Classifying Subcategories in Quotients of Exact Categories

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Classifying Triangulated and Thick

Triangulated Subcategories of an Algebraic Triangulated Category

Emilie B Arentz-Hansen

Master of Science in Mathematical Sciences Supervisor: Petter Andreas Bergh, IMF

Department of Mathematical Sciences Submission date: December 2017

Norwegian University of Science and Technology

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Abstract

The goal of this thesis is to prove a one-to-one correspondence between (thick) triangulated subcategories of the stable category associated with a Frobenius category and certain subcategories of the Frobenius category. The result is also generalized in the setting of an exact category and a quotient category. This thesis starts with an introduction to exact categories, quotient categories, Frobenius categories and the stable category of a Frobenius category. After this, the main results are explained and proved. Finally, a few applications are given.

Sammendrag

Målet med denne masteroppgaven er å konstruere en en-til-en korrespondanse mellom (tykke) triangulerte underkategorier av den stabile kategorien tilhørende en Frobenius-kategori og visse underkategorier av Frobenius-kategorien. Resultatet er videre generalisert slik at det holder for en eksakt kategori og en kvotientkategori. Oppgaven begynner med en introduksjon av eksakte kategorier, kvotekategorier, Frobenius-kategorier og den stabile kategorien av en Frobenius-kategori.

Deretter blir hovedresultatene forklart og bevist. Avslutningsvis gis det noen anvendelser.

Acknowledgments

I would like to thank my supervisor Professor Petter Bergh for all his help and support. I visited Professor Ryo Takahashi at the University of Nagoya in Nagoya, Japan for 3 weeks during my work, and I would like to thank him and others at the university for their help and hospitality during my stay.

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Notation

In this thesis we normally denote general categories byC, (pre)additive and abelian categories byA, exact categories by(E,S), Frobenius categories by(F,S)and triangulated categories by (T, T,∆), whereT is the autoequivalence and∆is the collection of distinguished triangles. We assume the reader to be familiar with additive, abelian and triangulated categories. However, a short introduction of the latter is given in Appendix B. We do not assume any prior knowledge about exact or Frobenius categories.

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Preface

This thesis was written in 2017 under the supervision of Professor Petter Bergh at the Norwegian University of Science and Technology. It treats exact categories, quotient categories, Frobenius categories and the stable category of a Frobenius category. The reader needs no prior knowledge about these subjects to understand this thesis. However, it is assumed that the reader is familiar with additive, abelian and triangulated categories. Nevertheless, a short introduction of the latter is given in Appendix B. This includes the definition and some results, but no proofs.

The stable category of a Frobenius category is triangulated, and several authors have studied the (thick) triangulated subcategories of such a category. In [7] Takahashi considers the category of maximal Cohen-Macaulay modules over a Gorenstein ring R, which is Frobenius, and the associated stable category. He classifies the thick triangulated subcategories of the stable category.

The motivation of this thesis was to prove a similar classification theorem for arbitrary Frobenius categories. This has been accomplished, and generalized further in the case of an exact category.

Chapter 1 introduces exact categories and quotient categories. Section 1.1 defines exact categories and gives some results. Our definition of an exact category is equivalent with the one given by Quillen in [9], and the prof of this is given. Section 1.2 treats quotient categories, especially what we mean withA/N for a preadditive categoryA and a subcategoryN . A Frobenius category is an exact category satisfying some extra assumptions, and the stable category of a Frobenius category is a quotient category. The purpose of this chapter is therefore to prepare for Chapter 2 on Frobenius categories, and Chapter 3 containing the main results of this thesis.

Chapter 2 introduces Frobenius categories and their stable categories. The definitions are given in section 2.1, while section 2.2 is concerned with the triangulated structure of the stable category.

Section 2.3 explains how the short exact sequences in a Frobenius category are closely related to the distinguished triangles in the stable category.

Chapter 3 contains the main results of this thesis. Section 3.1 treats the most general case in the setting of an exact category. First, the definitions of complete and thick subcategories of an exact categoryE and of a quotient categoryE/N are given. Then we provide the necessary assumptions to construct a one-to-one correspondence between complete/thick subcategories of E and complete/thick subcategories ofE/N . In section 3.2 we treat the special case when the exact category is Frobenius and the quotient category is the associated stable category. This results in a one-to-one correspondence between complete/thick subcategories of the Frobenius category and triangulated/thick triangulated subcategories of the stable category.

Chapter 4 gives some examples of Frobenius categories and applies the main results from Chapter 3 to these. Section 4.1 defines Gorenstein projective objects in an abelian categoryA and proves that the full subcategory consisting of these objects,GprojA, is Frobenius. Section 4.2 treats the special case where the abelian category ismodR, the category of finitely generated modules over a commutative ringR. In this case the Gorenstein projective objects are precisely the totally reflexiveR-modules. Furthermore, we consider the case where the ringRis a Goren- stein local ring. In this case the Gorenstein projective objects are the maximal Cohen-Macaulay modules overR. As mentioned, the classification of the thick triangulated subcategories of the stable category associated with this category is already given by Takahashi in [7]. While working on this thesis I visited Professor Ryo Takahashi at the University of Nagoya, Japan. Chapter 4 is a result of the work done under his guidance.

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1. Exact categories and quotient categories

1.1. Exact categories

The content of this section is taken from [9], [5] and [2]. We normally denote an additive category byA and an exact category byE. However, we will use the notationE leading up to the definition of an exact category.

Definition 1.1. Let E be an additive category andA −→^f B −→^g C a sequence in E. We call (f, g)akernel-cokernel pairiff is a kernel ofgandgis a cokernel off. LetSbe a family of kernel-cokernel pairs inE. If(f, g)∈ S, then we callf anadmissible monomorphismandg anadmissible epimorphism.

We use the notationsA ^f BandB ^g Cto specify thatf is an admissible monomorphism andgis an admissible epimorphism, respectively.

