New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math.21(2015) 369–382.

Noetherian property of infinite EI categories

Wee Liang Gan and Liping Li

Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result to the ab- stract setting of an infinite EI category satisfying certain combinatorial conditions.

Contents

1. Introduction 369

2. Generalities on EI categories 371

2.1. Quiver underlying an EI category 371

2.2. Finiteness conditions 371

2.3. Modules 372

2.4. Finitely generated modules 373

3. Noetherian property of finitely generated modules 373

3.1. Transitivity condition 374

3.2. Bijectivity condition 374

3.3. Main result 375

4. Proof of main result 378

4.1. Special case 378

4.2. Proof of Theorem 3.7 380

5. Further remarks 380

References 381

1. Introduction

Let FI be the category whose objects are finite sets and morphisms are injections. If an object of FI is a set withielements, then its automorphism group is isomorphic to the symmetric group Si; thus, a module of the cat- egory FI gives rise to a sequence of representations V_i of S_i. Our present

Received April 6, 2015.

2010Mathematics Subject Classification. 16P40.

Key words and phrases. EI categories, Noetherian property, FI-modules.

ISSN 1076-9803/2015

369

paper is motivated by the theory of FI-modules developed by Church, Ellen- berg, and Farb in [1] in order to study stability phenomenon in sequences of representations of the symmetric groups; see also [3]. The following theorem plays a fundamental role in the theory of FI-modules in [1].

Theorem 1.1. Any finitely generated FI-module over a field k of charac- teristic0 is Noetherian.

Theorem1.1 was proved by Church, Ellenberg, and Farb in [1, Theorem 1.3]. Subsequently, a generalization of Theorem1.1to any Noetherian ringk was given by Church, Ellenberg, Farb, and Nagpal in [2, Theorem A]. Using their result, Wilson [10, Theorem 4.21] deduced analogous theorems for the categories FIBC and FID associated to the Weyl groups of type B/C and D, respectively. The categories FI, FI_BC, and FI_D are examples ofinfinite EI categories, in the sense of the following definition.

Definition 1.2. An EI category is a small category in which every endo- morphism is an isomorphism. An EI category is said to be finite (resp.

infinite) if its set of morphisms is finite (resp. infinite).

L¨uck [5, Lemma 16.10] proved a version of Theorem 1.1 for finite EI categories whenkis any Noetherian ring. However, there is no such general result for EI categories which are infinite. The goal of our present paper is to find general sufficiency conditions on an infinite EI category so that the analog of Theorem 1.1 holds. After recalling some basic facts on EI categories in Section 2, we state our main result in Section 3 and give the proof in Section4. We make some further remarks in Section 5.

Our main result (Theorem 3.7) gives a generalization of Theorem 1.1 to the abstract setting of an infinite EI category satisfying certain simple combinatorial conditions. It is applicable to the categories FI, FI_BC, and FID. It is also applicable to the category VI of finite dimensional Fq-vector spaces and linear injections, whereFqdenotes the finite field withqelements.

In contrast to [1], the symmetric groups do not play any special role in our paper, and we do not use any results from the representation theory of symmetric groups in our proofs. We hope to extend the theory of FI- modules and representation stability to our framework. It should also be interesting to find conditions under which our main result extends to an arbitrary Noetherian ring k. The proof of [2, Theorem A] makes use of [2, Proposition 2.12] which does not hold when the category FI is replaced by the category VI.

Remark 1.3. A generalization of Theorem1.1 was proved by Snowden [9, Theorem 2.3] in the language of twisted commutative algebras; see [7, Propo- sition 1.3.5]. His result has little overlap with our Theorem 3.7. Indeed, to interpret the category of modules of an EI category C of type A∞ (in the sense of Definition2.2) as the category of modules of a twisted commutative algebra finitely generated in order 1, the automorphism group of any object iof C must necessarily be the symmetric group Si.

