong to the same connected component which is a contradiction. □

The above fact results in a corollary which is more convenient to use than Proposition 3.4.1 as a tool to narrow down the subcategory V\underline{R}.

Corollary 3.4.4. Let D be a full subcategory of C, relative to which \underline{R} is projective. Then Wess_{0}(C) ⊂ D. In particular Wess_{0}(C) ⊂ V\underline{R}.

Proof. We show if y ∉ Ob D, then y ∉ Wess_{0}(C). First of all since D contains all minimal objects in C, C_{<y} can not be empty because y ∉ Ob D. We claim C_{<y} is connected. Assume the opposite. Since D_{<y} ⊂ C_{<y} and C_{<y} is disconnected, by Lemma 3.4.3 D_{<y} must lie in only one of the components of C_{<y}. But then C_{<y} contains at least one minimal object x which doesn’t belong to Ob D_{<y}. Actually, x isn’t in Ob D either, because otherwise D_{<y} is disconnected. Now x ∉ Ob D contradicts with the fact that D contains all minimal objects. □

The following result extends Symonds [39] Proposition 3.10, saying that Wess_{0}(C) and C have the same numbers of connected components. Recall that there is a poset P (C) associated to each EI-category C. When C is finite we call the maximal length of chains of non-isomorphisms in P (C) the length of C, denoted by l(C).

Lemma 3.4.5. Let D ⊂ C be a full subcategory. If Wess_{0}(C) ⊂ D, then Wess_{0}(C) ⊂ D ⊂ C induce bijections on connected components.

Proof. Note that for any EI-category C there is a one-one bijection between the connected components of C and those of its underlying poset P (C), and thus one can mimic Symonds’ proof for finite posets by doing induction on the length of a category (or its underlying poset). □

Corollary 3.4.6. Let C be an EI-category and D a connected full subcategory, relative to which \underline{R} is projective. For every y ∉ Wess_{0}(C), Wess_{0}(C)_{<y}, Wess(C)_{<y} and D_{<y} are all connected.

Proof. By definition of Wess_{0}(C), C_{<y} is always connected. The results follow from the inclusions Wess_{0}(C_{<y}) = Wess_{0}(C)_{<y} ⊂ D_{<y} ⊂ C_{<y} and Wess_{0}(C_{<y}) = Wess_{0}(C)_{<y} ⊂ Wess(C)_{<y} ⊂ C_{<y}, combined with Lemma 3.4.5 and the fact that Wess_{0}(C) ⊂ D. □

We give a sufficient condition on the connectedness of overcategories and the relative projectivity of \underline{R}.

Proposition 3.4.7. Let C be an EI-category and D a full subcategory containing Wess_{0}(C). Let ι : D → C be the inclusion. Then every ι ↓_{y}, y ∈ Ob C, is connected if for any pair of objects x ∈ Ob D and y ∈ Ob C\ Ob D, Aut_{C}(x) acts transitively on Hom_{C}(x, y). When this is true, \underline{R} is relatively D-projective.

Proof. When y ∈ Ob D, ι ↓_{y} is connected because there is only one isomorphism class of maximal objects, of the form (y, g) where g ∈ Aut(y). Now we assume y ∈ Ob C\ Ob D. The objects of ι ↓_{y} are of the form (x, α), where x ∈ Ob D_{<y} and α ∈ Hom_{C}(x, y). If for every pair of objects x ∈ Ob D and y ∈ Ob C\ Ob D, Aut_{C}(x) acts transitively on Hom_{C}(x, y), then ι ↓_{y} has the same underlying poset as D_{<y} because (x, α) ≅ (x, β) for any two morphisms α, β ∈ Hom_{C}(x, y). This implies D_{<y} and ι ↓_{y} have the same number of connected components. Since Wess_{0}(C) ⊂ D, we know D_{<y} (hence ι ↓_{y}) is connected for any y ∈ Ob C by Corollary 3.4.6.

The second statement is a corollary of Proposition 3.4.1. □

The above proposition asserts that if C is a poset then Wess_{0}(C) is the smallest subposet, relative to which \underline{R} is projective.

