How Can We Understand an Abelian Category?

Home , AB5 category, Abelian category, Additive category, Biproduct, Category of modules, Field of fractions

HOW CAN WE UNDERSTAND AN ABELIAN CATEGORY?

Jackson Ryder

Supervisor: Associate Professor Daniel Chan

School of Mathematics and Statistics UNSW Sydney

November 2020

Submitted in partial fulfillment of the requirements of the degree of Bachelor of Science with Honours

Plagiarism statement

I declare that this thesis is my own work, except where acknowledged, and has not been submitted for academic credit elsewhere.

I acknowledge that the assessor of this thesis may, for the purpose of assessing it: • Reproduce it and provide a copy to another member of the University; and/or, • Communicate a copy of it to a plagiarism checking service (which may then retain a copy of it on its database for the purpose of future plagiarism checking).

I certify that I have read and understood the University Rules in respect of Student Academic Misconduct, and am aware of any potential plagiarism penalties which may apply.

By signing this declaration I am agreeing to the statements and conditions above.

Signed: Date:

Acknowledgements

Firstly, I would like to thank my supervisor Daniel Chan for exposing me to such a wonderful area of mathematics, and for his constant support and guidance throught this challenging year. I would also like to thank my highschool maths teacher, Phil Baillie, for ﬁrst introducing me to higher maths and taking the time to foster my interest in the sub- ject. Without him I would never have thought of pursuing a degree in mathematics to begin with. Last, but certainly not least, I thank my family, particularly my mother and brother, for their unconditional love and support. I truly would not be where I am today without them.

Abstract

A classical theorem of Morita theory states that if an abelian category has a progenerator then it is equivalent to the category of modules over some ring. However, not all abelian categories have a progenerator, and so there are some abelian categories to which the theorem of Morita can not be applied. We prove a Morita-type result, introducing conditions for an arbitrary abelian category to be equivalent to a quotient of the category of coherent modules over an indexed algebra.

iii

Contents

Chapter 1 Introduction 1 1.1 Introduction ...... 1 1.2 Assumed Knowledge ...... 2

Chapter 2 Abelian Categories 3 2.1 Additive Categories ...... 3 2.2 Abelian Categories ...... 6 2.3 Directed Systems and Limits ...... 10 2.4 Determining Equivalence with a Module Category ...... 11

Chapter 3 Cohomology in Abelian Categories 16 3.1 Cochain Complexes and Resolutions ...... 16 3.2 Exactness and Derived Functors ...... 18

Chapter 4 Quotients of Abelian Categories 20 4.1 The Field of Fractions of Z ...... 20 4.2 Quotient Categories ...... 21 4.3 Localising Subcategories ...... 26 4.4 Grothendieck Categories ...... 30

Chapter 5 I-Algebras and cohproj 33 5.1 I-algebras ...... 33 5.2 Cohomology of I-algebras ...... 42 5.3 Determining Equivalence with cohproj of a Coherent I-algebra . . . 45

References 52

Chapter 1 Introduction

1.1 Introduction Abelian categories arose in the mid-1950s in the attempt to unify cohomology the- ories for groups and for sheaves. They can be thought of as a generalisation of the category of modules over a ring, taking only the properties required to be able to perform homological algebra. For this reason, abelian categories appear frequently throughout fields such as algebraic topology and algebraic geometry. One downside to this generality is that we don’t know a lot more about an arbitrary abelian category than the properties it satisfies to be abelian. Compare this to the category of modules over a ring, where we have all of the knowledge of the theory of modules at our disposal. For this reason, results demonstrating equivalence of an abelian category with some more well-understood category are of significance. The aim of this thesis is to prove such a result. We begin by talking about abelian categories. We will introduce the basic theory of abelian categories in a similar fashion to [15], culminating in proving a theorem of Morita which gives necessary and sufficient conditions for an abelian category to be equivalent to the category ModR of modules over some ring R. As we mentioned earlier, abelian categories were introduced as a general setting for homological algebra. Chapter 3 will be devoted to a brief exposition of this theory, introducing the homological material necessary to prove the main theorem of the thesis. Chapter 4 will begin by presenting a notion of a quotient of an abelian category by a Serre subcategory, which will be the type of category our main result will identify an arbitrary abelian category with. The idea behind the construction of these quotients will be to describe the construction of the field of fractions of the integers, or more generally the localisation of modules over a commutative ring, in a way that can be applied to any abelian category. These quotient categories originally arose from the field of projective geometry, where it was shown in [14] and [7] that the category cohX of coherent sheaves on a projective scheme X is equivalent to a quotient qgrA of the category of finitely generated graded modules grA over the homogeneous coordinate ring A of X. This result gave way to the field of noncommutative projective geometry as this quotient construction works in the case that A is noncommutative, allowing us to view such a quotient as the category of coherent sheaves on some ‘noncommutative space’. The last portion

1 of this chapter will be devoted to study the properties of a specific type of abelian category, named after Grothendieck, which are intimately related to these quotients. These Grothendieck categories will appear in the final chapter as the category GrA of graded modules over a graded algebra A form a Grothendieck category. The goal of the last chapter of the thesis will be to prove an analogue of the theorem of Morita introduced in Chapter 2. The first section of this chapter will introduce indexed algebras; a special type of bigraded algebras that arose from noncommutative projective geometry, relating to the quotient qgrA. Modules over these algebras behave slightly differently to graded algebras, so some time will be spent studying such modules. We will then proceed to look at the cohomology of these algebras. In particular we will study the right derived functors of the internal Hom functor. Finally, we will introduce the properties that an arbitrary abelian category will have to satisfy to be equivalent to a quotient of a subcategory of the graded modules over an indexed algebra. Specifically, an abelian category must have an ample sequence of objects. We will finish this chapter by proving our Morita-type result, showing that an abelian category with such an ample sequence is equivalent to a quotient of a subcategory of graded modules over an indexed algebra. 1.2 Assumed Knowledge Throughout this thesis, basic results from category theory are assumed. Specifi- cally, notions such as categories, functors, natural transformations, diagram chas- ing, equivalence of categories, limits, universal properties and adjoint functors will not be formally introduced. More rigorous introductions to these topics can be found in [13]. Similarly, familiarity with the basic properties of graded rings and modules, which can be found in Chapter 1.5 of [5], will also be assumed.

2 Chapter 2 Abelian Categories

In this chapter we will give an introduction to abelian categories which are, in a sense, an abstraction of the category of modules ModR over a ring R, which take all of the properties necessary for homological algebra. We will present the definition of an abelian category in the usual segmented way, and then compare arbitrary abelian categories with module categories1, giving necessary and sufficient conditions for an abelian category to be equivalent to a module category. Our presentation of this material will follow similarly to Chapter 2 of [15]. 2.1 Additive Categories We start by considering our prototypical example: the category of right modules ModR over a ring R. We wish to abstract a number of special properties of this category to define an abelian category. Recall that if M and N are R-modules and f, f 0 : M → N are R-module homomorphisms we can define the homomorphism f + f 0 : M → N as (f + f 0)(m) := f(m) + f 0(m) for any m ∈ M, where the addition is that of N. The abelian group structure on N then induces an abelian group structure on HomR(M,N), with negatives given by (−f)(m) = −(f(m)) and identity being the zero morphism. Moreover, if L is also an R-module and g, g0 : L → M are homomorphisms, then (f+f 0)g = fg+f 0g and f(g+g0) = fg+fg0, so the group structure is bilinear with respect to composition of homomorphisms. This leads us to the first property that we want an abelian category to have. Definition 2.1.1. A category C is preadditive if for any X,Y ∈ C there is an abelian group structure on HomC(X,Y ) which is bilinear with respect to the composition of morphisms. Remark 2.1.2. As we have done in this definition, we will often abuse notation and refer to objects or morphisms as belonging to a category, when we mean they belong to the set of objects or set of morphisms of the category. Remark 2.1.3. If instead of abelian group structure we imposed that the sets of morphisms in our category have R-module structure for some ring R, then we say C is an R-linear category.

Notation 2.1.4. We will let 0XY denote the zero element of HomC(X,Y ), and will often drop the subscript when there will be no confusion about X and Y . 1The term module category is non-standard and often has another meaning, however we use it to mean the category of modules over a ring.

3 Example 2.1.5. If R is a ring, ModR and R-Mod are both preadditive categories, as we discussed above. Example 2.1.6. The category Ring of rings and ring homomorphisms is not preadditive. This follows from the fact that the image of a multiplicative identity under a ring homomorphism must also be a multiplicative identity, so a zero morphism cannot exist. Returning to abstracting properties of module categories, we recall that ModR has a categorical product and coproduct, given by the direct product and direct sum, respectively. Furthermore, we have that finite direct products are isomorphic to finite direct sums. This property is captured in the following definition and proposition. Definition 2.1.7. A preadditive category C is additive if it has all finite products and coproducts.

Proposition 2.1.8. Let {M1,...,Mn} be a ﬁnite set of objects in an additive cat- n n Q ∼ ` egory C. Then Mi = Mi. i=1 i=1

Proof. Let πi and ιi be the canonical projections and inclusions. We claim that there n n Q ` are maps αj : Mj → Mi and βj : Mi → Mj. Deﬁne the set of morphisms i=1 i=1 j j j {fi : Mi → Mj} by fi = 0MiMj if i 6= j and fj = idMj . Then we can invoke the i n universal properties of the product and coproduct on the pairs (Mj, {fj }i=1) and j n (Mj, {fi }i=1) to obtain the desired maps αj and βj. We then consider the following diagram

n ιj ` Mj Mi i=1

αj ϕ βj n Q πj Mi Mj, i=1 where ϕ is induced by the universal property of the coproduct applied to the maps n n n P P P αj. As the sum is ﬁnite, we see that πj( αiπi) = (πjαi)πi = πj, so αiπi is just i=1 i=1 i=1 n P the identity on the product. By duality ιiβi is the identity on the coproduct. i=1 To show ϕ is an isomorphism we will show it is monic and epic. Suppose we n ` have an object N ∈ C and morphism f : N → Mi such that ϕf = 0. Then i=1 n n P Q πjϕf = βjf = 0 for all j, so f = ( ιiβi)f = 0. Now suppose g : Mi → N is i=1 i=1 n P such that gϕ = 0. Then, similarly, gϕιj = gαj = 0 for all j, so g = g( αiπi) = 0. i=1 2

4 When the product and coproduct agree, as in the proposition, we refer to the product as a biproduct. So this proposition tells us that an additive category has all finite biproducts. Recall that a terminal object in a category is an object T such that for all other objects X there is exactly one morphism X → T , with the dual notion being that of an initial object. Another property of additive categories, often included in the definition, is that the category contains a zero object, which is an object that is both initial and terminal. However, this is in fact a consequence of the previous proposition, as the empty (co)product is an (initial) terminal object, so the empty biproduct must be a zero object. As we have just introduced some structure on a category, we should also introduce suitable constraints on a functor in order for this structure to be preserved. Definition 2.1.9. A functor F : C → D between preadditive categories is additive if the induced map on Hom groups is a group homomorphism. That is, for any

X,Y ∈ C the map HomC(X,Y ) → HomD(FX,FY ) is a group homomorphism. Again, we can more generally deﬁne an R-linear functor between R-linear categories to preserve the R-module structure on Hom modules. While it is clear that an additive functor preserves preadditive structure, it is not immediately clear that the same is true for additive structures. We will show now that, in fact, this is the case. Proposition 2.1.10. Let F be an additive functor and C,D be additive categories. If X,Y ∈ C then F (X Q Y ) ∼= FX Q FY . Q Q Proof. The projections F πX : F (X Y ) → FX, F πY : F (X Y ) → FY and Q Q Q inclusions F ιX : FX → F (X Y ), F ιY : FY → F (X Y ) show that F (X Y ) is a biproduct for FX and FY , so invoking universal properties gives the following commuting diagram FX ιFX FX Q FY

F ιX πFX F (X Q Y ) F πX FX.

As πFX ◦ιFX = F πX ◦F ιX = idFX , the two dotted arrows obtained by the universal properties must be inverse. 2

Example 2.1.11. If C is an additive category and X ∈ C, then the most important example of an additive functor is HomC(X, −): C → Ab. In particular the object HomC(X,X) is a unital ring with multiplication given by composition of morphisms. This ring is called the endomorphism ring of X, and is of particular interest when X is a progenerator, which we will introduce later. Before moving on to deﬁne abelian categories we introduce a categorical notion of kernels and cokernels, show that in an additive category they behave as one would hope with respect to mono and epimorphisms, and then introduce categorical interpretations of preimages, sums and intersections.

5 Deﬁnition 2.1.12. Let f : X → Y be a morphism of objects in an additive category. Then the kernel of f, denoted ker f, is the limit of the diagram 0

f X Y, and the cokernel of f, denoted coker f, is the colimit of the diagram f X Y

0 . Being deﬁned in terms of a categorical limit, we note that the kernel of a morphism f : X → Y is actually a pair (K, α), where K is an object and α : K → X is a morphism such that fα = 0 which is universal with respect to this property. That is, for any other pair (K0, α0) such that fα0 = 0 there is a unique morphism β : K0 → K such that α0 = αβ. The cokernel is then the dual notion of the kernel. We will frequently abuse notation and say that either the object or the morphism is the (co)kernel. Proposition 2.1.13. A morphism f : X → Y of objects in an additive category C is monic (resp. epic) if and only if ker f = (0, 0 → M) (resp. coker f = (0,N → 0)).

Proof. We prove only the monic case as the other follows by duality. Firstly suppose f is monic. Then f ◦ker f = f ◦0 implies that ker f = 0. On the other hand, suppose ker f = 0. Then if g1, g2 are morphisms in C with codomain X such that fg1 = fg2, then f(g1 −g2) = 0. By the universal property of the kernel we see that g1 −g2 = 0, hence f is monic. 2

With this in mind, we will often refer to monomorphisms in an additive category as injections and epimorphisms as surjections. Deﬁnition 2.1.14. Let f : X → Y be a morphism in a category C and let Y 0 be a subobject of Y such that Y/Y 0 := coker(Y 0 → Y ) exists. Then, if the kernel exists, the preimage of Y 0 is f −1Y 0 := ker(X → Y → Y/Y 0).