Remark 1.4. After this paper was written, we were informed by Putman and Sam that they have a very recent preprint [6] which proved the analog of Theorem 1.1when k is any Noetherian ring for several examples of linear- algebraic type categories such as the category VI. We were also informed by Sam and Snowden that they have a very recent preprint [8] which gives combinatorial criteria for representations of categories (not necessarily EI) to be Noetherian. Their combinatorial criteria are very different from our conditions; in particular, it does not seem that Theorem3.7will follow from their results.

2. Generalities on EI categories

LetC be an EI category. We shall assume throughout this paper that C is skeletal.

2.1. Quiver underlying an EI category. We denote by Ob(C) the set of objects ofC. For anyi, j∈Ob(C), we writeC(i, j) for the set of morphisms from itoj.

Definition 2.1. A morphismα ofC is called unfactorizable if:

• αis not an isomorphism;

• whenever α=β₁β₂ where β₁ and β₂ are morphisms of C, either β₁ orβ2 is an isomorphism.

We define a quiverQassociated to C as follows. The set of vertices ofQ is Ob(C). The number of arrows from a vertex i to a vertexj is 1 if there exists an unfactorizable morphism from ito j; it is 0 otherwise. We call Q the quiver underlying the EI category C.

Definition 2.2. LetZ+ be the set of nonnegative integers. We say thatC is an EI category of type A∞ if Ob(C) =Z+ and the quiver underlying C is:

0−→1−→2−→ · · ·.

2.2. Finiteness conditions. Recall that the EI categoryC is finite (resp.

infinite) if the set of morphisms of C is finite (resp. infinite).

Definition 2.3. We say that C is locally finite if C(i, j) is a finite set for all i, j∈Ob(C).

There is a partial order≤on Ob(C) defined byi≤jifC(i, j) is nonempty.

Definition 2.4. We say thatC is strongly locally finite if it is locally finite and for everyi, j∈Ob(C) withi≤j, there are only finitely manyl∈Ob(C) such thati≤l≤j.

Observe that C is strongly locally finite if and only if for every i, j ∈ Ob(C) with i ≤ j, the full subcategory of C generated by all objects l satisfying i≤l≤j is a finite EI category. Any locally finite EI category of type A∞ is strongly locally finite.

Remark 2.5. IfCis strongly locally finite, then any morphismαofCwhich is not an isomorphism can be written as α =β1· · ·βr whereβ1, . . . , βr are unfactorizable morphisms; moreover, in this case, for anyi, j∈Ob(C), one hasi≤j if and only if there exists a directed path inQ fromitoj.

2.3. Modules. Letk be a commutative ring.

For any setX, we shall write kX for the free k-module with basis X. If Gis a group, then kGis the group algebra ofGoverk.

Definition 2.6. The category algebra of C over k is the k-algebra C(k) defined by

C(k) = M

i,j∈Ob(C)

kC(i, j).

If α ∈ C(i, j) and β ∈ C(l, m), their product in C(k) is defined to be the compositionαβ ifi=m; it is defined to be 0 ifi6=m.

For any i∈ Ob(C), we denote by G_i the group C(i, i), and write 1_i for the identity element of the groupG_i.

Definition 2.7. A C(k)-moduleV is called akC-module if it isgraded, i.e.

ifV is equal as ak-module to the direct sum

V = M

i∈Ob(C)

Vi,

whereVi=1iV for all i∈Ob(C).

Equivalently, one can define akC-module to be a covariant functor from C to the category ofk-modules. We shall write Mod_k(C) for the category of kC-modules; in particular, for a group G, we write Modk(G) for the category of kG-modules. The category Mod_k(C) is an abelian category.

Definition 2.8. LetV be akC-module. An elementv∈V ishomogeneous if there exists i ∈ Ob(C) such that v ∈ Vi; we call i the degree of v, and denote it by degv.

For anyi∈Ob(C), we have the restriction functor Res : Mod_k(C)−→Mod_k(G_i), V 7→V_i.

The restriction functor is exact and it has a left adjoint, the induction functor Ind : Mod_k(G_i)−→Mod_k(C), U 7→ M

j∈Ob(C)

kC(i, j)⊗_kGi U.