Example 3.4.8. Suppose C is the following category

[figure: x; 1_{x}; f; τ; y; 1_{y}; α; β; z; 1_{z}; h] ,

with R arbitrary, Aut(x) acting trivially on Hom(x, y) = {τ}, Aut(z) interchanging α and β, and βτ ≠ ατ. From direct calculations, we can see that V\underline{R} = C. The following category is Wess(C)

[figure: x; 1_{x}; f; τ; y; 1_{y}] .

We can check \underline{R} is projective relative to the full subcategory D

[figure: x; 1_{x}; f; ατ; βτ; z; 1_{z}; h] ,

which is not convex and is the smallest full subcategory among all those relative to which \underline{R} is projective. Comparing these categories, we get Wess(C) ⊂ V\underline{R} = C, D ⊄ Wess(C) and Wess(C) ⊄ D. Note that Wess_{0}(C) = \{x\} is contained in V\underline{R}, Wess(C) and D.

3.5. Categories with subobjects. This part of the work grows out of our observation that if C is a finite poset then V\underline{R} is determined by Wess_{0}(C). In this section, we prove the same result for the categories with subobjects, which were introduced and studied by Oliver [32]. A category with subobjects (C, I) is a pair of categories I ⊂ C such that Ob I = Ob C, and such that the following two conditions are satisfied:

(1) | Hom_{I}(x, y)| ≤ 1 for any pair of objects x, y; and

(2) each morphism α ∈ Hom_{C}(x, y) can be written in a unique way as a composite α = α_{0} · f, where f ∈ Is_{C}(x, x′) for some x′, and α_{0} ∈ Hom_{I}(x′, y).

We note that Jackowski and Słomińska introduced the EI-categories with quotients in their paper [26], which is a concept dual to the EI-categories with subobjects in the sense that if (C, I) is a category with subobjects then (C^{op}, I^{op}) is a category with quotients, and vice versa. All results in this section have their counterparts for EI-categories with quotients.

Let (C, I) be a category with subobjects. We say (C, I) is a skeletal category with subobjects, if C is skeletal (see Mac Lane [30]). It is not true that if (C, I) is a category with subobjects, then the skeleton of C can be made into a category with subobjects. If we assume C is an EI-category, then we naturally have a definition of EI-categories with subobjects.

Lemma 3.5.1. Let (C, I) be a category with subobjects. The endomorphism group of each x ∈ Ob I is precisely {1_{x}}, and the only isomorphisms in Mor(I) are {1_{x}|x ∈ Ob I}. If furthermore C is EI, I is a poset.

Proof. It’s easy to see End_{I}(x) = Aut_{I}(x) = {1_{x}} since | End_{I}(x)| ≤ 1 and 1_{x} ∈ End_{I}(x). Now we show Is_{I}(x, y) = ∅ if x ≅ y in C and x ≠ y. If there were an α_{0} ∈ Is_{I}(x, y), then we would have two distinct factorizations α_{0} = 1_{y}α_{0} = α_{0}1_{x}, a contradiction to the definition of a category with subobjects.

If C is EI and Hom_{I}(x, y) ≠ ∅ for some x ≠ y, we show Hom_{I}(y, x) = ∅. This implies I is a poset. Indeed if there exists α ∈ Hom_{I}(x, y) and β ∈ Hom_{I}(y, x), then αβ = 1_{y} and βα = 1_{x} in I (and C). Hence α and β are isomorphisms, which is impossible because from above we know Is_{I}(x, y) = ∅ if x ≠ y. □

The above result implies that if (C, I) is a (not necessarily EI) category with subobjects, any non-empty set Hom_{I}(x, y) with x ≠ y will consist of a non-isomorphism.

Proposition 3.5.2. Let (C, I) be a category with subobjects. If every isomorphism class of objects in C is finite, then C is an EI-category.

Proof. Suppose there exists an α ∈ End_{C}(x)\ Aut_{C}(x) for some x ∈ Ob C. We show this assumption leads to a contradiction. By definition of a category with subobjects, α = α_{0}f for some α_{0} ∈ Hom_{I}(x′, x) and f ∈ Is_{C}(x, x′), where x′ ≅ x in C. We claim x′ ≠ x. If x = x′, then α_{0} ∈ Hom_{I}(x, x) = End_{I}(x) = {1_{x}} by preceding lemma. But then α = f is an isomorphism, a contradiction to our assumption. From x ≠ x′, we know α_{0} ∈ Hom_{I}(x′, x) is not an isomorphism.

