Classification of Subcategories in Abelian Categories and Triangulated Categories

Home , Localizing subcategory, Quotient category

CLASSIFICATION OF SUBCATEGORIES IN ABELIAN CATEGORIES AND TRIANGULATED CATEGORIES

ATHESIS

SUBMITTEDTOTHE FACULTY OF GRADUATE STUDIESAND RESEARCH

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FORTHE DEGREEOF

DOCTOROF PHILOSOPHY

MATHEMATICS

UNIVERSITYOF REGINA

Yong Liu

Regina, Saskatchewan

September 2016

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Yong Liu, candidate for the degree of Doctor of Philosophy in Mathematics, has presented a thesis titled, Classification of Subcategories in Abelian Categories and Triangulated Categories, in an oral examination held on September 8, 2016. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material.

External Examiner: Dr. Henning Krause, University of Bielefeld

Supervisor: Dr. Donald Stanley, Department of Mathematics and Statistics

Committee Member: Dr. Allen Herman, Department of Mathematics and Statistics

Committee Member: *Dr. Fernando Szechtman, Department of Mathematics and Statistics

Committee Member: Dr. Yiyu Yao, Department of Computer Science

Chair of Defense: Dr. Renata Raina-Fulton, Department of Chemistry and Biochemistry

*Not present at defense

Abstract

Two approaches for classifying subcategories of a category are given. We examine the class of Serre subcategories in an abelian category as our ﬁrst target, using the concepts of monoform objects and the associated atom spectrum [13]. Then we generalize this idea to give a classiﬁcation of nullity classes in an abelian category, using premonoform objects instead to form a new spectrum so that there is a bijection between the collection of nullity classes and that of closed and extension closed subsets of the spectrum. Additionally, we impose a natural sheaf structure induced by the center of a category on the atom spectrum, over which the sheaves of modules over the structure sheaf are also discussed.

The second approach is enlightened by the lattice structure implicitly shown in the s- tatements of classification of the subcategories in an abelian category. We introduce a new concept of classifying space of subcategories, those subcategories satifying finitely many closure operations, in an either abelian or triangulated category. We show that a class of subcategories is classified by a topological space if these subcategories are primely generated. Many well-known results fit into our framework, such as Neeman’s classification [19] of localizing subcategories of the derived category D(R) of a commutative Noetherian ring

R, etc.

i Acknowledgements

I would like to thank my supervisor Don for leading me into the ﬁeld of classiﬁcation of various subcategories, and for his support and patience as well. During my PhD program,

Adam, Ivo and Soumen helped me a lot for problem discussion etc. I also appreciate Allen,

Fernando and Yiyu for useful comments on my thesis.

ii Contents

Abstract i

Acknowledgements ii

Table of Contents iii

1 Introduction 1

2 Abelian category and its localization 8

2.1 Abelian category and its Serre subcategories ...... 9

2.2 The calculus of fractions ...... 13

2.3 Quotient category ...... 18

2.4 Adjoint functor ...... 21

2.5 Kan extension ...... 25

2.6 Localization functor ...... 32

3 Monoform and premonoform objects 42

3.1 Monoform objects ...... 43

iii 3.2 Premonoform objects ...... 47

3.2.1 Deﬁnitions and properties ...... 47

3.2.2 A premonoform but not monoform object ...... 50

3.3 Equivalence relation, support and topology ...... 53

3.4 Classiﬁcation of nullity classes ...... 55

4 A structure sheaf on the atom spectrum 65

4.1 Atom spectrum and a structure sheaf R+ ...... 66

4.2 Sheaf of modules ...... 75

4.2.1 A module structure over R+ ...... 75

4.2.2 Examples from quiver representations ...... 78

5 Classifying space of subcategories and its application 86

5.1 Classifying space ...... 87

5.1.1 Complete distributive lattices ...... 87

5.1.2 Points and topology ...... 91

5.1.3 Examples of classifying spaces ...... 95

5.2 Generally prime objects and classiﬁcation ...... 97

5.3 Irreducible vs. prime, and functoriality ...... 102

5.4 Primely generated subcategories ...... 106

5.4.1 Serre subcategories in a noetherian abelian category ...... 106

5.4.2 Thick subcategories in Dperf (R) ...... 107

5.4.3 Localizing subcategories in D(R) ...... 108

iv 5.4.4 Localizing subcategories in a stable homotopy category ...... 111

5.4.5 An example ...... 113

Bibliography 115

v Chapter 1

Introduction

We are motivated by the problem of classifying t-structures (see [4]) in a derived category Dperf (R) of perfect complexes over a ring R. It is known that each t-structure corresponds bijectively to an aisle, a full subcategory closed under positive suspensions and extensions such that the inclusion functor admits a right adjoint. Stanley’s idea [26] of classifying these aisles is to find an invariant given by perversity functions on the prime spectrum SpecR. However, it is efficient not to deal with aisles directly but a generalized notion of aisles called nullity classes, also known as torsion theories when in an abelian category. As a first step, I attempted to do and obtained a

c Generalization of Stanley’s method into the case when the derived category is Dfg(R).

c Here Dfg(R) is a full triangulated subcategory of D(R) consisting of those objects represented by coﬁbrant objects with each degree ﬁnitely generated, within which the nullity classes have invariants given by perversity functions as well, see [17] for detail.

In a general setting of a category and its subcategories, we focus on establishing a theory

1 of invariants to distinguish those subcategories of the same type, say those subcategories closed under ﬁnite coproducts, nullity classes (Deﬁnition 3.1.7) and Serre subcategories

(Definition 2.1.3) etc. This is successfully fulfilled historically in many different ways, for instance A. Neeman’s classification (see [19] and [21]) of (co)localizing subcategories of D(R) in terms of subsets of the prime spectrum SpecR, P. Balmer’s classification [3] of radical thick tensor ideals of a tensor triangulated category T via closed subsets of the

Hochster dual of spectrum of T , Benson’s classiﬁcation [5] of localizing tensor ideals in the homotopy category K(Inj kG) by subsets of SpecH∗(G, k) and so on.

Inspired by them and also R. Kanda’s paper [13] on the atom spectrum of an abelian category, we approach our problem of classifying nullity classes (or torsion classes) by looking into a new spectrum SpecA of the abelian category A, a topological space with equivalence classes of premonoform objects, shown as Corollary 3.4.12, so that

Theorem 1 (Result A). There is an order preserving bijection

{Nullity classes in A} →∼ {Closed and extension closed subsets of SpecA}.

Here the premonoform object is a slightly weaker notion of monoform object (see [28] e.g.), the latter is the same as strongly uniform module when the abelian category A is the category of modules over a ring. As a side issue and difﬁcult one, it is always interesting to

ﬁnd the corresponding elementary and optimal objects to build subcategories of a certain type, see [27] for example.

There is a common idea to develop the theory even further by considering the (atom) spectrum of an abelian category A together with the structure sheaf R induced by the

2 centers of its quotients by subcategories as a ringed space, hence the study of the sheaf of

modules naturally and so on. We found a similar idea later in A. Rosenberg’s papers [24]

and [23] when he was attempting to reconstruct a scheme X from the spectrum of its

derived category D(X) of coherent sheaves. See also Rouquier’s paper [25] on the example

of coherent sheaves on separated schemes. First we start from the simple example when

the category A is abelian with atom spectrum ASpecA, which consists of atom equivalence classes of monoform objects (Deﬁnition 3.1.1). With certain set theoretical assumptions, in fact, the Freyd-Mitchell Embedding Theorem always allows us to obtain a concretization of an abstract abelian category, though not the best choice. We show as Proposition 4.1.16 and Proposition 4.2.5 that

Theorem 2 (Result B). Let A be an abelian category and ASpecA its atom spectrum. Then

(1) there is a natural structure sheaf R of rings on ASpecA induced by the centers of A and its localization (or quotient, see [7]) with respect to Serre subcategories S, making

ASpecA a locally ringed space.

(2) there is an associated sheaf of modules over R, i.e. the existence of a faithfully exact functor ∼: A → R-mod, provided the concretization A → Ab is exact.

As a special case, when A is the category of R-modules with R a commutative Noetherian

ring with identity, the construction in (1) coincides with the one given by the afﬁne scheme

(SpecR, OSpecR). Notice that such a module structure depends on the choice of concretization F : A → Ab, and the restriction map of its sheaf of modules relies on the existence

of a localization functor LS : A → A. There is a more general approach to develop this

3 theory which we consider as a new project by studying the category of coherent sheaves.

Apart from Result A, we have another method to classify subcategories (of certain type)

by ﬁrst collecting the target subcategories and ordering them via inclusions thus giving a

poset Φ. In fact, Φ can be equipped with a complete distributive lattice structure naturally.

We then deﬁne the classifying space K(Φ) of subcategories of the given type to be those

subcategories which cannot be generated by their proper subcategories of the same type,

i.e. points in K(Φ). By restricting into respectively subsets of “irreducibles” of both K(Φ)

and Φ, we have shown as Theorem 5.2.8

Theorem 3 (Result C). There is an order preserving bijection

∼ {Closed subsets of Kgp(Φ)} → {Primely generated subcategories in Φ}

which is in fact an isomorphism of lattices.

This gives a more general setting of many famous results including P. Gabriel’s classiﬁca-

tion [7] of Serre subcategories, Neeman’s classiﬁcation of thick subcategories in Dperf (D)

and that of localizing subcategories in D(R) or in general that of localizing subcategories

in an axiomatic stable homotopy category [11], by showing that each subcategory is prime-

ly generated. For another comparison, in P. Johnstone’s book [12] on Stone spaces, he has

deﬁned a similar notion using lattices, which is used to develop the theory of classifying

subcategories in a derived category by J. Kock and W. Pitsch recently in [14].

A natural question to ask is, when can certain subcategories be classiﬁed by a topolog-

ical space X in the sense that there is an order preserving bijection (or an isomorphism of

4 lattices) from the collection of closed (or dually open) subsets of X to the lattice Φ of subcategories of given type? Then our Result C can be reinterpreted as certain subcategories

(in a given category) are classiﬁed by a topological space if and only if it is primely generated, where the topological space is constructed as our classifying space Kgp(Φ). However, the example of nullity classes occurring in Result A provides one in the exceptional list, see Section 5.4.5, while on the whole Result A and C compensate each other in a certain way if better the former theory can be generalized into the case of triangulated category and its subcategories.

To conclude, we solve the classification problem of nullity classes (or torsion theories) in an abelian category rather than a specific category of modules over a ring. Simultane- ously, from this approach we provide an idea how to give a classification by searching for suitable generators (or fundamental bricks in the sense of the obstruction of subcategories) of subcategories, i.e. premonoform objects v.s. nullity classes while monoform objects v.s. Serre subcategories, in my thesis. Also as a byproduct, we are able to develop a similar theory of sheaves of modules over the structure sheaf on the spectrum of an abelian category. Furthermore, we build a general framework (or a new viewpoint) of forming a topological space to classify subcategories and various criteria for the subcategories which can be classified by a topological space. Many well-known results fit into this framework.

Finally, one class of subcategories that cannot be classiﬁed by a topological space X, i.e. bijectively corresponding to the collection of closed subsets, can be classiﬁed by a different collection of subsets of a topological space, maybe different from X but can be chosen as

5 a spectrum consisting of equivalence classes of certain generators, such as (pre)monoform

objects.

Stemming from the main results presented earlier, there are quite a few questions and

related ones to ask considered as future research, which are summarized as follows:

(1) Can any of the ideas developed here give a classiﬁcation of t-structures under a more general circumstance, in other words, is there any classifying space that classiﬁes the t- structures? We may have a negative answer, see for instance [29].

(2) Given two collections ΦA and ΦB of subcategories of types A and B respectively (with

ﬁnitely many closure operations such as closed under extensions etc. to deﬁne), what is the relation between their classifying space K(ΦA) and K(ΦB)? Moreover, is there any

initial (or terminal) structure among them that deﬁnes a subcategory, so that any other

ones differ only by a K-theory, as Thomason measures his classiﬁcation of thick subcate-

gories from general triangulated subcategories in [30]. In other words, we are looking for

a Grothendieck group like structure K0(I) (a group or an algebra etc.) associated with the

subcategories determined via the initial structure I, so that there is a one-to-one correspon-

dence between the collection ΦA of subcategories for instance and (certain) subgroups (or

subalgebra etc.) of K0(I).

(3) Based on the structure sheaf R on the atom spectrum ASpecA of an abelian category A,

what more can we say about the sheaves of modules over that, comparing with the standard

scheme theory? We can also apply K-theory to the associated sheaf ∼ of modules and ask

what the difference means, since the functor ∼: A → R-mod is not an equivalence.

6 (4) Apply the theory to the category of coherent sheaves over the projective line or space.

7 Chapter 2

Abelian category and its localization

This is a preliminary chapter which brieﬂy introduces the basics of abelian categories

first. Then we will focus on the construction and properties of quotient category A/S of an abelian category A by its Serre subcategories S and next discuss the existence of a right adjoint functor especially for the projection functor Q : A → A/S, so that we have a localization functor. The Kan extension allows us to compute this right adjoint when it is pointwise defined, although in our particular example of abelian category, kA2-rep consisting of quiver representations of finite type, there is a special way to calculate it, since enough injective objects are provided there, by Neeman [20].

This chapter is organized as follows. The ﬁrst section introduces the notions of an abelian category A and its Serre subcategories S, which are the typical examples we work with. Then we give the construction of the quotient category A/S with projection Q :

A → A/S in two different ways, the calculus of fractions and Gabriel’s construction, in the following two sections. After the introduction of adjoint functors in Section 2.4, we

8 then focus on the existence and construction of a right adjoint of Q in the last three sections, which will be used later in Chapter 4.

The classical result presented here can be found in Gabriel [7], Weibel [31], MacLane [18],

Neeman [20] and Krause [15]. The Stack Project [22] is also good to refer to.

2.1 Abelian category and its Serre subcategories

In this section, we introduce the basic terminologies in the category theory, referring to

MacLane [18].

Deﬁnition 2.1.1. A category C consists of a collection of objects and a collection of morphisms satisfying the following properties

(1) for any object A in C, there is an identity morphism 1A;

(2) for any pair of morphisms f : A → B and g : B → C, there is a composition g ◦ f : A → C such that f ◦ 1A = f and 1B ◦ f = f;

(3) for any triple of morphisms f : A → B, g : B → C and h : C → D, the composition of morphisms is associative (h ◦ g) ◦ f = h ◦ (g ◦ f).

We call the assignment F : C → C0 a functor if it respects the identity morphisms and the compositions of morphisms, F (1C) = 1F (C) and F (gf) = F (g)F (f).

A subcategory D of a category C, is a subcollection of objects and arrows in C. In particular, the inclusion functor i : D → C is faithful, i.e. the assignment on the collection of morphisms

i : HomD(A, B) → HomC(iA, iB)

9 is injective. The subcategory D is full if such i is also surjective.

The category C is additive if it has all ﬁnite (co)products, and for any pair of objects the collection of morphisms HomC(A, B) forms an abelian group such that it is bilinear with respect to the compositions. In particular, an additive category has a zero object, considered as a (co)product over the empty diagram. A functor F : D → C between additive categories is additive if F respects the additions as well.

Deﬁnition 2.1.2. An abelian category A is an additive category such that

(1) for any morphism f : A → B there is a kernel Ker(f) and a cokernel Coker(f) as certain (co)limits (see Chapter III.3 and 4 in [18]);

(2) for any morphism f, the canonical morphism f : Coker(i) → Ker(p) in the diagram

i f p Ker(f) / A / B / Coker(f) O

Coker(i) / Ker(p) f induced by f is an isomorphism. In particular, the First Isomorphism Theorem holds in A.

A subobject A0 of A refers to the isomorphism classes of monics i : A0 ,→ A with

Ker(i) = 0. Dually, a quotient object A00 of A refers to the isomorphism classes of epics

00 p : A A with Coker(p) = 0.

There are many examples of abelian categories such as the category of ﬁnitely generated

R-modules when R is a commutative ring with an identity. In an abelian category, its Serre subcategories are of particular interest for classiﬁcation.

10 Deﬁnition 2.1.3. A Serre subcategory S of an abelian category A is a full subcategory

such that it is closed under subobjects, quotient objects and extensions, that is, for any

short exact sequence in A

0 → A → B → C → 0 both A, C ∈ S imply B ∈ S, called extension of C by A. This is also denoted as B ∈ A∗C.

In general, for two collections S1, S2 of objects in A, we deﬁne S1 ∗ S2 to be the collection

of objects that can be obtained as an extension of some Y ∈ S2 by some X ∈ S1.

Let S be a collection of objects in an abelian category A. Denote hSisub = {X ∈

A | X is a subobject of some M ∈ S}, hSiquot = {X ∈ A | X is a quotient of some M ∈

S n n n−1 0 S}, and hSiext = n≥0 S with S = S ∗ S recursively deﬁned over S of the zero

object. See also the discussion at the end of Section 3.1.

Example 2.1.4. Let A be an abelian category and S 6= ∅ a collection of objects in A.

Denote by hSi := hSiSerre the smallest Serre subcategory containing S. The objects in S

are called generators. Then by Proposition 2.4 in [13]

hSi = hhhSisubiquotiext.

If S = {X}, denoted hSi := hXi for simplicity. For the category Z-mod of ﬁnitely

generated abelian groups and S = {Z/2},

hZ/2i = hhhZ/2isubiquotiext = hZ/2iext.

There are extensions Z/2 ⊕ Z/2 and Z/4 that ﬁt into the short exact sequence

0 → Z/2 → E → Z/2 → 0

11 so that Z/2 ⊕ Z/2, Z/4 ∈ hZ/2i. Next, extend E by Z/2 and keep iterating it in this way.

Finally, we will obtain hZ/2i = Z-mod(2), the subcategory of ﬁnitely generated 2-torsion abelian groups.

Example 2.1.5. Consider the category Z-mod of ﬁnitely generated abelian groups, other than the trivial subcategory h0i = {0} the whole category Z-mod is also Serre since

hZi = hhhZisubiquotiext = hhZiquotiext = h{Z/p | p is prime} ∪ {Z}iext

For another, let S = {Z/2, Z/3}. Then

hZ/2, Z/3i = hZ/6i.

Indeed, Z/2 and Z/3 are subobjects of Z/6 so that hZ/2, Z/3i ⊆ hZ/6i, while the (split) exactness of

0 → Z/2 → Z/6 → Z/3 → 0 says Z/6 is an extension of Z/3 by Z/2 thus hZ/6i ⊆ hZ/2, Z/3i.

Proposition 2.1.6. Let S be a nonempty subcategory of an abelian category A. Then S is

Serre if and only if it is full and for any exact sequence A → B → C, both A, C ∈ S imply that B ∈ S.

Proof. Suppose the subcategory S is Serre. Then the exact sequence splits into three different ones, namely A → Im(f) → 0, 0 → Im(f) → B → Ker(g) → 0 and 0 →

Ker(g) → C. Since S is closed under subobject, quotient and extension, A, C ∈ S imply that Im(f), Ker(g) ∈ S hence B ∈ S. Conversely, suppose A ∈ S. Then the exact

12 sequence A → 0 → A implies that 0 ∈ S. In particular, this implies that S is closed under subobject and quotient from the exact sequences 0 → B → A and A → C → 0.

For a ﬁnitely generated R-module M and f ∈ R, deﬁne Sf = hR/(f)i to be the Serre

subcategory generated by R/(f), which has the following description. Recall that in a ring

2 3 R, the localization of R at the multiplicatively closed set {1, f, f , f , ...} is denoted by Rf

a in which every element is represented by a fraction of the form f n for some a ∈ R and

a b l m n n ∈ N, subject to the equivalence relation f n ∼ f m if f (af − bf ) = 0 for some l ∈ N.

n Proposition 2.1.7. Sf = {M ∈ R-mod | f M = 0 for some n ∈ N} = {M ∈

R-mod | M ⊗ Rf = 0}.