Deﬁnition 2.1.15. Let {Xi}i∈I be a family of subobjects of the object X. Then ` P • If the coproduct i∈I Xi exists, the sum of the subobjects i∈I Xi is the im- ` age im( i∈I Xi → X),

Q T • If the product i∈I Xi exists, the intersection of the subobjects i∈I Xi is the Q kernel ker(X → i∈I X/Xi). 2.2 Abelian Categories As we just discussed, categorical kernels and cokernels behave as we would expect them to with regards to our knowledge of kernels and cokernels in ModR. But an arbitrary additive category may not have all limits and colimits, hence morphisms

6 may exist that have no kernel or cokernel. This leads to the deﬁnition of a preabelian category. Deﬁnition 2.2.1. An additive category C is preabelian if every morphism in C has a kernel and cokernel. Example 2.2.2. Once again, the category ModR is preabelian, as the usual notion of kernels and cokernels for modules satisfy the universal properties of the categorical kernel and cokernel. Now suppose that we have a morphism f : X → Y in a preabelian category. Then we can consider the following diagram. f 0 ker f X Y cokerf 0 g

coker ker f h ker cokerf The arrow g is induced by the universal property of coker ker f applied to the composite ker f → X → Y . We then see that f = g ◦ coker ker f, and so coker f ◦ g ◦ coker ker f = 0. But coker ker f is epic, so coker f ◦ g = 0. Thus h is obtained by the universal property of ker coker f. So we see that in a preabelian category any morphism f factors canonically through its coimage coimf := coker ker f and its image imf := ker coker f. If we consider the case of ModR, then the first isomorphism theorem for modules tells us exactly that the map h is an isomorphism. This is the final property of ModR that we wish we wish an abelian category to have, meaning we can finally define an abelian category. Definition 2.2.3. A preabelian category A is abelian if for every morphism f ∈ A the canonical morphism coimf → imf is an isomorphism. Notation 2.2.4. To emphasise that abelian categories are similar to module categories we commonly refer to objects in an abelian category A as A-modules. As an abelian category A has all kernels and cokernels, we can speak of exact sequences within A, and functors which preserve exactness. In fact, the properties of ModR that we used to define an abelian category are precisely the properties needed to talk of such things. Definition 2.2.5. Let A be an abelian category, and let M 0,M and M 00 be A- modules. Then the sequence M 0 → M → M 00 is exact at M if ker(M → M 00) = im(M 0 → M). A short exact sequence is a sequence of the form 0 → M 0 → M → M 00 → 0 which is exact at M 0,M and M 00. If B is also an abelian category and F : A → B is an additive functor, then F is exact if for every short exact sequence in A

0 → M 0 → M → M 00 → 0

7 the image 0 → FM 0 → FM → FM 00 → 0 is exact in B. We will return to exactness when we talk about homological algebra later. For now, we will give some examples of abelian categories. Example 2.2.6. Let modR be the category of finitely generated modules over a ring R. If R is noetherian then this is an abelian category. To check that this full subcategory of the abelian category ModR is itself abelian, we only need to see that the (co)products and (co)kernels from the larger category agree, as the abelian group structure on Hom sets and the isomorphism between images and coimages are then induced. We call such a subcategory an abelian subcategory. In the case of modR, we need to verify that any finite direct sum of finitely generated modules is finitely generated (which is obvious) and that the kernel and cokernel of any homomorphism between finitely generated modules are both finitely generated. This follows from the standard facts that any module over a noetherian ring is finitely generated if and only if it is a noetherian module, and that for any short exact sequence of modules 0 → M 0 → M → M 00 → 0, if M is noetherian then so are M 0 and M 00. Remark 2.2.7. Not every subcategory of an abelian category that is itself abelian is an abelian subcategory. A trivial example is the subcategory of an abelian category

A consisting of any non-zero A-module M and the identity morphism idM . This is Qn obviously abelian where Hom(M,M) is the trivial group, i=1 M = M, ker idm = coker idM = M and coim idM = M = im idM . But clearly in this case the products, coproducts, kernels and cokernels cannot agree with the larger abelian category. Example 2.2.8. Let C be a category and A be an abelian category. Then the category Fun(C, A) of covariant functors from C to A, with natural transformations as morphisms, is also abelian. Let F,G : C → A be functors and σ, τ : F → G be natural transformations. We will deﬁne all of the abelian category structure on Fun(C, A) ‘pointwise’. We can view any natural transformation σ as a set of morphisms {σM : FM → GM | M ∈ C} in A such that for any morphism f : M → N in C, the following diagram commutes F f FM FN

σM σN Gf GM GN. We then deﬁne the sum of the natural transformations σ and τ by the maps

(σ + τ)M := σM + τM .

As A is an abelian category the abelian group structure on each of the HomA(F M, GM) induces an abelian group structure on HomFun(F,G).

8 Next, we define the product of the functors F and G by Y Y (F G)(M) := FM GM on objects and Y Y (F G)(f) := F f Gf for any morphism f ∈ C. Again the universal property satisfied by all FM Q GM in A shows that F Q G satisfies the universal property of a product in Fun(C, A). The coproduct is defined in the same way. Finally, we define the kernel of a natural transformation τ by

(ker τ)(M) := ker(τM ), on objects and if f : M → N, then (ker τ)(f) is the unique morphism making 0 (ker τ)M FM τM GM 0

F f Gf 0 (ker τ)N FN τN GN 0 commute. Then it is clear that the set of morphisms {(ker τ)M } define a natural transformation ker τ → F such that τ ◦ ker τ = 0. As A is abelian ker τ satisfies the universal property of a kernel because all of the ker(τM ) do. Cokernels are also defined in the same way, showing that Fun(C, A) is preabelian. The fact that A is abelian then induces isomorphisms coim τ(M) ∼= im τ(M) for every M ∈ C, but this says exactly that the functors coim τ and im τ are naturally isomorphic. Thus Fun(C, A) is abelian. This example encompasses a wide array of abelian categories, maybe most important of which would be the category GrR of graded modules over a graded ring. Suppose G is a group and R is a G-graded ring. We define the category C to have elements of G as objects, and if g, h ∈ G then we define HomC(g, h) := Rg−1h. It is then quite easy to identify GrR with Hom(C, Ab), the category of additive functors from C to A. It is clear that this is an abelian subcategory of Fun(C, A), so we see that GrR is abelian. Given that we introduced abelian categories as being, in a sense, a generalisation of module categories, one might understandably wonder if all abelian categories are just module categories. This, however, is not the case. Example 2.2.9. Returning to our earlier example, modR is abelian but not a module category. Module categories have infinite direct sums, but an infinite direct sum of finitely generated modules is no longer finitely generated. Thus there is no ring S such that ModS ' modR. The question of how close an abelian category is to a module category is an- swered by the Freyd-Mitchell Embedding Theorem, a proof of which can be found on page 150 of [6].

9 Theorem 2.2.10. Let A be a small abelian category. Then there exists a ring R and a fully faithful exact covariant functor F : A → ModR. This theorem allows us to think of every abelian category as ‘sitting inside of’ a larger module category, and thus we can use module theoretic arguments dependent on choosing elements in abelian categories. For example, all of the isomorphism theorems hold in any abelian category, with near identical proofs as in module category setting. Theorem 2.2.11. Let A be an abelian category and let M,M 0,M 00,N be A-modules such that M 0 ⊆ M and M 00 ⊆ M. Then 1. (First Isomorphism Theorem) if f : M → N, we have M/ ker f ∼= imf,

2. (Second Isomorphism Theorem) the intersection M 0 ∩ M 00 ⊆ M 00, and

M 0 + M 00/M 0 ∼= M 00/M 0 ∩ M 00,

3. (Third Isomorphism Theorem) if M 00 ⊆ M 0 we have M 0/M 00 ⊆ M/M 00, and

(M/M 00)/(M 0/M 00) ∼= M/M 0.

2.3 Directed Systems and Limits In this section we will introduce the notion of a direct limit, which is a special type of categorical limit over a directed index set. The utility of these limits will become obvious when we want to take any type of limit over the set of finite submodules of a given module (which we will want to do at multiple stages), in which case the set of finite submodules will form a directed set when ordered by inclusion. Definition 2.3.1. Let I be a set. Then I is directed, or filtered, if there is a binary relation ≤ on I such that ≤ is reflexive, transitive, and for any i, j ∈ I, there is a k ∈ I such that i ≤ k and j ≤ k. Example 2.3.2. Let I be a set and consider the set {F ⊆ I | F is finite}, with the binary operation ⊆. This is reflexive and transitive, and any two finite subsets are contained in their union, which is necessarily finite, so this set is in fact directed. To introduce some sort of ‘directed system’ in a category, we first interpret a directed set as a category itself. If (I, ≤) is a directed set, we can think of I as a category where the objects are elements of I, and for elements i, j ∈ I there is a single morphism i → j if i ≤ j. Definition 2.3.3. A directed system over a directed set I in a category C is a covariant functor I → C. More concretely, a directed system over I in C is a set of objects {Mi | i ∈ I} with morphisms ϕij : Mi → Mj when i ≤ j such that

ϕii = idMi and if i ≤ j ≤ k then ϕik = ϕjkϕij.

10 Deﬁnition 2.3.4. Let (Mi, ϕij) be a directed system in C. Then the direct limit of the system is an object lim M with morphisms ϕ : M → lim M such that −→ i i i −→ i 1. ϕi = ϕjϕij when i ≤ j, and

2. lim M is universal with respect to these properties. −→ i Example 2.3.5. Expanding upon the earlier example, let I be a set and J = {F ⊆

I | F is finite}. Let {Mi | i ∈ I} be a set of objects in a category C with direct limits. For any F ∈ J we define M = L M . Then L M = lim M . F i∈F i i∈I i −→F ∈J F Given the dependence on the directed set I in the definition of the direct limit, it is natural to wonder when does a subset J ⊆ I give the same direct limit. To this end, we say a subset J ⊆ I is cofinal in I if for every i ∈ I there is a j ∈ J such that i ≤ j. We then have the following result.

Proposition 2.3.6. Let (Mi, ϕii0 ) be a directed system over I in a category C, and suppose J be cofinal in I. Then lim M ∼= lim M . −→i∈I i −→j∈J j Proof. Firstly, we will denote the canonical maps associated to the direct limits ϕ and ϕ for lim M and lim M respectively. There is an obvious morphism i j −→i∈I i −→j∈J j lim M → lim M simply defined by inclusion. As J is cofinal, for any i ∈ I −→j∈J j −→i∈I I we have a j ∈ J such that i ≤ j, so we define morphisms ψ : M → lim M to i i −→j∈J j 0 0 0 be the composition ψi := ϕjϕij. Then if i ≤ i , i ≤ j and i ≤ j , we have that

ψi0 ϕii0 = ϕj0 ϕi0j0 ϕii0 = ϕj0 ϕij0 = ϕi = ϕjϕij = ψi, so (lim M , ψ ) satisfies the conditions to be a direct limit over I. Thus by −→j∈J j i the universal property we obtain a morphism lim M → lim M , which is −→i∈I i −→j∈J j necessarily inverse to the morphism lim M → lim M . 2 −→j∈J j −→i∈I i 2.4 Determining Equivalence with a Module Category Now that we know what an abelian category is and that every abelian category embeds into a module category, we are immediately faced with the question: how do we know when an abelian category is actually a module category? More precisely, how can we tell when an abelian category A is equivalent to the category of modules ModR over a ring R. We answer this question in this chapter by examining some of the unique properties of the module RR and the functor HomMod(RR, −). Firstly, we will introduce a category theoretic interpretation of a familiar finite- ness condition on modules. Definition 2.4.1. Let A be an abelian category and M an A-module. Then

M is finitely generated if for any directed family of subobjects {Mi}i∈I such that P i∈I Mi = M there is an index j such that M = Mj. While this definition of finite generation may appear strange at first, it in fact agrees with the usual definition in ModR.

11 Proposition 2.4.2. Let R be a ring and M a right R-module. Then M is ﬁnitely generated in the categorical sense if and only if it is ﬁnitely generated in the usual sense.

Proof. Suppose M is finitely generated in the categorical sense. The finitely generated (in the usual sense) submodules of M ordered by inclusion form a directed system and their sum is M, hence M must be equal to one of them. Conversely, P if M is finitely generated in the usual sense and M = i∈I Mi, then a finite set of generators for M must be contained in some Mj, and thus we must have that M = Mj. 2

This proposition tells us that the module RR is finitely generated in ModR. A standard fact from module theory is that every module is a homomorphic image of a direct sum of copies of the module RR. To capture this notion, we now define a generator in an abelian category. Definition 2.4.3. Let A be an abelian category and M an A-module. Then M is a generator if Hom(M, −) is faithful. More generally a set of A-modules {Mi} is a set of generators for A if for any two distinct morphisms f, f 0 : N → N 0 in A there 0 is an Mj and a g ∈ HomA(Mj,N) such that fg 6= f g. While it is not immediate that this definition captures the property that we desire, the following lemma shows that it in fact does. Lemma 2.4.4. Let A be a cocomplete abelian category and M an A-module. Then M is a generator if and only if every A-module N is a quotient of a coproduct of copies of M. ` Proof. Suppose M is a generator and let J = Hom(M,N). Let π : j∈J M → N be the unique morphism such that πιj = j, where j ∈ J and ιj is the canonical inclusion. Now let f : N → N 0 be a morphism such that fπ = 0. Then fj = 0 for all j ∈ J. But Hom(M, −) is faithful so we see that f = 0, implying that π is surjective. ` Conversely, suppose there exists a surjection π : j∈J M → N for some index set J, and let f : N → N 0 be a morphism such that Hom(M, f) = 0. Then fg = 0 for every g ∈ Hom(M,N). By composing with the canonical inclusions, we see that fπ = 0. But π is surjective so we see that f = 0, and thus Hom(M, −) is faithful. 2

Finally, we know that in ModR the functor Hom(RR, −) is exact. This is the last property of RR that we wish to abstract. Deﬁnition 2.4.5. Let A be an abelian category and M an A-module. Then M is projective if Hom(M, −) is exact. We say A has enough projectives if every A-module is a quotient of a projective module.