Definition 1.2. Let E be an additive category andS a family of kernel-cokernel pairs inE. Assume thatSis closed under isomorphisms and satisfies the following:

Ex0 For allA∈E the identity morphism1Ais an admissible monomorphism.

Ex0^op For allA∈E the identity morphism1Ais an admissible epimorphism.

Ex1 The class of admissible monomorphisms is closed under composition.

Ex1^op The class of admissible epimorphisms is closed under composition.

Ex2 Admissible monomorphisms are stable under pushout along arbitrary morphisms:

A B

A⁰ B⁰

f h P O h⁰

f⁰

Ex2^op Admissible epimorphisms are stable under pullback along arbitrary morphisms:

B⁰ C⁰

B C

g⁰ h⁰ P B h

g

In this caseSis anexact structureonE and(E,S)is anexact category. The elements ofSare calledshort exact sequences.

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Remark 1.3. (1)We will soon see that the axioms can be weakened and that they are equivalent to the classical definition given by Quillen in [9].

(2) By the duality of the axioms,Sis an exact structure onE if and only ifS^op is an exact structure onE^op.

(3) Isomorphisms are both admissible monomorphisms and admissible epimorphisms. As the diagram below illustrates, this follows from the fact that S is closed under isomorphisms and axiom Ex0 and Ex0^op, respectively.

A B 0

A A 0

f

∼= 1_A

∼= ∼= f⁻¹ ∼=

1A

(4) Assume that we haveA ^f B ^g C∈ Sandh:A→D. Then by axiom Ex2 there exists a pushout given by the left square of the following diagram

A B C

D P C

f h P O

g h⁰ f⁰ g⁰

By Lemma A.1 there exists a morphism g⁰ : P → C such that the diagram commutes and (f⁰, g⁰)∈ S. Dually, giveni:D→C, then we have a pullback

A P D

A B C

f⁰ g⁰

i⁰ P B i

f g

and a morphismf⁰:A→P such that(f⁰, g⁰)∈ S.

(5) An admissible epimorphism is always an epimorphism since it is a cokernel. Indeed, let A ^f B ^g C ∈ S and assume thatag =bgfor morphismsa, b:C →D. Thenagf = 0, so the cokernel property gives a unique morphismc:C→Dsuch thatcg=ag. However, botha andbsatisfies this, hencea=bandgis an epimorphism.

A B C

D

f g

ag

Dually, an admissible monomorphism is always a monomorphism since it is a kernel.

Example 1.4. LetA be an abelian category and define S:=n

X −→^f Y −→^g Z

0→X −→^f Y −→^g Z→0exacto .

ThenS is an exact structure onA and we call it thestandard exact structureonA. Another exact structure onA is given by

S⁰:=





 X

h1 0

i

−−−→X⊕Y 0 1

−−−−→Y

X, Y ∈A





 .

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1.1. Exact categories Let axiom Ex0’ be that10is an admissible epimorphism, where0denotes the zero object. In [5, Appendix A] Keller definesSto be an exact structure onE if itSis closed under isomorphisms and satisfies our axioms Ex0’, Ex1^op, Ex2 and Ex2^op. In our terminology, Quillen defines in [9]

that (E,S)is an exact category if S is a family of kernel-cokernel pairs inE which is closed under isomorphisms and satisfies the following:

a) For allA, BinE, A A⊕B B h1

0

i

0 1

is short exact.

b) The axioms Ex1, Ex1^op, Ex2 and Ex2^ophold.

c) Ifg : B → C has a kernel inE and if there exists a morphismh : D → B such that gh:D Cis an admissible epimorphism, thengis an admissible epimorphism.

c)^op Iff : A → B has a cokernel inE and if there exists a morphismi : B → E such that if :A Eis an admissible monomorphism, thenf is an admissible monomorphism.

Proposition 1.5. Our definition of an exact category as given in Definition 1.2, Keller’s definition and Quillen’s definition are all equivalent.

Before we prove the theorem we need the following lemma.

Lemma 1.6. Assume thatSis a family of kernel-cokernel pairs that satisfies Ex2 and Ex2^opand which is closed under isomorphisms. Then in the setting of Ex2 and respectively Ex2^op, i.e.

A B

A⁰ B⁰

f h P O h⁰

f⁰

resp.

B⁰ C⁰

B C

g⁰ h⁰ P B h

g

the sequence A B⊕A⁰ B⁰, h−f

h

i

h⁰f⁰

respectively B⁰ C⁰⊕B C

h−g⁰ h⁰

i

h g

is inS. Moreover, Lemma A.3 gives that the squares are both pushouts and pullbacks.

Proof. We prove it in the case of Ex2^op. Let(f, g)∈ S. From the dual part of Remark 1.3 (4) we have a commutative diagram

A B⁰ C⁰

A B C

f⁰ g⁰

h⁰ P B h

f g

with(f⁰, g⁰)∈ S. Using Remark 1.3 (4) again, we get the following commutative diagram, to the left, with rows and columns inS.

A B C

B⁰ E C

C⁰ C⁰

f f⁰ P O

g j⁰ j g⁰

e e⁰

A B

B⁰ E C

C⁰ C⁰⊕B

f f⁰ P O

j⁰ g

h 0 1B

i

j g⁰

h−g⁰ h⁰

i

e e⁰ α

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Since −g⁰

h⁰

f⁰= 0

1_B

f, the pushout property gives a morphismα:E→C⁰⊕Bwith αj⁰ =

0 1B

, αj= −g⁰

h⁰

.

Moreover, since(j⁰h⁰−j)f⁰= 0andg⁰is the cokernel off⁰, there exists a morphismγ:C⁰→E such that

γg⁰=j⁰h⁰−j.

Furthermore,αis an isomorphism with inverseβ = γ j⁰

. To see this, first note that we have αγg⁰=

1C 0t

g⁰. Sinceg⁰is an epimorphism by Remark 1.3 (5), this gives thatαγ =h

1_C 0

i . Thusαβ =

αγ αj⁰

= 1C⁰⊕B. To get the second part, note that βαj⁰ =

γ j⁰ 0

1_B

=j⁰= 1E◦j⁰, βαj=

γ j⁰ −g⁰

h⁰

=j= 1E◦j.