Proof. Let M be a ﬁnitely generated module. Then there is a ﬁnite uniform power to

annihilate all the generators. Thus the second equality holds. Now suppose fM = 0, then

M becomes an R/(f)-module which is also ﬁnitely generated. Hence M ∈ Sf . Now

suppose f nM = 0. Then the exact sequence for i < n

0 → f iM → f i−1M → f i−1M/f iM → 0

i−1 i i−1 i implies that M ∈ Sf as well since f(f M/f M) = 0 gives f M/f M ∈ Sf . There-

fore, by induction on n, we still have M ∈ Sf .

2.2 The calculus of fractions

We recall the localization of a category A in terms of left (or right) fractions. From

the next section we will see that it coincides with Gabriel’s quotient category when A is

abelian. See Chapter 10.3 in [31].

13 Deﬁnition 2.2.1. A left multiplicative system S is a collection of morphisms such that

(1) S contains the identity morphisms and it is closed under compositions;

(2) Any diagram

g X / Y

t Z with t ∈ S can be completed to a commutative diagram

g X / Y

t s Z / W f with s ∈ S.

(3) If ft = gt for t ∈ S, then there is s ∈ S such that sf = sg.

A right multiplicative system is dually deﬁned. We call it a multiplicative system if it is

both a left and a right multiplicative system.

Lemma 2.2.2. Let f : A → B and g : B → C be maps. Then we have two exact sequences

f (1) 0 → Ker(f) →i Ker(gf) → Ker(g) and

g j (2) Coker(f) → Coker(gf) → Coker(g) → 0.

Proof. For (1), it is clear that Ker(f) ⊆ Ker(gf) and that fi = 0 implies Im(i) ⊆ Ker(f).

Now for y ∈ Ker(gf) with f(y) = 0, we have y ∈ Ker(f) ∩ Ker(gf) = Ker(f), that is,

y = i(y) so that Ker(f) ⊆ Im(i). For (2), notice that the maps g and j are well-deﬁned

and jg = 0, thus Im(g) ⊆ Ker(j). Now suppose j(z + gf(A)) = g(B). Then z ∈ g(B) so that z = g(y) for some y ∈ B. Therefore, g(y + f(A)) = g(y) + gf(A) = z + gf(A), as

required.

14 Proposition 2.2.3. Let A be an abelian category and S a Serre subcategory. Denote by

ΣS = {f ∈ A | Kerf, Cokerf ∈ S}. Then ΣS is a multiplicative system.

Proof. (1) It contains the identity morphisms clearly. Thanks to the quotient functor

Q : A → A/S by Proposition 2.3.2, we have that Qf is an isomorphism if and only if

Kerf, Cokerf ∈ S. Thus given in A a sequence

f g X → Y → Z

such that f, g ∈ ΣS , Q(gf) = QgQf is an isomorphism in A/S. Thus Ker(gf), Coker(gf) ∈

S by Lemma 2.2.2, hence gf ∈ ΣS also.

(2) Next we claim that the pullback and the pushout diagrams give the Ore condition

by comparing the kernel and cokernel respectively. Suppose we have a diagram (sg, ft) as

a pullback

∃!g0 Ker t o K ∃!h i j | g X / B

t s C / A f Then (Ker t, gi) is the kernel of s, i.e. Ker t ∼= Ker s. Indeed, for any (K, j) such that

sj = 0, there exists a unique h : K → X such that th = 0 by assigning a zero map

K →0 C. It follows that there exists a unique map g0 : K → Ker t by the universal property of kernels. Also, we have an injection Coker t ,→ Coker s. In fact, there is a commutative

15 diagram

g (t) (s,−f) 0 / X / B ⊕ C / A O O O id id α g (t) (u,−f) ? 0 / X / B ⊕ C / A0 / 0 0 g in which the ﬁrst row is the pullback and (A , (u, −f)) is the cokernel of t . Since (ug, ht)

is a pushout diagram

g X / B

t u h C / A0 o s α f ' A The dual statement implies that Coker t ∼= Coker u. Hence by applying the snake lemma to the commutative diagram

0 / B B / 0 / 0

u s 0 0 / A α / A / Coker α / 0

we obtain a short exact sequence 0 → Coker u → Coker s → Coker α → 0, as required.

(3) Finally, given f : X → Y in A. Suppose sf = 0 for s : Y → Z. Then apply the

functor Hom(X, −) to the sequence 0 → Ker s →i Y →s Z, there is g : X → Ker s such

that f = i∗(g) = ig. Thus ft = 0 with inclusion t : Ker g → X and there is an injection

Coker t = X/Ker g ,→ Ker s so that Coker t ∈ S.

Conversely, suppose ft = 0, then apply the functor Hom(−,Y ) to the sequence W →t

p X → Coker t → 0, there is g : Coker t → Y such that f = p∗(g) = gp. Thus we have

sf = 0 with projection s : Y → Coker g and Ker s = g(Coker t) which lies in S as

well.

16 Proposition 2.2.4. Let R be a commutative ring with identity. Then there is a ring isomorphism

=∼ Rf → HomR-mod/Sf (R,R)

a −n given by [ f n ] 7→ af , where Sf = hR/(f)i is the Serre subcategory generated by R/(f).

a b l m n l+m Proof. Suppose f n ∼ f m . Then there is l ∈ N such that f (f a − f b) = 0 or f a = f l+nb. Thus we have a commutative diagram

R O f n a RRo / R _ ?

R so that af −n ∼ bf −m, hence the map is well-deﬁned. Now given any morphism αs−1 we can deﬁne a map R →t M by assigning t(1) ∈ s−1(f n1) so that the diagram

R t

n f M s α × RRw commutes. Since 2 out of 3 for quasi-isomorphisms holds in any abelian category, thus t is also a quasi-isomorphism. It is straightforward to check that the following diagram

R f n αt

~ f n αt RRo / R ` > s t α M

17 is commutative, hence the map is surjective. Furthermore, suppose af −n ∼ bf −m. That is, there is a commutative diagram

R O f n a u RRo / R _ f l c ? v f m b R

for some u, v and l ∈ N, so that uf n = f l = vf m and ua = c = vb. Thus we can choose

u, v appropriately making f m+la = f n+mua = f n+mvb = f n+lb, hence f l(af m − bf n) =

a b 0. Therefore, f n ∼ f m and the injectivity holds. It is straightforward to check that such a

map is indeed a ring map, by using the property of fractions.

2.3 Quotient category

This section gives the classical construction of quotient category by a Serre subcategory

from Gabriel’s paper [7]. See the appendix of Neeman’s book [20] as well.

Deﬁnition 2.3.1. Let A be an abelian category and S a Serre subcategory. The quotient category A/S consists of the same objects of A while the morphisms and their compositions are deﬁned as follows.

Any map f : A → X in A/S is represented by a map f : A0 → X/X0 in A with

A/A0,X0 ∈ S, so that for another representative f 0 : A00 → X/X00 with A/A00,X00 ∈ S,

18 the existence of a commutative diagram

f A / X G O

? f A0 / X/X0 ?

i p 4 / $ $ A00 / X/X00 f 0

with i monic and p epic, allows us to identify f 0 and f. In other words, the morphisms in

the quotient category A/S are given by the elements in the direct limit

Hom (A, X) ∼ lim Hom (A0, X/X0) A/S = −→ A A/A0,X0∈S

0 0 over the directed system {(A ⊆ A, X ⊆ X)}A/A0,X0∈S .

f g For a pair of morphisms A → X → Y in A/S, represented by f : A0 → X/X0 and

g : X00 → Y/Y 0 in A with A/A0,X0, X/X00,Y 0 ∈ S, the composition g ◦ f is represented

by the composition

f ∼ g A00 = f −1(X0+X00/X0) → X0+X00/X0 →= X00/X0∩X00 → Y/(Y 0+g(X0∩X00)) = Y/Y 00 observing that in the short exact sequence

0 → A0/A00 → A/A00 → A/A0 → 0 the term A0/A00 ∼= X/(X0 + X00) ∈ S whereas that X0,Y 0 ∈ S implies Y 00 ∈ S.

Proposition 2.3.2. Let A be an abelian category and S a Serre subcategory. Then there is a functor Q : A → A/S such that the following conditions hold.

19 (1) Q(X) = 0 for any X ∈ S. And Q is exact with A/S an abelian category. The functor Q is universal with respect to this property, i.e. any other exact functor F : A → B to an abelian category B such that F (X) = 0 for any X ∈ S, factors through Q uniquely .

(2) Q(f) = 0 if and only if Im(f) ∈ S.

(3) Q(f) is monic if and only if Ker(f) ∈ S; Q(f) is epic if and only if Coker(f) ∈ S;

Q(f) is an isomorphism if and only if both Ker(f), Coker(f) ∈ S.

Proof. See Chapter III.1 in Gabriel [7].

We refer to Weibel [31] for the construction of the category S−1A, recalling that the objects are those in A and the morphisms A → B are equivalence classes of left fractions s−1f : A → A0 ← B, or equivalently we can choose them as right fractions gt−1 : A ←

B0 → B, where s, t, g, f ∈ A and s, t ∈ S.

Proposition 2.3.3. Let A be an abelian category and S a multiplicative system. Then there is a functor q : A → S−1A such that

(1) q is exact and S−1A is abelian so that any other functor F : A → B to an abelian category B sending s ∈ S to an isomorphism factors through q uniquely.

(2) q(X) = 0 if and only if X ∈ S.

Proof. This is in fact a general construction using fractions. See Deﬁnition 10.3.1 and

Exercise 10.3.2 in Weibel [31].

−1 ∼ Proposition 2.3.4. There is an isomorphism ΣS A = A/S of categories, where ΣS =

{f ∈ A | Kerf, Cokerf ∈ S}.

20 Proof. By the universal properties of the functors Q and q, we have unique functors F :

−1 A/S → ΣS A : G such that FQ = q and Gq = Q. By the uniqueness, GF = idA/S

and FG = id −1 . Explicitly, the functor F is given by sending a representative f in the ΣS A

diagram

f A / X O i p ? A0 / X/X0 f

−1 −1 −1 −1 to q(p) q(f)q(i) and G sends fs to Q(f)Q(s) .

2.4 Adjoint functor

Before we move into the details of Kan extensions, we present some commonly used

properties of adjoint functors, and later we will see that the adjointness is a special type of

Kan extension. See Section IV.1 in MacLane [18].

Deﬁnition 2.4.1. A natural transformation η : F → G between two functors F,G : C → D

is called a natural equivalence if for every X ∈ C the arrow ηX : FX → GX is an

isomorphism.

Let F : C → D, G : D → C be functors. We say that F is left adjoint to G (or G is right adjoint to F ) if there is a natural equivalence called adjunction

∼ ϕ : D(F −, −) −→= C(−,G−) of Hom bifunctors, denoted by ϕ : F a G. In other words, for every pair of objects X ∈ C

21 and Y ∈ D, the map

=∼ ϕXY : D(FX,Y ) −→ C(X, GY ) is a bijection such that it is natural in both variables X and Y .

It is clear from the deﬁnition that the composition of adjunctions remains an adjunction.

Proposition 2.4.2. Let F : C → D and G : D → C be functors. Then a natural trans-

∼ formation ϕ : D(F −, −) −→= C(−,G−) is an equivalence if and only if there are natural transformations ε : 1 → GF and η : FG → 1 unique up to isomorphisms such that

ηF ◦ F ε = 1F and Gη ◦ εG = 1G

When this happens, we will call the natural transformations ε the unit and η the counit of the adjunction ϕ, respectively.

Proof. For the necessity, assume ϕ : F → G is an adjunction. Let α : X0 → X be a morphism in C and β : Y → Y 0 be a morphism in D. Then since ϕ is a natural transformation, we have commutative diagrams as follows

ϕXY D(FX,Y ) / C(X, GY )

(F α)∗ α∗

ϕX0Y D(FX0,Y ) / C(X0, GY )

β∗ (Gβ)∗ D(FX0,Y 0) / C(X0, GY 0) ϕX0Y 0

∗ ∗ Hence for each f ∈ D(FX,Y ) we have (Gβ)∗ ◦ α ◦ ηXY (f) = ηX0Y 0 ◦ β∗ ◦ (F α) (f), or equivalently, the naturality of η is the same as the following identity

Gβ ◦ ϕXY (f) ◦ α = ϕX0Y 0 (β ◦ f ◦ F α)

22 Let εX = ϕ(1FX ). Then by the naturality of ϕ the following diagram commutes

ϕ D(FX,FX) / C(X, GF X)

∗ (F α) (GF α)∗ ϕ D(FX,FY ) / C(X, GF Y ) O O ∗ (F α) α∗

D(FX,FY ) ϕ / C(X, GF Y )

where α : X → Y is in C. In particular, GF α ◦ εX = η(F α) = εY ◦ α,

εX X / GF X

α GF α Y / GF Y εY

−1 i.e. ε : 1 → GF is a natural transformation. By setting ηY = η (1GY ), we obtain similarly that η : FG → 1 is a natural transformation, such that ηF ◦ F ε = 1 and Gη ◦ εG = 1.

Indeed, for every X ∈ C we deduce that ϕ(ηFX ◦ F εX ) = ϕ(ηFX ) ◦ εX = εX = ϕ(1FX ).

Thus ηFX ◦ F εX = 1FX . The other property is proved similarly.

For the sufﬁciency, suppose we have two natural transformations ε : 1 → GF and

η : FG → 1 such that the two conditions hold. Then deﬁne

ϕ(f) = Gf ◦ εX for f ∈ D(FX,Y )

and

−1 ψ(g) = ϕ (g) = ηY ◦ F g

for g ∈ C(X, GY ). Firstly we have to check the naturality of ϕ and ψ. In fact, by deﬁnition

ϕ(β◦f◦F α) = G(β◦f◦F α)◦εX0 = Gβ◦Gf◦GF α◦εX0 = Gβ◦Gf◦εX ◦α = Gβ◦ϕ(f)◦α,

23 where the third equality uses the naturality of ε. Similarly, the naturality of η implies the naturality of ψ.

Next we show that such deﬁned natural transformations ϕ and ψ are inverses to each other. Indeed, ψϕ(f) = ψ(Gf ◦ εX ) = ηY ◦ F (Gf ◦ εX ) = ηY ◦ F Gf ◦ F εX = f ◦

ϕFX ◦ F εX = f, where the fourth equality is the naturality of η and the last equality is our assumption. Similarly, ϕψ = 1 so that ϕ is an adjunction.

For the uniqueness, suppose ε0 : 1 → GF and η0 : FG → 1 are the unit and counit

0 of the adjunction ϕ deﬁned as above. Then εX = η(1FX ) = G(1FX ) ◦ εX = εX , and

0 ηY = ξ(1GY ) = ηY ◦ F (1GY ) = ηY .

Proposition 2.4.3. The adjointness is unique up to natural equivalences. Explicitly, suppose ϕ : F a G and ϕ0 : F a G0 are adjunctions between the functors F : C → D and

G, G0 : D → C. Then there is a natural equivalence θ : G → G0.

Proof. For every Y ∈ D we have a natural equivalence

ϕ0ϕ−1 : C(−, GY ) → D(F −,Y ) → C(−,G0Y ).

Then the Yoneda Lemma (see e.g. Chapter III.2 in [18]) implies that there is a morphism

0 0 −1 θY : GY → G Y corresponding to the natural transformation ϕ ϕ . In particular, θY is an isomorphism since ϕ0ϕ−1 is a natural equivalence. Therefore, θ : G → G0 is a natural

0 −1 equivalence since ϕ ϕ is also natural in the variable Y .

24 2.5 Kan extension

The Kan extension is a more general construction than that of adjoint functors. In a

speciﬁc situation, we need not only its existence but also for it to be pointwise in the sense

that it can be expressed in a form of a (co)limit. As one application, we obtain a formal

criterion for the existence of a (right) adjoint functor. One can compare it with the Special

Adjoint Theorem, see Theorem V.8.1 in MacLane [18] for example.

Deﬁnition 2.5.1. Given a pair of functors F : A → B and K : A → C, a left Kan extension

of K along F is deﬁned to be a pair

K A / C ? F Lan K, or Ran K F F B

consisting of a functor LanF K : B → C and a natural transformation ε : K → LanF K ◦ F

called unit so that for any other pair (H : B → C, µ : K → H ◦ F ), there exists a unique

arrow ξ : LanF K → H making the diagram

ε K / LanF K ◦ F

ξF µ % H ◦ F commute. In other words, in appropriate functor categories the left Kan extension is left adjoint to the functor deﬁned by precomposing K, that is, a bijection

∼ HomCB (LanF K,H) = HomCA (K,H ◦ F )

natural in H, under which the unit ε is the image of 1LanF K . Thus when it exists the

left Kan extension is unique up to an isomorphism. Dually, a right Kan extension of K

25 along F is a pair consisting of a functor RanF K : B → C and a natural transformation

η : RanF K ◦F → K called counit such that for any other pair (S : B → C, δ : S◦F → K) there is a unique arrow ζ : S → RanF K such that the diagram

δ S ◦ F / K 9 ζF η RanF K ◦ F commutes. That is, a bijection

∼ HomAC (S ◦ F,K) = HomBC (S, RanF K)

natural in S, under which the counit η is the preimage of 1RanF K .

Proposition 2.5.2. Let F : A → B be a functor. Suppose it has a right adjoint G. Then F preserves all left Kan extensions which exist in A.

Proof. The adjunction B(F A, X) ∼= A(A, GX) allows us to obtain a bijection

BD(F L, S) ∼= AD(L, GS) for functors L : D → A and S : D → B by evaluating at the objects A = LA0 and

0 X = SX . Therefore, given any left Kan extension with unit ε : H → LanK H ◦ K

H F C / A / B > K Lan H K D we have

D ∼ D ∼ C B (F LanK H,S) = A (LanK H, GS) = A (H, GSK)

∼ C ∼ D = B (FH,SK) = B (LanK FH,S)

26 so that F LanK H = LanK FH with unit given by the image of the identity when S =

F LanK H under the composition of the ﬁrst three bijections, which turns out to be F ε.

Deﬁnition 2.5.3. Let K : A → C and F : A → B be functors. A left Kan extension

L = LanF K is pointwise if it is preserved by all corepresentable functors C(−,C) for all

C ∈ C.

Proposition 2.5.4. A functor K : A → C has a pointwise left Kan extension L = LanF K along the functor F : A → B if and only if the colimit of the composition functor KP :

(F ↓ X) → A → C exists for all X ∈ B.

Proof. Since C(−,C) preserves colimits, thus any left Kan extension given in Proposi- tion 2.5.5 as a colimit is pointwise. Conversely, suppose L = LanF K is pointwise and it is preserved by all C(−,C) for C ∈ C

K C(−,C) A / C / Set ? 7 L F C L B

C so that L = LanF C(K−,C) = C(L−,C). Then there is a bijection

[LC ,H] ∼= [C(K−,C),HF ] natural in H : B → Set and in particular when H = B(−,X) we have

LC X ∼= [LC , B(−,X)] ∼= [C(K−,C), B(F −,X)] ∼= Cone(KP,C) so that the set of colimit cones is representable. Hence the colimit of KP :(F ↓ X) →

A → C exists, where the ﬁrst isomorphism holds by the Yoneda Lemma and the last one is

27 shown as follows. Each cone τ : KP → C assigns A0 ∈ A and a map f 0 : FA0 → X an arrow τ(A0, f 0): KA0 → C subject to the naturality condition

τ(A, f 0 ◦ F h) = τ(A0, f 0) ◦ Kh for h : A → A0. And each natural transformation α : C(K−,C) → B(F −,X) assigns

0 0 0 0 0 A ∈ A and a map g : KA → C an arrow αA0 (g ): FA → X subject to the naturality condition

0 0 αA(g ◦ Kh) = αA0 (g ) ◦ F h

0 for h : A → A . The bijection follows immediately by assigning τ to α.

The following proposition allows us to compute the left Kan extension explicitly when it is pointwise.