12 The dual notion of a projective module is that of an injective module; that is a module I such that Hom(−,I) is exact. As with exact functors, we will discuss these modules in more depth when we talk about homological algebra. If an A-module P is ﬁnitely generated, projective and a generator we call P a progenerator. It turns out this is exactly what we need to identify when an abelian category is a module category. If P is a progenerator in our abelian category A and M is an A-module, then the natural ring structure on R := Hom(P,P ) turns the abelian group Hom(P,M) into f g a right R-module, with scalar multiplication given by g · f := g ◦ f : P −→ P −→ M, where f ∈ R and g ∈ Hom(P,M). Theorem 2.4.6 (Morita’s Theorem). Let A be a cocomplete abelian category with progenerator P , and let R = Hom(P,P ). Then the functor Hom(P, −): A → ModR deﬁnes an equivalence of categories.

Proof. Firstly, the fact that P is a progenerator tells us immediately that the functor F = Hom(P, −) is exact and faithful, so it remains to show that it is full and essentially surjective. First we will show that it is full. That is, we want to show that for any A-modules M and N, the map

HomA(M,N) → HomR(FM,FN) is surjective. We can view each of these sets of morphisms as contravariant functors of M, denoted by G := Hom(−,N) and H := Hom(F −,FN). Then in this way we see F is a natural transformation F : G → H. As we know F is faithful, we can show that it is full by showing F is a natural isomorphism of G and H.

That is, we will show that the map FM : GM → HM is an isomorphism for all A-modules M. To this end, we begin with the case that M = P . We then see ∼ that FP : GP = FN → HP = Hom(R,FN) = FN is the identity, and thus an isomorphism. Also, as F commutes with arbitrary coproducts, we see that FQ is an isomorphism where Q is any coproduct of copies of P . For an arbitrary A-module

M, we use Lemma 2.4.4 to obtain an exact sequence Q1 → Q0 → M → 0, where Q0 and Q1 are coproducts of copies of P and Q1 maps onto ker(Q0 → M). We then consider the following diagram.

0 GM GQ0 GQ1

F M FQ0 FQ1

0 HM HQ0 HQ1

As the rows are exact and the two right vertical arrows are isomorphisms, we have that FM must be an isomorphism too.

13 Now it remains to show that F is essentially surjective. Let S be an R-module, and take a sequence of the form

F1 → F0 → S → 0, L L where F0 = i∈I R and F1 = j∈J R are free. This is done by mapping a free module F0 onto S, and then mapping F1 onto the kernel of this homomorphism. ` ` If we let Q0 = i∈I P and Q1 = j∈J P it is clear that FQ0 = F0 and FQ1 = F1, and that as F is fully faithful there is a morphism f : Q1 → Q0 such that F (f) = F1 → F0. Letting M = coker f, and then applying F to the exact sequence ∼ Q1 → Q0 → M → 0 shows that S = FM. 2

Remark 2.4.7. We have also already seen that in ModR the module RR is a progenerator, so we have the converse of this result too. Example 2.4.8. A quiver is a directed graph (V,E) with a ﬁnite vertex set, where we allow loops and multiple edges. For example,

is a quiver. We have two maps s, t : E → V sending each edge to its source or its target, respectively. A path in a quiver Q = (V,E) is a finite sequence e0e1 . . . en of edges in Q such that t(ei−1) = s(ei) for i = 1, . . . , n. We can think of a quiver itself as a category Q, with objects given by the vertices of the quiver, and for two vertices v, v0, we define Hom(v, v0) := {paths from v to v0}.A quiver representation over a field k is a covariant functor F : Q → Modk from the quiver to the category of k-vector spaces. For example,

V1 V4

V0 where the Vi are vector spaces over k, and the arrows are linear transformations is a representation of the above quiver. The category of quiver representations of Q over k with natural transformations as morphisms is denoted RepkQ. We immediately see that this category is abelian, as by deﬁnition it is just Fun(Q, Modk), however it is not immediately obvious whether or not it is equivalent to a module category. It is, in fact, and it is easily shown using Morita’s Theorem.

14 We call the set of all paths in a quiver P . The path algebra of a quiver kQ is de- ﬁned to be the free vector space k[P ] with basis P , and multiplication given by con- ( p1p2, if s(p2) = t(p1), catenation of paths. That is, if p1, p2 ∈ kQ then p1 · p2 := 0 otherwise.

This is an associative k-algebra. We denote the empty path at a vertex v by ev. We can then deﬁne for each vertex v a quiver representation ρv, where we deﬁne

0 ρv(v ) := evkQev0 for any vertex v0 and if p : v0 → v00 is a path in Q we deﬁne

ρv(p)(x) := xp ∈ evkQev00

0 L for any x ∈ evkQev0 = ρv(v ). We then let ρ = v∈V ρv. Note that this representation exists as the vertex set is finite and RepkQ is abelian. We claim that this is in fact a progenerator in RepkQ. It is clear that ρ is finitely generated, as the vertex set is finite. To see that ρ is projective and a generator, we first show ∼ that Hom(ρv,V ) = V (v) for any representation V . There are inverse morphisms ϕ : Hom(ρv,V ) → V (v) given by ϕ(f) = f(ev), and ψ : V (v) → Hom(ρv,V ) defined by ψ(x) = (p 7→ xV (p)). These are clearly inverse, and thus define an isomorphism. Now if 0 → V 0 → V → V 00 → 0 is an exact sequence in RepkQ, then applying the functor Hom(ρv, −) gives the sequence 0 → V 0(v) → V (v) → V 00(v) → 0 which is necessarily exact. Thus each of the ρv are projective, hence any direct sum of them (such as ρ) is projective too. Now let f : V → V 0 be a morphism such that 0 Hom(ρ, f) = 0. For each v ∈ V , Hom(ρv, f): V (v) → V (v) is just the image of the vector space V (v) under f. As Hom commutes with finite direct sums, we see that all of these Hom(ρv, f) must be zero. But then f is just the zero map. So ρ is a progenerator with endomorphism ring M M Hom(ρ, ρ) = evkQev0 = kQ, v∈V v0∈V and thus by Morita’s theorem we see that RepkQ ' ModkQ.

15 Chapter 3 Cohomology in Abelian Categories

In this chapter we give a brief overview of the tools from homological algebra that we will need for later. For our purposes, homological algebra will be treated as a tool, and as such this section will be light on details. Proofs of any results which are not included in this chapter can be found in any introductory text on homological algebra, such as Appendix 3 in [5]. 3.1 Cochain Complexes and Resolutions Consider a sequence of objects

di+2 di+1 di di−1 ... −−→ Mi+1 −−→ Mi −→ Mi−1 −−→ ... in an abelian category A. We denote such a sequence M•, and if dpdp+1 = 0 for all p then we call M• a chain complex. The dp are referred to as the boundary maps of the chain complex. Similarly, a sequence of the form

i−2 i−1 i i+1 • d d d d M = ... −−→ Mi−1 −−→ Mi −→ Mi+1 −−→ ... is called a cochain complex if dp+1dp = 0 for all p. In this case the dp are called the coboundary maps. From here on out we will work with cochain complexes, but for everything we introduce there is a dual interpretation for chain complexes. Now we note that by the deﬁnition of a cochain we have that im dp ⊆ ker dp+1, so we can take the quotient hp := ker dp+1/im dp. We call this module the i-th cohomology module. Recalling our deﬁnition of an exact sequence from earlier, we see that a cochain complex is an exact sequence if hp = 0 for all p. So in some sense the cohomology of a cochain complex measures how far away from being exact it is. A morphism of cochain complexes f : M • → N • is a collection of maps f i : M i → N i which commute with the coboundary maps. The category of cochain complexes in A is denoted Ch•(A), and is in fact an abelian category. Proposition 3.1.1. Let 0 → L• → M • → N • → 0 be a short exact sequence of cochain complexes. Then there are induced maps on cohomology hi(L•) →

16 hi(M •) → hi(N •) for all i. Furthermore, there are natural maps δi : hi(N •) → hi+1(L•) and a long exact sequence

i ... → hi(L•) → hi(M •) → hi(N •) −→δ hi+1(L•) → ... called the long exact sequence of cohomology. The following lemma is a useful tool which we will repeatedly use later to prove that a certain category is abelian. Lemma 3.1.2 (Snake Lemma). Let A be an abelian category and consider a diagram of A-modules of the form L M N 0

f g h 0 L0 M 0 N 0 with exact rows. Then there is a long exact sequence

ker f → ker g → ker h → coker f → coker g → coker h.

The name Snake Lemma comes from the appearance of the morphism ker h → coker f in the extended diagram ker f ker g ker h

L M N 0

f g h

0 L0 M 0 N 0

coker f coker g coker h. Recall from earlier that an A-module I is injective if Hom(−,I) is an exact functor, and that A has enough injectives if for every A-module M there is an injective module I and a monomorphism f : M → I. If such a pair (I, f) has the additional property that for any other monomorphism f 0 : M → I0 to an injective module I0 there is a unique monomorphism ϕ : I → I0 such that f 0ϕ = f, then we say (I, f) is the injective envelope or injective hull of M. If every A-module has an injective envelope then we say A has injective envelopes. The dual notion of an injective envelope is a projective cover. Finally, we say a monic map f : M → N is essential if for any submodule N 0 ⊆ N, the intersection im f ∩ N 0 is non-zero. If such a monic exists then we say N is an essential extension of M.

17 Example 3.1.3. Let R be a ring. Then ModR has enough injectives and injective envelopes. On the other hand, while ModR has enough projectives it does not have projective covers in general. Now suppose we have a long exact sequence of the form

M • = ... 0 → 0 → M → I0 → I1 → ..., where all of the Ij are injective modules. Then we call M • an injective resolution of M. Proposition 3.1.4. Let A be an abelian category with enough injectives. Then every A-module has an injective resolution.

Proof. Let M be an A module. As A has enough injectives, we can ﬁnd an injective module I0 with a monomorphism M → I0. Let M 0 be the cokernel of this monomorphism. Then we can again embed M 0 into an injective module I1 and look at the cokernel of this embedding. Repeating this process deﬁnes an injective resolution M → I0 → I1 → ... of M. 2

3.2 Exactness and Derived Functors Let F : A → B be a covariant functor between abelian categories. Recall that F is exact if it preserves exact sequences. This is a very nice property for a functor to have, and in turn is not very common. More common are left and right exact functors, which preserve left exact and right exact sequences. Just as we can use cohomology to examine how close a sequence is to being exact, we can also see how close a functor is to being exact. Definition 3.2.1. Let F : A → B be a covariant left exact functor between abelian • categories. For each A-module M, we fix an injective resolution IM . Then for i ≥ 0 i i • we define the i-th right derived functor of F by R F (M) = h (FIM ). Theorem 3.2.2. Let A be an abelian category with enough injectives, and let F : A → B be a covariant, left exact functor to an abelian category B. Then 1. For each i ≥ 0, RiF is an additive functor, and is independent (up to natural isomorphism) of choices of injective resolutions,

2. R0F ∼= F ,

18 3. For any short exact sequence 0 → M 0 → M → M 00 → 0 in A, there is a long exact sequence

... → RiF (M 0) → RiF (M) → RiF (M 00) → Ri+1F (M 0) → ...,

4. For any injective A-module I, RiF (I) = 0 for all i > 0. Of course, we can dually deﬁne the left-derived functors of a right exact functor, and the dual of the previous theorem holds. Example 3.2.3. Let R be a ring and M an R-module. The functor Hom(M, −) is left exact, so we can consider its right derived functors Exti(M, −) := RiHom(M, −). The name Ext comes from the following. Let M and N be R-modules, and deﬁne the extension group

E1(M,N) := {α : 0 → M → L → N → 0 | α is short exact}/ ∼, where α ∼ α0 : 0 → M → L0 → N → 0 if there is a morphism L → L0 such that 0 M L N 0

f 0 M L0 N 0 commutes. Then E1(M,N) ∼= Ext1(M,N).

We also have the right exact functor M ⊗R −, so we can consider its left derived functors Tori(M, −) := LiM ⊗R −. The name Tor comes from the fact that if r ∈ R is a nonzerodivisor, then

Tor1(R/(r),M) = {m ∈ M | mr = 0}.

The fact that these two functors form an adjoint pair is no coincidence. They are a particular example of the following theorem. Theorem 3.2.4. Let (F,G) be a pair of adjoint functors between abelian categories. Then F is right exact and G is left exact. As adjoint functors will play an important role later we refer the unfamiliar reader to Chapter 4 of [13]. Remark 3.2.5. In ModR, just as we have the notion of a projective module M describing when Hom(M, −) is an exact functor, we have the notion of flat modules which are the modules M such that − ⊗R M is exact. For example, the field of rational numbers Q is a flat Z-module.

19 Chapter 4 Quotients of Abelian Categories

We have seen that if an abelian category contains a progenerator P that it is equivalent to the category of modules over the endomorphism ring of P . However, we have also seen that not every abelian category is equivalent to the category of modules over some ring. In this chapter we will introduce the notion of a quotient of an abelian category by a Serre subcategory, and all the relevant material on such quotients which will allow us to prove a Morita-type result for them in the next chapter. While we present them as a way to understand an abelian category, quotient categories were originally presented in [7], where they arose from algebraic geometry. Following the introduction of these quotients we discuss localising subcategories, which are Serre subcategories for which the corresponding quotient behaves similarly to category of modules over the ﬁeld of fractions of an integral domain. Our presentation of these ideas follows similarly to chapters 13 and 14 of [15]. Finally, we will introduce a special type of abelian categories, named after Grothendieck, which are closely related to quotients by localising subcategories. Before we deﬁne these quotient categories, however, we will present what will serve as a guiding example.

4.1 The Field of Fractions of Z Before we define a quotient of an abelian category we recall the field of fractions construction, which embeds an integral domain into the smallest field containing it. The general idea of this construction is to adjoin multiplicative inverses for all of the non-units of the integral domain. We will present this construction using the example of Z, however the same construction can be used for any integral domain. We first define an equivalence relation on the set Z × (Z − {0}) by

(m1, n1) ∼ (m2, n2) if n(m1n2 − m2n1) = 0 for some n ∈ Z − {0}.

We denote the equivalence class (m, n) as m/n. The ﬁeld of fractions Frac(Z) is then the quotient (Z × (Z − {0}))/ ∼ with addition given by

m1/n1 + m2/n2 = (m1n2 + m2n1)/n1n2

20 and multiplication

(m1/n1)(m2/n2) = (m1m2)/(n1n2).