Now since x′ ≅ x, there exists a β ∈ End_{C}(x′)\ Aut_{C}(x′), and β = β_{0}g for some β_{0} ∈ Hom_{I}(x′′, x′) and g ∈ Is_{C}(x′, x′′). As is shown in last paragraph, x′′ ≠ x′ and β_{0} is not an isomorphism. In fact x′′ cannot be x either, since if they were equal, we would have two morphisms in I: β_{0} : x → x′ and α_{0} : x′ → x, which implies α_{0}β_{0} = 1_{x} and β_{0}α_{0} = 1_{x′}. The two equalities assert that α_{0} and β_{0} are isomorphisms, inverse to each other, hence a contradiction to our assumptions on α_{0} and β_{0}. Thus any two of x, x′ and x′′ are not equal, and we can find a third non-isomorphism γ ∈ End_{C}(x′′)\ Aut_{C}(x′′) so that we can repeat what we’ve done for β ∈ End_{C}(x′)\ Aut_{C}(x′). Gradually, we’re going to produce an infinite list of isomorphic objects in C, x, x′, x′′, ⋯, while any two of them are not equal. This leads to a contradiction since we assume [x] is finite. Thus there is no such α ∈ End_{C}(x)\ Aut_{C}(x) for any x ∈ Ob C, and then End_{C}(x) = Aut_{C}(x) for all x ∈ Ob C, or C is EI. □

Some easy but useful facts about EI-categories with subobjects.

Lemma 3.5.3. Let (C, I) be an EI-category with subobjects. Then (1) all morphisms in C are monomorphisms;

(2) if C is a finite category with subobjects, then the number of objects z ∈ [x] for which Hom_{I}(z, y) ≠ ∅ equals | Hom_{C}(x, y)|/| Aut_{C}(x)|, for any y ≅ x̸ such that Hom_{C}(x, y) ≠ ∅; (3) if C is skeletal Aut_{C}(x) acts regularly on Hom_{C}(x, y), for any pair of objects

x, y ∈ Ob C.

Proof. Jackowski-Słomińska [26] proved the morphisms in any EI-category with quotients are epimorphism. Thus all the morphisms in (C, I) are monomorphisms.

By (1), Aut_{C}(x) acts freely on Hom_{C}(x, y). Statement (2) follows directly from counting the number of the Aut_{C}(x)-orbits on Hom_{C}(x, y).

Statement 3 is true by (2). □

Let (C, I) be a category with subobjects. Suppose D ⊂ C is a full subcategory satisfying the condition that if x ∈ Ob D then [x] ⊂ Ob D. Then we can naturally make D into a category with subobjects (D, I ∩ D), and can talk about full subcategories of a category with subobjects.

Lemma 3.5.4. Let (C, I) be an EI-category with subobjects, and ι : D ↪ C a full subcategory. Then for any y ∈ Ob C\ Ob D, the skeleton of ι ↓_{y} is isomorphic to a poset, which can be identified with the poset I_{<y} ∩ D. If D has a unique minimal object, then I_{<y} ∩ D, if not empty, has an initial object and is contractible. If C is skeletal, we have I_{<y} ∩ D ≅ P (D_{<y}).

Proof. If we fix an x ∈ Ob D_{<y} then every object (x′, α′) ∈ ι ↓_{y} with x′ ≅ x is isomorphic to some (x_{i}, α_{i}), where x_{i} ≅ x and α_{i} ∈ Hom_{I}(x_{i}, y). Since (x_{i}, α_{i}) ≅ (x_{j}, α_{j}) if and only if x_{i} = x_{j}, the skeleton of ι ↓_{y} is isomorphic to the full subcategory consisting of objects {(x, α)|x ∈ Ob D_{<y}, α ∈ Hom_{I}(x, y)}. Using the definition of a category with subobjects, it’s easy to see the full subcategory is a poset, and is isomorphic to I_{<y} ∩ D by our assumption.