The pushout property then gives thatβα= 1E.

Moreover,eγ=hsinceeγg⁰ =hg⁰andg⁰is an epimorphism. Hence we therefore have that eα⁻¹ =

eγ ej⁰

= h g

. BecauseS is closed under isomorphisms and(j, e)∈ S, we get (αj, eα⁻¹)∈ S, which equals

B⁰ C⁰⊕B C.





−g⁰ h⁰



 h

h g i

Proof of Proposition 1.5. Quillen’s definition gives our definition: axiom a) gives axiom Ex0 and Ex0^op by lettingBandAbe the zero object, respectively. Our definition clearly gives Keller’s definition. So the only thing remaining to prove is that Keller’s definition gives Quillen’s definition. We follow the proof given by Keller in [5, Appendix A].

1st step: axiom a) holds. First note that1_Ais an admissible epimorphism by axiom Ex0’ and Ex2^opbecause of the pullback

A A

0 0

1_A

P B

1₀

The fact that axiom a) holds follows now from Lemma 1.6 since

A A

B B

−1A

0 0

1_B

is a pullback, giving that A A⊕B B h

1 0

i

0 1

is short exact.

2nd step: axiom c) and c)^ophold. LetD −→^h B −→^g Cbe as in c) and letf be the kernel ofg.

By Lemma 1.6 g gh

:B⊕D→Cis an admissible epimorphism and so is g 0

= g gh

1B −h 0 1D

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1.1. Exact categories sinceSis closed under isomorphisms. The kernel of

g 0

isf⊕1D, sof⊕1Dis therefore an admissible monomorphism. Thus, because of the pushout

A⊕D B⊕D

A B

f⊕1 h

1 0i

P O

h 1 0i f

we can conclude thatf is also an admissible monomorphism. Since g 0

is the cokernel of f⊕1D,gis a cokernel off. Hence(f, g)∈ S. The proof of c)^opis done dually.

3rd step: axiom b) holds.Proving Ex1 is the only thing remaining. LetA ^f B ^g Cbe short exact and letB ^h B⁰be an admissible monomorphism. From Ex2 we get the following pushout

B B⁰

C C⁰

h

g g⁰

k

(1.1)

By Lemma 1.6 this is also a pullback and g⁰ k

is an admissible epimorphism. The following diagram is a pullback

B⁰⊕B B⁰⊕C

B C

1⊕g

h 0 1

i

P B

h 0 1

i g

Hence by axiom Ex2^op,1B⁰⊕gis an admissible epimorphism as well. So by Ex1^op g⁰ k

1B⁰ 0 0 g

= g⁰ kg is also an admissible epimorphism. Since

g⁰ kg

=g⁰

1_B⁰ h

, the second step gives thatg⁰ is an admissible epimorphism if it has a kernel. Moreover,hf is the kernel ofg⁰, following from the fact thatf is the kernel ofgand the pullback property of (1.1). Hence(hf, g⁰)∈ S, sohfis an admissible monomorphism.

Corollary 1.7. Let A ^f B be an admissible monomorphism and A ^g C be an admissible epimorphism and consider the pushout

A B

C D

f g _{P O} g⁰

f⁰

Theng⁰is an admissible epimorphism. Moreover, ifgis an isomorphism, so isg⁰.

Proof. The fact thatg⁰is admissible epimorphism follows directly from the 3rd step in the proof of Proposition 1.5. To get the moreover part, assume that g is an isomorphism with inverse h:C→A. Note thatf hg=f◦1_A= 1_B◦f, so the pushout property gives a unique morphism h⁰ : D → Bsuch thath⁰f⁰ =f handh⁰g⁰ = 1_B. We getg⁰h⁰g⁰ =g⁰◦1_B = 1_D◦g⁰, which implies thatg⁰h⁰ = 1_Dsinceg⁰is an (admissible) epimorphism. Henceh⁰is the inverse ofg⁰.

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Proposition 1.8. Let

K:

A B

A⁰ B⁰

i

f f⁰

i⁰

be commutative withiandi⁰admissible monomorphisms. Then the following are equivalent:

(i) The square K is a pushout.

(ii) The sequence A B⊕A⁰ B⁰

h−i f

i

f⁰ i⁰

is short exact.

(iii) The square K is both a pushout and a pullback.

(iv) The square K is part of a commutative diagram of the form

A B C

A⁰ B⁰ C

i

f f⁰

p

i⁰ p⁰

with short exact rows.

Proof. (i)⇒(ii) follows from Lemma 1.6, (ii)⇒(iii) follows from Lemma A.3 and (iii)⇒(i) is obvious.

(i)⇒(iv) follows from Lemma A.1: Sinceiis an admissible monomorphism it is the kernel of a morphismp : B → C with(i, p) ∈ S. Hence by the lemma, there existsp⁰ : B⁰ → C such that the diagram commutes and withp⁰ the cokernel of i⁰. Moreover, i⁰ is an admissible monomorphism by assumption, hence(i⁰, p⁰)∈ S.

(iv)⇒(ii) Sincep, p⁰ are admissible epimorphisms there exists a pullback as in the diagram below.

A A

A⁰ P B

A⁰ B⁰ C

j i

j⁰ q

q⁰

P B p

i⁰ p⁰

From the dual of the implication (i)⇒(iv), we get the rest of the commutative diagram above with short exact rows and columns. Our goal is to prove thatA ^j P ^q B⁰ ∈ Sis isomorphic to the sequence

A h−i

f

i

−−−−→B⊕A⁰ f⁰ i⁰

−−−−−→B⁰.

Sincep=p⁰f⁰:B →C, the pullback property gives a unique morphismk:B→Psuch that q⁰k= 1B and qk=f⁰.

Nowq⁰(1_P−kq⁰) = 0, so sincej⁰is the kernel ofq⁰, there exists a uniquel:P →A⁰with j⁰l= 1P−kq⁰.