Proposition 2.5.5. Let F : A → B and K : A → C be functors. Suppose the colimit of the functor KP exists in C for all X ∈ B, where P :(F ↓ X) → A is the projection and let

LX = Colim KP = Colim(F ↓ X) →P A →K C with colimit cone λ. Then each g : X → X0 induces a map Lg : LX → LX0 and with the

natural transformation deﬁned by εA = λ1FA , the pair (L, ε) is a left Kan extension of K along F .

Proof. Consider for each f : FA → X with f 0 = gf the diagram

λf KA / LX

Lg KA / LX0 λ0 f0

28 0 Since λf 0 is a cone and λ is universal, there is a unique arrow Lg making the diagram

commute.

Next for h : A → A0 in A form the following diagram

εA=λ1FA KA / LF 0 λF h Kh LF h 0 ( 0 KA 0 / LF A εA0 =λ 1FA0 By deﬁnition of L, the upper triangular commutes and the lower one commutes also thanks

to the naturality of λ0. Thus the outer square commutes which gives the naturality of ε.

Now suppose there is another pair (H, µ : K → HF ). Deﬁne ξ as in the following

diagram for f : FA → X and h : A → A0 with f = f 0 ◦ F h

λ0 Kh f0 KA / KA0 / LX

µA µA0 ξX $ HFA / HFA0 / HX HF h Hf 0 The left square is commutative by the naturality of µ. Since the diagonal composition

0 0 Hf ◦ µA0 is a cone while λ is universal, there is a unique arrow ξX making the diagram

commute. We need to show that ξ is natural and ε is universal with respect to this property.

In fact, consider the diagram

KA λf µA λ0 Lg z f0 LX s / LX0 HFA Hf ξ ξ 0 X X 0 z Hf HX s / HX0 Hg where f 0 = gf and f = f 0 ◦ F h. Notice that by deﬁnition of ξ the back square and the right hand square commute, and the top triangle commutes by deﬁnition of L. Thus the

29 front square commutes when precomposed with λf for every f, hence it is commutative

since λ is a colimit cone. Therefore, ξ is natural. It is clear from the deﬁnition of ξ that

µ = ξ ◦ ε by evaluating X = FA0. It is straightforward to check that such ξ is unique and

ε is universal.

The next proposition gives a formal necessary and sufﬁcient condition for the existence

of an adjoint functor. One can compare it with the (special) adjoint functor theorems.

Proposition 2.5.6. Let F : A → B be a functor. It has a right adjoint if and only if the left

Kan extension (LanF 1A, ε) exists and it is preserved by F . In this case, the unit ε coincides

with that of the adjunction F a G.

Proof. For necessity, suppose F has a right adjoint with unit ε : 1 → GF and counit

η : FG → 1. We ﬁrst show that there is for any functor H : B → C a bijection

∼ ϕ : HomAC (S,HF ) = HomBC (SG, H): ψ

natural in S : A → C, where ϕ(α) = Hη ◦ αG and ψ(β) = βF ◦ Sε for natural transfor-

mations α : S → HF and β : SG → H. In fact, the commutativity of the outer square of

the diagram

αGF HηF SGF / HF GF / HF O O Sε HF ε

S α / HF HF shows that ψϕ(α) = α, where the right square commutes because ηF ◦ F ε = 1H and the left square commutes by deﬁnition of the horizontal compositions α ◦ ε of natural transformations. The other identity ϕψ = 1 can be shown similarly.

30 The speciﬁcation of the value S leads us to the result. For S = 1A,

∼ ∼ Hom(LanF 1A,H) = Hom(1A,HF ) = Hom(G, H)

so that G = LanF 1A with unit the image ε of 1G. Similarly, for S = F

∼ ∼ Hom(LanF F,H) = Hom(F,HF ) = Hom(F G, H)

identiﬁes FG = LanF F together with unit F ε.

Conversely, suppose the left Kan extension (L = LanF 1A, ε) exists and is preserved by

F . In other words, we have natural bijections

∼ ξH : Hom(L, H) = Hom(1A,HF ), ξH (α) = αF ◦ ε

and

∼ δS : Hom(F L, S) = Hom(F,SF ), δS(β) = βF ◦ F ε

natural in H : B → A and S : B → B. Obverse that ε = ξL(1F ) and δFL(1FL) = F ε.

η = δ−1(1 ) δ Then we deﬁne 1B F as the candidate counit. It follows by the deﬁnition of that

1F = δ1B (η) = ηF ◦ F ε

which is one of the identities of adjunction. For the other one Lη ◦ εL = 1L, it is sufﬁcient

to show when ξL applied to it. That is,

−1 ε = ξL(1L) = ξL(Lη ◦ εL) = (Lη ◦ εL)F ◦ ε = LηF ◦ εLF ◦ ε = (LF ε) ◦ εLF ◦ ε

depicted as

ε 1 / LF

ε LF ε LF / LF LF εLF which is commutative by deﬁnition of the horizontal composition ε ◦ ε.

31 2.6 Localization functor

The existence of a localization functor plays an essential role in our theory which allows

us to deﬁne the restriction map of a sheaf of modules in Chapter 4. In general, the quotient

functor Q : A → A/S with a Serre subcategory S does not necessarily have a realization

as a localization functor, depending in fact on the existence of a right adjoint functor of the

quotient Q. We can obtain such a right adjoint easily in the example of category of quiver

representations of ﬁnite type, kA2-rep for example, by the fact that every object there has an injective resolution, although we can still use the pointwise Kan extension which applies to a more general situation. Most of the material here can be found in Neeman’s appendix [20] and Gabriel’s paper [7]. Krause’s paper [15] is also a good reference.

Lemma 2.6.1. For a pair of adjoint functors F a G : A → B, the counit η : FG → 1 is an isomorphism if and only if G is fully faithful.

Proof. Denote by ε : 1 → GF the unit of the adjunction. By the adjointness, the following diagram

∗ ηX B(X,Y ) / B(F GX, Y )

ϕ =∼ G ' A(GX, GY ) for any X,Y ∈ B is commutative, where ϕ is the adjunction of F a G. Indeed, for

f ∈ B(X,Y ) we have

G(f ◦ ηX ) ◦ εGX = G(f) ◦ G(ηX ) ◦ εGX = G(f) ◦ 1GX = G(f).

32 ∗ Hence G is fully faithful if and only if ηX is an isomorphism for all X and Y , that is, η is an isomorphism.

Deﬁnition 2.6.2. Let A be an abelian category and S a Serre subcategory. An object

1 X ∈ A is S-local if both A(A, X) = 0 and ExtA(A, X) = 0 hold for all A ∈ S.

Lemma 2.6.3. Let F : A → B be an exact functor between abelian categories. Suppose it has a right adjoint G. Denote S = {A ∈ A | FA ∼= 0}. Then any X ∈ A such that

X ∼= GY is S-local.

Proof. For A ∈ S, we have

A(A, X) ∼= A(A, GY ) ∼= B(F A, Y ) = 0

Now given an extension

0 → X →α E → A → 0

Since A ∈ S, then FX ∼= FE by applying the exact functor F . The commutative diagram

=∼ B(FE,Y ) / B(FX,Y )

=∼ =∼ A(E,X) / A(E, GY ) / A(X, GY ) / A(X,X) =∼ α∗ =∼

∗ implies that α is also an isomorphism. That is, α splits as required.

Lemma 2.6.4. Let A be an abelian category with a Serre subcategory S. Suppose X ∈ A is S-local. Then the natural map

A(A, X) ∼= A/S(A, X) is an isomorphism for all A ∈ A.

33 Proof. First observe that given a map f : A → X in A/S

A / X O

A0 / X/X0

with A/A0,X0 ∈ S, the S-locality of X implies that X0 ,→ X is zero thus X → X/X0 is an isomorphism. Hence any such map f in A/S can be represented by a map f : A0 → X

in A.

Now for the surjectivity, consider the pushout depicted as the lower left corner square in the diagram

f A / X O = ? 0 f A _ / X _ = A g / Y s / X Since A0 ,→ A is monic and A/A0 is S-local X → Y is also monic with cokernel isomor-

phic to A/A0, thus the short exact sequence

0 → X → Y → A/A0 → 0 splits and choose s : Y → X as one retract. Then the above commutative diagram implies that sg is the preimage of f.

34 For the injectivity, let f : A → X in A with image in A/S represented by g

g A / X O = ? g A0 / X _ O i = f A / X < p h A/A0

which is zero. Then the commutativity of the middle square says that fi = 0 so that f

factors through p. Again h = 0 since X is S-local. Therefore, f = 0.

Lemma 2.6.5. Let Q : A → A/S be the projection of an abelian category A with respect

to a Serre subcategory S. Assume Q admits a right adjoint G. Then the counit η of the

adjunction is an isomorphism.

Proof. Let A ∈ A and X ∈ A/S. Then since GX is S-local, we have in the commutative

diagram

Q A(A, GX) / A/S(QA, QGX) =∼ =∼ (η ) u X ∗ A/S(QA, X) by the adjunction an isomorphism (ηX )∗, thanks to Lemma 2.6.4. Since every object in

A/S has the form QA, η is an isomorphism.

Proposition 2.6.6. Let A be an abelian category S its Serre subcategory. Suppose the projection Q : A → A/S has a right adjoint G. Then G is fully faithful, the counit η is an isomorphism, and the image of G consists of those S-local objects. Furthermore, any

35 exact functor between abelian categories admitting a right adjoint can be realized as a

projection.

Proof. By Lemma 2.6.5 and Lemma 2.6.1, the counit η is an isomorphism and G is fully

faithful.

Next we will show for an S-local object X the unit of the adjunction

εX : X → GQX

−1 is an isomorphism. In fact, it has a two-sided inverse given by the preimage Q ηQX of the

counit ηQX via the isomorphism

A(GQX, X) ∼= A/S(QGQX, QX)

−1 thanks to Lemma 2.6.4. Indeed, since Q(Q ηQX ◦ εX ) = ηQX ◦ QεX = 1QX = Q(1X )

and an isomorphism

A(X,X) ∼= A/S(QX, QX)

−1 −1 by Lemma 2.6.4, then Q ηQX ◦ εX = 1X . Similarly, Q(εX ◦ Q ηQX ) = QεX ◦ ηQX =

1QGQX = Q(1GQX ). Also since GQX is also S-local by Lemma 2.6.3, there is an isomor-

phism

A(GQX, GQX) ∼= A/S(QGQX, QGQX)

−1 by Lemma 2.6.4 again, which implies that εX ◦ Q ηQX = 1GQX .

The proof of the last statement is omitted here which can be found as Proposition A.2.12

in Neeman [20].

36 Here is an efﬁcient criterion for the existence of right adjoint functor, provided in the

abelian category A that every object X ∈ A can be embedded into an injective object.

Proposition 2.6.7. Let Q : A → A/S be the projection with respect to a Serre subcategory

S. Suppose A has enough injectives. Then Q has a right adjoint G if and only if every

injective object I ∈ A has a maximal S-subobject.

Proof. Suppose Q a G is an adjoint pair and X ∈ A. We claim that the kernel KerεX

of the unit εX : X → GQX is a maximal S-subobject. Clearly KerεX ∈ S since QεX :

0 QX → QGQX is an isomorphism with inverse ηQX : QGQX → QX. Suppose X ⊆ X

and X0 ∈ S. Then the composition X0 ,→ X → GQX is zero since X0 is S-local, thus

0 X ⊆ KerεX which gives the necessity.

For the sufﬁciency, ﬁrst observe that the condition implies in fact every object X in A has a maximal S-subobject since we can always embed X into an injective object I so that

I0 ∩ X ⊆ X will do the job, where I0 is a maximal S-subobject of I.

Next we will construct the adjoint G explicitly as follows. For X ∈ A, consider its injective resolution of length 2

0 → X/X0 → I → J

where X0 ⊆ X is a maximal S-subobject. Then we deﬁne G in the following commutative diagram

0 / X/X0 / I / J

α 0 / GX / I/I0 / J/J0

37 and it is clearly functorial. Applying Q to the commutative diagram, we deduce that the

induced map X/X0 → GX becomes an isomorphism in A/S by the 5-Lemma. So does

the composition β : X → X/X0 → GX. Thus we assign for any f : A → GX in A a

map

Q(f) Q(β)−1 QA −→ QGX −→ QX = X

in A/S. We claim that such assignment

A(A, GX) → A/S(QA, X)

is a natural bijection. For surjectivity, let’s represent a morphism f : QA → X in A/S by

f : A0 → X/X0

X0 / X _ 0_

f = A / X / X O

? A0 / X/X0 / / X/X f 0

0 0 0 with A/A ,X ∈ S. Since X ∈ S we can embed it into the maximal S-subobject X0 of X.

Then the commutativity of the diagram allows us to represent f by the bottom composition

0 A → X/X0. Now by the injectivity of I, there is a map A → I hence an induced map

A/A0 → J making the following extended diagram

0 / A0 / A / A/A0 / 0

g 0 / X/X0 / I / J

α Ô 0 / GX / I/I0 / J/J0

38 commutative in which the rows are exact. Notice that the right column composition A/A0 →

0 J/J0 is zero since A/A is S-local, thus so is A → I → I/I0 → J/J0. Therefore, the middle composition A → I/I0 factors through a map g : A → GX as the preimage of f. In fact, Q(g) = Q(β) ◦ f holds as the following commutative diagram shows

Q(β)◦f A / GX G O

= ? 0 f α = Ao / X/X0 / GX/Im(β) O f f

A g / GX where the middle row A0 → GX/Im(β) represents the composition Q(β) ◦ f.

For the injectivity, suppose Q(β)−1 ◦ Q(f) = 0 for a given f : A → GX. Then

Q(f) = 0 so that Im(f) is an S-subobject of GX and in particular an S-subobject of I/I0.

Thus Im(f) = 0 and f = 0.

Now we are ready to realize the quotient functor Q : A → A/S with respect to a Serre subcategory as a localization functor in the sense of an endofunctor L from A to itself such that the essential image of L is equivalent to A/S. This material can be found in Krause’s paper [15].

Deﬁnition 2.6.8. A localization is a functor L : A → A with a natural transformation

ε : 1A → L such that Lε is a natural equivalence and εL = Lε.

Lemma 2.6.9. The pair (L : A → A, ε : 1 → L) is a localization if and only if there exists an adjoint pair F a G : A → B with G fully faithful such that L = GF and ε is the unit of the adjunction.

39 Proof. Suppose (L, ε) is a localization. Let B be a full subcategory of A consisting of the

objects

B = {A ∈ A | εA is an isomorphism}

−1 Thus for any X ∈ B we have an inverse εX . Then deﬁne F : A → B by FA = LA and

G : B → A the inclusion. It is straightforward to check

ϕ : B(F A, X) → A(A, GX): ψ

−1 given by ϕ(f) = Gf ◦ εA and ψ(g) = εX ◦ F g is an adjunction. Indeed, the commutative diagram

f FA / X

F εA εX F GF A / F GX F Gf due to the naturality of ε and Lε = εL implies ψϕ(f) = f, and the commutative diagram

g A / GX

εA GεX GF A / GF GX GF g thanks to the naturality of ε again and Gε = εG implies ϕψ(g) = g.

Conversely, suppose F a G : A → B is a adjoint pair in which G is fully faithful with

unit ε and counit η. Then by Proposition 2.4.2 we have ηF ◦ F ε = 1F and Gη ◦ εG = 1G.

Furthermore, by Lemma 2.6.1 the counit η is an equivalence. Thus

Lε = GF ε = G(ηF )−1 = (GηF )−1

is also an equivalence. Now we check

(GηF ) ◦ (εL) = (GηF ) ◦ (εGF ) = (Gη ◦ εG)F = 1GF = 1GF

40 which implies Lε = εL, as required.

41 Chapter 3

Monoform and premonoform objects

In this chapter, we attempt to obtain a classification of nullity classes in an abelian category A by studying the premonoform object, which is a generalization of monoform object. Specifically, we show that the nullity classes generated by premonoform objects are classified by the closed and extension closed subsets of the spectrum SpecA consisting of

equivalence classes of premonoform objects, via the support.

We organize the chapter as follows. After introducing the concept of monoform objects

in Section 3.1, we move into the area of premonoforms and use examples to give a com-

parison with similar notions in Section 3.2. Later, a new spectrum of an abelian category

is deﬁned in Section 3.3 with underlying space of equivalence classes of premonoform

objects, which allows us to classify nullity classes in the last section.

In the category of modules over a commutative noetherian ring with identity, monoform

and premonoform coincide. The earliest references on these two concepts we can ﬁnd are

Storrer [28] in 1972 and Gordon and Robson [8] in 1973. We also refer to Kanda [13] and

42 Lam [16].

3.1 Monoform objects

Recall that a subcategory in an abelian category is called a Serre subcategory if it is a full abelian subcategory closed under taking subobjects, quotients and extensions. See Sec- tion 2.1. It is natural to use monoform objects to give a classiﬁcation of Serre subcategories, see [13], whose deﬁnition relies on the construction of these categories.

Deﬁnition 3.1.1. A nonzero object M in an abelian category A is monoform if for every nonzero subobject N ≤ M there is no nonzero common subobjects between M and M/N.

As monoform object is introduced abstractly in the context of abelian category, we will brieﬂy describe it in the context of category of modules over a ring as a special case, which are called strongly uniform modules in [28].

Recall that a nonzero module M is strongly uniform if it is a rational extension (see

Section 1 in [28]) of every nonzero submodule. In a rational extension 0 $ N ⊆ M of modules, the submodule N is also called dense in M. See Section 8 in [16].

Deﬁnition 3.1.2. Let N ⊆ M be two R-modules. We say that it is a rational extension if that for every N ⊆ A ⊆ M and every f : A → M with N ⊆ Kerf implies f = 0. In this case, we also call N a dense submodule of M. A nonzero module M is strongly uniform if it is a rational extension of every nonzero submodule of itself.

43 We have several equivalent descriptions of rational extensions, in which some of them

have the advantage to be phrased using categorical language.

Lemma 3.1.3. For the following statements,

(1) N ⊆ M is a rational extension;

(2) HomR(M/N, E(M)) = 0 where E(M) is an injective hull of M;

(3) For every m, m0 ∈ M with m0 6= 0, there is r ∈ R such that mr ∈ N and m0r 6= 0;

(4) For every N ⊆ A ⊆ M, HomR(A/N, M) = 0;

(5) There are no common nonzero submodules between M and M/N,

(1)-(4) are equivalent, and they all imply (5). In particular, rational extensions are essential extensions.

Proof. The equivalence of (1), (2), (3) is proved as Lemma 1.1 in [28], while that of (1),

(2), (4) is proved as Proposition 8.6 in [16]. It is easy to see (4) implies (5) since if there is a common nonzero subobject beween M and M/N, say A/N = H0 ∼= H,→ M, then the composition of these maps gives a nonzero f : A/N → M.

Proposition 3.1.4. In the category of modules over a ring, a module is monoform if and only if it is strongly uniform.

Proof. Assume M is monoform. Suppose there is submodule N ⊆ M giving a nonrational extension. That is, by proposition 8.6 in [16], for every N < A < M there is a nonzero map f : A/N → M with Kerf ∼= A0/N for some A0 < A. Then A/A0 ∼= (A/N)/Kerf ,→ M provides a common nonzero object between M and M/A0, a contradiction. For the converse, suppose M is not monoform so that A/N = H0 ∼= H,→ M for some submodules

44 N < A of M, then the composition of these maps gives a nonzero f : A/N → M, so that

M has a nonrational extension N ⊆ M.

As it is shown in Lemma 1.5 in [28] that for a commutative ring R, the strongly uniform modules thus monoforms, are those of the form R/p with p a prime ideal, noticing that a monoform module is necessarily cyclic (i.e. has a single generator).

Proposition 3.1.5. Let R be a commutative ring. Then an R-module R/p is strongly uniform if and only if p is a prime ideal.

In the category of graded modules over a graded ring, we have a similar description.

Proposition 3.1.6. For a graded commutative ring S, the quotient S/p is strongly uniform if and only if p is a homogeneous prime ideal.