When D has a unique minimal object, so does I_{<y} ∩ D. Hence it’s contractible because in the poset the unique minimal object is indeed an initial object. If C is skeletal, every isomorphism class of objects contains only one object. So the identification I_{<y} ∩ D ≅ P (D_{<y}) follows. □

Definition 3.5.5. Let (C, I) be a finite category with subobjects. Then we denote the two full subcategories of C which share the same object sets with Wess_{0}(I) and Wess(I), respectively, by C_{Wess_{0}}^{I} and C_{Wess}^{I}. Obviously C_{Wess_{0}}^{I} ⊂ C_{Wess}^{I}.

We comment here that in general Wess_{0}(C) ⊊ C_{Wess_{0}}^{I} and Wess(C) ⊊ C_{Wess}^{I}. But when (C, I) is skeletal, we do have Wess_{0}(C) = C_{Wess_{0}}^{I} and Wess(C) = C_{Wess}^{I}. Our next two propositions show the importance of these two new full subcategories of C.

Proposition 3.5.6. Let C be a finite category with subobjects. Then C_{Wess_{0}}^{I} is the smallest full subcategory among all full subcategories of C, relative to which \underline{R} is projective. Consequently, \underline{R} is projective relative to C_{Wess}^{I}, and V\underline{R} is the smallest convex subcategory (or ideal) of C that contains C_{Wess_{0}}^{I}.

Proof. By Proposition 3.4.1 and Lemma 3.5.4, we know \underline{R} is relatively C_{Wess_{0}}^{I}-projective. Hence if D contains C_{Wess}^{I} then \underline{R} is relatively D-projective. Now suppose D is a full subcategory and C_{Wess_{0}}^{I} ⊄ D. Then by Lemma 3.5.4 again, there exists an overcategory associated with ι : D ↪ C which is disconnected. Thus \underline{R} won’t be relatively D-projective, and we’ve proved that C_{Wess_{0}}^{I} is the smallest full subcategory among all full subcategories of C, relative to which \underline{R} is projective.. □

Our next result is a generalization of a Bouc’s theorem [6] on finite posets.

Proposition 3.5.7. Let (C, I) be a finite category with subobjects. Given a full subcategory D with C_{Wess}^{I} ⊂ D, the inclusions C_{Wess}^{I} ⊂ D ⊂ C induce homotopy equivalences.

Proof. Following Bouc’s idea, we’re going to use Quillen’s Theorem A to prove C_{Wess}^{I} ⊂ D induces an equivalence. Let’s take any object y ∈ Ob D. We want to show the overcategory category ι ↓_{y} is always contractible, where ι : C_{Wess}^{I} ↪ D is the inclusion. If y ∈ Ob C_{Wess}^{I}, then this category has a terminal object (y, 1_{y}), and hence is contractible. If y ∉ Ob C_{Wess}^{I}, we claim ι ↓_{y} is still contractible. By our assumption and Lemma 3.5.4, we know the skeleton of ι ↓_{y} is isomorphic to the poset (I ∩ D)_{<y} ∩ Wess(I), because (C_{Wess}^{I}, Wess(I)) ⊂ (D, I ∩ D) is a full subcategory. Since (I ∩ D)_{<y} ∩ Wess(I) = Wess(I)_{<y} = Wess(I_{<y}) ⊂ I_{<y} and I_{<y} is contractible by definition, Bouc’s original result on posets (see Bouc [6] or Benson [5] Proposition 6.6.5) implies Wess(I)_{<y} is contractible. Hence so is ι ↓_{y} and we’re done. □

The above two propositions result in the following reduction of higher limits for several types of categories with subobjects.

Corollary 3.5.8. Let (C, I) be a category with subobjects and D a full subcategory satisfying either one of the following conditions:

(1) C_{Wess}^{I} ⊂ D; or

(2) C_{Wess_{0}}^{I} ⊂ D and RC is a right flat RD-module,

then we have \underleftarrow{lim}_{C}^{∗} N ≅ \underleftarrow{lim}_{D}^{∗}(N ↓_{D}^{C}). If N = \underline{R}, this isomorphism gives rise to a ring isomorphism.