Of course we all know that this is just the field of rational numbers Q. We can also use this same construction to send any Z-module, otherwise known as an abelian group, to a Q-vector space, and in fact this assignment is functorial. We have a field of fractions functor L : Ab → ModQ, which is just the tensor functor − ⊗Z Q. We call the field of fractions functor L to allude to the fact that it is a specific example of a more general construction originating from algebraic geometry: localisation of a commutative ring. This process is essential in algebraic geometry, as it is how we understand the local behaviour of rational functions on algebraic varieties. Grothendieck’s philosophy was that to do algebraic geometry one should work not with a variety itself, but with the category of quasi-coherent sheaves over that variety, which form an abelian category. This category of quasi-coherent sheaves is even more important in noncommutative algebraic geometry, as in most cases we do not have any tangible notion of a ‘noncommutative space’, and instead work solely with the category of quasi-coherent sheaves. We will touch on this idea more in the last chapter. For this reason we would like to have some sort of notion of ‘localisation of an abelian category’. This is what we aim to introduce with our notion of a quotient category. 4.2 Quotient Categories In this section we will construct a quotient of an abelian category A, which will essentially add more morphisms to A in order to make a specific subcategory of objects isomorphic to zero. This will allow us to eventually interpret the category of modules over a field of fractions as a categorical quotient. We now start by introducing a specific type of subcategory which we will be ‘quotienting out by’. Definition 4.2.1. Let A be an abelian category and T be a non-empty, full subcategory of A. We say T is a Serre subcategory if for every short exact sequence 0 → M 0 → M → M 00 → 0 in A, M ∈ T if and only if M 0 ∈ T and M 00 ∈ T . If T is a Serre subcategory of an abelian category A we will call the objects of T torsion modules, and any module who’s only submodule belonging to T is 0 is called torsion-free. Also, as kernels and cokernels agree in T , and T is closed under extensions, we can immediately see that T is abelian and the inclusion functor must be exact. Example 4.2.2. Returning to our field of fractions example, we compute the kernel of the field of fractions functor and show that it is a Serre subcategory. Suppose G ∈ Ab and that LG = 0. Then for any g ∈ G there is a non-zero a/b ∈ Q such that g ⊗Z a/b = 0. But then ga ⊗Z 1/b = 0, so we see that there is an a ∈ Z such that ga = 0. This is true of all g ∈ G, so G is a torsion abelian group. Recall that

21 an abelian group G is torsion if every element of G has ﬁnite order. We denote the full subcategory of torsion abelian groups by Tors. We claim that ker F = Tors. We know that ker F ⊆ Tors, so we only have to show the reverse inclusion. Suppose G ∈ Tors and that g ∈ G is of order n. Then g has the image 0 under the ﬁeld of fractions functor, as for any g ⊗Z a/b ∈ LG we have that

g ⊗Z a/b = g ⊗Z na/nb = gn ⊗Z a/nb = 0 ⊗Z a/nb.

So Tors ⊆ ker L. Now we show that Tors is a Serre subcategory. Let 0 → G0 → G → G00 → 0 be an exact sequence of abelian groups, and ﬁrst suppose G is torsion. Then G0 is a subgroup of G, so is clearly torsion too. Then it is immediate that G00 is also torsion as every element of G00 can be represented by a sum g + G0 for some g ∈ G. Conversely, suppose G0 and G00 are torsion and that g ∈ G. As G00 is torsion, the image g of g in G00 is such that ng = 0 for some n ∈ Z, which is equivalent to saying ng ∈ G0. But G0 is also torsion, so there is an m ∈ Z such that m(ng) = (mn)g = 0. Thus G is also torsion. As this example suggests, one (and in fact the only) way in which Serre subcategories arise is as the kernel of exact functors. Proposition 4.2.3. Let F : A → B be an exact functor between abelian categories. Then ker F is a Serre subcategory of A.

Proof. Let 0 → M 0 → M → M 00 → 0 be an exact sequence. First suppose M 0 and M 00 are in the kernel of F . Then applying F to the exact sequence gives the exact sequence 0 → 0 → FM → 0 → 0 in B. So FM is also in the kernel of F . Conversely suppose M ∈ ker F . Then again by applying F to our exact sequence we obtain

0 → FM 0 → 0 → FM 00 → 0, and so both FM 0 and FM 00 must be in the kernel of F . 2

We can now finally define the quotient of an abelian category by a Serre subcategory. Definition 4.2.4. Let A be an abelian category1 and T be a Serre subcategory of A. Then the quotient category A/T is the category defined by: • ob(A/T ) = ob(A),

If M and N are A-modules, then Hom (M,N) := lim Hom (M 0, N/N 0), • A/T −→ A 0 0 0 0 where the direct limit runs over the set IMN := {(M ,N ) | M ⊆ M and N ⊆

1We must impose that A has a small set of generators in order for the indexing set of the direct limit to be a set, and not a proper class.

22 N such that M/M 0,N 0 ∈ T }, and where (M 0,N 0) ≤ (M 00,N 00) if M 00 ⊆ M 0 and N 0 ⊆ N 00,

• Composition of morphisms in A/T is induced by composition in A. The next proposition will show that this is a well-defined category. Before we prove this, however, we note that unlike other quotient constructions such as for groups, rings, modules, etc., where the quotient is defined to be the set of equivalence classes under some equivalence relation, the quotient category is constructed by adding more morphisms to the category in order to make every module in T isomorphic to 0. This means that while the objects of the quotient category stay the same, the skeleton of the category becomes smaller. We now check that the claims we made in the definition of the quotient are actually true. Proposition 4.2.5. Let A be an abelian category and T a Serre subcategory of A. Then the index set defining the morphisms in A/T is directed, and we can compose morphisms in the quotient.

0 0 00 00 0 Proof. Suppose (M ,N ) and (M ,N ) are elements of IMN . We claim that (M ∩ 00 0 00 M ,N + N ) ∈ IMN , which suﬃces to show that IMN is directed. As T is closed under subobjects and quotients, we see that N 0 ∩ N 00 ∈ T . By the second isomorphism theorem we have that N 0/(N 0 ∩ N 00) ∼= (N 0 + N 00)/N 00, and so as N 00 ∈ T we see that the sum N 0 + N 00 must be too. Similarly, we know that M 0/(M 0 ∩ M 00) ∼= (M 0 + M 00)/M 00 ∈ T , as it is a submodule of M/M 00. Then by the third isomorphism theorem we have

(M/(M 0 ∩ M 00))/(M 0/(M 0 ∩ M 00)) ∼= M/M 0 ∈ T ,

0 00 0 00 0 00 hence M/(M ∩ M ) ∈ T . Thus (M ∩ M ,N + N ) ∈ IMN . We can then see that if (M 0,N 0) ≤ (M 00,N 00) the maps M 00 → M 0 and N/N 0 → N/N 00 deﬁne a morphism 0 0 00 00 HomA(M , N/N ) → HomA(M , N/N ), forming a directed system over IMN in A. Now it remains to see that we can compose morphisms in the quotient category. We will only give an outline of this proof, as the particular details are tedious and provide little insight into the quotient. A detailed proof can be found in [10]. Let f : L → M and g : M → N be morphisms in the quotient category. Then we can view these as the images of morphisms f : L0 → M/M 0 and g : M 00 → N/N 0 in the direct limit, where L/L0,M 0, M/M 00,N 0 ∈ T . Then we let L00 = f −1((M 0 +M 00)/M 0). The quotient L/L00 is torsion, and we have a map

f 0 : L00 → (M 0 + M 00)/M 0 induced by f. On the other end, we have that N 00 := N 0 + g(M 0 ∩ M 00) is torsion and there is a morphism

g0 : M 00/(M 0 ∩ M 00) → N/N 00

23 which is induced by g to M 00. We then say the composition of f and g is the composite f 0 g0 L00 −→ (M 0 + M 00)/M 0 −→∼ M 00/(M 0 ∩ M 00) −→ N/N 00.

We also note ﬁnally that the identity morphism in HomA/T (M,M) is just the image of the identity idM in A in the direct limit. 2

Definition 4.2.6. Let T be a Serre subcategory of an abelian category A. Then the quotient functor π : A → A/T is the functor which sends any A-module M to itself, and any morphism f to the image of f in the direct limit, as in the construction of A/T . We now have the following proposition, which tells us that our quotient category does in fact make every torsion module isomorphic to 0. Moreover, this proposition and the following theorem show that the converse of Proposition 4.2.3 is true. Proposition 4.2.7. Let A/T be the quotient of an abelian category A by a Serre subcategory T . For any A-module M, πM ∼= 0 if and only if M ∈ T . ∼ Proof. (⇒) Suppose πM = 0. Then HomA/T (πM, πM) = 0. In particular the image of the identity morphism in the direct limit defining this set of morphisms must be 0. So there are M 0,M 00 ⊆ M such that M/M 0,M 00 ∈ T and the morphism M 0 → M/M 00 induced by the identity is the zero map on M. The image of this map is just (M 0 + M 00)/M 00, so M 0 ⊆ M 00 ∈ T . But T is closed under extensions, so the exact sequence 0 → M 0 → M → M/M 0 → 0 shows that M ∈ T . (⇐) Suppose M ∈ T . Then the set {(0,M 0) | M 0 ⊆ M and M 0 ∈ T } is cofinal in the directed set defining HomA/T (πM, πM). Thus any morphism in 0 HomA/T (πM, πM) is the image of some morphism HomA(0,M ) in the direct limit. 0 But HomA(0,M ) = 0, so all the morphisms in HomA/T (πM, πM) are 0. In particular, the identity on πM is the zero map, hence πM ∼= 0. 2

It is immediate that A/T and the quotient functor are additive as Ab has direct limits but, perhaps surprisingly, this quotient category is abelian and the quotient functor is exact. Theorem 4.2.8. Let A be an abelian category and T a Serre subcategory of A. Then A/T is abelian and π : A → A/T is exact, and A/T is universal with respect to these properties. That is, if F : A → B is an exact functor to an abelian category B such that F T = 0 then there is an exact functor G : A/T → B such that F = Gπ. Moreover, if T = ker F then G is faithful.

24 Before we begin the proof we have the following two lemmas, the proofs of which are in a similar vein to that of deﬁning composition of morphisms in the quotient category. For this reason we direct the reader to Propositions 3.2.7 and 3.2.8 of [10] to see the details. Lemma 4.2.9. If f is a morphism in A, then πf = 0 if and only if imf ∈ T . Lemma 4.2.10. π commutes with kernels and cokernels. That is, for any morphism f ∈ A, ker(πf) = π(ker f) and coker(πf) = π(coker f).

Proof of Theorem 4.2.8. By the earlier discussion we know that π and A/T are additive. The previous lemma says exactly that π is exact and that A/T is preabelian. By the functoriality of π the isomorphisms of coimages and images in A hold in A/T , and so A/T is abelian. We now prove that A/T is universal. Precisely we show that if B is another abelian category with an exact functor F : A → B such that F T = 0, then there is an exact functor G : A/T → B such that A F B G π A/T commutes. As πM = M for all A-modules M, we let G(πM) = FM. Now let f : πM → πN be a morphism in A/T , which is the image of the morphism f : M 0 → N/N 0 in A with M/M 0,N 0 ∈ T . Consider the two exact sequences

g 0 → M 0 −→ M → M/M 0 → 0 and 0 → N 0 → N −→h N/N 0 → 0. Then the morphisms F g and F h are isomorphisms as F is exact and F T = 0, so we deﬁne Gf to be the composite (F h)−1F f(F g)−1 : FM → FN. It is clear that this G is functorial and that Gπ = F by construction. As Gπ = F we see that G must be exact. Now suppose ker F = T . To see that G is faithful suppose that πf : πM → πN is a morphism in A/T such that G(πf) = 0. We know Gπ = F , so we see that F f = 0. F is exact, so im(F f) = F (imf) = 0. But ker F = T so f has image in T , hence πf = 0 by Lemma 4.2.9. 2

Corollary 4.2.11. Let A be an abelian category with Serre subcategory T , and let B be a full subcategory of A closed under submodules and quotients. Then 1. The inclusion functor ι : B → A is exact,

2. B ∩ T is a Serre subcategory of B,

3. There is an exact functor ι : B/B ∩ T → A/T such that

25 B ι A

π1 π2 ι B/B ∩ T A/T commutes. Proof. 1. As kernels are submodules and cokernels are quotient modules, and B is closed under submodules and quotients, we see that the inclusion is exact.

2. This is true by Proposition 4.2.3, as B ∩ T = ker π2ι.

3. This is immediate by invoking the universal property of the quotient B/B∩T . 2

Example 4.2.12. As we showed earlier, Tors is a Serre subcategory of Ab. We claim that the quotient Ab/Tors is just the category ModQ of Q-vector spaces. We know that Q is ﬂat over Z so the ﬁeld of fractions functor L is exact. Thus the universal property of the quotient gives an exact functor F : Ab/Tors → ModQ. We have also seen that ker L = Tors, hence F is faithful by Theorem 4.2.8. Essential surjectivity is easy to see, as every Q-vector space V can be seen as an abelian group by restriction of scalars. In this case, we have that LV ∼= V , so we see that F πV = LV ∼= V , so F is essentially surjective.

Lastly, we show that F is full. Suppose f ∈ HomModQ(LG, LH) for some abelian groups G and H. We can assume both are torsion-free, which means they are submodules of LG and LH, respectively. Now if g ∈ G, then f(g) = h/n ∈ LH for some h ∈ H and n ∈ Z − {0}, in which case we see ng ∈ f −1(H). Then it is clear that G/(f −1(H) ∩ G) is a torsion abelian group, which in turn shows that the restriction f|f −1(H)∩G is in the direct limit defining HomAb/Tors(M,N). As F π(f|f −1(H)∩G) = L(f|f −1(H)∩G) = f, we see that F is full. So we see that Ab/Tors ' ModQ, and so we have successfully managed to describe a process that captures the idea of the field of fractions construction in a manner that can be applied to any abelian category. 4.3 Localising Subcategories Continuing with our field of fractions example we note the field of fractions functor, being defined in terms of a tensor product − ⊗Z Q, must have a right adjoint by the tensor-hom adjunction. As the quotient category Ab/Tors is equivalent to the category of Q-vector spaces the quotient functor π must also have a right adjoint. Definition 4.3.1. Let T be a Serre subcategory of an abelian category A. We say T is a localising subcategory if the quotient functor π : A → A/T has a right adjoint ω : A/T → A. We call ω the section functor. If this section functor exists then for an A/T -module πM, the A-module ωπM is a torsion-free representative of the equivalence class of πM in A/T , which we will now show.