Notation 2.9. For anyi∈Ob(C), letM(i) = Ind(kG_i).

Thus,

M(i) = M

j∈Ob(C)

M(i)_j, where M(i)_j =kC(i, j).

In particular, note thatM(i) =C(k)1i.

Remark 2.10. It is easy to see that M(i) is a projective kC-module. In- deed, since Ind has an exact right adjoint functor, it takes projectives to projectives.

2.4. Finitely generated modules. IfV is akC-module andsis an ele- ment ofV homogeneous of degreei, we have a homomorphism

πs:M(i)−→V

defined by πs(α) =αsfor allα ∈C(i, j), for allj ∈Ob(C). IfS is a subset ofV and all elements ofS are homogeneous, then we have a homomorphism

(2.11) πS :M

s∈S

M(degs)−→V

whose restriction to the component corresponding to sis πs. The image of π_S is the kC-submodule of V generated by S.

Notation 2.12. IfV is akC-module andSis a set of homogeneous elements of V, let

M(S) =M

s∈S

M(degs).

Definition 2.13. A set S is called a set of generators of a kC-module V if S ⊂ V and the only kC-submodule of V containing S is V itself. A kC-moduleV isfinitely generated if it has a finite set of generators.

A setSof generators of akC-moduleV is said to be a set of homogeneous generators if all the elements of S are homogeneous. Clearly, V is finitely generated if and only if it has a finite set of homogeneous generators. Hence, we have:

Lemma 2.14. AkC-moduleV is finitely generated if and only if there exists a finite set S of homogeneous elements of V such that the homomorphism πS :M(S)→V of (2.11) is surjective.

Note that ifC is locally finite and V is a finitely generated kC-module, thenV_i is finite dimensional for alli∈Ob(C).

Definition 2.15. AkC-moduleV isNoetherian if everykC-submodule of V is finitely generated.

Equivalently, akC-module is Noetherian if it satisfies the ascending chain condition on itskC-submodules.

3. Noetherian property of finitely generated modules

We assume in this section that C is a locally finite EI category of type A∞.

3.1. Transitivity condition. We say thatCsatisfies thetransitivity con- dition if for each i∈Z+, the action of Gi+1 onC(i, i+ 1) is transitive.

Lemma 3.1. If C satisfies the transitivity condition, then for any i, j∈Z+

with i < j, the Gj action on C(i, j) is transitive.

Proof. Letα, α⁰∈C(i, j). SinceCis of type A∞, there exists two sequences of morphisms

αr, α_r⁰ ∈C(i+r−1, i+r) for r= 1, . . . , j−i,

such that α = αj−i· · ·α₁ and α⁰ = α_j−i⁰ · · ·α⁰₁. There exists g₁ ∈ G_i+1 such that g1α1 =α⁰₁. We find, inductively, an element gr ∈Gi+r such that g_rα_r=α⁰_rgr−1 forr = 2, . . . , j−i. Then one has gj−iα=α⁰. 3.2. Bijectivity condition. Suppose thatCsatisfies the transitivity con- dition.

Notation 3.2. For eachi∈Z+, we choose and fix a morphism αi ∈C(i, i+ 1).

Moreover, for anyi, j ∈Z+ withi < j, let

αi,j =αj−1· · ·αi+1αi ∈C(i, j).

LetHi,j = Stab_Gj(αi,j), and define the map

(3.3) mi,j :C(i, j)−→C(i, j+ 1), γ 7→αjγ.

Suppose h ∈ H_i,j. By the transitivity condition, there exists g ∈ G_j+1 such thatgαj =αjh. We have

gαi,j+1 =gαjαi,j =αjhαi,j =αjαi,j =αi,j+1, and henceg∈Hi,j+1. Now, for any γ∈C(i, j),

m_i,j(hγ) =α_jhγ=gα_jγ =gm_i,j(γ).

It follows that m_i,j maps eachH_i,j-orbit O inC(i, j) into a H_i,j+1-orbit in C(i, j+ 1); in particular, we get a map on the set of orbits:

(3.4) µi,j :Hi,j\C(i, j)−→Hi,j+1\C(i, j+ 1), whereµi,j(O) is theHi,j+1-orbit that containsmi,j(O).