Proof. If the condition in (1) is satisfied, then every overcategory associated to the inclusion ι : D ↪ C is contractible as is shown in the proof of Proposition 3.5.7. So we can use Proposition 2.3.4 to establish the isomorphisms.

If the conditions in (2) are satisfied, then we can use Proposition 3.5.6 and Lemma 3.3.20 to get the isomorphism. □

4. Resolutions and their applications

Let C be a finite EI-category and R a field or a complete discrete valuation ring. Given an RC-module M, we study the structure of its projective cover and the minimal projective resolution. By a key property of the minimal projective resolution P → M → 0, the Ext group Ext_{RC}^{n}(M, N) is equal to Hom_{RC}(P_{n}, N) provided N is semisimple.

4.1. The minimal projective resolution of an RC-module. We describe the projective cover of an RC-module M in this section. The full subcategories C^{M} and C_{M}, for any RC-module M, are defined in 3.3.9.

Lemma 4.1.1. Suppose M is an RC-module. If P_{M} ≅ ⊕_{y,U}P_{y,U} is the projective cover of M, then every such y belongs to C_{M}, which means C_{P_{M}} ⊂ C_{M}. Thus C_{M/Rad M} ⊂ C_{M}.

If D is an ideal of C with D ∩ C_{M} ≠ ∅, then P_{M} ↓_{D}^{C} is the projective cover of M ↓_{D}^{C}. Particularly, if x is an M-minimal object, then P_{M}(x) is the projective cover of M(x).

Proof. Since M is generated by its values on objects in Ob C_{M}, so is P_{M} and hence every y belongs to C_{M}. We have C_{M/Rad M} ⊂ C_{M} because C_{M/Rad M} = C_{P_{M}}.

Now let D be an ideal in C with D∩C_{M} ≠ ∅. Then P_{M} ↓_{D}^{C} is a projective RD-module which admits a surjection onto M ↓_{D}^{C}. In order to show P_{M} ↓_{D}^{C} is the projective cover of M ↓_{D}^{C}, we need to prove (P_{M} ↓_{D}^{C})/Rad(P_{M} ↓_{D}^{C}) ≅ (M ↓_{D}^{C})/Rad(M ↓_{D}^{C}).

Since D is an ideal of C, for any RC-module N we have N ↓_{D}^{C} = 1_{D} · N as an RDmodule. If P is a projective RC-module, we can easily see Rad(P) ↓_{D}^{C} ≅ Rad(P ↓_{D}^{C}) from the structure theorem of projective modules. Particularly, we get Rad(RC) ↓_{D}^{C} ≅ Rad(RD). Now we claim Rad(N) ↓_{D}^{C} ≅ Rad(N ↓_{D}^{C}) is true for any RC-module N. In fact it follows from some simple calculations: (Rad N) ↓_{D}^{C} = 1_{D} · Rad N = 1_{D} · Rad(RC) · N = Rad(RD) · N = Rad(RD) · 1_{D} · N = Rad(RD) · N ↓_{D}^{C} = Rad(N ↓_{D}^{C}). We have the following short exact sequences

0 → Rad N → N → N/Rad N → 0,

and

0 → Rad(N ↓_{D}^{C}) → N ↓_{D}^{C} → (N ↓_{D}^{C})/Rad(N ↓_{D}^{C}) → 0,

of RC-modules and RD-modules, respectively. The first short exact sequence restricts to a short exact sequence of RD-modules

0 → (Rad N) ↓_{D}^{C} → N ↓_{D}^{C} → (N/Rad N) ↓_{D}^{C} → 0.