26 Proposition 4.3.2. Let T be a localising subcategory of an abelian category A, with section functor ω : A/T → A. Then for any πM ∈ A/T , ωπM is torsion- free. Furthermore, the composite πω : A/T → A/T is naturally isomorphic to the identity functor idA/T . Proof. Suppose M 0 is a torsion submodule of ωπM. Then by the adjunction isomorphism ν we have

0 ∼ 0 ∼ HomA(M , ωπM) = HomA/T (πM , πM) = HomA/T (0, πM) = 0.

In particular the inclusion of M 0 in ωπM is the zero morphism, so M 0 = 0. We already have a natural transformation ε : πω → id given by the counit of the adjunction, so to prove the second statement of the proposition we need to show that the maps επM : πωπM → πM are isomorphisms for each πM ∈ A/T . Yoneda’s

Lemma says the functor πN 7→ HomA/T (πN, −) is fully faithful, thus it is suﬃcient to prove that HomA/T (πN, επM ) : HomA/T (πN, πωπM) → HomA/T (πN, πM) is an isomorphism for all πN ∈ A/T . We consider the following diagram

ν HomA(N, ωπM) HomA/T (πN, πM)

HomA/T (πN, πωπM) HomA/T (πN, πM), where the diagram commutes as the adjunction isomorphism ν must satisfy

ν(−) = ε ◦ π(−).

The top row is an isomorphism. We claim the left vertical morphism is too, in which case the bottom row must also be an isomorphism. To prove this claim, we ﬁrst note that as ωπM is torsion-free,

Hom (πN, πωπM) = lim Hom (N 0, ωπM), A/T −→ A where the direct limit runs over the submodules N 0 of N such that N/N 0 is torsion. We also note that for such a submodule N 0, we can apply π to the exact sequence

0 → N 0 → N → N/N 0 → 0

0 ∼ 0 00 to see that πN = πN. Now suppose HomA(N , ωπM) → HomA(N , ωπM) is a map in the direct limit and consider the commutative diagram

00 ν 00 HomA(N , ωπM) HomA/T (πN , πωπM)

0 ν 0 HomA(N , ωπM) HomA/T (πN , πM).

27 The two horizontal arrows are isomorphisms and by the previous discussion the right vertical arrow is too, so we see that the left vertical arrow is an isomorphism. As this ∼ holds for any morphism in the direct limit, we can say that HomA/T (πN, πωπM) = HomA(N, ωπN). 2

There is also another functor related to the ﬁeld of fractions construction. We have the functor τ : Ab → Tors which sends an abelian group G to the largest torsion subgroup. Explicitly, τ(G) = {g ∈ G | gn = 0 for some n ∈ Z}. We call τ the torsion functor. Proposition 4.3.3. Let T be a Serre subcategory of an abelian category A. Then the following are equivalent: 1. Every A-module has a largest torsion submodule,

2. The inclusion functor ι : T → A has a right adjoint τ : A → T .

Proof. (1) ⇒ (2) We deﬁne the right adjoint τ to the inclusion functor ι as follows. For any A-module M we let τM be the largest torsion submodule of M. Now suppose f : M → N is a morphism of A-modules. Then we have an exact sequence

0 → ker f|τM → τM → f(τM) → 0.

Thus f(τM) is torsion and a submodule of N, hence is contained in τN. We then deﬁne τf to be the restriction f|τM : τM → τN. Now we must show that this functor τ is right adjoint to ι. If M is torsion and f : M → N, then the image of f is torsion, and hence contained in τN. That is, ∼ HomA(M, τN) −→ HomA(M,N). Then we see

∼ HomA(ιM, N) = HomA(M,N) = HomA(M, τN) = HomT (M, τN).

(2) ⇒ (1) Let τ be the right adjoint to the inclusion ι, and recall that the adjunction has a counit ε : ιτ → idA. We claim that the map εN : ιτN → N is monic. A standard result about adoint functors says that a left adjoint is fully faithful if and only if the unit of the adjunction is a natural isomorphism. This is the case for ι, so then we consider the exact sequence

0 → K → ιτN −→εN N, where K is the kernel of the counit εN . Then we apply the functor τ to obtain

0 → τK → τιτN → τN, which is exact as τ is left exact. But, as we mentioned before, the transformation τιτ → τ is naturally isomorphic to the identity functor, so τK must be 0. Then we

28 see ∼ 0 = HomT (K, τK) = HomA(ιK, K) = HomA(K,K), so K = 0. We then immediately have that τN is a torsion submodule of N. We now show that this is the largest torsion submodule of N. Suppose M is another torsion submodule of N. Then we can view the inclusion of M in N as an element of

HomA(ιM, N). The adjunction isomorphism ν : HomT (M, τN) → HomA(ιM, N) must satisfy the identity ν(−) = ε ◦ ι(−), which means every morphism f : ιM → N in A factors through ιτN. In particular, the inclusion of M in N must factor through ιτN, which shows that M must be a submodule of τN. 2

Lemma 4.3.4. Let T be a Serre subcategory of an abelian category A, M and N be A-modules and suppose τN exists. Then if M is torsion, HomA(M, N/τN) = 0. Proof. Let f : M → N/τN be a morphism and deﬁne N 0 to be the kernel of the composition N → N/τN → coker f. Then N 0/τN ∼= im f, so there is an exact sequence 0 → τN → N 0 → im f → 0. As M is torsion so is im f, hence N 0 must be torsion too. But τN is the largest torsion submodule of N, hence N 0 ⊆ τN. So we see im f = 0, and thus f = 0. 2

If a torsion functor exists, then it is clear that for any A-module N, N/τN is torsion-free. As all torsion submodules of N are contained in τN, the set J := {(M 0, τN) | M 0 ⊆ M and M/M 0 torsion} is coﬁnal in the directed set deﬁning the morphisms in the quotient category, and thus we see that HomA/T (M,N) = lim Hom (M 0, N/τN). −→J A The existence of a torsion functor and a section functor are closely related, as we will now see. The idea for the proof of the next proposition comes from [15]. Proposition 4.3.5. Let T be a Serre subcategory of an abelian category A. If T is localising, then there is a torsion functor τ : A → T . Conversely, if a torsion functor τ exists and A has injective envelopes, then T is localising.

Proof. Suppose T is localising. Let M be an A-module and ηM : M → ωπM be the morphism deﬁned by the unit of the adjunction. We deﬁne τ(M) := ker ηM . By Proposition 4.3.2 we know ωπM is torsion-free and hence ker ηM contains all torsion submodules of M, meaning τ(M) is the largest torsion submodule of M. Then by Proposition 4.3.3 a torsion functor exists. Conversely suppose a torsion functor exists and A has injective envelopes. To show that the quotient functor π has a right adjoint, we show that the functor M 7→

HomA/T (πM, πN) is representable for all A-modules N. We deﬁne the representing object now. Let α : N/τN → E be the inclusion of N/τN in an injective envelope,

29 and deﬁne H to be the kernel of the composition E → cokerα → coker α/τ(coker α). This will be our representing object. We obtain an exact sequence

f 0 → τN → N −→ H → coker f → 0 in which coker f is torsion. Thus πf is an isomorphism, and hence

∼ HomA/T (πM, πN) = HomA/T (πM, πH).

So we have reduced the problem to showing that H is a representing object for ∼ the functor M 7→ HomA/T (πM, πH). To do so we show that HomA(M,H) = HomA/T (πM, πH). Now H is an essential extension of N/τN, so for any torsion submodule H0 of H we have that H0 ∩ N/τN is a torsion submodule of N/τN. But N/τN is torsion-free, hence H must be too. This in turn tells us that Hom (πM, πH) = lim Hom (M 0,H), where the limit runs over the sub- A/T −→ A modules M 0 of M such that M/M 0 is torsion. We want to show that the natural 0 map induced by the directed system ϕM 0 : HomA(M,H) → HomA(M ,H) is an isomorphism for any such M 0. From the exact sequence

0 → M 0 → M → M/M 0 → 0

0 we immediately see that ϕM 0 is injective as M/M is torsion, H is torsion-free 0 and HomA(−,H) is left exact. To see that it is surjective, take a map f ∈ 0 HomA(M ,H). Then we have the diagram 0 M 0 M M/M 0 0

f 0 0 H E E/H 0. As E is an injective module and the composite M 0 → H → E is an injection, f 0 extends to a map f : M → E. Thus we see there is a unique map f 00 : M/M 0 → E/H making the diagram commute. But recall M/M 0 is torsion and E/H ∼= coker α/τ(coker α), hence f 00 = 0 by Lemma 4.3.4. This shows that im f ⊆ H, and 0 thus f|M 0 = f . Thus ϕM 0 is surjective. 2

4.4 Grothendieck Categories In this last section of this chapter we will introduce a special type of abelian category, known as a Grothendieck category. This will be a relatively short chapter, as we aim mostly to give a brief exposition of the properties of these categories which will be useful to us in the next chapter. Any proofs or details omitted from proofs can be found in [15] and [7]. Deﬁnition 4.4.1. A Grothendieck category is an abelian category A such that A has a set of generators, all colimits and exact direct limits.

30 Remark 4.4.2. The condition that an abelian category A has all colimits is called AB3, and if additionally A has exact direct limits we say A satisﬁes AB5. So a Grothendieck category is an AB5 category with a set of generators. In very much the same vein as Example 2.2.8, using pointwise constructions we can show that if C is a category and A is a Grothendieck category then the category of covariant functors Fun(C, A) is again a Grothendieck category. This in turn tells us that the category of G-graded modules GrR over a G-graded ring R is a Grothendieck category; a fact which we will use in the next chapter. With our most important example in mind, we now want to introduce the useful properties of Grothendieck categories which we can later apply to GrR. We start by showing that quotients of Grothendieck categories by localising subcategories behave similarly to quotients of abelian categories by Serre subcategories. Proposition 4.4.3. Let A be a Grothendieck category and T a localising subcategory. Then both T and A/T are Grothendieck categories.

Sketch of Proof. As T is localising the inclusion functor has a right adjoint by Proposition 4.3.5. This implies that the inclusion commutes with direct limits. As

A is AB5 we then see that T is too. If {Mi} is a set of generators for A then the collection of quotient modules of the Mi, which we call Q, also forms a set. Then {M ∈ Q | M ∈ T } is a set of generators for T . Similarly the quotient functor π has a right adjoint as T is localising, so A/T is also AB5, and the set {πMi} form a set of generators for A/T . 2

Just as we could think of any abelian category as being embedded into some larger module category by way of the Freyd-Mitchell Embedding Theorem, the following theorem of Gabriel and Popescu allows us to think of every Grothendieck category as sitting inside a quotient of a module category by a localising subcategory. A short proof of this theorem can be found in [16]. Theorem 4.4.4 (Gabriel-Popescu Theorem). Let A be a Grothendieck category and M be a generator in A with endomorphism ring R = Hom(M,M). Then the functor HomA(M, −): A → ModR deﬁnes an equivalence between A and a quotient of ModR by a localising subcategory. This theorem immediately gives us useful information about any Grothendieck category. Proposition 4.4.5. A Grothendieck category A has both enough injectives and injective envelopes.

Proof. The fact that A has enough injectives follows from the Gabriel-Popescu The- orem, as ModR has enough injectives, and the fact a quotient of any abelian category with enough injectives by a localising subcategory also has enough injectives. This fact is Theorem 2.14.14 in [15]. For a proof that A has injective envelopes see Corollaire 2 on page 375 of [7]. 2

31 An immediate corollary is that in a Grothendieck category the existence of a torsion functor and a section functor are equivalent. We will use this result later to prove a Serre subcategory is localising by showing that every module has a largest torsion submodule. Corollary 4.4.6. Let T be a Serre subcategory of a Grothendieck category A. Then the following are equivalent: 1. T is localising,

2. There is a torsion functor τ : A → T .

Proof. This is immediate by the previous proposition and Proposition 4.3.5. 2

32 Chapter 5 I-Algebras and cohproj

The goal of this chapter is to prove an analogue of Morita’s Theorem, which will allow us to determine when an abelian category A is equivalent to a quotient of a subcategory of graded right modules over a special type of bigraded algebra. While we motivate this result as a way of understanding an abelian category this theory actually has its roots in algebraic geometry, particularly projective geometry. These types of results were originally used to determine when an abelian category could be the category of coherent sheaves on a projective scheme. For more on this viewpoint one can see [1] and [12]. We will start by introducing the special type of algebra we mentioned earlier: indexed algebras, and the basic theory for modules over an indexed algebra, which behave a little differently than modules over a graded ring. Then we will introduce a suitable cohomology theory necessary to define an analogue of the condition χ1 from [1]. Finally, we will introduce a notion of an ample sequence in an abelian category, which will be analogous to the existence of a progenerator in determining equivalence with a module category. We will then conclude by showing that if an abelian category has such an ample sequence, then it is equivalent to a quotient of the category of graded modules over an indexed algebra that satisfies the condition χ1. All of the material that we will present in this chapter is a generalisation to the case of algebras indexed by any directed set from the material presented in [9], [11] and [12], where all the algebras considered were indexed by Z. 5.1 I-algebras In this section we will give an overview of indexed algebras and the graded modules over such algebras. Our overarching goal for the chapter will be to define the specific type of quotient category that we will eventually be able to compare an arbitrary abelian category to. Definition 5.1.1. Let I be a set. An I-indexed algebra A is a pre-additive category whose objects {O(i)}i∈I are indexed by I. We denote the morphisms of this category Aij := Hom(O(j), O(i)). If A is k-linear, for some field k, then we call A an I-indexed k-algebra, or an I-algebra for short. The reason we use the term I-algebra is as follows. If A is an I-algebra then L the ring A = i,j∈I Aij, with multiplication induced by composition of morphisms, is an associative k-algebra. When we say the multiplication is induced by the

33 composition of morphisms, we mean that the only non-zero multiplication in our algebra is of the form

Aij ⊗k Ajl → Ail. We will frequently abuse the terminology and refer to this ring as an I-algebra throughout. If A is an I-algebra, then a graded right A-module is a graded k-vector space L M = i∈I Mi with scalar multiplication of the form Mi ⊗k Aij → Mj satisfying the module axioms. We denote the category of graded right A-modules GrA, and A−Gr for graded left modules, both of which have morphisms being degree zero graded module homomorphisms. Recall from our earlier discussion in Chapter 4 that these are both Grothendieck categories. From here on out we will only consider graded right modules, unless otherwise speciﬁed. Example 5.1.2. As a ﬁrst example we consider the quantum polynomial algebra × kq[x, y] := khx, yi/(xy−qyx) for some q ∈ k . This is a Z-graded algebra, where the grading is given by degree. If q = 1 then kq[x, y] is just the polynomial algebra in two commuting variables k[x, y], and if q 6= 1 then kq[x, y] is a noncommutative algebra. From a quantum polynomial algebra we obtain a -algebra k [x, y] = L A , Z q i,j∈Z ij where

Aij = {f ∈ kq[x, y] | degf = j − i}. This is a particular case of a more general construction which shows that every -graded algebra A = L A and graded right A-module M = L M deﬁnes a Z i∈Z i i∈Z i -indexed algebra A and a graded right A-module M. We let A = L A , where Z i,j∈Z ij A = A , and M = L M . This is a -indexed algebra as the multiplication ij j−i i∈Z i Z is given by

Aij ⊗k Ajl = Aj−iAl−j ⊆ Al−i = Ail, and M is a module over A as

M i ⊗k Aij = MiAj−i ⊆ Mj = M j.