We say thatCsatisfies thebijectivity conditionif, for eachi∈Z+, the map µi,j is bijective for allj sufficiently large. It is clear that this is independent of the choice of the mapsα_i.

Remark 3.5. For i < j, we have a bijection

Gj/Hi,j −→C(i, j), gHi,j 7→gαi,j.

Identifying C(i, j) withG_j/H_i,j via this bijection, the maps (3.3) and (3.4) are, respectively,

m⁰_i,j :Gj/Hi,j →Gj+1/Hi,j+1, gHi,j 7→uHi,j+1,

and

µ⁰_i,j :H_i,j\G_i,j/H_i,j →H_i,j+1\G_i,j+1/H_i,j+1, H_i,jgH_i,j 7→H_i,j+1uH_i,j+1, where u ∈ Gj+1 is any element such that αjg =uαj. The bijectivity con- dition is equivalent to the condition that, for each i∈ Z+, the map µ⁰_i,j is bijective for allj sufficiently large. (The setH\G/H whereH is a subgroup of a finite groupGappear naturally in the theory of Hecke algebras, see [4].) Lemma 3.6. Assume thatC satisfies the transitivity and bijectivity condi- tions. Then for each i∈Z+, the map mi,j is injective for all j sufficiently large.

Proof. Let j be an integer such that µi,j is injective. Suppose γ1, γ2 ∈ C(i, j) and m_i,j(γ₁) = m_i,j(γ₂). By Lemma 3.1, there exists g₁, g₂ ∈ G_j such thatγ1 =g1αi,j and γ2 =g2αi,j. There also existsg∈Gj+1 such that gαj =αjg1. One has

gαjαi,j =αjg1αi,j =αjγ1=αjγ2 =αjg2αi,j =gαjg⁻¹₁ g2αi,j,

and hence m_i,j(α_i,j) =m_i,j(g₁⁻¹g₂α_i,j). It follows, by the injectivity ofµ_i,j, thatα_i,j andg₁⁻¹g₂α_i,jare in the sameH_i,j-orbit. Thus, there existsh∈H_i,j such thathα_i,j =g₁⁻¹g₂α_i,j. Buthα_i,j =α_i,j, soα_i,j =g₁⁻¹g₂α_i,j, and hence

γ₁ =g₁α_i,j =g₂α_i,j =γ₂.

3.3. Main result. We shall give the proof of the following theorem in the next section.

Theorem 3.7. Assume thatC satisfies the transitivity and bijectivity con- ditions, and k is a field of characteristic 0. Let V be a finitely generated kC-module. Then V is a NoetheriankC-module.

Let us give some examples of categories C with Ob(C) =Z+, where the conditions of the theorem are satisfied. For anyi∈Z+, we shall denote by [i] the set {r∈Z|1≤r ≤i}; in particular, [0] =∅.

Example 3.8. Let Γ be a finite group. We define the category C = FIΓ

as follows. For any i, j ∈ Z+, let C(i, j) be the set of all pairs (f, c) where f : [i] → [j] is an injection, and c : [i] → Γ is an arbitrary map. The composition of (f₁, c₁)∈C(j, l) and (f₂, c₂)∈C(i, j) is defined by

(f₁, c₁)(f₂, c₂) = (f₃, c₃) where

f3(r) =f1(f2(r)), c3(r) =c1(f2(r))c2(r), for all r ∈[i].