Since (Rad N) ↓_{D}^{C} ≅ Rad(N ↓_{D}^{C}), comparing the second and the third short exact sequences we obtain (N ↓_{D}^{C})/Rad(N ↓_{D}^{C}) ≅ (N/Rad N) ↓_{D}^{C} for any RC-module N. If we choose N to be M and P_{M}, respectively, we have the following isomorphisms (M ↓_{D}^{C})/Rad(M ↓_{D}^{C}) ≅ (M/Rad M) ↓_{D}^{C} and (P_{M} ↓_{D}^{C})/Rad(P_{M} ↓_{D}^{C}) ≅ (P_{M}/Rad P_{M}) ↓_{D}^{C}, which imply (P_{M} ↓_{D}^{C})/Rad(P_{M} ↓_{D}^{C}) ≅ (M ↓_{D}^{C})/Rad(M ↓_{D}^{C}) because (M/Rad M) ↓_{D}^{C} ≅ (P_{M}/Rad P_{M}) ↓_{D}^{C}.

Finally if x is an M-minimal object, then we take D = C_{≤x}, and the last statement follows. □

With the above result we can go on to describe the minimal projective resolution of an arbitrary RC-module. Given an RC-module M and its projective cover P_{M}, from the previous lemma we know P_{M} ≅ ⊕_{y,U}P_{y,U} with y ∈ Ob C_{M}. When we look at the minimal resolution of M

P_{M} : ⋯ → P_{n} → ⋯ → P_{1} → P_{0} → M → 0,

we know P_{0} is simply P_{M} and P_{1} is the projective cover of K_{0}, the kernel of the map P_{0} = P_{M} → M. Since C_{K_{0}} ⊂ C_{P_{0}} ⊂ C_{M}, we have C_{P_{1}} ⊂ C_{M} too by the lemma. Hence we conclude the following result on the minimal projective resolution by repeating the same argument for every P_{n}.

Corollary 4.1.2. Let M be an RC-module and P_{M} its minimal projective resolution. Then C_{P_{n}} ⊂ C_{M} for each module P_{n} in the projective resolution. Suppose D is an ideal of C. Then P_{M} ↓_{D}^{C} is the minimal projective resolution of M ↓_{D}^{C}.

If M is an atomic module whose support is {[x]} for some x ∈ Ob C, then in the minimal projective resolution of M, P_{1} ≅ ⊕_{y,U}P_{y,U} with y ≅ x or Irr_{C}(x, y) ≠ ∅ if y ≅ x̸.

Proof. The first statement is a direct consequence of the preceding lemma and the discussion after it, and so we only prove the second part here. Indeed, it’s easier to see this using a larger projective resolution of M. Suppose P̃_{M} is constructed in a way such that P̃_{0} = (RC · 1_{x})^{i} for some positive integer i, and P̃_{1} is the projective cover of the kernel K̃_{0} of the map P̃_{0} → M → 0. Then K̃_{0} contains all non-isomorphisms in P̃_{0}, and thus Rad(RC) · K̃_{0} = Rad(K̃_{0}) contains all reducible morphisms in P̃_{0}. This implies K̃_{0}/Rad K̃_{0} is isomorphic to a direct sum of the form ⊕_{y,U}S_{y,U}, where y ≅ x or y ≅ x̸ and Irr_{C}(x, y) ≠ ∅. Thus P̃_{1} is isomorphic to ⊕_{y,U}P_{y,U}, where y ≅ x or y ≅ x̸ and Irr_{C}(x, y) ≠ ∅. Since P_{M} is the minimal projective resolution of M, P_{M} must be isomorphic to a direct summand of P̃_{M}. In particular, P_{1} is isomorphic to a direct summand of P̃_{1}. Hence the statement follows. □

Let M be an RC-module. The support of M is defined to be the full subcategory of C consisting of objects x such that M(x) ≠ 0.

Proposition 4.1.3. Let M be an RC-module which is relatively D-projective for a full subcategory D. Then M/Rad(M) has a support contained in D. As a consequence, the projective cover P_{M} is relatively D-projective.

Proof. Since M is D-projective, for any y ∈ Ob C\ Ob D, we have

M(y) = ∑_{x∈Ob D_{≤y}} RHom_{C}(x, y) · M(x) ⊂ Rad(RC) · M = Rad(M).