Proof. Let a, b ∈ S − p be homogeneous elements and suppose (b) + p/p ⊆ S/p is a rational extension. Take 1+p, a+p ∈ S/p, then there is s ∈ S such that s(1+p) ∈ (b)+p and s(a + p) 6= p, that is, s − tb = p for some t ∈ S, p ∈ p and sa∈ / p. Hence sa = tab + pa∈ / p, and particularly, ab∈ / p. Conversely, it is sufﬁcient to show that

(b) + p/p ⊆ S/p is a rational extension for every b∈ / p. Indeed, for every a∈ / p, s ∈ S, we always have (a + p)b∈ / p and (s + p)b ∈ (b) + p/p since p is prime.

Now we extend the notion of monoform to study a certain kind of subcategory of an abelian category, called a nullity class. See for example [27]. Furthermore, if we require that the nullity class is cocomplete in the sense that it is closed under arbitrary coproducts, it is then usually called a torsion class.

45 Deﬁnition 3.1.7. Let A be an abelian category and N a full subcategory of A. Then N is called a nullity class if

(1) for any exact sequence M → N → 0, M ∈ N implies N ∈ N ;

(2) for any exact sequence 0 → N 0 → M → N → 0, N 0 ∈ N and N ∈ N imply M ∈ N .

Notice that as a special quotient, any nullity class N is closed under taking retract, that is whenever A ⊕ B ∈ N , A ∈ N as well thanks to the projection pA : A ⊕ B → A.

For every collection S of objects in an abelian category A, we can generate a new subcategory by taking quotients and extensions of objects in S. Deﬁne a full subcategory

hSiquot = {X ∈ A | X is a quotient of some M ∈ S}.

For another, we set S0 as the class of the zero object and deﬁned for S, S0 collections of

objects in A,

S ∗ S0 = {E ∈ A | 0 → X → E → X0 → 0, for some X ∈ S,X0 ∈ S0}.

as a full subcategory. In particular, we denote S1 = S and deﬁne Sn = S∗Sn−1 recursively.

Also deﬁne as a full subcategory

[ n hSiext = S . n≥0

Lemma 3.1.8. Let A be an abelian category and M,N ∈ A. Then

(1) the operation ∗ on collections of objects is associative;

(2) hM ∗ Niquot ⊆ hMiquot ∗ hNiquot;

(3) the category hhSiquotiext is a nullity class.

46 Proof. Both properties hold thanks to the snake lemma. For a proof see (2) and (4) in

Proposition 2.4 in [13]. For (3), it is sufﬁcient to show that this category is closed under

taking quotients, and the result follows immediately from (2).

3.2 Premonoform objects

In this section, we introduce the premonoform objects and their basic properties in an

abelian category, then use several concrete examples to illustrate the relationship between

this concept and those in the category of modules over a (non)commutative ring with iden-

tity.

3.2.1 Deﬁnitions and properties

A monoform object in an abelian category is deﬁned as the one that admits no non-

trivial subquotient (the subobject quotiented by should be nonzero) as its subobject, or

equivalently, there is no nontrivial map from any nontrivial subquotient to itself.

Similarly, we deﬁne a premonoform object in an abelian category by replacing subquo-

tient by simply quotient.

Deﬁnition 3.2.1. An object M 6= 0 in an abelian category A is premonoform if it contains no nontrivial quotient as its subobject, meaning that there is no injection from its nontrivial quotient M/N to M, i.e. 0 N M so that M/N 6= 0 or M.

Clearly, the monoform objects are in particular premonoform by deﬁnition.

47 Lemma 3.2.2. An object M is premonoform if and only if there are no nonzero maps

from its nontrivial quotients to M, i.e. HomA(M/N, M) = 0 for every nonzero subobject

N M.

Proof. Suppose M is not premonoform, then there is a nontrivial quotient embedding into

M. Conversely, if there is a nonzero map f : M/N → M for 0 6= N < M, then by the

∼ ∼ isomorphism theorem there is N < A M such that M/A = (M/N)/(A/N) = Imf ,→

Lemma 3.2.3. An object M is premonoform if and only if for any endomorphism f of M, it is either zero or injective.

Proof. Let M be a premonoform object and f a nonzero endomorphism of M. Then N = ker f is a nonzero subobject of M such that f induces an injection M/N ∼= Imf ,→ M.

Conversely, suppose for any subobject N < M there is an injection M/N → M. Then the endomorphism f : M → M/N → M is either zero or an injection. Hence either N = M or N = 0. Therefore, M is premonoform.

There are many examples of premonoform objects in the category of modules over a ring.

Proposition 3.2.4. In the category of finite dimensional representations of quivers of finite type over a field of characteristic zero, the three concepts monoform, premonoform and indecomposable coincide.

Proof. Thanks to the theory of Auslander-Reiten quiver, any nontrivial (sub)quotients of an indecomposable object appear only on its right hand side so that there are no arrows

48 backwards, where the AR quiver is drawn in the way that the AR translations start from

left to right, see Chapter IV.4 in [1] for example.

Hence in the representation theory of nice quivers, the indecomposable projectives, in-

jectives and simples provide many examples of premonoform objects. Also in commutative

ring theory, the premonoform and the monoform are identical.

Proposition 3.2.5. In the category of ﬁnitely generated modules over a commutative noethe-

rian ring with identity, a module M is premonoform if and only if it is monoform.

Proof. It is clear that monoform implies premonoform. Now let M be premonoform and

suppose it is not monoform. Then there exists an H 6= 0 as a common submodule of both

M and X = M/N for some nonzero N < M by deﬁnition. Thus there is an associated

prime ideal p ∈ AssH such that R/p ,→ H. In particular, Xp 6= 0 since p ∈ AssH ⊆

∼ AssX ⊆ SuppX. Passing to the local case at p, we obtain on the one hand that Xp ⊗R/p =

∼ ∼ L Xp ⊗ Rp ⊗ R/p = Xp ⊗ k(p) = k(p) since Xp is ﬁnitely generated, and on the other

∼ hand Xp ⊗ R/p = Xp/pXp which is nonzero by Nakayama’s Lemma. Therefore, we have

an exact sequence

f M 0 → pXp → Xp → k(p) → 0

∼ in which f is nonzero. Moreover, composed with a projection onto k(p) = (R/p)p, f can

be lifted into a nonzero map X → R/p since X is ﬁnitely generated. Hence there is a

nonzero map M/N = X → R/p ,→ M, a contradiction by Lemma 3.2.2.

Therefore, Section 3.1 provides lots of examples of premonoform objects as well.

49 3.2.2 A premonoform but not monoform object

It is interesting to ﬁnd in an abelian category a premonoform object that is not mono-

form. Let T (V ) be the tensor algebra of a vector space V over a ﬁeld k, with basis {a, b}.

L∞ ⊗i Consider T (V ) as a left module over itself. It is known that T (V ) = i=0 V is a free associative algebra on the generators a, b, also not noetherian. In this subsection, we will show that T (V ) is a premonoform but not monoform object in the category of left T (V )-

modules.

Lemma 3.2.6. Any map T (V ) →·x T (V ) of right multiplication by a nonzero x ∈ T (V ) is injective. Therefore, every endomorphism of T (V ) is either zero or injective.

Proof. Let y ∈ T (V ). Suppose x = u1 + ... + un with each ui 6= 0 and y = v1 + ... + vm

⊗ni ⊗mj for ui ∈ V and vj ∈ V , where ni < ni+1 and mj < mj+1 for all i, j. If yx = 0, then the highest term is vmun = 0 by linearly independence of basis vectors. Therefore, it sufﬁces to show that with respect to the product by concatenation

V ⊗s × V ⊗t → V ⊗(s+t)

there are no zero divisors, so that vm = 0 and y = v1 + ... + vm−1. Then induction implies that y = 0. Indeed, in general the tensor product of vector spaces has a basis given by the tensor product of basis elements respectively. Thus for any two vectors α ∈ V ⊗s and

⊗t β ∈ V , we can assume α = a1e1 + ... + apep and β = b1f1 + ... + bqfq with nonzero ai, bj ∈ k and ei, fj basis elements, then αβ = 0 implies that in particular a1bj = 0 for all j. Hence bj = 0 and β = 0, as we required.

50 Therefore, T (V ) is premonoform by Lemma 3.2.3 as a left module over itself.

Lemma 3.2.7. Let T (V )/T (V )b be the cokernel of the right multiplication T (V ) →·b T (V ) by b. Then the map T (V ) →·a T (V )/T (V )b is injective.

Proof. Let x ∈ T (V ) with x = x1e1 + ... + xnen the linear combination of basis elements.

Suppose xa + T (V )b = T (V )b, then there is y ∈ T (V ) with y = y1f1 + ... + ymfm the linear combination of basis elements such that xa − yb = 0. Then

x1e1a + ... + xnena − y1f1b − ... − ymfmb = 0

implies that x1 = ... = xn = y1 = ... = ym = 0 since {e1a, ..., ena, f1b, ..., fmb} is part of the basis of T (V ) with no repetition by word concatenation. Therefore, x = 0.

Hence T (V ) is not monoform as a left module over itself since the nontrivial quotient

T (V )/T (V )b has T (V ) as its submodule.

To conclude this section, we compare some interesting notions in the category of modules over a (non)commutative ring with that of premonoform.

A module is uniform if any two nonzero submodules have nonzero intersection, or e- quivalently, it is an essential extension of every nonzero submodule. Thus, strongly uniform implies uniform since any rational extension is also essential. See Lemma 1.1 in [28].

Comparing with the notions of simple and indecomposable modules, we have the following relationship in the category of modules over a (non)commutative ring with identity.

Proposition 3.2.8. Let M be a module over a (non)commutative ring with identity. Then

51 for the following concepts, M is (1) simple; (2) strongly uniform; (3) monoform; (4) premonoform; (5) uniform; (6) indecomposable, we have

(4) :B

$ (1) +3 (2) ks +3 (3) (6) :B

$ (5) where (4) and (5) are not comparable.

Proof. The equivalence of (2) and (3) is Proposition 3.1.4. (1) ⇒ (3) holds because the simplicity of M gives no nontrivial subquotient of M. (2) ⇒ (5) and (3) ⇒ (4) are true by deﬁnitions. For (4) ⇒ (6), any decomposition M = M1 ⊕ M2 gives an

∼ injection M/M1 = M2 ,→ M. (5) ⇒ (6) is true since such a decomposition gives a trivial intersection M1 ∩ M2 = 0. The incomparable relation between (4) and (5) is given in

Example 3.2.9

The following examples show that the implications in Proposition 3.2.8 are all strict, and that (4), (5) are not comparable. Some of them can be found in [16].

Example 3.2.9. (1) Any simple R-module is of the form R/m for some maximal ideal m. Thus by Proposition 3.1.5, the quotient R/(x) of the polynomial ring R = Q[x, y] gives a strongly uniform R-module but not simple. Indeed, it has (x, y)/(x) as a nontrivial submodule.

(2) The cyclic group Z/4 is uniform but neither premonoform nor monoform, or in general we can consider the cyclic groups Z/pn of order pn for n > 1.

52 (3) The tensor algebra T (V ) in this section is a premonoform module over itself but not uniform since T (V )a ∩ T (V )b = 0 by the form of basis elements given by monomials in a, b, such as {a2, ba, ab, b2} for the basis of degree 2 monomials.

(4) The commutative Q-algebra R = Q[x, y]/(x2, xy, y2) is indecomposable as an R-

module but not uniform since it has a trivial intersection Rx ∩ Ry = 0. It is not pre-

monoform either since there is an R-module map f : R → R deﬁned by f(1) = x such that f(y) = 0, i.e. f is not injective.

3.3 Equivalence relation, support and topology

Let A be an abelian category. Denote by Spec0A the collection of isomorphism classes of premonoform objects in A. Denote by C(A) the smallest nullity class containing A ∈ A, that is, the intersection of all nullity classes containing A. We also say that C(A) is the nullity class generated by A.

Deﬁnition 3.3.1. Deﬁne A ∼ B if and only if they generate the same nullity class, i.e.

C(A) = C(B).

It is clear that this relation is an equivalence relation on Spec0A, and we denote by

SpecA = Spec0A/ ∼ the collection of equivalence classes H of premonoform objects,

called the spectrum of A. A sufﬁcient condition for the spectrum being a set is that the category A is essentially small, i.e. the isomorphism classes of objects A form a set.

For any M ∈ A, deﬁne the support of an object

SuppM = {H ∈ SpecA | ∃H0 ∈ H s.t. H0 ∈ C(M)}.

53 The support of a subcategory N ⊆ A is deﬁned as the union

[ Supp N = Supp M. M∈N

We can also deﬁne

Supp−1Φ = {M ∈ A | SuppM ⊆ Φ}

for any subclass Φ ⊆ SpecA. However, this is not always a nullity class for arbitrary sub-

class Φ but for those closed and extension closed ones, it is as we will see in Theorem 3.4.7.

Deﬁnition 3.3.2. A subclass Φ ⊆ SpecA is closed if for every M ∈ Φ, SuppM ⊆ Φ. It is

extension closed if whenever there is a short exact sequence

0 → M → X → N → 0,

SuppM ⊆ Φ and SuppN ⊆ Φ imply SuppX ⊆ Φ.

Lemma 3.3.3. Assume M,N are objects in an abelian category A. Then

(1) if M is premonoform, then M ∈ SuppM;

(2) if M is premonoform, then C(M) ⊆ C(N) if and only if SuppM ⊆ SuppN. In particular, if both M and N are premonoform, then M ∼ N if and only if SuppM =

SuppN;

(3) if M → N → 0 is exact, then SuppN ⊆ SuppM;

(4) the subclass SuppM is closed;

Proof. (1) Since M is premonoform and M ∈ C(M), thus M ∈ SuppM by deﬁnition.

For (2), M ∈ SuppM ⊆ SuppN implies that there is M 0 ∼ M such that M 0 ∈ C(N)

54 so that C(M) = C(M 0) ⊆ C(N). The converse is true by deﬁnition. (3) holds because

nullity class is closed under surjections. For (4), take M 0 ∈ SuppM. Then there is H ∈ M 0 such that H ∈ C(M), thus SuppH ⊆ SuppM by (2).

Lemma 3.3.4. A class Φ ⊆ SpecA is closed if and only if for every M ∈ Φ there is H ∈ A

such that M ∈ SuppH ⊆ Φ.

Proof. Necessity holds by deﬁnition. Conversely, suppose M ∈ SuppH ⊆ Φ for some H.

Then there is M 0 ∈ M such that M 0 ∈ C(H), hence SuppM = SuppM 0 ⊆ SuppH ⊆ Φ by Lemma 3.3.3.

Proposition 3.3.5. Assume A is an abelian category such that SpecA forms a set. Then the collection of closed subsets indeed forms a topology of closed subsets.

Proof. It is clear by deﬁnition that the empty set and the whole set SpecA is closed. Sup- pose Φ1, Φ2 are closed and M ∈ Φ1 ∪ Φ2. Then M ∈ Φi for some i = 1, 2 implies

that SuppM ⊆ Φi ⊆ Φ1 ∪ Φ2. In particular, it is specialization closed. Now suppose T M ∈ i∈I Φi. Then M ∈ Φi for each i implies that SuppM ⊆ Φi by closedness, hence T SuppM ⊆ i∈I Φi.

3.4 Classiﬁcation of nullity classes

We characterize the premonoform objects in a slightly different way due to its construction in order to give a classiﬁcation of nullity classes. This classiﬁcation resembles that of the Serre subcategories of a noetherian abelian category given by Kanda in [13].

55 Lemma 3.4.1. Given an object M in an abelian category A. To say that M is not pre- monomoform is equivalent to saying that M lies in the nullity class N generated by all quotients M/N for 0 6= N < M.

Proof. Suppose M is not premonoform. Then it has a nontrivial quotient M/N as its subobject say N 0 < M, i.e. M/N ∼= N 0 ,→ M. Then the short exact sequence

0 → M/N → M → M/N 0 → 0 implies that M ∈ N since both M/N, M/N 0 ∈ N and N is closed under extensions.

Conversely, suppose M ∈ N such that it is an n-extension but not an (n − 1)-extension of the quotients of the objects M/N. Then if n = 0 we have M = 0 hence it is not premonoform. Otherwise, there is a short exact sequence

0 → M/N → M → M 0 → 0 such that both ends are nonzero, where M 0 is an (n − 1)-extension of quotients of the objects M/N. Therefore, M has a nontrivial quotient M/N as its subobject so that M is not premonoform.

An object in an abelian category is noetherian if every ascending chain of subobjects becomes stationary after ﬁnitely many stages. A category is noetherian if it is essentially small and in which every object is noetherian.

Proposition 3.4.2. Given any nullity class N in an abelian category A and M ∈ A any noetherian object. Then M ∈ N if and only if SuppM ⊆ SuppN .

56 Proof. One implication is obtained by deﬁnition of support. Indeed, M ∈ N implies

S SuppM ⊆ N∈N SuppN = SuppN . Now suppose SuppM ⊆ SuppN but M/∈ N .

Then consider the collection of objects N < M such that M/N does not lie in N . Notice

that this collection is nonempty since M/∈ N so that N = 0 gives one element. Hence

there is a maximal subobject N0 < M such that M/N0 ∈/ N , thanks to the Noetherian

property of M. Then M/N0 is premonoform by Lemma 3.4.1 since otherwise M/N0 must

lie in the nullity class generated by the nontrivial quotients of M/N0 which all belong to

N by maximality, so that M/N0 ∈ N as well. Hence M/N0 ∈ SuppM ⊆ SuppN . Hence

0 there is X ∈ N such that M/N0 ∈ SuppX, and thus there is M ∈ M/N0 such that

M 0 ∈ C(X). Therefore,

0 M/N0 ∈ C(M/N0) = C(M ) ⊆ C(X) ⊆ N ,

a contradiction.

We say that a nullity class (or more generally, a subcategory satisfying ﬁnite many clo-

sure operations such as Serre subcategories etc.) N is generated by premonoform objects if

it is the intersection of all nullity classes containing every premonoform object in N . That

is, N = C(⊕M∈N ,MpremM), recalling that nullity class is closed under retract.

Corollary 3.4.3. Let A be a abelian category. Suppose N , N 0 are nullity classes and N is

generated by premonoform objects. Then N ⊆ N 0 if and only if SuppN ⊆ SuppN 0.

Proof. By assumption, N = C(⊕M∈N ,MpremM) for all M premonoform. Thus M ∈

SuppN ⊆ SuppN 0 so that there is M 0 ∈ M and X ∈ N 0 with M 0 ∈ C(X), hence

0 0 0 M ∈ C(M) = C(M 0) ⊆ C(X) ⊆ N . Therefore, N ⊆ N since N is a nullity class.

57 We state a standard result in the following, which is Theorem 2.9 in [13]. This enables

us to conclude that every subcategory of the abelian category A is generated by premono-

form objects if A is noetherian.

Proposition 3.4.4. Let A be a noetherian abelian category. For every object M ∈ A, there

is a chain

0 = M0 ⊆ M1 ⊆ ... ⊆ Mn−1 ⊆ Mn = M

such that Mi/Mi−1 is premonoform for every i = 1, 2, ..., n.

The following statement is an easy generalization of Proposition 6.3 in [2], which re- lates noetherian modules.

Lemma 3.4.5. Let A be an abelian category. Suppose there is a short exact sequence

f g 0 → N 0 → M → N → 0

in A. Then M is noetherian if and only if both N,N 0 are noetherian. In particular, for an

atom equivalent pair A ∼ B of objects, A is noetherian if and only if B is noetherian.

Proof. Since any chain of subjects in N 0 or N also gives a chain of subobjects in M, the

−1 necessity follows immediately. Conversely, suppose {Mi} is a chain in M. Then {f Mi}

0 and {g(Mi)} are chains in N ,N respectively, so that for large enough index i that the

−1 stationary of both {f Mi}, {g(Mi)} implies that of {Mi}.

Notation 3.4.6. For convenience, we denote by N a nullity class in a given abelian cate-

gory A, Nnoeth a nullity class consisting of noetherian objects, and Nprem a nullity class

58 ext ext generated by premonoform objects. Also we denote by Cc a subclass or Sc a subset

that is closed and extension closed if the spectrum SpecA is a class or a set, respective-

ly. Finally, denote by SpecnoethA the subclass of SpecA in which every equivalence class

A ∈ SpecnoethA has a noetherian representative A.