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1.1. Exact categories

By Remark 1.3 (5)j⁰is a monomorphism, hence

j⁰lk= (1_P−kq⁰)k= 0 =⇒ lk= 0, j⁰lj⁰= (1P −kq⁰)j⁰=j⁰ =⇒ lj⁰= 1A⁰. Similarly, sincei⁰is a monomorphism, we get

i⁰lj= (qj⁰)lj=q(1_P−kq⁰)j =qj−(qk)(q⁰j) =−f⁰i=−i⁰f =⇒ lj=−f.

The morphisms

k j⁰

:B⊕A⁰ →P and

q⁰ l

:P →B⊕A⁰ are inverses of each other:

k j⁰ q⁰

l

=kq⁰+j⁰l= 1_P and q⁰

l

k j⁰

=

q⁰k q⁰j⁰ lk lj⁰

=

1_B 0 0 1_A⁰

. Note that

f⁰ i⁰

=q k j⁰

and

i −f

= q⁰ l

j.

Hence we get an isomorphism

A P B⁰

A B⊕A⁰ B⁰

j

−1A

q

hq⁰ l

i h−i

f

i

f⁰i⁰

ThusA h−i

f

i

−−−−→B⊕A⁰

f⁰ i⁰

−−−−−→B⁰∈ S.

Definition 1.9. LetE⁰be a subcategory of an exact category(E,S). ThenE⁰isextension closed if wheneverX ^f Y ^g Zis inSwithX, Z∈E⁰, thenY ∈E⁰.

Proposition 1.10. LetE⁰be a full subcategory of an exact category(E,S)with0∈E⁰. Assume thatE⁰is extension closed and define

S⁰:=

X ^f Y ^g Z∈ S

X, Y, Z∈E⁰ .

Then(E⁰,S⁰) is an exact category andE⁰ closed under isomorphisms. We say that the exact structure onE⁰isinduced by the exact structure onE.

Proof. E⁰is additive: SinceE⁰is a full subcategory of an an additive category and contains the zero object,E⁰ is preadditive. For any objects X, Y inE⁰,X X⊕Y Y is short exact.

HenceX⊕Y ∈E⁰sinceE⁰is extension closed. ThusE⁰is additive.

By Proposition 1.5 it is enough to show that the axioms Ex0’, Ex1^op, Ex2 and Ex2^op are satisfied. Axiom Ex0’ follows immediately from0∈E⁰and the definition ofS.

Ex2: LetA ^f B ^g Cbe inS⁰and leth:A→ Dbe inE⁰. SinceE is exact, the pushout alongf andhexists, thus we get the following commutative diagram inE:

A B C

D P C

f h P O

g h⁰ f⁰ g⁰

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By Proposition 1.8 the second row is inS. SinceE⁰is extension closed andD, Care inE⁰, so is P. Thus the pushout alongfandhlies inE⁰. Ex2^opis done dually.

Ex1^op:LetA ^f B ^g CandP ^p B⁰ ^h Bbe inS⁰. Note thatghis an admissible epimorphism in(E,S)since the category is exact. Henceghis an admissible epimorphism in(E⁰,S⁰) as well if it has a kernel which lies inE⁰. By Ex2^opthere exists a pullback alonghandf with objects inE⁰:

A⁰ B⁰ C

A B C

f⁰ h⁰ P B

gh h

f g

By the dual of Lemma A.2f⁰is the kernel ofgh, so(f⁰, gh)∈ S⁰.

Closed under isomorphisms: Letf : X → Y be an isomorphism and assume thatX ∈ E⁰. ThenX ^f Y 0is inS, henceY ∈E⁰sinceE⁰is extension closed.

The next proposition consider the special case when the exact category is an abelian category.

Proposition 1.11. LetE be a full additive subcategory of an abelian categoryA. Suppose that E is extension closed and define

S:=

X ^f Y ^g Z inE

0 X ^f Y ^g Z 0exact inA

. Then(E,S)is an exact category.

Conversely, if(E,S)is an essentially small exact category, then there exists an abelian cate- goryA such thatE is an extension closed, full additive subcategory ofA and withSas above, i.e. consisting of the sequences inE that are exact inA.

Proof. Part I: This follows immediately from Proposition 1.10 whereA has the standard exact structure as defined in Example 1.4.

Part II: LetA be the category of left-exact contravariant functors fromE into the category of abelian groups. ThenA is abelian, andE becomes a subcategory ofA via the Yoneda embed- ding. It can be shown thatE is extension closed, and that a sequenceX →Y →Z inE is inS if and only if0→X →Y →Z →0is exact inA. The detailed proof of this can be found in both [5, Appendix A] and [2, Appendix A].

1.2. Quotient categories

The content of this section is mostly taken from [4].

Definition 1.12. LetA be a preadditive category. A collection of morphismsI is a (two-sided) idealofA if

(i) I(X, Y) :=I ∩Hom(X, Y)is a subgroup of the abelian groupHom(X, Y), and (ii) wheneverf ∈Hom(X, Y),g∈ I(Y, Z),h∈Hom(Z, W), thenhgf ∈ I(X, W).

Remark 1.13. Givenf, g∈Hom(X, Y), we say thatf is related togiff−g ∈ I(X, Y). It is clear that this is an equivalence relation onHom(X, Y).

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1.2. Quotient categories Definition 1.14. LetA be a preadditive category andIan ideal ofA. We define thequotient categoryA/Iby

objA/I := objA

Hom_A_/I(X, Y) := Hom_A(X, Y)/I(X, Y).

This means thatA/Ihas the same objects asA and that the morphisms inA/I are the equivalence classes of the morphisms inA. The equivalence class of a morphismf : X → Y inA will be denoted byf inA/I. When it is clear from the context, the notationHom_A(X, Y), or simplyHom(X, Y), will be used instead ofHom_A_/I(X, Y).