Hence M/Rad(M) is non-zero only on some objects in D. This implies P_{M} ≅ ⊕_{y,U}P_{y,U} for some P_{y,U} with y ∈ Ob D, and then P_{M} is D-projective because D contains the vertex of every P_{y,U}. □

Note that if M is indecomposable, then M and P_{M} usually have different vertices. One can consider P_{x,V} → S_{x,V} when they aren’t equal. From the same example we can see the above proposition cannot be strengthened: M being D-projective doesn’t imply M/Rad M is D-projective. Furthermore the kernel of the surjection P_{M} → M does not have to be D-projective if M is. Thus given a D-projective module M and its minimal resolution P_{M} → M → 0, usually we cannot expect any P_{n}, n ≥ 2 to be D-projective, except D = C_{M}.

4.2. Applications.

Proposition 4.2.1. Given two RC-modules M and N, we have

Ext_{RC}^{∗}(M, N) = Ext_{RC_{M}^{N}}^{∗} (M ↓_{C_{M}^{N}}^{C} , N ↓_{C_{M}^{N}}^{C}),

where C_{M}^{N} = C_{M} ∩ C^{N}.

Proof. We take the minimal RC-resolution of M

P_{M} : ⋯ → P_{1} → P_{0} → M → 0.

It is supported on C_{M} hence is an RC_{M}-resolution of M ↓_{C_{M}}^{C}. It’s obvious that

Hom_{RC}(P_{n}, N) = Hom_{RC_{M}^{N}} (P_{n} ↓_{C_{M}^{N}}^{C} , N ↓_{C_{M}^{N}}^{C}),

for all n, which furthermore give rise to an isomorphism of cochain complexes

{Hom_{RC}(P_{M}, N)} ≅ {Hom_{RC_{M}^{N}} (P_{M} ↓_{C_{M}^{N}}^{C} , N ↓_{C_{M}^{N}}^{C})}.

If we can show P_{M} ↓_{C_{M}^{N}}^{C} is still a projective resolution of M ↓_{C_{M}^{N}}^{C} as an RC_{M}^{N}-module then we’re done. But this comes from Lemma 4.1.1 since C_{M}^{N} is an ideal in C_{M}. □

Corollary 4.2.2. Let C_{x}^{y} = C_{≥x} ∩ C_{≤y}. Then

Ext_{RC}^{∗}(S_{x,V} , S_{y,W}) ≅ Ext_{RC_{x}^{y}}^{∗}(S_{x,V} , S_{y,W}).

In particular we have Ext_{RC}^{∗}(S_{x,V} , S_{x,W}) ≅ Ext_{R Aut(x)}^{∗}(V, W).

Note that in Corollary 4.2.2 if Hom(x, y) = ∅ then C_{x}^{y} = ∅ hence Ext_{RC}^{∗}(S_{x,V} , S_{y,W}) vanish. Let A be a finite-dimensional algebra and M, N two A-modules. There is a standard result by S. Eilenberg ([13] Proposition 10) asserting that if N is semisimple, then Ext_{A}^{n}(M, N) ≅ Hom_{A}(P_{n}, N), where ⋯ → P_{n} → ⋯ → P_{0} → M → 0 is the minimal projective resolution of M.

Proposition 4.2.3. Let S_{x,V} and S_{y,W} be two simple RC-modules. If Irr_{C}(x, y) = ∅, then Ext_{RC}^{1}(S_{x,V} , S_{y,W}) = 0.

Proof. Suppose ⋯ → P_{1} → P_{0} → S_{x,V} → 0 is the minimal projective resolution of S_{x,V}. Then since S_{y,W} is simple, we have Ext_{RC}^{1}(S_{x,V} , S_{y,W}) ≅ Hom(P_{1}, S_{y,W}). But P_{1} ≅ ⊕P_{z,V} for z ≅ x or z ∈ Ob C_{≥x} such that Irr(x, z) ≠ ∅. By our assumption and Corollary 4.1.2 the degree one Ext group has to be zero because any indecomposable projective module of the form P_{y,W} cannot be a direct summand of P_{1}. □

Recall that there is a poset P (C) associated to every EI-category C. When C is finite we call the maximal length of chains of non-isomorphisms in P (C) the length of C, denoted by l(C). Recall that the global dimension of a finite dimensional algebra A is the projective dimension of A/Rad(A).