Theorem 3.4.7. Let A be an abelian category. Then the nullity classes of noetherian objects are classiﬁed by the closed and extension closed subclasses of the spectrum SpecnoethA.

More precisely, there is an order preserving bijection

ext −1 Supp : {Nnoeth ⊆ A} → {Cc ⊆ SpecnoethA} : Supp

in which Supp−1Φ = {M ∈ A |M is noetherian and SuppM ∈ Φ}.

Proof. Suppose N is a nullity class of noetherian objects in A and M,N ∈ A are noethe-

rian such that SuppM, SuppN ⊆ SuppN . In particular, M,N ∈ N by Proposition 3.4.2.

Suppose X is an extension

0 → M → X → N → 0

of N by M. Then X ∈ N since N is closed under extensions. Hence SuppX ⊆ SuppN .

Also, SuppN is closed since for every H ∈ SuppN there is X ∈ N such that H ∈

SuppX ⊆ SuppN .

On the other hand, given any such closed and extension closed subset Φ, Supp−1Φ is extension closed since whenever M,N ∈ Supp−1Φ any extension X of N by M yields

SuppX ⊆ Φ so that X ∈ Supp−1Φ. Also, it is closed under quotients by Lemma 3.3.3.

Therefore, both Supp and its inverse Supp−1 are well-deﬁned.

59 Now suppose N is a nullity class of noetherian objects. Then since SuppM ⊆ SuppN

if and only if M ∈ N by Proposition 3.4.2, hence Supp−1SuppN = N .

In turn, suppose M ∈ SuppSupp−1Φ, i.e. there is M 0 ∈ M such that M 0 ∈ C(X) for some X ∈ Supp−1Φ. Hence SuppM 0 ⊆ SuppX ⊆ Φ so that M 0 ∈ Supp−1Φ. Therefore,

M ∈ C(M) = C(M 0) ⊆ Supp−1Φ, i.e. M ∈ SuppM ⊆ Φ. Conversely, suppose

M ∈ Φ. Then there is H ∈ A such that M ∈ SuppH ⊆ Φ by closedness of Φ, or

equivalently, there is M 0 ∈ M such that M 0 ∈ C(H) for some H ∈ Supp−1Φ. Therefore,

−1 M ∈ SuppSupp Φ.

Here is an immediate consequence of Theorem 3.4.7, noticing that the spectrum SpecA

becomes a set under the condition that the abelian category A is noetherian, and N =

Nnoeth. See Notation 3.4.6.

Corollary 3.4.8. Suppose A is a noetherian abelian category. Then the nullity classes

are classiﬁed by the closed and extension closed subsets of the spectrum SpecA. In other

words, there is an order preserving bijection

ext −1 Supp : {N ⊆ A} → {Sc ⊆ SpecA} : Supp .

We compute the spectrum of the category kA2-rep consisting of ﬁnite dimensional

1 2 representations of the quiver A2 :◦←◦ over an algebraically closed ﬁeld, as an example.

Example 3.4.9. For the ﬁnite dimensional representation of type A2, there are indecompos-

ables, including projectives P1,P2, simples S1 = P1,S2 and injectives I1 = S2,I2 = P2,

which give all the premonoform objects by Proposition 3.2.4. Since no pair of them is

60 equivalent, the underlying set of the spectrum is

Spec(kA2-rep) = {P1,P2,S2}.

whose topology of closed subsets is given by the empty set and SuppP1 = {P1}, SuppS2 =

{S2}, SuppP2 = {P2,S2}, Spec(kA2-rep) and {P1,S2}. Except {P1,S2}, each closed

subset is extension closed, thus corresponds to a nullity class uniquely. This also implies

that the support is different from the usual ones since SuppP2 * SuppP1 ∪ SuppS2 al-

though there is a short exact sequence 0 → P1 → P2 → S2 → 0. See Section 5.4.5 for

further discussion.

Example 3.4.10. Let us consider the nullity classes in the category of modules over the

polynomial ring k[x, y] in two variables. Notice that the nullity classes are the same as the

thick subcategories, and they are classiﬁed via the normal support, see [27].

In our context, the nullity class generated by the ideal (x, y) coincides with that gener-

ated by k[x, y], i.e. C((x, y)) = C(k[x, y]). In fact, they can generate each other by using extensions, retracts and quotients as follows. The free resolution given by projection on two generators

k[x, y] ⊕ k[x, y] → (x, y) → 0 implies (x, y) ∈ C(k[x, y]) while the self-multiplication by x

0 → (x, y) →x (x, y) → (x, y)/x(x, y) → 0

together with the isomorphism

(x, y)/x(x, y) ∼= k[x, y]/(x, y) ⊕ k[x, y]/(x)

61 implies k[x, y]/(x, y) ∈ C((x, y)), so that k[x, y] ∈ C((x, y)) since we have an extension

0 → (x, y) → k[x, y] → k[x, y]/(x, y) → 0.

We need a stronger equivalence relation on the spectrum Spec0A to distinguish the two.

Deﬁne for any subclass Φ ⊆ SpecA a nullity class C(Φ) = C(⊕A∈ΦA). Notice that

C(A) = C(B) whenever A ∼ B, thus C(Φ) is independent of the chosen representative,

hence it is well-deﬁned, recalling that nullity class is closed under retract.

Lemma 3.4.11. Let Φ be a closed and extension closed subclass of SpecA. Suppose M is premonoform. Then M ∈ C(Φ) if and only if M ∈ Φ. In particular, C(Φ) = Supp−1Φ if

every nullity class is generated by premonoforms.

Proof. Let S = {A ∈ A | A ∈ Φ}. Then by Lemma 3.1.8 we have C(Φ) = hT iext,

where T = hSiquot. For any X ∈ T there is a surjection A → X for some A ∈ S, so

that SuppX ⊆ SuppA ⊆ Φ, since Φ is closed. In particular, if X = M is premonoform,

then M ∈ SuppM ⊆ Φ. Notice that by induction for each n-extension T n and X ∈ T n,

SuppX ⊆ Φ since Φ is extension closed. In particular, if there is a short exact sequence

0 → B → M → A → 0

with A ∈ T and B ∈ T n, then M ∈ SuppM ⊆ Φ. The last statement follows immediate-

ly.

ext Recall in Notation 3.4.6 that Cc represents the closed and extension closed subclasses

of SpecA.

62 Corollary 3.4.12. Suppose A is an abelian category. Then there is an order preserving bijection

ext Supp : {Nprem ⊆ A} → {Cc ⊆ SpecA} : C.

Proof. Thanks to Theorem 3.4.7 and the deﬁnition of the map C, it is sufﬁcient to show that C(SuppN ) = N and SuppC(Φ) = Φ for every nullity class N generated by premonoforms and closed and extension closed subclass Φ. Suppose M ∈ N is premonofor- m, then M ∈ SuppN , thus M ∈ C(⊕A∈SuppN A) = C(SuppN ). For M ∈ SuppN ,

there is M 0 ∈ M such that M 0 ∈ N , thus M ∈ C(M) = C(M 0) ⊆ N . Hence

C(SuppN ) ⊆ N . Now suppose M ∈ Φ. Then M ∈ C(⊕A∈ΦA) = C(Φ) so that

M ∈ SuppM ⊆ SuppC(Φ). Conversely, for M ∈ SuppC(Φ), there is M 0 ∈ M such

that M 0 ∈ C(A) for some A ∈ C(Φ). Thus M ∈ C(M) = C(M 0) ⊆ C(A) ⊆ C(Φ).

Therefore, M ∈ Φ by Lemma 3.4.11.

We end this section with two examples of nullity classes, which demonstrate that Corol-

lary 3.4.12 is indeed a generalization of Corollary 3.4.8, and also that nullity classes can be

out of control.

Example 3.4.13. By reviewing the example of tensor algebras in Section 3.2.2, we indeed

have a nullity class C(T (V )) generated by the premonoform object T (V ) but it is not a

nullity class of noetherian objects since T (V ) is not noetherian, in the category of modules over the tensor algebra T (V ) for V = kha, bi. Nevertheless, there are many nullity classes not generated by premonoforms either, such as nullity classes C(W ) of vector spaces gen-

erated by an inﬁnite dimensional vector space W (with basis of cardinality say ℵ0) so that

63 the objects are those vector spaces with basis of cardinality ≤ ℵ0.

64 Chapter 4

A structure sheaf on the atom spectrum

Following Kanda’s theory [13] on atom spectrum of atom equivalence classes of monoform objects in an abelian category A, we build on a ringed space structure thereon in

Section 4.1, expecting that this idea can also be carried into the case when the spectrum is replaced by that of equivalence classes of premonoform objects, or in more general cases.

Later in Section 4.2, we associate a sheaf Mf of modules to any object M in A, showing that based on the existence of a localization functor given in Section 2.6, the association

∼ is a faithfully exact functor if there is an exact concretization of A. We then compute explicitly the examples of quiver representations of type An and the ﬁnite dimensional representations over the 2-Kronecker quiver.

65 4.1 Atom spectrum and a structure sheaf R+

The atom spectrum ASpecA is deﬁned in general for an abelian category A, which

consists of atom equivalence classes of monoform objects, see Section 3.1. Here the mono-

forms M ∼ N are atom equivalent if they share a common nonzero subobject. We will

focus on the category of ﬁnitely generated modules over a commutative Noetherian ring,

recalling that these monoform objects are exactly those strongly uniform modules.

Let R be a ring and let A denote the R-module category. For an R-module M, the atom

support as in [13] is deﬁned as

ASuppM = {H ∈ ASpecA | there is H0 s.t. H0 is a subquotient of M}

In particular, when R is a commutative Noetherian ring with identity we have

Lemma 4.1.1. ASuppM = {R/p | p ∈ SuppM}.

Proof. Take R/p ∈ ASuppM. Then there is H ∈ R/p such that H is a subquotient of

M. Since H ∼ R/p, there is a common subobject H between R/p and H, which can be

chosen as a principal submodule without loss of generality, say R/p itself. Hence R/p is a

0 subquotient of M of the form M /N. Now suppose p ∈/ SuppM, i.e. Mp = 0. Then the

exact sequence

0 0 → Np → Mp → R/p ⊗ Rp → 0

implies that k(p) = R/p ⊗ Rp = 0, a contradiction. Conversely, notice that SuppM =

AssM. For primes q with properties R/q ,→ M, it is clear that R/q ∈ ASuppM by

66 deﬁnition. Now if p ⊇ q, then there is a surjection R/q → R/p so that R/p ∈ ASuppM as well.

Proposition 4.1.2. Let R be a commutative noetherian ring with identity and A = R-mod.

Then ASpecA = {R/p | p ∈ SpecR}.

Proof. For any monoform M ∈ A, AssM 6= ∅ and we take p ∈ AssM so that there is an

injection R/p ,→ M. In particular, R/p is monoform and R/p ∼ M. For the converse, it

is sufﬁcient to check that R/p is premonoform for any prime p. Indeed, suppose there is a

nonzero endomorphism f : R/p → R/p sending 1+p to x+p with x∈ / p. Then whenever ax + p = bx + p, we have a − b ∈ p thus a + p = b + p, hence f is injective.

Lemma 4.1.3. p ⊆ q if and only if there is a nonzero map f : R/p → R/q.

Proof. Suppose there is y ∈ p − q, then yf(1 + p) = y(x + q) implies that q = yx + q so that yx ∈ q, or x ∈ q. Hence f = 0.

We point out that the topology of open subsets introduced here is that of closed subsets used by Kanda [13] in his atom spectrum.

c Lemma 4.1.4. Denote the complement by Uf = (ASuppR/(f)) for f ∈ R. Then

{Uf }f∈R forms a basis of open subsets for ASpecA. Moreover, there is another basis

c {UM }M∈A with UM = (ASuppM) , so that they generate the same topology if A is the category of ﬁnitely generated modules over a commutative Noetherian ring.

Proof. Notice that U0 = ∅ and U1 = ASpecA. By Lemma 4.1.1, we can check easily that ASuppR/(fg) = ASuppR/(f) ⊕ R/(g), hence Ufg = Uf ∩ Ug. For M = 0, UM =

67 c ASpecA and for M = R, UM = ∅. Also, UM ∩ UN = (ASuppM ∪ ASuppN) =

c (ASuppM ⊕ N) = UM⊕N . Hence they also form a basis. Furthermore, recall that for

every ﬁnite generated module M ∈ A, there is a ﬁltration 0 = M0 ⊆ M1 ⊆ ... ⊆ Mn = M

∼ such that Mi/Mi−1 = R/pi for some prime ideal pi and i = 1, 2, ..., n. Hence ASuppM =

n ∪i=1ASuppR/pi. Notice that every prime ideal is ﬁnitely generated say pi = (fi1, ..., fimi )

n mi in R. Thus ASuppM = ∪i=1 ∩j=1 ASuppR/(fij) thanks to Lemma 4.1.1.

Proposition 4.1.5. Let A = R-mod. Then there is a homeomorphism φ : ASpecA →

SpecR sending R/p to p.

Proof. We claim that R/p ∼ R/q if and only if p = q, so that φ is well-deﬁned. In fact,

by choosing a principal submodule as their common subobject, we can thus assume that

∼ c R/p = R/q. Hence p = q by Lemma 4.1.3. Notice that the collection of Df = (V (f))

also forms a basis of open subsets for SpecR and Lemma 4.1.4 allow us to identify the open

subsets correspondingly. In fact, we can check easily that ψ deﬁned by ψ(p) = R/p gives

an inverse of φ such that φ(ASuppR/(f)) = SuppR/(f) = V (f) while ψ(SuppR/(f)) =

−1 −1 −1 −1 ASuppR/(f). Hence φ (∪Df ) = ∪φ Df and ψ (∪Uf ) = ∪ψ Uf . Therefore, φ is a

homeomorphism.

Deﬁnition 4.1.6. Let A be an abelian category. We deﬁne Z(A) = Hom(1A, 1A) to be the

center of A, consisting of natural transformations from the identity functor 1A to itself.

Lemma 4.1.7. For A = R-mod, there is a ring isomorphism

∼ ∼ Z(A) = HomA(R,R) = Z(R)

68 where Z(R) is the center of the ring R. In particular, Z(A) ∼= R if R is commutative.

Proof. The addition is given by

0 0 0 (τ + τ )M (x) = (τM + τM )(x) = τM (x) + τM (x) and the multiplication is given by composition of maps

0 0 0 (ττ )M (x) = (τM ◦ τM )(x) = τM (τM (x))

0 0 where M ∈ A and x ∈ M. Notice that the naturality of τ implies that τM τM = τM τM .

The following diagram

τR R / R

αx αx M / M τM for a morphism αx deﬁned by αx(1) = x. The naturality of τ implies that τM (x) =

∼ τM αx(1) = αxτR(1) = xτR(1), which is uniquely determined by τR ∈ HomA(R,R) =

Z(R). Thus we deﬁne φ : Z(A) → Z(R) by φ(τ) = τR(1). Then it is straightforward to show that φ respects the ring structures.

Corollary 4.1.8. Let Sf = hR/(f)i be the Serre subcategory generated by R/(f) and

∼ deﬁne R(Uf ) = Z(A/Sf ). Then R(Uf ) = Z(Rf ).

∼ Proof. This follows immediately from the fact that A/Sf = Rf -mod by Proposition 2.1.7 and Proposition 2.3.2.

The following deﬁnition of certain (pre)sheaf comes naturally in most of our examples.

We refer to the material provided by the Stack Project online, see Section 6.30 Basis and sheaves in [22].

69 Deﬁnition 4.1.9. Let X be a topological space with a basis of open subsets B. Suppose there is an assignment F deﬁning for each U ∈ B a ring F(U) and for any pair U 0 ⊆ U in

0 B a ring map ρUU 0 : F(U) → F(U ) such that

(1) ρUU = idF(U) for each U ∈ B.

00 0 (2) For U ⊆ U ⊆ U in B, ρU 0U 00 ρUU 0 = ρUU 00 .

(3) If U = ∪iUi with Ui ∈ B and s ∈ F(U) such that s|Ui∩U = 0 ∈ F(Ui) for each i, then s = 0.

(4) Given U = ∪iUi with Ui ∈ B and si ∈ F(Ui) such that for every W ⊆ Ui ∩ Uj in

B, si|Ui∩W = sj|Uj ∩W , then there is s ∈ F(U) such that s|Ui = si for each i.

Then we call F is a presheaf of rings on the basis B if it satisﬁes (1) and (2), a sheaf of rings on the basis B if it satisﬁes (1)-(4).

Alternatively, combining (3) and (4) we can replace these conditions by

(3’) Given U = ∪iUi in B and si ∈ F(Ui) such that si|Ui∩W = sj|Uj ∩W for any i, j and

W ⊆ Ui ∩ Uj in B, there is a unique s ∈ F(U) such that s|Ui = si for all i.

Proposition 4.1.10. Given a (pre)sheaf F of rings on a basis B of a topological space X, there is a unique (pre)sheaf of rings F + on X such that F +(U) = F(U) for every U ∈ B.

Proof. We can deﬁne F +(W ) = lim F(U 0), the inverse limit over all the basis elements

0 U ⊆ W , for an open subset W of X.

Deﬁnition 4.1.11. Let SM = hMi be the Serre subcategory generated by M. Deﬁne

0 R (UM ) = Z(A/SM ) as the center of the quotient category A/SM .

70 Lemma 4.1.12. Let A be an abelian category such that ASpecA forms a set. Then R0 is a presheaf on the basis {UM }M∈A of ASpecA.

Proof. It is sufﬁcient to show that for the inclusion UM ⊆ UM 0 , the restriction map

0 0 ρMM 0 : R (UM 0 ) → R (UM )

deﬁned by (ρMM 0 (τ))X = F (τX ) respects the compositions of natural transformations,

where F is the quotient map FMM 0 : A/SM 0 → A/SM and X ∈ A/SM . In fact, for

UM ⊆ UM 0 ⊆ UM 00 , we have SM ⊇ SM 0 ⊇ SM 00 such that the following diagram

FM0M00 A/SM 00 / A/SM 0

FMM0 F 00 MM % A/SM of quotient categories commutes thanks to the uniqueness of quotient functors, which fur-

ther implies that ρMM 0 ρM 0M 00 = ρMM 00 .

Lemma 4.1.13. Suppose F is a presheaf on X, B a basis of X and P ∈ X. Then the stalk

FP at P becomes the direct limit FP = lim F(V ) over the neighborhoods V of P in B. −→

Proof. Recall that every germ hU, si in FP is represented by a pair (U, s) of open neigh-

borhood U of P and a section s ∈ F(U), so that any two such pairs (U, s), (U 0, s0) are

0 equivalent if and only if there is another open neighborhood W of P such that s|W = s |W .

Thus for any hU, si, there is V ∈ B such that hU, si = hV, s|V i ∈ lim F(V ). The converse −→ holds since every representative in lim F(V ) also represents an element in the stalk FP . −→ In fact, a more general result is true. We include it here for convenience, see Chapter

IX.3 in MacLane [18]. Recall that for a functor F : A → B and an object x ∈ B, the

71 comma category (x ↓ F ) consists of arrows f : x → F a with a ∈ A as objects and for another object f 0 : x → F a0, the morphism from f to f 0 is deﬁned to be an arrow g : a → a0 making the diagram

X f f 0 } ! F a / F a0 F g commute.

Deﬁnition 4.1.14. A functor K : J → I is called coﬁnal if each i ∈ I, the comma category

(i ↓ K) is nonempty and connected, i.e. there exist arrows with appropriate directions

making the diagram

i i ··· i i

Kj1 / Kj2 o ··· / Kjn−1 o Kjn

commute for any two objects i → Kj1 and i → Kjn in (i ↓ K).

Lemma 4.1.15. Let F : J → I be a functor. Suppose K : I → A is a functor and both

K and KF has colimit. If F is ﬁnal, then there is a natural isomorphism ColimKF →

ColimK.