Note that composition in the quotient categoryA/I is in fact well defined. Indeed, assume thatf−f⁰ ∈ I(X, Y)andg−g⁰∈ I(Y, Z). Then

gf−g⁰f⁰=gf−gf⁰+gf⁰−g⁰f⁰=g(f −f⁰) + (g−g⁰)f⁰∈ I(X, Z),

which implies thatgf =g⁰f⁰. Associativity of composition in the quotient categoryA/Ifollows directly from the associativity inA. ThusA/Iis indeed a category.

Proposition 1.15. LetA be a (pre)additive category andI an ideal ofA. Then the quotient categoryA/Iis (pre)additive.

Proof. SinceI(X, Y)is a subgroup of the abelian groupHom(X, Y), the factor group Hom(X, Y) = Hom_A(X, Y)/I(X, Y)

is also an abelian group. The bilinearity of composition inA/Ifollows directly from the bilinearity of composition inA. If 0 is a zero object inA, then it is a zero object inA/Ias well sinceHom(0, X)andHom(X,0)will contain only one morphism. If a biproduct ofX, Y inA is given by

X

i₁

p₁

X⊕Y

i₂

p₂

Y

then it is trivial to see that a biproduct ofX, Y inA/Iis given by X

i1

p₁

X⊕Y

i2

p₂

Y.

Definition 1.16. We defineΣ :A →A/Ito be the functor given by X 7→X

(f :X →Y)7→(f :X →Y) We may refer tofas the image off.

It is trivial to see thatΣis in fact an additive functor.

Example 1.17. We say that a morphismf :X →Y factors throughthe objectN if there exist morphisms such that the diagram below commutes.

X Y

N

f

α β

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LetA be a preadditive category andN a (full) subcategory which is closed under finite direct sums. DefineI(X, Y)⊆Hom(X, Y)to be the collection of all morphisms which factor through some object inN , and denote byI the union of allI(X, Y). ThenI is an ideal ofA. To see this, first assume thatf₁, f₂∈ I(X, Y)factors throughN₁, N₂, respectively, withf_i =β_iα_ifor i= 1,2. Then the diagram

X Y

N₁⊕N₂

f1−f2

h α₁

−α2

i

β1β2

commutes. SinceN is closed under finite direct sums, this means thatf1−f2∈ I(X, Y). Thus I(X, Y)is a subgroup ofHom(X, Y). Now assume that we are in the setting of axiom (ii). Then we have

X Y Z W

N

f g

α

h β

Hencehgf∈ I(X, W)since it factors throughNviahgf= (hβ)(αf).

Definition 1.18. LetA be a preadditive category. Assume thatN is a (full) subcategory which is closed under finite direct sums and letIbe as in the example above. Then we defineA/N to be the quotient categoryA/I.

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2. Frobenius categories and the associated stable categories

The main sources of this chapter are [3] and [4].

2.1. Definition

In this section, we define what it means for an exact category to be a Frobenius category. We normally denote a general exact category by(E,S)and a Frobenius category by(F,S). How- ever, we use the notation(F,S)in the next definitions leading up to the definition of a Frobenius category.

Definition 2.1. LetS be a family of kernel-cokernel pairs in an additive categoryF. An object P inF isS-projectiveif for all admissible epimorphisms g:Y Z and for all morphisms a:P →Zthere exists a (not necessarily unique) morphismb:P→Y such thata=gb:

P

X Y Z ∈ S

∃b a

f g

Dually, an objectIinF isS-injectiveif for all admissible monomorphismsf :X Y and for all morphismsa:X →Ithere exists a morphismb:Y →Isuch thata=bf:

X Y Z ∈ S

I

f

a ∃b

g

Example 2.2. Any initial object isS-projective and any terminal object isS-injective. In partic- ular, the zero object is bothS-projective andS-injective.

Definition 2.3. LetS be a family of kernel-cokernel pairs in an additive categoryF. We define projF, resp.injF, to be the full subcategory ofF consisting of allS-projective objects, resp.

S-injective objects.

Definition 2.4. An exact category (F,S) has enoughS-projectives if for all objects X in F there exists an admissible epimorphism g:P X with P an S-projective object in F, and enoughS-injectives if for all objectsX inF there exists an admissible monomorphism f :X IwithIanS-injective object inF.

Definition 2.5. An objectAis adirect summandofXif there exist morphismsA−→^f X −→^g A such thatgf = 1A.

Lemma 2.6. The direct sum of twoS-injective objects isS-injective and a direct summand of an S-injective object isS-injective. Dually, the direct sum of twoS-projective objects isS-projective and a direct summand of anS-projective object isS-projective.

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Proof. The first part about direct sum is trivial, so we only prove the second part about direct summand. LetJ be a direct summand of anS-injective objectI. Then there exist morphisms J

i

p

Iwithpi= 1J.

X Y

J I

∀f

∀a ∃b

i p

Since I is S-injective we have that for all admissible monomorphisms f :X Y and morphismsa: X →J there exists a morphismb:Y →Isuch thatia=bf. Letc :Y →J with c=pb. Thencf =pbf =pia=a. HenceJis anS-injective object.

Remark 2.7. A sequenceX ^f Y ^g Z ∈ S is right split wheneverZ isS-projective and left split wheneverX isS-injective. To see this, simply letain the definition ofS-projective and S-injective objects be the identity morphism.

Definition 2.8. An exact category(F,S)is aFrobenius categoryif it has enoughS-projectives, enoughS-injectives and ifprojF = injF.

Let(F,S)be a Frobenius category unless otherwise stated. For a pair of objectsX,Y inFlet I(X, Y)denote the additive subgroup ofHom(X, Y)consisting of the morphisms which factor through anS-injective object. Thestable categoryF associated withF is the category whose objects are the objects ofF and the set of morphisms fromX toY isHom_F(X, Y)/I(X, Y).

In other words,F is the quotient categoryF/injF as defined in Definition 1.18. We use the notationHom(X, Y)instead ofHom_F(X, Y)to denote the set of morphisms fromX toY in F. The equivalence class of a morphismf :X →Y inF is denoted byfinF.