Theorem 4.2.4. Let C be a finite EI-category. Then RC has finite global dimension if and only if for all x ∈ Ob C, | Aut(x)|^{−1} ∈ R. In fact for any RC-module M, proj.dim(M) ≤ l(C_{M}). Particularly gl.dim(RC) ≤ l(C).

Proof. Since C is finite, by Theorem 3.1.2 RC has finitely many simple modules. Note that RC having finite global dimension is equivalent to the statement that every simple RC-module has a finite projective resolution.

Suppose | Aut(x)|^{−1} ∈ R for all x ∈ Ob C. We do an induction on l(C), the length of C. Fix a simple S_{y,W}. It has a minimal projective resolution P written as

⋯ → P_{n} → ⋯ → P_{1} → P_{0} → S_{y,W} → 0.

Since P(y) → S_{y,W} (y) = W → 0 is the minimal projective resolution of W and | Aut(y)|^{−1} ∈ R, we have P_{0}(y) ≅ W since R Aut(y) is semi-simple. It implies that all P_{n}, n > 0 are supported on C_{>y} for P_{n}(y) = 0 when n > 0. Therefore if we take K_{0} as the kernel of the map P_{0} → S_{y,W}, K_{0} must be an RC-module supported on C_{>y}, and

⋯ → P_{n} → ⋯ P_{1} → K_{0} → 0

becomes a minimal projective resolution of K_{0}. Because C_{>y} has smaller length than C, the resolution of K_{0} is finite. So is P, the resolution of S_{y,W}.

On the other hand if any S_{y,W} has a finite projective resolution P, then P(y) is a finite resolution of the simple R Aut(y)-module S_{y,W} (y) = W. Since projective R Aut(y)-modules are the same as injective R Aut(x)-modules, the finite exact sequence P(y) → W → 0 splits. Hence each and every W is projective which means R Aut(y) is semi-simple, or equivalently | Aut(y)|^{−1} ∈ R.

Under the circumstance if we consider an arbitrary RC-module M, we can show proj.dim(M) ≤ l(C_{M}). Let

P_{M} : 0 → P_{n} → ⋯ → P_{1} → P_{0} → M → 0

be the minimal projective resolution of M. Then P_{0} is supported on C_{M}. Just like what we’ve done above, the kernel of P_{0} → M is supported on C_{M}\{M-minimal objects} ⊂ C_{M} so the support of P_{1} is contained in C_{M}\{M-minimal objects} as well. Inductively the size of C_{P_{i}} decreases strictly when i grows bigger and bigger. This means proj.dim(M) ≤ l(C_{M}). □

Abusing the notation, when R is understood we say C has finite global dimension if RC has finite global dimension.

Corollary 4.2.5. Let C be a finite EI-category with the property that every morphism is an epimorphism. Then RC has finite global dimension if and only if \underline{R} has finite projective dimension.

Proof. If RC has finite global dimension then \underline{R} certainly has a finite projective resolution. On the other hand if \underline{R} has a finite minimal projective resolution P → \underline{R} → 0 then by our assumption we can show, for any x ∈ Ob C, P(x) → R → 0 is a finite projective resolution for the R Aut_{C}(x)-module R. The reason is that we can produce a (larger) projective resolution P̃ of \underline{R} such that every P̃_{n} is a direct sum of some representable functors of the form R Hom_{C}(z, −). Since Aut_{C}(x) acts freely on R Hom_{C}(z, x) whenever Hom_{C}(z, x) ≠ ∅, any non-zero P̃_{n}(x) is a projective R Aut_{C}(x)- module, and so is P_{n}(x) as a direct summand of P̃_{n}(x). In the end the finite projective resolution of the R Aut_{C}(x)-module R is split for all x ∈ Ob C so R has to be projective, or equivalently | Aut_{C}(x)|^{−1} ∈ R for all x. It implies RC has finite global dimension by the preceding proposition. □

We note that the finite categories with quotients (see Section 3.6) satisfy the condition in Corollary 4.2.5 that every morphism is an epimorphism.

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School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Current Address:

Department of Mathematical Sciences, University of Aberdeen, Meston Building, Aberdeen AB24 3UE, Scotland, UK

 E-mail address: fxu@maths.abdn.ac.uk