It is easy to see that C is a locally finite EI category of type A∞ with isomorphisms

G_i−→^∼ S_inΓⁱ, (f, c)7→(f, c(1), . . . , c(i)),

where Si denotes the symmetric group on [i]. We chooseαi to be the pair (fi, ci) where fi is the natural inclusion [i],→ [i+ 1] andci : [i]→Γ is the

constant map whose image is the identity elementeof Γ. Clearly,Csatisfies the transitivity condition. Observe thatHi,j is the subgroup ofGj consisting of all pairs (f, c) satisfying f(r) = r and c(r) = e for all r ∈ [i]. For any (f, c)∈C(i, j), denote byf⁻¹[i] the set of r ∈[i] such that f(r)∈[i]. Two pairs (f, c), (g, d)∈C(i, j) are in the sameHi,j-orbit if and only if

f⁻¹[i] =g⁻¹[i], f |_f⁻¹_[i]=g|_g⁻¹_[i], and c|_f⁻¹_[i]=d|_g⁻¹_[i]. Let G⁰_i denote the set of all triples (U, a, b) whereU ⊂[i], a:U → [i] is an injection, andb:U →Γ is an arbitrary map. We have an injective map

θi,j :Hi,j\C(i, j)−→G⁰_i, Hi,j(f, c)7→(f⁻¹[i], f |_f⁻¹_[i], c|_f⁻¹_[i]), Whenj≥2i, the mapθi,j is surjective. One hasθi,j+1µi,j =θi,j. Therefore, µ_i,j is bijective whenj ≥2i, and henceC satisfies the bijectivity condition.

Remark 3.9. When Γ is the trivial group, the category FIΓis equivalent to the category FI; in this case, the observations in Example3.8are essentially contained in the proof of [4, Lemma 3.1]. When Γ is the cyclic group of order 2, the category FIΓ is equivalent to the category FIBC in [10]. One can similarly show that the category FI_D in [10] satisfies the conditions of Theorem3.7.

Example 3.10. Let Fq be the finite field with q elements. We define the categoryC as follows. For anyi, j∈Z+, letC(i, j) be the set of all injective linear maps fromFⁱqtoF^jq. The groupGiis the general linear groupGLi(Fq).

We chooseαi to be the natural inclusionFⁱ_q ,→Fⁱ⁺¹_q . Clearly, C is a locally finite EI category of type A∞ satisfying the transitivity condition. Let us check the bijectivity condition.

The mapα_i,j is the natural inclusion Fⁱq ,→ F^jq, and H_i,j is the subgroup of GL_j(Fq) consisting of all matrices of form:

I_i X 0 Y

whereI_idenotes thei-by-iidentity matrix,X is anyi-by-(j−i) matrix, and Y is any invertible (j−i)-by-(j−i) matrix. We write the elements ofC(i, j) as j-by-i matrices. For any A ∈ C(i, j), we denote by U_A the subspace of Fⁱ_q spanned by the lastj−irows of A, and define f_A:Fⁱ_q →Fⁱ_q/U_A to be the linear map that sends ther-th standard basis vectore_r ofFⁱqto ther-th row of A modulo U_A, for each r ∈ [i]. Two elements A, B ∈ C(i, j) are in the sameHi,j-orbit if and only if UA=UB and fA=fB. LetG⁰_i be the set of all pairs (U, f) where U is any subspace of Fⁱq and f : Fⁱq → Fⁱq/U is a surjective linear map. We have an injective map

θ_i,j :H_i,j\C(i, j)−→G⁰_i, H_i,jA7→(U_A, f_A).

Whenj≥2i, the mapθi,j is surjective. One hasθi,j+1µi,j =θi,j. Therefore, µ_i,j is bijective whenj≥2i.

Remark 3.11. The category C of Example 3.10 is equivalent to the cat- egory VI of finite dimensional Fq-vector spaces and linear injections. In Example3.12 below, we consider a variant VIC whose objects are also the finite dimensionalFq-vector spaces but the morphisms are pairs (f, Z) where f is an injective linear map andZ is a complementary subspace to the image of f. The Noetherian property of VI and VIC over a Noetherian ring kare proved by Putman and Sam in [6]; for VI, it is also proved by Sam and Snowden in [8].

Example 3.12. As before, let Fq be the finite field with q elements. We define the category C as follows. For any i, j ∈ Z+, let C(i, j) be the set of all pairs (f, Z) where f is an injective linear map from Fⁱq to F^jq and Z ⊂F^jq is a subspace complementary to the image off. The composition of morphisms is defined by

(f, Z)◦(f⁰, Z⁰) = (f ◦f⁰, Z +f(Z⁰)).