Remark 2.9. Assume that f : X → Y factors through some S-injective objectJ as in the diagram below. Letµ:X I be an admissible monomorphism,IanS-injective. Then there existsα:I→Y such thatαµ=f.

X Y

I J

f µ

g

α

β h

Namely, since J is an S-injective object and µis an admissible monomorphism, there exists β: I→ J such thatβµ=g. Letα:=hβ. Thenαµ=hβµ=hg=f.

2.2. Triangulation of the stable category

The stable category of a Frobenius category is triangulated in a very natural way, as we will see in this section. The triangulated categories that arise in this way are calledalgebraic, and the ones that appear naturally in homological algebra (homotopy categories of complexes, derived categories) are all of this form. Given a stable category F, we construct an autoequivalence T :F →F, a collection of triangles∆, and prove that this give a triangulated structure onF. Appendix B gives a short introduction to triangulated categories. It is assumed that the reader is

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2.2. Triangulation of the stable category familiar with the content of that appendix. We will therefore refer to the axioms (TR1) to (TR4) without explaining their content.

Lemma 2.10. LetX ^µ I ^π X⁰andY ^µ I⁰ Y⁰

0 π⁰

be exact in(F,S)withI⁰(but not neces- sarilyI) anS-injective object. Given any morphismf :X →Y there exist morphisms such that the following diagram commutes.

X I X⁰

Y I⁰ Y⁰

µ f

π I(f) f⁰

µ⁰ π⁰

Proof. We have thatµis an admissible monomorphism and I⁰ isS-injective, so there exists a morphismI(f) : I → I⁰ such thatI(f)µ = µ⁰f. Since0 = π⁰µ⁰f = π⁰I(f)µandπis the cokernel ofµ, there exists a morphismf⁰:X⁰→Y⁰such thatf⁰π=π⁰I(f).

Lemma 2.11. LetX ^µ I ^π X⁰andX ^µ I⁰ X⁰⁰

0 π⁰

be inS, whereIandI⁰areS-injectives.

ThenX⁰andX⁰⁰are isomorphic inF.

Proof. From Lemma 2.10 we get morphisms such that the following diagram commutes

X I X⁰

X I⁰ X⁰⁰

X I X⁰

µ π

f g

µ⁰ π⁰

f⁰ g⁰

µ π

Since0 = (f⁰f−1_I)µandπis the cokernel ofµ, there existsh:X⁰→Isuch thathπ=f⁰f−1_I. Thusπhπ=π(f⁰f−1I) =πf⁰f−π=g⁰gπ−π= (g⁰g−1X⁰)π, and sinceπis an epimorphism it follows thatπh=g⁰g−1X⁰. This means thatg⁰g−1X⁰ factors through theS-injective object I, henceg⁰g= 1_X0. Similarly,gg⁰= 1_X00.

For an objectX inF, let[X]denote the isomorphism class ofX inF. LetX I X⁰ be inSwithIanS-injective object. Then the lemma states that[X⁰]is independent of the choice of X I X⁰. For each object X inF choose a sequence X ^µ(X)I(X) ^π(X) T X ∈ S withI(X)anS-injective object, and defineT(X) := T X. Givenf : X → Y we get from Lemma 2.10 a commutative diagram

X I(X) T X

Y I(Y) T Y

µ(X) f

π(X)

I(f) T(f)

µ(Y) π(Y)

Moreover, the next lemma shows that the equivalence class ofT(f)is independent of the choice ofI(f).

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Lemma 2.12. Given a commutative diagram

X I(X) T X

Y I(Y) T Y

µ(X) f

π(X)

I_i(f) Ti(f) µ(Y) π(Y)

fori= 1,2, thenT1(f) =T2(f)inF.

Proof. We have[I1(f)−I2(f)]µ(X) =µ(Y)f−µ(Y)f = 0, so the cokernel property ofπ(X) gives that there exists a uniqueg:T(X)→I(Y)such thatgπ(X) =I1(f)−I2(f). Hence

π(Y)◦g◦π(X) =π(Y)◦[I1(f)−I2(f)] = [T1(f)−T2(f)]◦π(X).

Sinceπ(X)is an epimorphism, this implies thatπ(Y)g=T1(f)−T2(f). ThusT1(f)−T2(f) factors through theS-injective objectI(Y), soT1(f) =T2(f)inF.

As a consequence of this,T :F →F is a well-defined functor, and the next theorem shows thatT is an autoequivalence. Furthermore,T is an automorphism if we are able to chooseT X such thatT : [X]→[X⁰]is a bijection for allX. Readers who only want to consider triangulated categories where the functor is an automorphism would have to make this assumption throughout this thesis.

Theorem 2.13. T :F →F is an autoequivalence. Moreover, ifT : [X]→[X⁰]is a bijection for allX, thenT is an automorphism.

Proof. T is dense: LetY ∈F. SinceF has enoughS-projectives there existsX ^µ P ^π Y inSwithPanS-projective object. TheS-projectives coincide with theS-injectives, soPis also injective. HenceT(X)andY are isomorphic inF by Lemma 2.11. Moreover, ifT : [X]→[Y] is a bijection, then there exists a uniqueX⁰∈[X]withT(X⁰) =Y.

T is full: Assume that we haveg : T(X)→ T(Y). We want to constructf : X →Y such thatT(f) =g.

X I(X) T X

Y I(Y) T Y

µ(X) f

π(X)

g⁰ g

µ(Y) π(Y)

SinceI(X)isS-projective there existsg⁰ : I(X)→I(Y)such thatπ(Y)g⁰ =gπ(X). Hence π(Y)g⁰µ(X) =gπ(X)µ(X) = 0, so by the kernel property ofµ(Y), there existsf : X →Y such thatg⁰µ(X) =µ(Y)f. Thus by Lemma 2.12,T(f) =ginF andT is full.

T is faithful:Assume thatT(f1) =T(f2). We want to prove thatf1=f2.