Even further, this construction gives a G-indexed algebra from a G-graded algebra, where G is any group. We can also define a graded A-module M 0 from a graded A-module M: we simply let M 0 = M. This module is a graded A-module as it has multiplication 0 0 Mi Aj = MiAi,(j+i) ⊆ Mj = Mj. Then we see that in fact GrA = GrA. Returning to our example of quantum polynomial algebras, we can now get a ∼ feel for the utility that indexed algebras afford us. If q 6= 1 then we know kq[x, y] 6= k[x, y] as kq[x, y] is not commutative. Surprisingly, however, the Z-algebras k[x, y] × and kq[x, y] are isomorphic for any q ∈ k ! Let A = k[x, y] and Aq = kq[x, y]. Then the -algebra A = L A has generators x , y ∈ A with the relations Z i,j∈Z j−i i i i,i+1 0 0 xiyi+1 = yixi+1 for all i ∈ Z. Similarly Aq has generators xi, yi ∈ (Aq)i,i+1 with 0 0 0 0 relations xiyi+1 = qyixi+1. To define an isomorphism between A and Aq we define

34 a pair of inverse algebra homomorphisms. Let ϕ : A → Aq and ψ : Aq → A be as follows: ( x 7→ x0 ϕ := i i −i 0 yi 7→ q yi, and ( x0 7→ x ψ := i i 0 i yi 7→ q yi. It is immediate that these homomorphisms are inverse, so we only need to show that they preserve the relations deﬁning each algebra. So we check

−i−1 0 0 −i q xiyi+1 = ϕ(xiyi+1) = ϕ(yixi+1) = q yixi+1,

0 0 0 0 which shows that xiyi+1 = qyixi+1, and

i+1 0 0 0 0 i q xiyi+1 = ψ(xiyi+1) = ψ(qyixi+1) = q(q yixi), ∼ so xiyi+1 = yixi+1. Hence ϕ and ψ define an isomorphism of Z-algebras A = Aq. So while at first glance we may have imagined that the categories of graded modules over these two algebras would be very different, we have been able to define an isomorphism of the associated Z-algebras, which then tells us that we have an × equivalence Grkq[x, y] ' Grk[x, y] for any q ∈ k . An immediate difference from the graded algebra case is that we can no longer realise an I-algebra A as a module over itself, due to the bigrading. In turn, the free modules in GrA are different, as they are no longer just sums of shifts of the graded algebra. Recall that the components of an I-algebra are defined by sets of morphisms. We will let ei denote the identity element of Aii. We call any module which is direct sum of the modules eiA free. If this direct sum is also finite we call the module free and finitely generated. Definition 5.1.3. Let A be an I-algebra, and M an A-module. Then: • M is finitely generated if there is a surjection F → M, where F is free and finitely generated,

∼ • M is ﬁnitely presented if M = coker(F1 → F0), where F0 and F1 are free and ﬁnitely generated.

Definition 5.1.4. Let A be an I-algebra and let M ∈ GrA be finitely generated. Then we say M is coherent if the kernel of any homomorphism F → M, where F is free and finitely generated, is also finitely generated. We denote the full subcategory of coherent A-modules by cohA. It has many desirable properties, one of which being that it is abelian.

35 Proposition 5.1.5. cohA is an abelian subcategory of GrA which is closed under extensions.

Proof. We prove this by showing that coherent modules satisfy the ‘2 out of 3’ property. That is, if the sequence of graded right modules 0 → M 0 → M → M 00 → 0 is exact and any two of the three modules are coherent, then the third is also. This will show that all kernels, cokernels and finite direct sums of coherent modules, which exist in GrA, are in fact coherent. Furthermore, it will show that cohA is closed under extensions. First, assume M and M 00 are coherent. As M 0 ⊆ M, any homomorphism F → M 0, where F free and finitely generated, can also be seen as a homomorphism F → M, which must have finitely generated kernel as M is coherent. So we just have to show M 0 is finitely generated. Let F → M be a surjection expressing finite generation of M, and let K be the kernel of the composite F → M → M 00. Then K is finitely generated, as M 00 coherent. But K also maps onto M 0 = ker(M → M 00) by the universal property of kernels, so M 0 is also finitely generated. Now suppose M 0 and M 00 are coherent. It is clear that M is finitely generated, as any element of M is the sum of elements of M 0 and lifts of elements of M 00, both of which are finitely generated. Now let g : F → M be a homomorphism, where F is free and finitely generated. Then we have the commutative diagram

0 0 F F 0

g h 0 M 0 M M 00 0, where h = F → M → M 00. By the Snake Lemma we have the exact sequence 0 → ker g → ker h → M 0. We know ker h is finitely generated, as M 00 is coherent. Let K denote its image in M 0, which is finitely generated as M 0 coherent. Then 0 → ker g → ker h → K → 0 is exact, and so by the first part of the proof ker g is finitely generated. Finally, suppose M 0 and M are coherent. Then M 00 is finitely generated, as M is. Let F 0 and F 00 be free and finitely generated, f : F 0 → M 0 be a surjection, and let h : F 00 → M 00 be any homomorphism. Then we consider the following commutative diagram.

0 F 0 F 0 ⊕ F 00 F 00 0

f g h 0 M 0 M M 00 0 By the Snake Lemma we have an exact sequence ker g → ker h → 0. We know ker g is ﬁnitely generated by the coherence of M, hence ker h is too. 2

36 For a graded k-algebra B, we say that B is right coherent if BB is a coherent module. This coherence condition is to make every finitely presented module coherent, similarly to the case of noetherian modules and finitely generated modules over a noetherian algebra. Specifically, if B is coherent then FinPresB = cohB, where FinPresB is the full subcategory of GrB consisting of finitely presented modules. Also akin to the case of finitely generated modules over a noetherian ring, FinPresB is abelian when B is coherent, which it is not in general. Things are slightly more complicated in the indexed case as, due to the bigrading, we cannot define a coherent I-algebra in the same way. To remedy this we introduce the notion of a bimodule, and then present a notion of right coherence for I-algebras. Definition 5.1.6. Let A and B be I-algebras. The category of A-B bimodules Bimod(A − B) is defined as follows: the objects of Bimod(A − B) are bigraded k- L vector spaces i,j∈I Mij with commuting homomorphisms µijl : Mij ⊗k Bjl → Mil and ψijl : Aij ⊗k Mjl → Mil. Morphisms in this category are k-linear homomorphisms compatible with all the homomorphisms µijl and ψijl. We can now naturally think of an I-algebra A as an A-A bimodule. From now on, we will assume I will be a directed set with partial order ≤. This directed condition is required in order for us to be able to take direct limits over sets of submodules of A-modules, which is what we will use to define a torsion functor for a Serre subcategory later. We will now introduce a number of different conditions on I-algebras and graded modules, which are analogues of the graded case. Definition 5.1.7. Let A be an I-algebra and M an A-module. Then:

• A is connected if Aii = k for every i ∈ I.

• A is positively indexed (resp. negatively indexed) if Aij 6= 0 ⇒ i ≤ j (resp. i ≥ j). Now let A be positively indexed. L • A tail M≥n of M is a submodule of the form i≥n Mi for some n ∈ I.

• M is upper bounded by n ∈ I if M≥n = 0. M is lower bounded by n if Mn 6= 0 implies that i 6≤ n. If M is upper and lower bounded by u and l respectively, and l ≤ u, we say M is bounded. While it seems strange, the asymmetry in our deﬁnition of upper and lower bounded is necessary to capture the desired property for a positively indexed I- algebra, due to the lack of a total order on I. Take, for example, a module M over a positively indexed Z2-indexed algebra A, where the partial order on Z2 is the product order. Suppose that M is lower bounded by (2, 2) and upper bounded by (3, 3). If we took the naive deﬁnition of lower bounded, the non-zero degrees in M would be represented by the shaded lattice points in the diagram below.

37 (3, 3) (2, 2)

As the product order on Z2 is not a total order our ‘bounded’ module actually has very little restriction on it, whereas if we use the deﬁnition as in 5.1.7, the resulting non-zero region

(3, 3) (2, 2)

appears much more contained. More importantly, the naive definition of a bounded module is not desirable as if the order on the index set is not a total order then the full subcategory of bounded modules does not form a Serre subcategory of cohA. From now on, all I-algebras will be connected and positively indexed. In the L graded algebra case, A≥n was just defined to be i≥n Ai, as the algebra is also a module over itself. In the case I = Z, as considered in [9], this notion was recaptured L by letting A≥n = j−i≥n Aij. If I is a directed set without a group structure we cannot construct this the same way, which leads us to the following definition. We L let A∗,≥n denote the subobject i∈I (eiA)≥n of A in Bimod(A − A). While this is not a direct generalisation in the sense that we don’t recover A≥n if I = Z, it does retain the desired properties of A≥n, such as giving a description of the torsion functor. We now return to right coherence, giving a definition for I-algebras. Definition 5.1.8. Let A be an I-algebra. We say A is right coherent if for all i ∈ I and all n ∈ I, the modules eiA and eiA/eiA≥n are coherent A-modules. In line with the modules we consider, we will only work with right coherent I-algebras, although left coherent algebras can be defined similarly. As in the k- algebra case we have the following. Proposition 5.1.9. Let A be a coherent I-algebra. Then an A-module M is finitely presented if and only if M is coherent.

Proof. If M is coherent, then it is obviously also finitely presented. On the other ∼ hand, if M is finitely presented then M = coker(F1 → F0), where F0 and F1 are free and finitely generated. As A is coherent all of the eiA are coherent modules, hence F0 and F1 are too. Then, as cohA is abelian, M must also be coherent. 2

38 We note that if A is coherent then so are all the eiA≥n, as for each i ∈ I we have the exact sequence

0 → eiA≥n → eiA → eiA/eiA≥n → 0, and so cohA being abelian implies that eiA≥n is a coherent module. The following Lemma, while seemingly simple, will become essential when we begin to talk about the cohomology of I-algebras.

Lemma 5.1.10. Let A be an I-algebra, and M ∈ cohA. If A is coherent, then M≥n is coherent for any n ∈ I.

Proof. We first let M = eiA. Then eiA≥n is coherent for any n ∈ I, by the definition of a coherent I-algebra. Thus F≥n is also coherent when F is any free and finitely generated module. For the general case, we take a finite presentation

F1 → F0 → M of M, where F0 and F1 are free and ﬁnitely generated. Then we have that

(F1)≥n → (F0)≥n → M≥n → 0 is exact, and thus M≥n is coherent as cohA is abelian. 2

Remark 5.1.11. This method of proof involving showing a property is true of any eiA and then showing the general case by taking a ﬁnite presentation and using that cohA is abelian will occur quite frequently. As in the Z-graded case of [9], we say a module M is torsion if for every m ∈ M, mA is upper bounded. If m ∈ Mi, this is equivalent to saying that meiA≥n = 0 for some n ∈ I. Any element of a module satisfying this condition is called a torsion element. If M has no torsion elements we say M is torsion-free. We call the subcategory of torsion modules TorsA. We can also characterise torsion in terms of the largest torsion submodule τ(M), should one exist. In this case M is torsion if and only if τ(M) = M, and M is torsion-free if and only if τ(M) = 0. Proposition 5.1.12. Let A be an I-algebra. Then an A-module M is torsion if and only if it is a direct limit of upper bounded modules. Before we prove this proposition we introduce the internal Hom functor, which we will use to give an explicit description of τ(M). The reason we introduce this internal Hom is that, due to the bigrading once again, we cannot naively think of Hom(A, M) as an A-module. Deﬁnition 5.1.13. Let A and B be I-algebras, M ∈ Bimod(A − B), and N ∈

GrB. Then the internal Hom is the graded right A-module HomB(M,N) := L i∈I Hom(eiM,N), with right A action induced by the left A-module structure on M.

39 From the deﬁnition, we have immediately that Hom(A, M) ' idM , as each L Hom(eiA, M) are the set of homomorphisms from j∈I Aij to M, which we can easily identify with the elements of Mi. Proof of Proposition 5.1.12. First, assume M is a direct limit of upper bounded modules. Then every m ∈ M is the image of an element of one of these modules, hence mA is upper bounded. Thus M is torsion. To prove the converse, we describe the torsion functor τ. We show that τ(M) = lim Hom(A/A ,M), where we take the direct limit over I. We note ﬁrstly that −→ ∗,≥n lim Hom(A/A ,M) is actually a submodule of M, as we obtain it by applying −→ ∗,≥n the contravariant functor lim Hom(−,M) to the exact sequence −→

A → A/A∗,≥n → 0.