Proof. Let a = ColimKF and σ : KF → a the universal cone. For i ∈ I, there is j ∈ J and an arrow α : i → F j since F is ﬁnal. Thus we have the composition τi :

Ki → KF j → a as our candidate cone for ColimK. Notice that it is a well-deﬁne natural transformation since the comma category (i ↓ F ) is connected. We need to show that τ : K → a is universal. Given any other cone µ : K → b, the precomposition

72 µF : KF → b is a cone for ColimKf = F , thus there is a unique arrow f : a → b making

the corresponding diagram commute. Then combined with the commutative diagram

Kα Ki / KF j

µi µF j = b / b due to the naturality of µ, we thus deduce that fτ = µ so that τ is a universal cone.

Here it is clear by deﬁnition that the inclusion functor from the category of basis ele-

ments to that of all open subsets is coﬁnal, and the lemma applies.

Proposition 4.1.16. Let G be a sheaf of rings on X. Suppose F be a presheaf of rings

on the basis B of X such that there is an isomorphism φ : F ∼= G of presheaves on B.

Then it induces an isomorphism φ+ : F + ∼= G of sheaves on X such that the diagram of

presheaves on B

φ F / G >

+ φ F + commute.

Proof. The construction of F + is similar to the one in Proposition-Deﬁnition 1.2 of [9].

Thanks to Lemma 4.1.13, we can deﬁne for any open subset U of ASpecA a set F +(U)

` of sections τ : U → P ∈U FP such that (1) for any P ∈ U, τ(P ) ∈ FP and (2) for any P ∈ U, there is a neighborhood V ∈ B contained in U and σ ∈ F(V ) such that for any Q ∈ V the germ of σ at Q is σQ = τ(Q). Equipped with these local properties one can check immediately that F + with the natural restriction maps is a sheaf of rings on X satisfying the required properties.

73 Proposition 4.1.17. Let A = R-mod. Then R deﬁned by R(Uf ) = Hom(1A/Sf , 1A/Sf ) is

a presheaf of rings, with basis B = {Uf }f∈R of ASpecA.

Proof. Thanks to Proposition 4.1.10, it is sufﬁcient to show that R is a presheaf of rings

on the basis B. For every pair Uf ⊆ Ug, we have Sf ⊇ Sg so that there is a further quotient

functor F : A/Sg → A/Sf . Then the restriction

ρgf : R(Ug) → R(Uf )

is deﬁned by (ρgf (τ))M = F (τM ). It is indeed well-deﬁned because F is a functor, τ

is a natural transformation and every morphism in A/Sf can be written as a sequence

of alternative back and forth morphisms in A/Sg. Hence the properties (1) and (2) of

Deﬁnition 4.1.9 hold.

Recall that for a commutative ring with identity, the prime spectrum SpecR is an afﬁne

scheme with the structure sheaf O deﬁned as for any open subset U ⊆ SpecR the set of

L functions s : U → p∈U Rp such that s(p) ∈ Rp for every p and s is locally a quotient of

element of R. One can show that for each open subset Df = {p ∈ SpecR | f∈ / p} in the

∼ basis of SpecR, we have O(Df ) = Rf . See Chapter II.2 in [9] for detail.

Proposition 4.1.18. Let A = R-mod. Then there is a isomorphism of ringed spaces

(φ, φ]) : (ASpecA, R+) → (SpecR, O) defined by φ(R/p) = p, where R+ is the sheafification of the presheaf R of rings on the basis {Uf }f∈R of ASpecA.

74 Proof. Notice that via the homeomorphism φ, the assignment φ∗R is a presheaf of rings

∼ on the basis {Df }f∈R of SpecR such that O = φ∗R, Proposition 4.1.16 implies that the

] + induced map φ : O → (φ∗R) is an isomorphism of sheaves of rings on SpecR.

4.2 Sheaf of modules

In an abstract abelian category A, the Freyd-Mitchell Embedding Theorem (see Section

1.6 in Weibel [31]) guarantees us a concretization (see Deﬁnition 4.2.1) that there is a (fully) faithful and exact functor from A to a certain category of modules over a ring, in particular, the category of abelian groups. We thus may consider each object of A set theoretically, although this might not be the best choice. This occurs often when we study the sheaf of modules over (ASpecR-mod, R+) deﬁned in Section 4.1, where there is also a natural concretization given by the forgetful functor R-mod → Ab.

4.2.1 A module structure over R+

We will use the localization functor introduced in Section 2.6 to deﬁne the sheaf of modules over the structure sheaf R+ of rings on the atom spectrum ASpecA of an abelian category A.

Deﬁnition 4.2.1. An (abelian) category A is concrete if there is an (additively) faithful functor F : A → Ab, called the concretization of A. We require the target of F to be the category of abelian groups instead of that of sets.

Lemma 4.2.2. Let F : A → C be a functor between additive categories. Then F is

75 additive if and only if it preserves ﬁnite coproducts. In particular, F is additive if it is an equivalence.

Proof. It is straightforward to check both directions. See Proposition VIII.2.4 in [18].

Lemma 4.2.3. Let L : A → A be a localization functor with respect to a Serre subcategory S of an abelian category A. Suppose F : A → Ab is a concretization. Then the composition

Z(A/S) → EndL(M) → EndFL(M)

gives a Z(A/S)-module structure for FL(M), i.e. τx = FL(τM )(x) for x ∈ FL(M).

Proof. Notice that the composition G = FL is an additive functor. Then

(τ + σ)x = G(τ + σ)M (x) = G(τM + σM )(x)

= G(τM )(x) + G(σM )(x) = τx + σx.

(τσ)x = G(τσ)M (x) = G(τM )G(σM )(x) = τ(σx).

Also

τ(x + y) = G(τM )(x + y) = G(τM )(x) + G(τM )(y) = τx + τy.

Finally, 1x = G(1M )(x) = 1G(M)(x) = x.

Thanks to the existence of localization functors, such assignment Mf indeed gives a sheaf of modules over R for every object M ∈ A.

76 Lemma 4.2.4. Let F : A → Ab be a concretization and UN ⊇ UP basis elements with a projection TNP : A/SN → A/SP . Then

rNP : Mf(UN ) → Mf(UP )

deﬁned by rNP = F ηLN (M) becomes a restriction map, where the pair LN : A/SN →

A/SN and η : 1A/SN → LN is the localization functor equivalent to TNP .

Proof. Recall that the restriction map ρNP : R(UN ) → R(UP ) for R is deﬁned by

(ρNP (τ))X = TNP (τX ). We need to show that r is a module map satisfying

rNP (τx) = ρNP (τ)rNP (x)

for all τ ∈ Z(A/SN ) and x ∈ Mf(UN ). In fact, since ηLN = LN η thus

rNP (τx) = F (ηLN (M))(FLN (τM )(x))

= F (LN ηM )(FLN (τM )(x)) = FLN (ηM τM )(x).

For the other side

ρNP (τ)rNP (x) = FLN (ρNP (τ)M )(F ηLN (M)(x))

= FLN (LN (τM ))(FLN ηM (x)) = FLN (LN (τM )ηM )(x).

Thus the formula follows immediately from the naturality of η

τM M / M

ηM ηM LN M / LN M LN (τM ) applied to the morphism τM .

77 Proposition 4.2.5. The functor ∼= FL : A → R-mod is faithfully exact if the concretization F : A → Ab is exact.

Proof. The exactness follows from that of the direct limit and the localization functor

L : A → A. It is faithful since both F and the quotient Q are.

4.2.2 Examples from quiver representations

The theory of quiver representations provides a rich source of examples to work with.

See for instance Volume 1 of [1] for the basic properties of quiver theory and especially the

1 2 Auslander-Reiten theory presented in the Section IV.4 therein. Let A2 be the quiver ◦←−◦

and k a ﬁeld. Denote by A2 the category kA2-rep of kA2-representations of ﬁnite type.

Since there are only 3 indecomposables P1 = S1, P2 = I1 and S2 = I2, the underlying

set of the spectrum is

ASpecA2 = {P1, S2}.

Here P2 = P1 are atom equivalent since P1 ,→ P2 is a subobject. The topology is

c c discrete since we have U0 = (ASupp0) = ASpecA2, UP1 = (ASuppP1) = {S2},

c c US2 = (ASuppS2) = {P1} and UP2 = (ASuppP2) = ∅, drawn as

U0 = a

. 0 P UP1 US2 a =

0 P . UP2

Notice that this diagram gives all the open subsets of ASpecA2. The presheaf R of rings

78 thus becomes

k 1 1

k Ð k

0 Ð whose sheaﬁﬁcation R+ is

π1 π2

k k

Now for UP1 , the corresponding localization functor LP1 = GQP1 : A2 → A2 can be

computed by Proposition 2.6.7, where G is the right adjoint of the quotient functor QP1 :

A2/hP1i → A2. Thus we have LP1 (P1) = 0 and LP1 (P2) = LP1 (S2) = S2. Similarly, for

US2 , we have LS2 (P1) = LS2 (P2) = P2 and LS2 (S2) = 0.

Use the obvious forgetful functor as the concretization F : A2 → Ab on the category

of representations, induced by the diagram

R(UN ) / EndFLN (M) 6

FG & EndQN (M)

79 where Mf(UN ) = FLN (M), we thus have presheaf of R-modules

Pf1 = Pf1(U0) = P1 p

y % 0 P2 Pf1(UP1 ) Pf1(US2 )

% y 0 ~

Pf1(UP2 )

Pf2 = Pf2(U0) = P2 1 y % ~ ~ S2 P2 Pf2(UP1 ) Pf2(US2 )

% y 0 ~

Pf2(UP2 ) and

Se2 = Se2(U0) = S2 1

y % ~ S2 0 Se2(UP1 ) Pf1(US2 )

% y 0

Se2(UP2 ) + Observe that Se2 is a sheaf already so that Se2 = Se2. Since the topological space ASpecA is discrete, the sheaﬁﬁcation of Pf1 and Pf2 can be computed as the product of stalks, namely

P2 S2 ⊕ P2 1 π1 π2

{ $ 0 P2, S2 P2

0 } $ 0 z + + denoted by Pf1 and Pf2 , respectively. This shows in particular that the functor ∼ is not an

equivalence of categories.

80 In general, for the representation of type An

1 2 n−1 n ◦←−◦←− · · · ←− ◦ ←−◦

we can compute as the following. Denote by An = kAn-rep. Then the atom spectrum becomes

ASpecAn = {S1, S2, ..., Sn}

In fact, thanks to the Auslander-Reiten quiver we have atom equivalences as

S1 = [1, 1] ∼ [1, 2] ∼ [1, 3] ∼ ... ∼ [1, n]

S2 = [2, 2] ∼ [2, 3] ∼ [2, 4] ∼ ... ∼ [2, n] . .

Sn−1 = [n − 1, n − 1] ∼ [n − 1, n]

Sn = [n, n] where [i, j] represents the indecomposable representation

1 i−1 i 1 1 j j+1 n 0←− · · · ←− 0 ←−k←− · · · ←−k←− 0 ←− · · · ←−0 that has vector space k from position i to j and 0 elsewhere.

Lemma 4.2.6. The atom spectrum ASpecAn has a discrete topology with basis given by

L c {USI }I where I ⊆ {1, 2, 3, ..., n} is a subset, SI = i∈I Si and USI = (ASuppSI ) .

Proof. This follows immediately from the fact that ASuppSi = {Si} and that the atom

support ASupp respects ﬁnite coproduct.

81 + Lemma 4.2.7. The structure sheaf R deﬁned on the atom spectrum ASpecAn is

+ I R (USI ) = k where I ⊆ {1, 2, 3, ..., n} is a subset.

Proof. One observes first that the quotient category of An by a Serre subcategory generated by SI is isomorphic to Am for some n < m, by applying the Auslander-Reiten quiver which is in fact a connected tree. Thus the result follows from the sheafification of the presheaf

R(USI ) = k on ASpecAn.

Therefore, by Proposition 2.6.7 we can compute in general the associated sheaf of modules as follows

Proposition 4.2.8. For an indecomposable representation [i, j] of type An, the associated presheaf of modules over the presheaf R of rings is

[gi, j](USI ) = FLI [i, j]

where LI : An → An is the localization with respect to the Serre subcategory generated by SI and F : An → Ab is the natural concretization. In particular, the associated sheaf of modules is + M [gi, j] (USI ) = FLI−{k}[i, j] k∈ /I

Proof. Consider the localization Lk : An → An with respect to the Serre subcategory generated by the simple [k, k]. For an indecomposable [i, j], if k = i then the injective resolution

0 → [i, j]/[k, k] = [i + 1, j] → [i + 1, n] → [j + 1, n] → 0

82 implies that Lk([i, j]) = [i + 1, j]. If k 6= i, the injective resolution

0 → [i, j] → [i, n] → [j + 1, n] → 0 implies that.

The last statement holds since ASpec has discrete topology so that the sheafification becomes the coproduct of stalks at the points in the open subset, and the stalks is a colimit over a finite directed system.

For the 2-Kronecker quiver

1 α 2 ◦⇒◦ β we denote its category of representation of ﬁnite type by Ae2. There are 3 types of indecomposables described in the Auslander-Reiten quiver as preprojective component consisting of representations [i, i + 1], tubes consisting of [i, i] and preinjective component consisting of [i + 1, i], among which there are simples S1 = [1, 0],S2 = [0, 1], projectives

P1 = [1, 2],P2 = S2 and injectives I1 = S1,I2 = [2, 1], here [i, j] denotes the representa-

α i j tion (k ⇒ k ). β

Lemma 4.2.9. The atom spectrum of Ae2 is

ASpecAe2 = {S1, S2} with discrete topology.

Proof. Obverse ﬁrst that the among the indecomposables, the representations in the tubes other than the ones [1, 1] are not monoform since [i, i] for i > 1 are obtained as extensions of

[1, 1] by itself iteratively so that it contains a nontrivial quotient as a subobject. Notice also

83 that except the simple S1, there is an obvious injection S2 → [i, j] to an indecomposable

[i, j], thus these representations are atom equivalent. The open subsets are US1 = {S2} and

US2 = {S1}, hence it is discrete.

+ Lemma 4.2.10. The structure sheaf R on ASpecAe2 is

π1 π2

k k

Proof. It is sufﬁcient to ﬁgure out the quotient categories of Ae2 by the Serre subcategories

generated by S1 and S2 respectively, which becomes clear from the natural short exact

sequence

j i 0 → S2 → [i, j] → S1 → 0 for any indecomposable [i, j]. Hence each of the quotient categories is isomorphic to

A1.

Proposition 4.2.11. For an indecomposable [i, j], the associated presheaf of modules over

R on ASpecAe2 is

[i, j] Nn

j ~ i I2 S1

! 0 }

84 with sheaﬁﬁcation

j i I2 ⊕ S1

π1 π2

j | " i I2 S1

# 0 { Proof. For any indecomposable [i, j], there is an injective resolution

j 2j−i 0 → [i, j] → I2 → I1 → 0

j j i i Thus LS1 ([i, j]) = I2 . For LS2 , notice that [i, j]/S2 = S1. Hence LS2 ([i, j]) = S1.

However, there is a more general way to obtain this functor ∼: A → R+-mod when the abelian category A has neither obvious (exact) concretization nor right adjoint functor to the quotient functor, such as the category CohP1 of coherent sheaves on the projective line P1, which we will leave as a new project.

85 Chapter 5

Classifying space of subcategories and its

application

This chapter provides a different approach from Chapter 3 to study various subcategories, claiming that those generalized prime subcategories are classiﬁed by a topological space, called classifying space of subcategories, which applies to both abelian and triangulated categories, see Theorem 5.2.8. As an interesting example in Section 5.4.5, it shows that nullity classes are not able to be classiﬁed by a topological space, which is however explained in Chapter 3, see for instance Theorem 3.4.7.

In the first section we introduce the classifying space of subcategories using the language of lattices and then compute several examples. Then we give a classification theorem in Section 5.2 using the classifying space. We also compare in a lattice prime elements and irreducible elements in Section 5.3, and establish the functoriality of our construction. In the last section we try to fit some of the classical results into our theory of classifying space of subcategories, such as the classification of Serre subcategories, thick subcategories and

86 localizing subcategories etc.

We will use the language of lattices to state our result, and refer the reader to John- stone [12]. Also, Kock and Pitsch [14] have a similar idea, producing a classiﬁcation theorem of localizing subcategory of the derived category D(R) of a ring R, where the theory of frames discussed in Johnstone [12] is heavily used, which we do not use. For the theory of triangulated categories we refer to Weibel [31], and Neeman [20].

5.1 Classifying space

The classiﬁcation theorems such as Theorem 3.4.7 and its corollaries are in fact isomorphisms of lattices. In this section and the following, we study the classiﬁcation of subcategories in a different way by constructing a topological space K(Φ), with the aid of the theory of lattices.

5.1.1 Complete distributive lattices

We start from the basics of the theory of lattices (see e.g. [6] and [12]) and then introduce at the end of this section the concept that subcategories are classiﬁed by a topological space.

Deﬁnition 5.1.1. A partial order ≤ on a set S is a relation that satisﬁes the following properties: for a, b, c ∈ S,

(1) a ≤ a;

(2) a ≤ b and b ≤ a imply a = b;

87 (3) a ≤ b and b ≤ c imply a ≤ c.

A set with a partial order is called a partially ordered set, or simply a poset.

Definition 5.1.2. Let (Φ, ≤) be a poset. For a subset S ⊆ Φ, the join (or supremum) of S is an element b = ∨S such that a ≤ b for any a ∈ S and that a ≤ c for all a ∈ S implies b ≤ c. Dually, the meet (or infimum) of S is an element d = ∧S such that d ≤ a for any a ∈ S and that c ≤ a for all a ∈ S implies c ≤ d.A lattice is a poset such that any finite subset has both join and meet.

In particular, the join of the empty set denoted by 0 is called the bottom element while the meet of the empty set denoted by 1 is called the top element.

We call Φ a distributive lattice if additionally one of the following properties holds for any a, b, c in Φ. In fact, that one holds implies the other, see Lemma 1.5 in [12].

(1) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c);

(2) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).

Furthermore, we say that the join is distributive over arbitrary meet if for any a ∈ Φ and S ⊆ Φ that a ∨ (∧S) = ∧b∈S(a ∨ b) and dually the meet is distributive over arbitrary join if a ∧ (∨S) = ∨b∈S(a ∧ b).

Finally, a lattice Φ is complete if it has arbitrary join. Or equivalently, if Φ has arbitrary meet by Proposition I.4.3 in [12].

The most commonly used examples of lattices here are the following two, either from a category or from a topological space.

Example 5.1.3. (1) Let A be an abelian category such that the collection Φnull of nullity

88 classes form a set. Then with the inclusion as a partial order, and the intersection ∩ as meet, generating a nullity class as join we have a complete lattice. However, it is not always distributive as we will see later.

(2) Let X be a topological space and ΦX the set of closed subsets of X. Then ΦX is a complete distributive lattice with the inclusion as partial order, the intersection as meet and the closure of union as join.

Deﬁnition 5.1.4. Let Φ1, Φ2 be posets. A homomorphism of posets is a map f :Φ1 → Φ2 such that f(a) ≤ f(b) whenever a ≤ b in Φ1. A homomorphism f is called an isomorphism of posets if it is a bijection whose inverse also respects the partial order.

Furthermore, if Φ1, Φ2 are lattices. A homomorphism f of posets becomes a homomorphism of lattices if additionally f(a ∨ b) = f(a) ∨ f(b) and f(a ∧ b) = f(a) ∧ f(b) for any a, b ∈ Φ1. Such f is called an isomorphism of lattices if it is a bijection such that its inverse respects the partial order and join, meet as well.

An immediate consequence is the following

Lemma 5.1.5. Suppose f :Φ1 → Φ2 is an isomorphism of posets. If additionally that each

Φi is a (complete, resp.) lattice, then f respects (arbitrary, resp.) joins and meets, i.e. f is an isomorphism of lattices.