X I(X) T X

Y I(Y) T Y

µ(X) f₁−f2

π(X)

h I(f₁)−I(f2) T(f₁)−T(f₂)=0

µ(Y) π(Y)

Note thatπ(Y)[I(f1)−I(f2)] = [T(f1)−T(f2)]π(X) = 0, so the kernel property of µ(Y) gives a unique morphismh : I(X) → Y such that I(f1)−I(f2) = µ(Y)h. Furthermore, π(Y)[I(f1)−I(f2)]µ(X) = 0as well, so there exists a unique morphismα : X → Y such that[I(f1)−I(f2)]µ(X) =µ(Y)α.Bothf1−f2andhµ(X)satisfies the condition ofα, hence f1−f2=hµ(X), givingf1=f2inF.

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2.2. Triangulation of the stable category Remark 2.14. (1) The construction of the functorT involves a choice for each objectX. If the functorsT1andT2are obtained from different choices, then it is possible to show thatT1andT2

are naturally isomorphic functors. See for example Happel’s proof in [3, Section 2.2] for details.

(2) To simplify notation we will useX ^x I(X) ^x T Xinstead ofX ^µ(X) I(X) ^π(X) T X. Take a morphismu:X →Y inF and letX ^µ I ^π X⁰ ∈ SwithIanS-injective object.

Consider the solid part of the following diagram inF

X I X⁰

Y Cu X⁰ T X

µ u P O

π u

v w⁰

w:=g⁰w⁰ g⁰

whereCuis the pushout alongµandu, andw⁰is the cokernel ofvas given in Lemma A.1. Note thatY ^v Cu ^w⁰ X⁰∈ Sby Proposition 1.8. Lemma 2.11 provides a morphismg⁰:X⁰ →T X such thatg⁰is an isomorphism inF. Definew:=g⁰w⁰:Cu→T X.

Definition 2.15. Using the notation as above, we call the triangleX −→^u Y −→^v Cu

−w→T X and its image inF astandard triangle. We define∆to be the collection of all triangles inF which are isomorphic to a standard triangle, and we call such trianglesdistinguished triangles.

Remark 2.16. Some authors have a different definition of a standard triangle, which we call strictly standard: A triangleX −→^u Y −→^v Cu

−w→T X isstrictly standardif it is obtained from the pushout

X I(X) T X

Y Cu T X

x u P O

x u

v w

We will see in Corollary 2.18 that every standard triangle is isomorphic to the strictly standard triangle constructed from the same morphismu:X →Y. Hence the collection of all triangles which are isomorphic inF to astrictlystandard triangle equals∆. Thus both definitions of a standard triangle will give the same triangulation onF.

We shall now prove thatF is a triangulated category. However, we will need the Triangu- lated Five Lemma (Lemma B.4) in order to prove (TR4). Therefore, we first prove that F is pretriangulated.

Theorem 2.17. The triple(F, T,∆)is a pretriangulated category.

Proof. (TR1). It is clear from the definition that∆is closed under isomorphisms and that every morphism is part of a triangle. Consider the following commutative diagram of triangles:

X X I(X) T X

X X C1_X T X

1

x 1 P O

x

1 1

1 v w

Since1X : X → X is an isomorphism, so is1X by Corollary 1.7. Hence we have in fact an isomorphism of triangles. The bottom row is a standard triangle, thus the image inF of the top

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row is distinguished. It is clear thatX −→¹ X −→^x I(X)−→^x T X ∈∆andX −→¹ X →0 →T X are isomorphic inF, making the latter triangle distinguished.

(TR2’). Note that (TR2’) is half of (TR2), i.e. rotating only one way: see Appendix B. We only need to prove the axiom for standard triangles. LetX −→^u Y −→^v C_u −^w→T Xbe a standard triangle given by the following diagram

X I X⁰

Y Cu X⁰ T X

µ u P O

π u

v w⁰

w=g⁰w⁰ g⁰

From Lemma 2.10 we getu⁰ :I → I(Y)andu⁰⁰ :X⁰ →T Y such that the following diagram commutes

X I X⁰

Y I(Y) T Y

µ u

π

u⁰ u⁰⁰

y y

Sinceyu=u⁰µ, the pushout property ofCugives a unique morphismθ:Cu →I(Y)such that the following diagram commutes

X Y

I Cu

I(Y)

µ u

P O v

y u

u⁰ θ

Note thatyθv=yy = 0 =u⁰⁰w⁰vandyθu=yu⁰ =u⁰⁰π=u⁰⁰w⁰u. Henceyθ=u⁰⁰w⁰ by the pushout property ofCu. We get the following commutative diagram

Y Cu X⁰

I(Y) I(Y)⊕X⁰ X⁰

T Y T Y

v

y

w⁰

hθ w⁰

i h

1 0

i

y

0 1

y−u⁰⁰

(2.1)

As mentioned before Definition 2.15, the first row is exact. It follows from Quillen’s axiom a) that the second row is exact. Hence from Proposition 1.8 the upper left square is a pushout. Since

y −u⁰⁰ θ

w⁰

=yθ−u⁰⁰w⁰= 0, Diagram 2.1 yields that

Y −−→^v Cu





θ w⁰





−−−−→I(Y)⊕X⁰

hy −u⁰⁰ⁱ

−−−−−−−→T Y

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2.2. Triangulation of the stable category is a standard triangle.

Now consider the following diagram:

Y Cu I(Y)⊕X⁰ T Y

Y C_u T X T Y

v

hθ w⁰

i

y−u⁰⁰

0g⁰

v w −T u

We have 0 g⁰

θ w⁰

=g⁰w⁰=w, making the middle square commutative. It is possible to use Lemma 2.12 to prove that(T u)g⁰ =u⁰⁰inF. Hence the square to the right commutes inF since

−T u 0 g⁰

= 0 −(T u)g⁰

=

y −u⁰⁰

. Moreover, we know thatg⁰ is an isomorphism inF, hence the diagram above is an isomorphism of triangles, thusY −→^v Cu

−w→T X −−−→^{−T u} T Y is distinguished.