The group G_i is the general linear group GL_i(Fq). Clearly, C is a locally finite EI category of type A∞satisfying the transitivity condition. We choose α_i to be the pair (f_i, Z_i) where f_i is the natural inclusion Fⁱq ,→ Fⁱ⁺¹q and Z_i ={(0, . . . ,0, z) |z∈ Fq}, so H_i,j is the subgroup of GL_j(Fq) consisting of all matrices of the form:

I_i 0 0 Y

where Y ∈GLj−i(Fq).

The map

µ⁰_i,j :Hi,j\GL_j(Fq)/Hi,j −→Hi,j+1\GL_j+1(Fq)/Hi,j+1

is induced by the standard inclusion of GL_j(Fq) into GL_j+1(Fq), see Re- mark 3.5. To check the bijectivity condition, it suffices to show that for each i∈Z+, the mapsµ⁰_i,j are surjective for allj sufficiently large.

Let us show thatµ⁰_i,jis surjective whenj>3i. LetX∈GLj+1(Fq) where j>3i. We need to show that for someg, g⁰∈Hi,j+1, the only nonzero entry in the last row or last column ofgXg⁰ is the entry in position (j+ 1, j+ 1) and it is equal to 1. First, it is easy to see that for suitable choices of h₁, h⁰₁ ∈H_i,j+1, the entries ofh₁Xh⁰₁ in positions (r, s) are 0 if r > 2i and s6i, or ifr 6 iand s >2i. Indeed, we may first perform row operations on the last j−irows of X to change the entries in positions (r, s) to 0 for r >2iands6i, then perform column operations on the last j−icolumns of the resulting matrix to change the entries in positions (r, s) to 0 forr6i and s > 2i. Set Y = h₁Xh⁰₁. Since Y has rank j + 1, and j+ 1 > 3i, there exists r, s > 2i such that the entry in position (r, s) of Y is nonzero.

Swapping row r with rowj+ 1, and then swapping columns with column j+ 1, we may assume that the entry ofY in position (j+ 1, j+ 1) is nonzero, and by rescaling the entries in the last row we may assume that this entry

is 1. It is now easy to see that for someh₂, h⁰₂∈H_i,j, the matrixh₂Y h⁰₂ has the required form.

4. Proof of main result

Recall thatC denotes a locally finite EI category of type A∞. We assume in this section thatCsatisfies the transitivity and bijectivity conditions, and kis a field of characteristic 0.

4.1. Special case. Leti∈Z+. We first prove Theorem3.7 in the special case when V is M(i). By Lemma 3.6, there exists N ∈ Z+ such that the maps mi,j of (3.3) are injective and the maps µi,j of (3.4) are bijective for all j≥N. We fix such a N.

We shall need the following simple observation.

Lemma 4.1. Let H be a finite group and let O1 be a set on which H acts transitively. Let O2⊂O1 be any nonempty subset. Then one has:

1

|H|

X

h∈H

h



 1

|O2| X

x∈O2

x



= 1

|O1| X

y∈O1

y

in the kH-module kO1.

Proof. Let r be the order of StabH(y) for any y∈O1. Then one has:

1

|H|

X

h∈H

h



 1

|O2| X

x∈O²

x



= 1

|O2| X

x∈O²

1

|H|

X

h∈H

hx

!

= 1

|O2| X

x∈O2



 r

|H|

X

y∈O1

y





= 1

|O1| X

y∈O1

y.

Now suppose j ≥N. For each Hi,j-orbit O in C(i, j), we define a kGj- module endomorphism

f_O :M(i)j →M(i)j

by

f_O(gα_i,j) = 1

|O| X

γ∈O

gγ, for any g∈G_j.

The elementsf_OforO∈Hi,j\C(i, j) form a basis for End_kGj(M(i)j). Hence, from the bijectivity of µ_i,j, we have a linear bijection

ν_i,j : End_kGj(M(i)_j)→End_kGj+1(M(i)_j+1), f_O 7→f_µi,j(O).