Direct limits are exact in GrA as it is a Grothendieck category, so we obtain an injection lim Hom(A/A ,M) ,→ lim Hom(A, M) ∼ M. −→ ∗,≥n −→ = We now show that an element of M is torsion if and only if it belongs to lim Hom(A/A ,M). Suppose m ∈ M is torsion. Then we can consider m as an −→ ∗,≥n d element of Hom(edA, M). But as m is torsion, there is an l ∈ I for which medA≥l = 0. In this case, we can also view m as an element of Hom(ed(A/A∗,≥l),M). Thus m is an element of lim Hom(A/A ,M) . Conversely, suppose m ∈ lim Hom(A/A ,M) . −→ ∗,≥n d −→ ∗,≥n d We can then also view m as an element of Hom(A, M)d = Hom(edA, M) = Md such that medA≥l = 0 for some l ∈ I. But then m is torsion. Now the result follows easily. All of the Hom(A/A∗,≥n,M) are upper bounded L by n, due to the positive indexing on A. Explicitly, ei(A/A∗,≥n) = j6≥n Aij = 0 if i ≥ n. Now if M is torsion we know M = τ(M), so it is a direct limit of upper bounded modules. 2

Proposition 5.1.14. If A is coherent, then TorsA is a localising subcategory.

Proof. We prove TorsA is a Serre subcategory from which the result follows from Corollary 4.4.6, as GrA has enough injectives and we have shown a torsion functor exists. One direction is easy, as any submodule or quotient of a torsion module is torsion. Now assume 0 → M 0 → M → M 00 → 0 is exact, and M 0,M 00 are both torsion. 00 0 Let m ∈ Mi. As M is torsion, there is an n ∈ I such that meiA≥n ⊆ M . As we mentioned earlier eiA≥n is coherent. In particular it is ﬁnitely generated, say 0 by a1, . . . , as. Then all of the maj are torsion as they are elements of M , there are ﬁnitely many and I is directed, so we can choose an l ∈ I such that for all j, majApl = 0, where aj ∈ Aip. But then mAil = 0, so m is torsion. 2

As TorsA is localising, we can construct the quotient category QGrA := GrA/TorsA, with quotient functor π : GrA → QGrA and section functor ω : QGrA → GrA.

40 We now further restrict ourselves to coherent modules. Note that all coherent modules are lower bounded, as the eiA are by the positive indexing. When wanting to deﬁne a quotient of cohA there are two obvious possibilities. Firstly, we let cohbA be the full subcategory of bounded coherent A-modules, which is Serre by Corollary 4.2.11 as it is the intersection of cohA and TorsA. We call the corresponding quotient category cohprojA. We could also, however, consider cohprojA to be the image of cohA in the quotient QGrA. While not immediately obvious, these two notions are actually equivalent. Theorem 5.1.15. Let A be a coherent I-algebra, and let ι : cohA → GrA be the inclusion functor. Then there is an induced functor ι : cohA/cohbA → QGrA which is exact and fully faithful. That is, we can identify the quotient cohA/cohbA with the image of cohA in QGrA.

Proof. As cohA is an abelian subcategory of GrA and cohbA = cohA ∩ TorsA, Corollary 4.2.11 says that there is an exact functor ι : cohA/cohbA → QGrA. As cohbA = ker πι, we see that ι is faithful. To see that it is full, let M,N ∈ cohA and 0 0 0 b f ∈ HomQGr(ιπ M, ιπ N), where π : cohA → cohA/coh A is the quotient functor. Then f is the image of a map f : M 0 → ι(N)/N 0 in GrA, where M 0 ⊆ ι(M),N 0 ⊆ ι(N) and ι(M)/M 0,N 0 ∈ TorsA. As ι(M) is coherent it is finitely generated, so the quotient ι(M)/M 0 is also finitely generated. But the quotient is torsion, hence must ∼ 0 be upper bounded. Then we see that ι(M)≥d = M≥d for some d ∈ I. So we can 0 0 0 0 consider ι(M)≥d as a submodule of M , hence we see that (M ,N ) ≤ (ι(M)≥d,N ) 0 0 in the index set defining HomQGr(ιπ M, ιπ N). This tail ι(M)≥d is coherent by Lemma 5.1.10 so, without loss of generality, we may assume M 0 is coherent. We can then take a free and finitely generated module F such that M 0 ∼= F/K, where K = ker(F → M 0). This K is necessarily coherent. As F is projective our map f 0 0 0 lifts to a map f : F → ι(N). The image of K under f is coherent, as f |K : K → N 0 0 0 is a map between coherent modules. Thus f ∈ Homcohproj(π M, π N), showing ι is full. 2

We will also refer to the quotient functor for the quotient cohprojA as π, as now we have seen that it is simply π|cohA. This quotient cohprojA will be the quotient category which we will ultimately determine equivalence of an arbitrary abelian category with. Remark 5.1.16. If instead of an I-algebra we considered a Noetherian Z-graded algebra A, then cohprojA = qgrA, as defined in [1]. This is due to the fact that a module being finitely generated, finitely presented and coherent are all equivalent if A is Noetherian. We will now show that every module over a coherent algebra has a resolution by free and finitely generated modules, which will be useful as we will later show that these modules are projective.

41 Proposition 5.1.17. Let A be a coherent I-algebra. Then every M ∈ cohA has a resolution of the form

... → F1 → F0 → M → 0, where the Fi are all free and ﬁnitely generated.

Proof. As M ∈ cohA there is a surjection F0 → M, where F0 is free and ﬁnitely generated. Then A being coherent means F0 is a coherent module, thus the kernel of this map is also coherent. We then again surject a free and ﬁnitely generated module onto this kernel, and repeating this process generates the desired resolution. 2

5.2 Cohomology of I-algebras In this section we prove some results about the cohomology of coherent I-algebras, as introduced in [9], which allows us to deﬁne an analogue of the χ condition used to formulate the main result of [1]. Proposition 5.2.1. Let A be a coherent I-algebra. Then the category cohA has enough projectives.

Proof. We ﬁrst show that this is true for GrA, and then that ﬁnitely generated projectives in GrA are coherent. We claim the modules eiA are projective. Indeed, Hom(eiA, M) is a map from eiA into M, which we can naturally view as an element 0 00 of Mi. So if we have an exact sequence 0 → M → M → M → 0, applying the 0 00 functor Hom(eiA, −) gives a sequence 0 → Mi → Mi → Mi → 0, which is obviously exact. So eiA is projective. The sum of these projectives is equal to A, and colimits are exact, which means GrA has enough projectives. We then immediately have that cohA has enough projectives, as A being coherent means that the eiA are Ls coherent, and for any coherent A-module M there is a surjection j=1 eij A → M. 2

Remark 5.2.2. The proof of this proposition also tells us that the resolution deﬁned in Proposition 5.1.17 is a projective resolution. We now return to study the internal Hom functor. This acts exactly as the Hom(−, −) functor described in [1]. In particular, being described in terms of Hom functors means it is left exact in both arguments, and if I is an injective graded right B-module then HomB(−, I) is exact. As it is left exact we can consider the right derived functors of the internal Hom functor, which are the internal Ext j L j functors given by ExtB(M,N) := i∈I ExtB(eiM,N). The following lemmas and proposition generalise the content of Theorem 6.8 in [11] to our context. Lemma 5.2.3. Let A be a coherent I-algebra. Then the right derived functors of the torsion functor τ : TorsA → GrA commute with direct limits.

Proof. As A is coherent, the modules ejA/ejA≥n all have a resolution by free and ﬁnitely generated modules. We have seen that τ(−) ∼ lim Hom(A/A , −), so = −→ ∗,≥n

42 we know that Riτ(−) ∼ lim Exti(e A/e A , −). Then we can compute each of j = −→ j j ≥n these as the i-th cohomology of Hom(−, −) applied to this resolution. Thus Riτ(−) commutes with direct limits. 2

Lemma 5.2.4. Suppose A is a coherent I-algebra. If T is a torsion module then Riτ(T ) = 0 for i > 0.

Proof. By the previous lemma and Proposition 5.1.12 we can assume that T is upper bounded, say by u ∈ I. By applying lim Hom(−,T ) to the exact sequence −→

0 → A∗,≥n → A → A/A∗,≥n → 0 (5.2.1) and considering the corresponding long exact sequence

... → 0 → lim Exti(A ,T ) → lim Exti+1(A/A ,T ) → 0 → ..., −→ ∗,≥n −→ ∗,≥n we see that it is suﬃcient that show that lim Exti(A ,T ) = 0 for i ≥ 0. For −→ ∗,≥n each j ∈ I, ejA≥n has a resolution F• by free and ﬁnitely generated modules as A is coherent. Clearly all of the elA in this resolution are such that l ≥ n. So i as T is upper bounded by u and Ext (ejA≥n,T ) is a submodule of a quotient of i Hom(Fi,T ), we see that, for each j ∈ I, Ext (ejA≥n,T ) = 0 if n ≥ u. Thus lim Exti(A ,T ) = 0 for i ≥ 0. 2 −→ ∗,≥n Proposition 5.2.5. Let A be an I-algebra, and let ω : QGrA → GrA denote the section functor. Then if A is coherent we have that the saturation of a module M := ωπM = lim Hom(A ,M). f −→ ∗,≥n Proof. First we show that ωπM ∼ lim Hom(A , M/τ(M)). To see this we note = −→ ∗,≥n

ωπM = Hom(A, ωπM) M = HomGr(eiA, ωπM) i∈I M = HomQGr(πeiA, πM). i∈I

We claim that the set {(eiA≥n, τM) | n ∈ I} is cofinal in the index set defining each 0 0 of the HomQGr(πeiA, πM). Indeed, we have seen that {(N , τM)}, where N ⊆ eiA 0 0 such that eiA/N is torsion, is cofinal. Now suppose N is such a submodule of eiA. As A is coherent eiA is a coherent module. In particular it is finitely generated, so the quotient is too. But the quotient is also torsion, hence is upper 0 0 bounded. Let the upper bound be n ∈ I. Then eiA≥n ⊆ N , and so (N , τM) ≤

43 ∼ (eiA≥n, τM). Thus {(eiA≥n, τM)} is coﬁnal, and hence HomQGr(πeiA, πM) = lim Hom (e A , M/τM). Returning to our earlier computation, we see that −→ Gr i ≥n M ωπM ∼ lim Hom (e A , M/τM) ∼ lim Hom(A , M/τM). = −→ Gr i ≥n = −→ ∗,≥n i∈I

Now we apply the functor lim Hom(A , −) to the exact sequence −→ ∗,≥n 0 → τ(M) → M → M/τ(M) → 0, and show that lim Exti(A , τ(M)) vanishes when i = 0, 1. The fact that −→ ∗,≥n lim Hom(A , τ(M)) = 0 is immediate as τ(M) is torsion, hence is a direct limit −→ ∗,≥n of upper bounded modules, and lim Hom(A , −) commutes with direct limits. −→ ∗,≥n For the case when i = 1, we apply the functor lim Hom(−, τ(M)) to (5.2.1) and −→ consider the corresponding long exact sequence. We then see that

lim Ext1(A , τ(M)) ∼ lim Ext2(A/A , τ(M)) = R2τ((τ(M)), −→ ∗,≥n = −→ ∗,≥n which is 0 by the previous Lemma. Thus

ωπM ∼ lim Hom(A , M/τ(M)) ∼ lim Hom(A ,M). = −→ ∗,≥n = −→ ∗,≥n 2

Corollary 5.2.6. For every M ∈ GrA there is an exact sequence

0 → τ(M) → M → M → lim Ext1(A/A ,M) → 0, (5.2.2) f −→ ∗,≥n and for every i ≥ 0 there are isomorphisms Ri+1τ(M) ∼= Riωπ(M).

Proof. We apply the functor lim Hom(−,M) to (5.2.1). Then we obtain the desired −→ sequence by considering the long exact sequence of cohomology, using the previous Proposition and the fact that Exti(A, M) vanishes for i > 0. 2

With these results in mind, we can now introduce the χ condition and begin to adapt the necessary cohomological results from [1]. This is a rather technical condition to impose on an algebra, and we introduce it here for the following reason. We have the following diagram: cohA GrA π ω π cohprojA QGrA. Unlike GrA, cohA in most cases does not have enough injectives, and so the section functor ω : QGrA → GrA does not necessarily restrict to a section functor ω : cohprojA → cohA. As we have the inclusion of cohprojA in QGrA, as shown in

44 Theorem 5.1.15, we can still consider the saturation Mf of a coherent module M by defining Mf := ωιπM. However, this saturation may not be coherent. This is where we introduce the condition χ, as we will be able to show that this condition is equivalent to the tails of the saturation of a coherent module being coherent in high degrees. Definition 5.2.7. Let A be an I-algebra, M ∈ cohA and i ≥ 0. Then we say the condition χ (M) holds for M if for all j ≤ i, lim Extj(A/A ,M) is upper i −→ ∗,≥n bounded. If χi(M) holds for all M ∈ cohA, then we say χi holds for A, and if χi holds for all i ∈ I, then we say A satisfies χ. As mentioned before, our goal now is to prove that that a coherent I-algebra A satisfies χ1 if and only if for some d ∈ I the saturation Mf≥d is coherent for every M ∈ cohA.

Proposition 5.2.8. Let A be a coherent I-algebra satisfying χ0 and M ∈ cohA. ∼ Then χ1(M) holds if and only if M≥d = Mf≥d for some d ∈ I. Proof. Taking the tails of the sequence (5.2.2) gives the exact sequence

M → M → lim Ext1(A/A ,M) → 0. (5.2.3) ≥d f≥d −→ ∗,≥n ≥d Now if χ (M) holds then lim Ext1(A/A ,M) is upper bounded, say by u. Then 1 −→ ∗,≥n ≥d ∼ ∼ M≥d = Mf≥d for any d > u. Conversely, if there is a d ∈ I such that M≥d = Mf≥d, then lim Ext1(A/A ,M) must be 0. Thus χ (M) is satisﬁed. 2 −→ ∗,≥n ≥d 1

This shows that if χ1 holds, the tails of the saturation of a coherent module M are coherent in high degrees, as the tails M≥n are coherent by Lemma 5.1.10. Remark 5.2.9. We believe, as is hinted at in [12], that all coherent algebras satisfy

χ0, however we have not been able to show this. 5.3 Determining Equivalence with cohproj of a Coherent I-algebra We now want to be able to determine when an abelian category A is equivalent to the quotient cohprojA for some I-algebra A. Recall that when we wanted to determine equivalence with ModR earlier we looked for an object in our category that behaved similarly to the module RR. We can no longer do the same, as the bigrading on the indexed algebra A means that is not a graded A-module. However, we do have the modules eiA for each i ∈ I, so instead we turn our attention to these modules. We now let A be an abelian k-linear category, with a sequence of objects E =

(Ei)i∈I indexed by I. Such a sequence deﬁnes a connected, positively indexed I- L algebra A(E) = i≤j Aij where A(E)ij = Hom(Ej,Ei) if i < j, and A(E)ii = k. We will simply call this algebra A when E is clear. We also have a functor Γ∗ : A → L GrA(E) which sends X ∈ A to the graded right A(E)-module i∈I HomA(Ei,X).