(TR3). Consider two standard triangles

X I X⁰

Y Z X⁰ T X

µ u

π u

v s τ

and

A I⁰ A⁰

B C A⁰ T A

µ⁰ u⁰

π⁰ u⁰

v⁰ s⁰ λ

withτ , λ isomorphisms. Let w := τ s and w⁰ := λs⁰. Assume that we have the following commutative diagram inF

X Y Z T X

A B C T A

u ϕ

v ψ

w

T ϕ

u⁰ v⁰ w⁰

From Lemma 2.10 and 2.11 we know that we can construct the following commutative diagram withν :T X →X⁰the inverse ofτ :X⁰→T X

X I(X) T X

X I X⁰

A I⁰ A⁰

A I(A) T A

µ(X) π(X)

ν⁰ ν

µ ϕ

π

ϕ⁰ ϕ⁰⁰

µ⁰ π⁰

λ⁰ λ

µ(A) π(A)

We get thatT ϕ=λϕ⁰⁰νfrom Lemma 2.12. By assumptionψu=u⁰ϕ, hence there is a morphism α:I→Bsuch thatψu=u⁰ϕ+αµ. Now

v⁰ψu=v⁰(u⁰ϕ+αµ) =v⁰u⁰ϕ+v⁰αµ=u⁰µ⁰ϕ+v⁰αµ=u⁰ϕ⁰µ+v⁰αµ= (u⁰ϕ⁰+v⁰α)µ.

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SinceZis a pushout we get a morphismθ:Z→Csuch thatθu=u⁰ϕ⁰+v⁰α, θv=v⁰ψas in the diagram below.

X I

Y Z

C

µ

u u

u⁰ϕ⁰+v⁰α v

v⁰ψ θ

The pushout property of Z gives thats⁰θ=ϕ⁰⁰ssince

(s⁰θ−ϕ⁰⁰s)u=s⁰(u⁰ϕ⁰+v⁰α)−ϕ⁰⁰π=π⁰ϕ⁰−ϕ⁰⁰π= 0, (s⁰θ−ϕ⁰⁰s)v=s⁰θv−ϕ⁰⁰sv=s⁰v⁰ψ−ϕ⁰⁰sv= 0−0 = 0.

Hence we get a commutative diagram

X Y Z X⁰

A B C A⁰

u ϕ

v ψ

s

θ ϕ⁰⁰

u⁰ v⁰ s⁰

giving us the following morphism of triangles

X Y Z T X

X Y Z X⁰

A B C A⁰

A B C T A

u v w=τ s

ν

T ϕ=λϕ⁰⁰ν u

ϕ

v ψ

s

θ ϕ⁰⁰

u⁰ v⁰ s⁰

λ u⁰ v⁰ w⁰=λs⁰

Thus (TR3) holds for standard triangles.

Corollary 2.18. Letu:X →Y be a morphism inF. The two standard triangles given by

X I X⁰

Y Z X⁰ T X

µ u _{P O}

π u

v s τ

and

X I⁰ X⁰⁰

Y Z⁰ X⁰⁰ T X

µ⁰ u P O

π⁰ u⁰

v⁰ s⁰ τ⁰

are isomorphic inF. Moreover, every standard triangle is isomorphic to the strictly standard triangle constructed from the same morphismu:X →Y.

Proof. This follows immediately from (TR3) and the Triangulated Five Lemma (Lemma B.4) applied to the following commutative diagram:

X Y Z T X

X Y Z⁰ T X

u v w

u v⁰ w⁰

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2.2. Triangulation of the stable category

Theorem 2.19. The triple(F, T,∆)is a triangulated category.

Proof. (TR4). We only need to consider the case of standard triangles. By Corollary 2.18 we may take the triangles to be strictly standard, as defined in Remark 2.16. Assume that we have three standard triangles given by

X Y

I(X) Z⁰

T X T X

u

x i

u

x i⁰

,

Y Z

I(Y) X⁰

T Y T Y

v

y j

v

y j⁰

and

X Z

I(X) Y⁰

T X T X

w:=vu

x k

w

x k⁰

Our goal is to prove that there exist morphisms such that the diagram below commutes inF with Z⁰ −→^f Y⁰ −→^g X⁰ ^{(T i)j}

0

−−−−→T Z⁰∈∆.

X Y Z⁰ T X

X Z Y⁰ T X

X⁰ X⁰ T Y

T Y T Z⁰

u i

v

i⁰ f

w=vu k

j

k⁰

g T u

j⁰ (T i)j⁰ j⁰

T i

(2.2)

LetZ⁰ ^l I(Z⁰) ^l T Z⁰∈ SwithI(Z⁰)S-injective. Sinceiandlare admissible monomorphisms, so isli, thus there existsp:I(Z⁰)→M such that(li, p)∈ S. Now, by Corollary 2.18, we may take Y ^li I(Z⁰) ^p M instead of Y ^y I(Y) ^y T Y. Hence we may assume that I(Y) =I(Z⁰)andy =li. Thenyu=liu=lux. DefineIu:=lu:I(X)→I(Y)as in the diagram below. Then there existsT u:T X →T Y withT ux=yI_u=ylu. Let1 :I(Y)→I(Z⁰), giving a morphismT i : T Y → T Z⁰ withT iy = l. In other words the following diagrams commute:

X I(X) T X

Y I(Y) T Y

x u

x

Iu=lu T u

y y

Y I(Y) T Y

Z⁰ I(Z⁰) T Z⁰

y i

y

1 T i

l l

Sincewx=kw=kvuandjw=jvu=vyu=vliu=vlux, we get the following pushouts:

X Y

I(X) Z⁰

Y⁰

x u

i

kv u

w f

X Z

I(X) Y⁰

X⁰

w

x k

j w

vlu g

Classifying Subcategories in Quotients of Exact Categories - Classifying Triangulated and Thick Triangulated Subcategories of an Algebraic Triangulated Category