Notation 4.2. Define e_i,j ∈kH_i,j by ei,j = 1

|H_i,j| X

h∈H_i,j

h.

Lemma 4.3. For any f ∈End_kGj(M(i)_j), one has:

(4.4) ν_i,j(f)(α_i,j+1) =e_i,j+1α_j(f(α_i,j)).

Proof. By linearity, it suffices to verify (4.4) when f = f_O, for each O ∈ Hi,j\C(i, j).

The injectivity of m_i,j implies that {α_jγ | γ ∈ O} is a subset of µ_i,j(O) consisting of|O|elements. Thus, by the preceding lemma, one has:

ei,j+1αj(f_O(αi,j)) = 1

|H_i,j+1| X

h∈Hi,j+1

h



 1

|O| X

γ∈O

αjγ





= 1

|µ_i,j(O)|

X

y∈µi,j(O)

y

=f_µi,j(O)(α_i,j+1)

=νi,j(f_O)(αi,j+1).

For anykG_j-submoduleU ⊂M(i)_j, we have a natural inclusion Hom_kGj(M(i)_j, U)⊂End_kGj(M(i)_j).

By Maschke’s theorem, ifU (U⁰ arekGj-submodules of M(i)j, then (4.5) Hom_kGj(M(i)_j, U)(Hom_kGj(M(i)_j, U⁰).

Lemma 4.6. LetXbe akC-submodule ofM(i). Iff ∈HomkGj(M(i)j, Xj), then

ν_i,j(f)∈Hom_kGj+1(M(i)_j+1, X_j+1).

Proof. Since f ∈Hom_kGj(M(i)_j, X_j), one has f(α_i,j) ∈ X_j. It follows by (4.4) that ν_i,j(f)(α_i,j+1) ∈ X_j+1. By the transitivity condition, one has

νi,j(f)(α)∈Xj+1 for all α∈C(i, j+ 1).

Definition 4.7. For anykC-submodule X ofM(i), let Fj(X) = HomkGj(M(i)j, Xj).

By Lemma4.6, we have a commuting diagram:

(4.8) Fj(X)_{ _} ^{νi,j //}

Fj+1(X)_{ _} ^{νi,j+1 //}

Fj+2(X)_{ _} ^{νi,j+2 //}

· · ·

F_j(M(i)) ^{νi,j //}F_j+1(M(i)) ^{νi,j+1 //}F_j+2(M(i)) ^{νi,j+2 //}· · · . The maps in the bottom row of (4.8) are bijective. Thus,

(4.9) The maps in the top row of (4.8) are injective.

Proposition 4.10. If X is a kC-submodule of M(i), then X is finitely generated.

Proof. Ifn∈Z+, we shall writeX(n) for thekC-submodule ofXgenerated byL

i≤nX_i. It suffices to prove thatXis equal toX(n) for somen. Suppose not. This means that for eachn, there existsrn> nsuch thatX(n)rn6=Xrn, and henceX(n)_rn 6=X(r_n)_rn. Consider the sequence

n1 =N, n2 =rn1, n3=rn2, . . . . We have

N =n₁< n₂ < n₃ <· · ·, and

X(ni)ni+1 (X(ni+1)ni+1 for all i.

By (4.5) and (4.9), we have

Fn2(X(n1))(Fn2(X(n2)),→Fn3(X(n2))(Fn3(X(n3))

,→F_n4(X(n₃))(F_n4(X(n₄)),→ · · · This is an increasing chain which is strictly increasing at every other step.

But the dimension of each term in this chain is at most dimFN(M(i)).

Therefore, we have a contradiction.

4.2. Proof of Theorem 3.7. Let V be a finitely generated kC-module.

LetY be akC-submodule ofV. We shall show that Y is finitely generated.

By Lemma 2.14, there exists a finite set S of homogenous elements of V such that the homomorphism πS : M(S) → V of (2.11) is surjective.