Proof. We show that f respects joins. The other for meets is similarly. In fact, for any a, b ∈ Φ1 we have f(a) ∨ f(b) ≤ f(a ∨ b) since f preserves the partial order. So does its inverse f −1, that is, a ∨ b = f −1(f(a)) ∨ f −1(f(b)) ≤ f −1(f(a) ∨ f(b)). It follows that

−1 −1 a ∨ b ≤ f (f(a) ∨ f(b)) ≤ f (f(a ∨ b)) = a ∨ b. Therefore, f(a ∨ b) = f(a) ∨ f(b).

89 We introduce the following notion in Deﬁnition 5.1.6 to represent one of the classes

of subcategories in a given category, namely, the class Φnull of nullity classes in a given

abelian category, the class of Serre subcategories of a given abelian category, the class of

localizing subcategories in a triangulated category and so on.

Deﬁnition 5.1.6. Let A be a category. A class ΦA of subcategories of type A (or roughly

subcategories of certain type) in A is a collection of subcategories in which every subcate-

gory satisﬁes ﬁnitely many conditions.

Deﬁnition 5.1.7. Let A be a category. A class ΦA of subcategories of type A in A is

∼ classiﬁed by a topological space X if there is an isomorphism of posets f :ΦX → ΦA.

Since both posets ΦX and ΦA have a lattice structure, this is equivalent to requiring that

∼ f :ΦX → ΦA is an isomorphism of lattices thanks to Lemma 5.1.5.

This is in fact a common phenomenon. For example, Gabriel’s result [7] in 1962 says

that the class of Serre subcategories in R-mod is classiﬁed by the prime spectrum SpecR of the ring R with specialization topology. Neeman [19] showed in 1992 that the class of localizing subcategories in D(R) is classiﬁed by SpecR with discrete topology. Another

one given by Balmer in 2005 says that the class of radical thick tensor ideals in a tensor

triangulated category A is classiﬁed by the Hochster’s dual (SpcA)h of Balmer’s spectrum

consisting of prime thick tensor ideals, see [3] and [10]. Also, Benson et al. proved [5] in

2011 that the class of localizing tensor ideals in the homotopy category K(InjkG) is classi-

ﬁed by the spectrum SpecH∗(G, k) of cohomology of the group G with discrete topology.

90 5.1.2 Points and topology

In this subsection, we construct a topological space K(Φ) associated with a lattice Φ that is a candidate to classify our target class of subcategories of certain type. Usually Φ comes from a category or a topological space and we will use the capital letters such as C to denote an element therein, for convenience. Recall the structures of these lattices given in Example 5.1.3.

Deﬁnition 5.1.8. Let Φ be a lattice. Assign to each element C ∈ Φ a pairing (C,Co),

o called a point and denoted by PC , if C 6= C , where

o _ 0 0 0 C = {C | C C,C ∈ Φ}.

Notice that the collection of points in Φ may be empty. The collection K(Φ) = {PC | C ∈

Φ} is called the classifying space associated to the lattice Φ.

Due to the source of lattices, from a category or a topological space, we have the following properties about the points PC .

Lemma 5.1.9. (1) Let ΦA be the lattice of subcategories of type A in a category. Then for each PC ∈ K(ΦA), there is x ∈ C such that C = hxi ∈ ΦA, where hxi refers to the intersection of all C ∈ ΦA that contain x.

(2) Let ΦX be the lattice of closed subsets of a topological space X. Then for each

PC ∈ K(ΦX ), there is x ∈ C such that C = {x} ∈ ΦX , where {x} denotes the closure of

{x} in X.

91 0 W 0 Proof. We show (2) and (1) is similar. Suppose C − C C C 6= ∅ and x an element in C0∈Φ their difference. Then if the closure {x} 6= C, we then deduce that it has to be one of the

0 0 W 0 C s, and in particular, x ∈ C C C , a contradiction. C0∈Φ In a category A, a subcategory C is called principal or cyclic if it is generated by a single object, i.e. C = hxi for some x ∈ C. In topology, such a point is usually called generic. Thus Lemma 5.1.9 claims that the representative C of a point PC is principal or generic.

Notation 5.1.10. Let K(Φ) = {PC | C ∈ Φ} be the classifying space associated to a lattice

0 Φ. Denote by K(C) = {PC0 | C ≤ C} a collection of points, for each C ∈ Φ.

Recall that a topological space is T0 (or Kolmogorov) if for any two distinct points there is an open neighborhood containing one point but not the other.

Theorem 5.1.11. Suppose Φ is a complete distributive lattice. Then the collection {K(C)}C∈Φ deﬁnes a topology of closed sets on K(Φ) so that K(Φ) is Kolmogorov.

Moreover, if the poset Φ = ΦX comes from a collection of closed subsets of a topological space X such that the closure C = {x} of any x ∈ X represents a point PC in K(ΦX ), then K(ΦX ) is homeomorphic to the Kolmogorov quotient of X.

Proof. Notice that the empty set is a ﬁnite supremum and the maximal element gives the whole space. Also it is clear that arbitrary intersection of closed remains closed. Now suppose given C1,C2 ∈ Φ, we need to show that K(C1 ∨ C2) ≤ K(C1) ∨ K(C2), that is, for any C ≤ C1 ∨ C2 that represents a point PC ∈ K(Φ), either C ≤ C1 or C ≤ C2.

Suppose not, then the distributivity implies that C = C ∧ (C1 ∨ C2) = (C ∧ C1) ∨ (C ∧ C2)

92 gives a nontrivial decomposition as a union of proper closed subsets, so that PC cannot

represent an actual point in K(Φ), a contradiction.

It is clear that for any two distinct points PC1 ,PC2 ∈ K(Φ), we can choose for either

c i = 1 or 2 an open neighborhood K(Ci) , the complement to separate them. Indeed,

suppose PC1 ,PC2 ∈ K(C1). Then C2 C1, so that we can choose the complement

c K(C2) as a separation instead since otherwise we would have C1 ≤ C2, a contradiction.

Hence K(Φ) is T0.

Now suppose given a space X and a collection ΦX of closed sets of X. Recall that the

Kolmogorov quotient KQ(X) = X/ ∼ is deﬁned as a quotient space by the equivalence

relation x ∼ y iff {x} = {y}. Thus there is a natural map

φ : KQ(X) → K(ΦX )

W 0 by assigning an equivalence class [x] to P , which is an actual point since {x} = 0 C {x} C {x} 0 C ∈ΦX implies one of C0 is not properly contained in {x}. It is clear that φ is injective. Now

0 W 0 suppose PC ∈ K(ΦX ). Then φ([x]) = P{x} = PC for some x ∈ C − C C C by Lem- C0∈Φ ma 5.1.9. Finally, it is straightforward to check that such φ is closed and continuous by observing that φ[C] = K(C) and p−1φ−1K(C) = C, where [C] are the closed in KQ(X)

and p : X → KQ(X) is the projection.

Remark 5.1.12. The classifying space K(ΦX ) can also be deﬁned via a collection of open

subsets in a given topological space X instead of closed ones, and it gives the same space.

closed In fact, if we denote our previous deﬁnition of lattice of closed subsets by (ΦX , ≤) and

open the new one by (ΦX , ≥), then there is an isomorphism of lattices induced by taking the

93 complement, noticing that the lattices have the opposite partial order. Speciﬁcally, denote

^ 0 0 open Do = {D D | D ∈ ΦX }

and then deﬁne

open open K(ΦX ) = {PD | D ∈ ΦX }

0 in which every point is a pair PD = (D,Do) such that D 6= Do, and K(D) = {PD0 | D ≥

open D} for every D ∈ ΦX . Then the correspondence is established by the observation that

o c c c c o c c c c (C ) = (C )o, (Do) = (D ) and K(D) = K(D ), K(C) = K(C ) for closed subset

C and open subset D, where the supscript c refers to taking complement.

Remark 5.1.13. Notice that in K(ΦX ) with X a topological space each point PC is repre-

sented by a single point x ∈ X by Lemma 5.1.9, so that {K(C)}C∈ΦX automatically gives a topology of closed subsets on K(ΦX ). In particular, each point PC is prime as we will

see later. However, it is not the case when Φ is a lattice of subcategories of certain type.

Lemma 5.1.14. Let f :Φ1 → Φ2 be an isomorphism of lattices. Then

(1) C ∈ Φ1 represents a point in K(Φ1) if and only if f(C) represents a point in K(Φ2);

(2) f induces a homeomorphism K(Φ1) ≈ K(Φ2).

Proof. (2) follows from (1). For (1), notice that Co 6= C if and only if f(C)o 6= f(C)

because f is an isomorphism.

94 5.1.3 Examples of classifying spaces

We study some examples in this subsection. Let Φr denote the lattice of replete subcat-

egories of a category A, that is the subcategory closed under isomorphisms.

Proposition 5.1.15. Let A be an essentially small category so that the isomorphism classes

X of objects in A forms a set. Then the classifying space K(Φr) is homeomorphic to X

with discrete topology.

∼ o Proof. Take an element C ∈ Φr and objects x, y ∈ C. Then either x = y or C = C .

c Suppose x y. Then we have a decomposition C = C1 ∪ (C1) with C1 = [x]r the

c smallest replete subcategory containing x and its complement in C. Notice that (C1) is

also a replete subcategory and they are both properly contained in C by assumption. Then

the space K(Φr) is homeomorphic to the space of isomorphism classes of objects in A with

discrete topology.

In fact, each isomorphism class [x] for x ∈ A gives a replete subcategory [x]r such that there are no proper replete subcategories contained in [x]r, thus it represents a point

P[x]r ∈ K(Φr). Hence each point PC in K(Φr) corresponds to exactly one isomorphism class of objects in A. This topology is discrete since both [x]r and its complement are

replete thus every singleton K([x]r) = {P[x]r } is both closed and open.

Now consider in a Krull-Schmidt category A, i.e. every object can be written as a coproduct of ﬁnite many indecomposables, the lattice Φ` of replete subcategories that are also closed under ﬁnitely coproducts and retracts.

Proposition 5.1.16. Let A be an essentially small category. Then the classifying space

95 K(Φ`) is homeomorphic to the space of isomorphism classes of indecomposable objects

in A with the discrete topology.

Proof. Since each point PC can be represented by one object x ∈ A, then decompose ` x = x1 x2 without loss of generality so that either [x1]` or [x2]` is [x]`, where [x]` represents the smallest subcategory in Φ` containing x. Otherwise, [x]` can written as a join of two properly contained subcategories so that it cannot represent a point PC in

K(Φ`). In particular, each representative x for PC can be chosen to be indecomposable.

Thus, the topology is also discrete.

Let Φs denote the lattice of Serre subcategories in an abelian category A.

Proposition 5.1.17. Suppose A is a noetherian abelian category. Then the points in the classifying space K(Φs) correspond bijectively to the isomorphism classes of monoform objects in A.

Proof. It sufﬁces to show that the representative of every point PC = P[x]s can be chosen to be monoform. In fact, suppose x is not monoform, then it has a monoform subobject x1 such that either [x1]s or [x/x1]s is [x]s since otherwise [x]s would not represent a point in

K(Φs), where [x]s represents the smallest Serre subcategory containing x. If [x1]s = [x]s then it’s done, else [x/x1]s = [x]s so that if x/x1 is not monoform, then we can replace x by x/x1 and repeat the procedure again, say a subobject x2 of x such that x1 ,→ x2. Thus we can obtain a chain of subobjects of x

x1 ,→ x2 ,→ x3 ,→ ... ,→ x

96 which has to be stationary after ﬁnitely many stages, since A is noetherian. Therefore,

[x]s = [x/x1]s = ... = [x/xn]s in which x/xn is monoform.

Conversely, each monoform object x indeed represents a point P[x]s in K(Φs) in the sense that [x]s cannot written as a join of proper subcategories since otherwise x ∈ [x]r would be contained in one proper Serre subcategory C [x]r, which is a contradiction.

5.2 Generally prime objects and classiﬁcation

The classiﬁcation theorem is based on the two crucial concepts, namely, prime and generally prime elements in a lattice Φ of subcategories. In fact, they resolve the problem of K(Φ) that it may not form a topological space, or we do not require the distributivity of the lattice Φ. We will study the relation between the two before we prove the main theorem.

Deﬁnition 5.2.1. Let A be a category such that there is a lattice Φ of subcategories in A of certain type. A subcategory C ∈ Φ or a point PC ∈ K(Φ) is called prime (generally prime

or simply g-prime, resp.) if whenever C ≤ C1 ∨ C2 with Ci ∈ Φ we have either C ≤ C1 or W C ≤ C2 (C ≤ i∈I Ci implies C ≤ Ci for some i ∈ I with I an arbitrary index set, resp.).

In particular, if PC is represented by a single object x ∈ A, that the category C is prime

is equivalent to that whenever x ∈ C1 ∨ C2 we have either x ∈ C1 or x ∈ C2. We denote

Kp(Φ) = {PC ∈ K(Φ) | C is prime}

the set of prime points.

97 Now choose a collection of subsets {Kp(C)}C∈Φ in which

0 Kp(C) = {PC0 ∈ Kp(Φ) | C ≤ C}.

Proposition 5.2.2. Let A be a category and Φ a lattice of subcategories in A of certain type such that Kp(Φ) forms a set. Then the collection {Kp(C)}C∈Φ forms a topology of closed subsets for Kp(Φ). Moreover, Kp(Φ) is Kolmogorov.

Proof. Since ∅, A ∈ Φ, we have Kp(∅) = ∅ and Kp(A) = Kp(Φ). It is clear that V V Kp(Ci) = Kp( Ci) since one can take the prime points of both sides from the equality V V K(Ci) = K( Ci). Finally, we need to show that Kp(C1) ∨ Kp(C2) = Kp(C1 ∨ C2)

for each Ci ∈ Φ. In fact, one can take the primes of both sides from K(C1) ∨ K(C2) ≤

K(C1 ∨ C2) to obtain one containment. Hence it is sufﬁcient to show the converse. Let

PC ∈ Kp(C1 ∨ C2). Thus C ≤ C1 ∨ C2 so that C ≤ C1 or C ≤ C2 by primeness of C. The

same argument of Proposition 5.1.11 applies to the last assertion.

Similar to Lemma 5.1.14 we also have

Lemma 5.2.3. Let f :Φ1 → Φ2 be an isomorphism of lattices. Then it induces a homeo-

morphism Kp(Φ1) ≈ Kp(Φ2).

We can deﬁne Kgp(Φ) similarly

Kgp(Φ) = {PC ∈ K(Φ) | C is generally prime}

the set of generally prime points and show without difﬁculty that it has a topology of closed

subsets given by

0 Kgp(C) = {PC0 ∈ Kgp(Φ) | C ≤ C}.

98 Proposition 5.2.4. The space Kgp(Φ) is a Kolmogorov space with the topology of closed

∼ subsets Kgp(C) for C ∈ Φ. Moreover, any isomorphism of lattices Φ1 → Φ2 induces a homeomorphism Kgp(Φ1) ≈ Kgp(Φ2).

Example 5.2.5. The whole category is usually prime but not g-prime. For instance, let

b A = Dfg(Q) and Φ the poset of nullity classes therein. Since the generators are of the form ΣiQ for some i ∈ Z and every nullity class becomes C(ΣiQ) ∩ A, it is clear that A as a nullity class is prime. However, it is not g-prime because we have

_ A = C(ΣiQ) ∩ A i∈Z while none of C(ΣiQ) ∩ A contains A.

Proposition 5.2.6. Kgp(ΦX ) = K(ΦX ) if and only if X has specialization-closed topology.

Proof. Suppose X has specialization-closed topology. Then the arbitrary supremum of

closed subsets is exactly their union, hence any point in K(ΦX ) represented by x ∈ X has to be g-prime as well. For the converse, suppose every point in K(ΦX ) is g-prime and S S S S i∈I Ci 6= i∈I Ci with Ci ∈ ΦX . Then there is x ∈ i∈I Ci − i∈I Ci, so that x ∈ {x} ⊆

Ci for some i ∈ I by g-primeness of P{x} ∈ K(ΦX ), which is a contradiction.

S Proposition 5.2.7. For a well-ordered index I, suppose the union i∈I Ci of any chain in

... ≤ Ci−1 ≤ Ci ≤ Ci+1 ≤ ...

again lies in Φ. Then Kgp(Φ) = Kp(Φ)

99 W 0 Proof. Assume C ∈ Φ is prime and C ≤ Ci with Ci ∈ Φ. Then we have a chain Ci :=

S W S 0 0 j∈I,j≤i Ci such that Ci = Ci is exactly the union of Ci by assumption. Suppose C is

0 represented by an object x and let i ∈ I be the least element such that x ∈ Ci. Then we

0 0 claim x ∈ Ci, hence C is g-prime. Indeed, if i has a successor, then Ci = Ci−1 ∨ Ci so that the primeness of C implies that x ∈ Ci since i is chosen to be the least one. For those i which has no successor,

0 _ 0 _ Ci = (∨j

0 0 in which ∪j

0 0 C. However, x ∈ ∪j

Let Φ be a complete lattice of certain subcategories of a category A. Notice that every general prime subcategory necessarily represents a point in K(Φ) by deﬁnition. We call a subcategory of A is generated by primes or primely generated if it can be written as a join of subset of prime objects.

Theorem 5.2.8. Assume that Φ is a complete lattice of subcategories of certain type in A.

Then there is a bijection

θ : {Kgp(C)}C∈Φ → {C ∈ Φ | C = Cb} : ξ

W 0 where Cb = C0≤C C . In particular, there is an isomorphism of lattices C0 g-prime

∼ {closed subsets in Kgp(Φ)} → {subcategories of certain type generated by g-primes}.

100 Proof. Deﬁne θ as

_ 0 θ(Kgp(C)) = C := D.

PC0 ∈Kgp(C) Observe that every g-prime C ∈ Φ represents a point in K(Φ) since otherwise C = Co

implies C ≤ C0 C for some C0 ∈ Φ by g-primeness, which is absurd. In particular,

θ(Kgp(C)) = Cb = D.

Then by comparing the index sets we have

_ _ _ D = C0 = C0 ≤ D0 = Db ≤ D 0 0 PC0 ∈Kgp(C) C ≤C D ≤D C0 g-prime D0 g-prime since every such C0 ≤ Cb = D, so that θ is well-deﬁned for Db = D.

For the inverse map, deﬁne ξ as ξ(C) = Kgp(C). The order preserving property follows immediately from the deﬁnitions of θ, ξ.

Next, for every C with C = Cb,

θξ(C) = θ(Kgp(C)) = Cb = C.

Also for every Kgp(C) with C ∈ Φ,

ξθ(Kgp(C)) = ξ(Cb) = Kgp(Cb) = Kgp(C)

since Cb ≤ C and every point in Kgp(C) is represented by some g-prime E ∈ Φ contained

W 0 in C so that E ≤ C = Cb. C0≤C g-prime C0∈Φ In other words, we can restate Theorem 5.2.8 so as to give applications in Section 5.4.

Corollary 5.2.9. Let Φ be a complete lattice of subcategories of a certain type in a category

A. Then these subcategories are classiﬁed by Kgp(Φ) if they are primely generated.

101 5.3 Irreducible vs. prime, and functoriality

In this section, we will brieﬂy discuss in a lattice irreducible and prime elements in various versions. In fact, the two notions are the same correspondingly if the lattice is distributive. Furthermore, we show that our construction of Kgp(Φ) for a complete lattice Φ is functorial in a non-obvious way, shedding light on the point-free construction in John- stone [12], where the point in the lattice Φ is described as a homomorphism p :Φ → {0, 1} of lattices with the range lattice {0, 1} consisting of only the top and the bottom elements so that it is functorial in an obvious way. See Section II.1.3. ibid.

Deﬁnition 5.3.1. Let Φ be a (complete) lattice and let C ∈ Φ be an element.

(1) C is join irreducible if whenever C = C1 ∨ C2 for C1,C2 ∈ Φ, we have either

C = C1 or C = C2. W (1’) C is completely join irreducible if C = Ci with Ci ∈ Φ implies C = Ci for some i.