Let X be the preimage π_S⁻¹(Y) of Y. It suffices to prove that any kC- submodule X of M(S) is finitely generated. We shall use induction on|S|.

If |S|= 1, the result follows from Proposition 4.10. Suppose |S|> 1. We choose anys∈S and let S⁰ =S− {s}; so M(S) =M(S⁰)⊕M(degs). Let ps :M(S)→ M(degs) be the projection map with kernelM(S⁰). We have a short exact sequence

0−→X∩M(S⁰)−→X −→^ps ps(X)−→0.

Since X∩M(S⁰) ⊂ M(S⁰) and ps(X) ⊂ M(degs), it follows by induction hypothesis that X∩M(S⁰) and p_s(X) are both finitely generated. Hence, X is finitely generated. This concludes the proof of Theorem 3.7.

Remark 4.11. After this paper was written, Andrew Putman informed us that he had also found a similar proof for the categories FI and VI.

5. Further remarks

In this section, we discuss generalizations of the injectivity and surjectivity properties of finitely generated FI-modules; see [1, Definition 3.3.2]. Let C be a locally finite EI category of type A∞, and kbe any commutative ring.

Proposition 5.1. Assume that C satisfies the transitivity condition. Let V be a Noetherian kC-module. Then the map αj :Vj → Vj+1 is injective for allj sufficiently large.

Proof. Let Xj be the kernel of the map αj :Vj → Vj+1. Suppose g ∈Gj. By the transitivity condition, there existsg₁ ∈G_j+1such thatg₁α_j =α_jg. If x∈Xj, then one hasαj(gx) =g1αj(x) = 0. Hence,Xj is akGj-submodule of V_j. For any β ∈ C(j, j+ 1), there exists g₂ ∈ G_j such that g₂α_j = β;

hence, for any x∈X_j, one has β(x) =g₂α_j(x) = 0. It follows thatX_j is a kC-submodule ofV. Let

X = M

j∈Z+

X_j.

ThenXis akC-submodule ofV (called thetorsion submoduleofV, see [2]).

By hypothesis,X is finitely generated. Since eachX_j is a kC-submodule of X, it follows thatXj must be zero for allj sufficiently large.

For anykC-moduleV, we shall denote byρ_j(V) the image of thekG_j+1- module map

kC(j, j+ 1)⊗_kGjVj −→Vj+1, α⊗v7→αv.

Proposition 5.2. Assume that C satisfies the transitivity condition. Let V be akC-module. Suppose V_j is a finitely generated k-module for all j∈Z+. ThenV is finitely generated as a kC-module if and only ifρ_j(V) =V_j+1 for allj sufficiently large.

Proof. Suppose that V is finitely generated. By Lemma 2.14, there exists a finite set S of homogeneous elements ofV such that the homomorphism π_S : M(S) → V of (2.11) is surjective. Let n be the maximal of degs for s∈S. Supposej≥n. Then one has

ρ_j(V) =π_S(ρ_j(M(S))) =π_S(M(S)_j+1) =V_j+1.

Conversely, suppose there exists n ∈ Z+ such that ρ_j(V) = V_j+1 for all j ≥n. For each i≤n, letBi be a finite subset of Vi which generatesVi as a k-module. Let S =B₀∪ · · · ∪B_n. ThenS is a finite set of homogeneous elements of V. Let V⁰ be thekC-submodule of V generated byS. Clearly, V_i⁰ =V_i for all i≤n. Suppose, for induction, that V_j⁰ =V_j for some j≥n.

Then one has

V_j+1⁰ ⊃ρ_j(V⁰) =ρ_j(V) =V_j+1.

Hence,V_j+1⁰ =V_j+1. It follows thatV⁰=V, soV is finitely generated.

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(Wee Liang Gan)Department of Mathematics, University of California, River- side, CA 92521, USA.

wlgan@math.ucr.edu

(Liping Li)Department of Mathematics, University of California, Riverside, CA 92521, USA.

lipingli@math.ucr.edu

This paper is available via http://nyjm.albany.edu/j/2015/21-18.html.