(2) C is join prime if whenever C ≤ C1 ∨ C2 for C1,C2 ∈ Φ, we have either C ≤ C1 or C ≤ C2. W (2’) C is completely join prime if C ≤ Ci with Ci ∈ Φ implies C ≤ Ci for some i, which we also call generally prime in Section 5.2.

By dualization of the lattice (Φ, ≤, ∨, ∧), we refer to the lattice (Φ, ≥, ∧, ∨) with the same underlying set but with reversed order and join, meet interchanged. In other words, we turn the graph of Φ upside down. Thus we can deﬁne similarly (complete) meet irreducible and (completely) meet prime elements in Φ, respectively.

102 Notice that completely join (meet resp.) irreducible implies ﬁnite join (meet resp.) irreducible, and similarly for primeness.

Lemma 5.3.2. Let Φ be a (complete) lattice and C ∈ Φ. Then

(1) if C is (completely) join (meet resp.) prime, then it is (completely) join (meet resp.) irreducible;

(2) if additionally Φ is (completely) distributive, then that C is (completely) join (meet resp.) irreducible implies it is (completely) join (meet resp.) prime.

Proof. We show only for the ﬁnite versions and the complete cases are similar.

(1) Call C ∈ Φ iprime if C = C1 ∨ C2 implies C ≤ C1 or C ≤ C2. Clearly, prime implies iprime. Now suppose C is iprime. Then C = C1 ∨ C2 implies that C ≤ C1 or

C ≤ C2, so that C1 ≤ C ≤ C1 or C2 ≤ C ≤ C2. Hence iprime implies irreducible.

(2) Suppose C ≤ C1 ∨ C2. Then C = (C ∧ C1) ∨ (C ∧ C2) so that either C = C ∧ C1 or C = C ∧ C2 by irreducibility of C. Hence C ≤ C1 or C ≤ C2.

The problem of functoriality becomes clearer if we use the point-free description, see

Section II.1.3 in Johnstone [12]. Denote by

∧ Kp (Φ) = {C ∈ Φ | C is meet prime}.

∧ 0 ∧ 0 Let Kp (C) = {C ∈ Kp (Φ) | C ≥ C} for C ∈ Φ. For a homomorphism f :Φ → Ψ of

∧ ∧ ∧ complete lattices, we deﬁne Kp (f): Kp (Ψ) → Kp (Φ) as a supremum

∧ _ −1 Kp (f)(C) = f (↓ (C)), where ↓ (C) denotes the principal ideal generated by C so that it contains all elements

∧ below C, for every meet prime C in Ψ. Such Kp (f) is well-deﬁned. Indeed, suppose

103 W f −1(↓ (C)) ≥ A ∧ B. Then C ≥ f(A) ∧ f(B) applied f so that C ≥ f(A) or

W −1 W C ≥ f(B) and we assume the former. It follows that f (↓ (C)) = C≥f(D) D ≥ A.

∧ Proposition 5.3.3. For a complete distributive lattice Φ, the collection {Kp (C) | C ∈ Φ}

∧ ∧ forms a topology of closed subsets on Kp (Φ), so that Kp is a contravariant functor.

Proof. This is essentially Lemma II.1.3 and Theorem II.1.4 in [12]. For the topology,

∧ since Kp (0) = ∅ for the bottom element 0 here we require the primes are nonzero, and

∧ ∧ ∧ V V ∧ Kp (1) = Kp (Φ) for the top element 1 and Kp ( Ci) = Kp (Ci), we only need to show

∧ ∧ ∧ ∧ ∧ that Kp (C1) ∪ Kp (C2) = Kp (C1 ∧ C2) for each Ci ∈ Φ. Clearly, Kp (Ci) ⊆ Kp (C1 ∧ C2)

∧ ∧ ∧ by deﬁnition so that Kp (C1) ∪ Kp (C2) ⊆ Kp (C1 ∨ C2). Conversely, if C ≥ C1 ∧ C2, then

C ≥ Ci for some i by primeness.

∧ Given a homomorphism f :Φ → Ψ. We need to show that Kp (f) is continuous.

Indeed,

∧ −1 ∧ ∧ Kp (f) (Kp (C)) = Kp (f(C)),

∧ −1 which follows immediately from the fact that B ≥ f(A) if and only if Kp (f) (B) ≥

Remark 5.3.4. We can deﬁne spaces consisting of completely meet prime elements, (com-

∧ pletely) meet irreducible elements, and their dual cases respectively, denoted by Kcp(Φ),

∧ ∧ ∨ Ki (Φ), Kci(Φ) and dually Kp (Φ) etc. respectively. With a slightly different argument, we can show similar results to Proposition 5.3.3.

Notice that in the case of completely join primes, this space coincides with our Kgp(Φ) =

∨ Kcp(Φ) since every such prime element deﬁnes a point in K(Φ). See Theorem 5.2.8.

104 ∨ Also, under the dualization of a lattice Φ, Kp(Φ) becomes a subspace of Kp (Φ) which consists of all join prime elements in Φ. In Kp(Φ), we require extra that each prime C also deﬁnes a point in K(Φ), different from the complete case above.

∨ Similarly to Theorem 5.2.8 (which is in fact the case for Kcp(Φ)) and its proof, we have parallel results.

Theorem 5.3.5. Let Φ be a complete lattice of subcategories of certain type in a category

A. Then the subcategories generated by join primes are classiﬁed by the closed subsets of

∨ Kp (Φ). More precisely, there is a bijection

∨ θ : {Kp (C)}C∈Φ → {C ∈ Φ | C = Cb} : ξ

W 0 ∨ ∨ where Cb = C0≤C C , deﬁned by θ(Kp (C)) = Cb and ξ(C) = Kp (C). C0 join prime

Proof. It is straightforward to check that θ and ξ are well-deﬁned and inverses to each

b ∨ ∨ other, noting that Cb = Cb and Kp (Cb) = Kp (C).

With a slight modiﬁcation we can deduce immediately that

Corollary 5.3.6. Let Φ be a complete lattice of subcategories of certain type in a category

A. Then the subcategories generated by (complete resp.) join irreducibles are classiﬁed by

∨ ∨ the closed subsets of Ki (Φ) (Kci(Φ) resp.). The dual statements for the meet primes and irreducibles also hold accordingly.

These resemble the classical results appearing in Gabriel [7]. See also Section 4.1 in

Rouquier [25].

105 5.4 Primely generated subcategories

There are several examples that ﬁt into the context of Theorem 5.2.8 in the sense that

each subcategory is primely generated. The cases when we already know the generators

would be easier to handle; for example, the Serre subcategories in a noetherian abelian

category, the thick subcategories in Dperf (R) and the localizing subcategories in D(R), of a commutative noetherian ring R.

We will simply use the brackets h−i to denote a subcategory of certain type generated by a collection of objects correspondingly.

5.4.1 Serre subcategories in a noetherian abelian category

By Kanda’s theory [13], the Serre subcategories of a noetherian abelian category A is classiﬁed by its atom spectrum ASpecA via atom supports ASupp. Speciﬁcally, for any

two monoform objects H1 and H2, they are atom equivalent H1 ∼ H2 if there is a nonzero

common subobject. The atom spectrum is deﬁned as the collection of equivalence classes

of monoform objects. Furthermore, the atom support is given by

ASuppM = {H ∈ ASpecA | ∃H0 s.t. H0 is a subquotient of M}

It is shown in [13] that the atom support respects short exact sequences.

Proposition 5.4.1. Let A be a noetherian abelian category. Then every Serre subcategory

is primely generated. In particular, for every principle Serre subcategory hXi, there is a monoform object H such that hXi = hHi.

106 Proof. Notice that each Serre subcategory S in a Noetherian abelian category can be generated by monoform objects in the sense that S = ASupp−1ASuppS = {X | ASuppX ⊆

ASuppS} = hH0 ∈ H | H ∈ ASuppSi since every Noetherian object has a ﬁltration such that each consecutive quotient is a monoform object, see Theorem 2.9 in [13]. Therefore, to show that each Serre subcategory is primely generated it is sufﬁcient to show that the principal Serre subcategory generated by a monoform H, or equivalently that H is prime.

Indeed, suppose H ∈ hC1,C2i with each Ci ∈ Φ Serre. Then H ∈ ASuppC1 ∪ ASuppC2

0 0 so that we can assume H ∈ ASuppC1. Thus there is an H ∈ H such that H is a subquo-

0 tient of X ∈ C1. In particular, ASuppH = ASuppH ⊆ ASuppX ⊆ ASuppC1. Hence

H ∈ C1.

5.4.2 Thick subcategories in Dperf (R)

Recall that a thick subcategory of a triangulated category is a full triangulated subcategory such that it is closed under taking retracts. See [19] for the classiﬁcation of such thick subcategories in Dperf (R) of a commutative noetherian ring R.

Proposition 5.4.2. In the derived category Dperf (R) of perfect complexes, every thick subcategory is primely generated.

Proof. Since the specialization closed subsets of SpecR corresponds to thick subcategories of Dperf (R), each thick subcategory has a set of generators of the form R/p. In fact, one

−1 can show that Supp (∪i∈I V (pi)) = hR/piii∈I gives all the thick subcategories. Thus it is sufﬁcient to show every R/p is prime. This is straightforward by properties of support and

107 notice that SuppR/p = V (p) is closed.

5.4.3 Localizing subcategories in D(R)

A localizing subcategory in a triangulated category is a full triangulated subcategory that is closed under taking coproducts. In particular, it is thick. We will show that every localizing subcategory in D(R) is primely generated. However, this does not imply that all smashing subcategories are primely generated because being primely generated is a relative concept, working within the lattice of subcategories of the same type. All the necessary properties can be found in Neeman [19].

For convenience, we denote the following subset of SpecR by the small support

suppM = {p ∈ SpecR | k(p) ⊗ M 6= 0} for an object M ∈ D(R). We can also deﬁne its inverse in a natural way

supp−1W = {X ∈ D(R) | suppX ⊆ W } for any subset W ⊆ SpecR. The small support has similar properties as that of the normal support in commutative algebra.

Lemma 5.4.3. The small support has the following properties,

(1) k(p) ⊗ k(q) = 0 if q 6= p. In particular, supp k(p) = {p};

(2) supp ⊕i∈I Xi = ∪i∈I suppXi for Xi ∈ D(R);

(3) suppΣX = suppX for X ∈ D(R);

108 (4) For any distinguished triangle in D(R)

X → Y → Z → ΣX, we have suppY ⊆ suppX ∪ suppZ;

(5) supp(X ⊗ Y ) = suppX ∩ suppY for X,Y ∈ D(R).

− (6) SuppX = suppX for every bounded below complex X ∈ Dfg(R);

(7) suppX 6= ∅ for every nontrivial X ∈ D(R).

Proof. For (1), it is sufﬁcient to consider the tensor product of modules. So suppose q 6= p and thus there is an x ∈ p − q. Then k(p) ⊗ k(q) = (R/p ⊗ Rq) ⊗ (R/q ⊗ Rq) = 0 since

1 1 (1 + p) ⊗ s = (x + p) ⊗ xs =0. Since the derived tensor product commutes with coproducts, suspensions and is also exact, the assertions (2), (3), (4) follow immediately. For (5), notice that either X ⊗ k(p) = 0 or Y ⊗ k(p) = 0 implies X ⊗ Y ⊗ k(p) = 0. Conversely, take p ∈ suppX ∩ suppY . Then

X ⊗ Y ⊗ k(p) = X ⊗ (⊕Σik(p)) = ⊕Σi(X ⊗ k(p))

= ⊕Σi+jk(p) which is nontrivial.

Lemma 5.4.4. supp−1W = hk(p) | p ∈ W i, that is, it is the intersection of all localizing subcategories containing every k(p) with p ∈ W .

Proof. Suppose X ∈ D(R) such that suppX ⊆ W . Then for every p ∈ W c we have 0 =

X ⊗ k(p) = Γp/p−{p}(X) ⊗ k(p), thus Γp/p−{p}(X) = 0 by Lemma 2.14 in [19]. Therefore,

109 X ∈ hk(p) | p ∈ W i by Lemma 2.10 in [19]. Conversely, since suppk(p) = {p}, it is

sufﬁcient to show that supp−1W is a localizing subcategory, i.e. it is closed under taking

arbitrary coproducts, suspensions and extensions, which follows from Lemma 5.4.3.

Recall that every localizing subcategory D(R) is an ideal by Lemma 1.4.6 in [11]. In

fact, the subcategory consists of the objects absorbing D(R) forms a localizing subcategory

also containing R so that it becomes the whole category.

Lemma 5.4.5. Every localizing subcategory L in D(R) is generated by k(p) with p ∈ S ⊆

SpecR, where S = suppL. In other words, L = supp−1suppL.

Proof. Suppose X ∈ L. Then for any p ∈/ suppL, we have 0 = X ⊗k(p) = Γp/p−{p}(X)⊗ k(p). Thus Γp/p−{p}(X) = 0 by Lemma 2.14 in [19]. Hence X ∈ hk(p) | p ∈ Si by

Lemma 2.10 in [19]. Conversely, take p ∈ suppL. Then k(p) ⊗ X 6= 0 for some X ∈ L.

In particular, ⊕Σk(p) = k(p) ⊗ X ∈ L because L is an ideal. Hence k(p) ∈ L since L is localizing.

Corollary 5.4.6. X ∈ L if and only if suppX ⊆ suppL.

Proof. This is the exact statement of L ⊇ supp−1suppL in Lemma 5.4.5, thanks to the

−1 characterization of supp in Lemma 5.4.4.

Proposition 5.4.7. In D(R) with R a commutative noetherian ring, every localizing subcategory is primely generated.

Proof. It is sufﬁcient to investigate the primeness of the principal category generated by k(p). Suppose k(p) ∈ hC1,C2i for some localizing subcategories Ci = hk(p) | p ∈ Sii,

110 i=1,2. Then {p} ⊆ S1 ∪ S2 by taking supp simultaneously, thus either p ∈ S1 or p ∈ S2 so that k(p) ∈ C1 or k(p) ∈ C2 by Corollary 5.4.6. Therefore, every localizing subcategory in D(R) is primely generated.

5.4.4 Localizing subcategories in a stable homotopy category

As a matter of fact, a general statement holds in the axiomatic stable homotopy theory [11]. A stable homotopy category A is a triangulated category with a compatible closed symmetric monoidal structure, in which arbitrary coproducts exist, every cohomology functor is representable and there is a set of strongly dualizable objects generating A.

The following description of localizing subcategories enables us to do such a generalization.

Lemma 5.4.8. Let T ⊆ SpecR be an arbitrary subset. Then we can characterize the localizing subcategory as

hk(p) | p ∈ T i = {X ∈ D(R) | X ⊗ KT = 0}

L where KT = p∈T c k(p).

Proof. One containment is easy to deduce from Lemma 5.4.3 and the fact that tensor

product respects coproducts. Now suppose X ⊗ KT = 0 so that X ⊗ k(p) = 0 for every

c p ∈ T . Thus Γp/p−{p}(X) ⊗ k(p) = X ⊗ k(p) = 0, hence Γp/p−{p}(X) = 0 by Lemma

2.14 in [19]. Therefore, X ∈ hk(p) | p ∈ T i by Lemma 2.10 in [19].

Thus it becomes much more easier to deduce the primeness of principal localizing subcategory generated by k(p), as it is shown in Proposition 5.4.10. Recall from Section

111 6 of [11] that a noetherian stable homotopy category is a monogenic stable homotopy category (i.e. the generating set consists of only the unit object which is also compact) with a single compact generator S such that the graded commutative ring π∗(S) = [S,S]∗ is noetherian. The derived category D(R) of a commutative noetherian ring R gives an example of noetherian stable homotopy category.

Lemma 5.4.9. Let A be a noetherian stable homotopy category such that every monochromatic category hK(p)i is minimal among nonzero localizing subcategories of A. Then every localizing subcategory has the form

hK(p) | p ∈ T i = {X ∈ A | X ∧ KT = 0}

` for some subset T ⊆ SpecR, where KT = p∈T c K(p).

Proof. This is in fact Corollary 6.3.4 in [11], we give a proof of the equality here. Since

c K(p) ∧ KT = 0 for any p ∈ T by Proposition 6.1.7 in [11], thus hK(p) | p ∈ T i ⊆

{X ∈ A | X ∧ KT = 0}. Conversely, suppose X ∈ A such that X ∧ KT = 0. Notice

c that X ∧ KT = 0 if and only if X ∧ K(p) = 0 for every p ∈ T . It follows that X ∈ hX ∧ K(p) | p ∈ SpecRi by Proposition 6.3.2 in [11], so that X ∈ hX ∧ K(p) | p ∈ T i ⊆ hK(p) | p ∈ T i since every localizing subcategory is an ideal.

Proposition 5.4.10. Let A be a noetherian stable homotopy category such that every monochromatic category hK(p)i is minimal among nonzero localizing subcategories of

A. Then every localizing subcategory is primely generated.

Proof. Thanks to Lemma 5.4.9, we can assume hK(q) | q ∈ T1i, hK(q) | q ∈ T2i are

112 localizing subcategories such that K(p) ∈ hK(q) | q ∈ T1i ∨ hK(q) | q ∈ T2i. Then

K(p) ∈ hK(q) | q ∈ T1i ∨ hK(q) | q ∈ T2i = hK(q) | q ∈ T1 ∪ T2i

= {X ∈ A | X ∧ KT1∪T2 = 0},

hence K(p) ∧ KT1∪T2 = 0 so that p ∈ T1 ∪ T2 and it has to be in one of them. In particular,

K(p) ∈ hK(q) | q ∈ Tii for some i = 1, 2.

The same procedure applies to the thick subcategories of small objects in a more general setting. Also see Section 6 of [11].

Proposition 5.4.11. In a noetherian stable homotopy category, every thick subcategory of small objects is primely generated.

Proof. Notice that every thick subcategory is generated by S/p with p ∈ T in a specialization closed subset of SpecR, and we have the same properties as in the case of D(R).

For example, suppS/p = V (p), and X ∈ C in a thick subcategory of A if and only if suppX ⊆ suppC, where suppX = {p ∈ SpecR | K(p) ∧ X 6= 0}. Then one can show that every S/p is prime hence the thick subcategories are primely generated.

5.4.5 An example

Nevertheless, we do have examples of subcategories that are not primely generated.

1 2 One can compare this example with Example 3.4.9. Consider the quiver A2 :◦←−◦ and the category A of ﬁnite dimensional representations of A2 over an algebraically closed ﬁeld k. For convenience, we denote a = P1, b = P2 and c = S2.

113 Let Φ be the lattice of nullity classes in A. We can specify Φ, K(Φ), Kp(Φ), Kgp(Φ) and their topology Kp(C) explicitly thanks to the Auslander-Reiten quiver of A2

a o c @ @

Thus Φ = {h0i, hai, hci, hb, ci, A} and K(Φ) = {Ph0i,Phai,Phci,Phb,ci} so that Kgp(Φ) =

Kp(Φ) = {Ph0i,Phai,Phci}, with 5 closed subsets ∅, {Ph0i}, {Ph0i,Phai}, {Ph0i,Phci} and

Kgp(Φ). This example demonstrates that nullity classes cannot be classiﬁed by a topological space but only those primely generated ones. Notice that Phb,ci is not prime since we have hb, ci ≤ hai ∨ hci by the relation given in AR quiver of A2.

However, nullity classes are classiﬁed by closed subsets with extra conditions in a d- ifferent topological space. For example, in the spectrum SpecA = {a, b, c} of premonoforms, they correspond to the closed and extension-closed subsets, see Example 3.4.9.

Finally, it is interesting to point out that the lattice Φ does not come from a topological space by computing its point space ptΦ (see the deﬁnition of point space in Chapter II.1 of [12]), on which there is no valid topology. As a matter of fact, by depicting the diagram of the lattice Φ

A hb, ci hai hci

h0i and a general theory of lattices (see e.g. Chapter 4 of [6]), we conclude that the lattice Φ is not even distributive since it contains a pentagon, see Theorem 4.10 (ii) in [6].

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