45 Note that this is in fact a graded right A(E) module where the scalar multiplication is deﬁned by composition, as if m ∈ Γ∗(X)i and a ∈ A(E)ij then

ma = m ◦ a : Ej → Ei → X ∈ Γ∗(X)j.

It is also useful to consider the truncated versions of this functor Γ≥m = Γ∗(−)≥m, for m ∈ I. In other words, these are the tails of the module Γ∗(X). We denote the full subcategory of objects X such that there is a d ∈ I for which Hom(Ei,X) = 0 for i ≥ d as A0. Note that these are just the objects X such that Γ∗(X) is upper bounded. We now introduce the conditions on an I-indexed sequence which correspond to the conditions we used to deﬁne a progenerator.

Deﬁnition 5.3.1. Let E = (Ei) be an I-indexed sequence as above. Then • E is projective if for every surjection X → Y in A, there is an n ∈ I such that Hom(Ei,X) → Hom(Ei,Y ) is surjective for i ≥ n,

• A projective sequence E is coherent if the modules Γ≥m(X) are ﬁnitely generated for every X ∈ A and every m ∈ I,

• A coherent sequence E is ample if A0 = 0. It is clear that the projective and coherent conditions for an I-indexed sequence are analogous to the projective and ﬁnite generation conditions for an A-module. To see that the ampleness condition corresponds to the generator condition for modules we have the following results. Ls Lemma 5.3.2. Let E = (Ei) be an I-indexed sequence in A, and f : j=1 eij A → Γ≥m(X) be a morphism in GrA(E) with all ij ≥ m. Then f is the image of some Ls morphism g : j=1 Eij → X. ∼ Proof. As A(E) is positively indexed and ij ≥ m, we see that Γ≥m(Eij ) = Γ∗(Eij ) = eij A. Then, as ij ≥ m, we have that ∼ ∼ HomGr(Γ≥m(Eij ), Γ≥m(X)) = HomGr(eij A, Γ≥m(X)) = Γ≥m(X)ij = HomA(Eij ,X).

Proposition 5.3.3. A coherent sequence E = (Ei) is ample if and only if for every X ∈ A and every m ∈ I there are some i1, ..., is ∈ I, with all ij ≥ m, such that Ls there exists a surjection j=1 Eij → X.

Proof. First, assume E is ample. By the deﬁnition of coherence all of the Γ≥m Ls are ﬁnitely generated, thus we have a surjection j=1 eij A → Γ≥m(X) with all the Ls ij ≥ m. As eij A = Γ≥m(Eij ), Lemma 5.3.2 says there is a morphism j=1 Eij → X. As the image of this morphism is a surjection its cokernel must be contained in A0, as E projective. But ampleness says A0 = 0, so the cokernel is 0.

46 Ls Conversely, if there is a surjection f : j=1 Eij → X as above, then projectivity says Γ≥mf is surjective for m large enough, in which case it is immediate that 0 A0 = 0. Here ‘m large enough’ means that there is some index m such that Γ≥m is surjective for all m ≥ m0. 2

This reformulation is also the reason why we deﬁne an ample sequence to be coherent, and a coherent sequence to be projective. So just as before when Morita’s theorem showed us that an abelian category with a progenerator P is equivalent to the category of modules over the endomorphism ring of P , we are now ready to show that an abelian category with an ample sequence of objects E is equivalent the quotient category cohprojA(E). Before we prove this we have one more result, which will show from where the coherent condition on an ample sequence derives its name.

Proposition 5.3.4. If E = (Ei) is a coherent sequence then A is a coherent I- algebra, and the modules Γ≥m(X) are coherent for every X ∈ A and every m ∈ I.

Proof. Firstly, we note that by deﬁnition all of the Γ≥m(X) are ﬁnitely generated. Ls Let f : j=1 eij A → Γ≥m(X) be any homomorphism. We can clearly take all of Ls the ij ≥ m. Now by Lemma 5.3.2, f is the image of a map g : j=1 Eij → X in A. Then if n ≥ ij for every j = 1, . . . s, we have

(ker f)≥n = ker(Γ≥ng) = Γ≥n(ker g), so ker f is ﬁnitely generated. This shows that Γ≥m(X) is in fact coherent. Finally, we know that the modules eiA = Γ≥i(Ei) and eiA≥n = Γ≥n(Ei) are coherent, as E is coherent. Then we have the exact sequence

0 → eiA≥n → eiA → eiA/eiA≥n → 0, which shows eiA/eiA≥n is coherent, as the ﬁrst two terms are. Thus A is a coherent I-algebra. 2

With these results we can now prove our main theorem, generalising Theorem 2.4 of [12]. Theorem 5.3.5. Let k be a field, I be a directed set and A an abelian k-linear category. If A has an ample sequence of objects E = (Ei)i∈I such that A(E) satisfies χ0, then A' cohprojA(E) and A(E) satisfies χ1. Conversely, if A is a coherent I-algebra satisfying χ1, then the sequence (eiA)i∈I is ample in cohprojA. Proof. Our proof follows the general structure of the proof of Theorem 4.5 in [1]. We break the proof into a number of steps. First, we will suppose that A has an ample sequence E = (Ei)i∈I . For convenience we will refer to A(E) simply as A. Let Γ and its truncations be defined as above.

(Step 1). πΓ is exact.

47 Proof of Step 1. The functor Γ is left exact, as it is deﬁned by Hom functors, and π is exact, so πΓ is left exact. Then projectivity of the sequence E means that Γ preserves surjections in suﬃciently large degree. Hence the cokernel of Γ(f) is upper bounded for any surjection f in A. Thus π(coker Γ(f)) = 0, so πΓ is exact.

(Step 2). If Y is a subobject of any X ∈ A, then Γ(X)/Γ(Y ) is torsion-free. In particular Γ(X) is torsion-free.

Proof of Step 2. Take m to be a coset representative for an element of Γ(X)/Γ(Y ).

Then the element represented by m is torsion in the quotient if mA≥n ⊆ Γ(Y ) for n large enough. Let m ∈ Γ(X)d = Hom(Ed,X) be such that mA≥n ⊆ Γ(Y ) for some n. We want to show that in fact m ∈ Γ(Y ). As E is ample, Proposition 5.3.3 Ls says we can choose a surjection f : j=1 Eij → Ed such that ij ≥ d and ij ≥ n for j = 1, . . . , s. The map f is a sum of maps fj : Eij → Ed, so the map m ◦ f is the sum of maps m ◦ fj : Eij → X. Hence m ◦ fj is an element of Γ(X)ij . But ij ≥ n, so all of the maps m ◦ fj are elements of Γ(Y ), which means m ◦ f is too. As f is a surjection we see that m is a map from Ed → Y , thus is an element of Γ(Y )d. (Step 3). πΓ is faithful.

Proof of Step 3. As we have already shown that πΓ is exact, it suﬃces to show that Ls πΓ(X) 6= 0 if X 6= 0. As E is ample, there is a non-zero morphism j=1 Eij → X, so Γ(X) 6= 0. By step 2, Γ(X) has no torsion submodules, hence πΓ(X) 6= 0.

(Step 4). πΓ is essentially surjective.

Proof of Step 4. Let M ∈ cohA. Then we can take a ﬁnite presentation

s r M M eij A → eik A → M → 0 j=1 k=1

Ls Lr of M. Let X be the cokernel of the corresponding morphism j=1 Eij → k=1 Eik ∼ in A. Then, as πΓEi = πeiA for any i ∈ I, we see that πΓ(X) = πM, so πΓ is essentially surjective.

(Step 5). Let X ∈ A. If Γ(X) ,→ M and M torsion-free, then M/Γ(X) is torsion- free. (Corollary). Γ(X) ∼= Γ(]X).

48 Proof of Step 5. Let m ∈ Md, and suppose mA≥n ⊆ Γ(X) for n large enough. We will show m ∈ Γ(X)d = Hom(Ed,X). We can view m as a map edA → M. Then, as E is ample, we can take an exact sequence of the form

r s M M Eil → Eij → Ed → 0 (5.3.1) l=1 j=1

Lr Ls in A. We call these maps f = l=1 fjl and g = j=1 gj, where the fjl ∈ Aij il and gj ∈ Adij . We can choose these maps such that ij ≥ d and ij ≥ n for all j, and P such that edA/ gjA is bounded. We achieve this second condition by choosing the {gj} to contain the generators for some A∗,≥n, which is ﬁnitely generated as A(E) is a coherent algebra. We can see that mgj ∈ Γ(X) for all j, so mg deﬁnes a Ls map ϕ : j=1 Eij → X such that ϕf = 0. Applying Hom(−,X) to (5.3.1) gives us the exact sequence

s r M M 0 → Hom(Ed,X) → Hom( Eij ,X) → Hom( Eil ,X), j=1 l=1

0 0 which implies the existence of a map m : Ed → X such that m g = ϕ = mg. Thus 0 0 we can view m as an element of Γ(X)d. As m gj = mgj for every j, the fact that P P 0 edA/ gjA is bounded shows that A∗,≥n ⊆ gjA, which in turn shows that m −m 0 is annihilated by A∗,≥n. But M was torsion-free, so we see that m = m ∈ Γ(X). Proof of Corollary. To prove the corollary, we ﬁrst note that by Step 2, Γ(X) is torsion-free. So (5.2.2) becomes 0 → Γ(X) → Γ(]X) → R1τ(Γ(X)) → 0. But R1τ(Γ(X)) is torsion, so Step 5 says it must be 0.

(Step 6). πΓ is full, which completes the proof that it is an equivalence.

Proof of Step 6. We need to show that the canonical map

Hom(X,Y ) → Hom(πΓ(X), πΓ(Y )) is a surjection for any X,Y ∈ A. We already know it is injective, so we show it is surjective by showing that is an isomorphism. By the adjoint isomorphism and the Corollary of Step 5,

Hom(πΓ(X), πΓ(Y )) ∼= Hom(Γ(X), Γ(]Y )) ∼= Hom(Γ(X), Γ(Y )).

∼ Then Hom(Ei,Y ) = Γ(Y )i = Hom(eiA, Γ(Y )), hence the result holds when X = Ei. It must then also hold when X is any ﬁnite direct sum of the Ei. By the ampleness of E, every X ∈ A has a resolution by these ﬁnite sums, so we apply Hom(−,Y ) to this resolution and the result follows from left exactness of Hom(−,Y ).

(Step 7). A satisﬁes χ1.

49 Proof of Step 7. A satisﬁes χ0 by assumption. By essential surjectivity, for any M ∈ cohA there is an X ∈ A such that πΓ(X) ∼= πM. Then applying ω and using Step 5 shows that Mf ∼= Γ(]X) ∼= Γ(X). By Lemma 5.1.10 and Proposition 5.3.4 the modules M≥d and Γ≥dX are coherent for all d ∈ I, so by considering the exact sequence M → Γ(X) → lim Ext1(A/A ,M) → 0 ≥d ≥d −→ ∗,≥n ≥d we see that lim Ext1(A/A ,M) must be coherent. It is also torsion, hence is −→ ∗,≥n ≥d upper bounded. Thus χ1(M) is satisﬁed, and M was arbitrary so χ1 holds for A.

(Step 8). If A coherent and satisﬁes χ1, then (eiA)i∈I is an ample sequence in cohprojA.

Proof of Step 8. As A is coherent, all of the modules eiA are coherent. Hence if (eiA)i∈I is a projective sequence, then it is coherent too. Furthermore, the deﬁnition of coherence for a module, combined with Proposition 5.3.4 shows that this sequence is also ample. So it remains to show that (eiA)i∈I is projective. Let f : πM → πN be a surjection in cohprojA. Then we can represent this as a surjective homomorphism f 0 : M 0 → N 0 for some M 0,N 0 ∈ cohA, such that ∼ 0 ∼ 0 0 M≥d = M≥d and N≥d0 = N≥d0 for some d, d ∈ I. Then for i large enough, 0 ∼ 0 ∼ Homcoh(eiA, M ) = Homcoh(eiA, M) and Homcoh(eiA, N ) = Homcoh(eiA, N). The 0 eiA are projective objects, so f : Homcoh(eiA, M) → Homcoh(eiA, N) is a surjection. As χ1 holds, Proposition 5.2.8 and the sequence (5.2.3) tell us that for m ∼ ∼ large enough M≥m = Mf≥m and N≥m = Ne≥m. Note that we can choose a single 0 index for both M and N as I is directed. Thus we have that f : Homcoh(eiA, Mf) → Homcoh(eiA, Ne) is a surjection for i large enough. Finally, we use the adjunction isomorphism to translate this to a surjection Homcohproj(πeiA, πM) → Homcohproj(πeiA, πN), showing that (eiA)i∈I is a projective sequence. 2

So we now have an answer to the question ‘how can we understand an abelian category?’ We look for an ample sequence of objects, and if we ﬁnd one then we can conclude our abelian category is the quotient cohprojA of some indexed algebra A. As a closing note, we appeal to the ﬁeld of algebraic geometry for an example of how our result can be applied. Example 5.3.6. We will now present the original example which led to the introduction of quotient categories: the category of coherent sheaves coh P1 on the projective line. The example holds more generally for any projective scheme, however we stick with the case of P1 for simplicity. A sheaf of modules on P1 is a geometric object, however we have an algebraic interpretation, courtesy of Serre. We can think of a sheaf of modules on P1 as the sheaf associated to a Z-graded A = k[x, y]-module via the Proj construction, found on page 116 of [8]. Here A has

50 the usual grading given by degree. We then define an equivalence relation ∼ on graded A-modules, where ∼ M ∼ N if M≥n = N≥n for some n ∈ Z. Then we say a sheaf of modules is coherent if the corresponding graded A-module is equivalent under ∼ to a finitely generated module, and two graded A-modules define isomorphic sheaves if they are equivalent under ∼. This algebraic interpretation was first introduced by Serre in [14], and was later realised to be a quotient construction in [7]. In fact, the quotient that coh P1 is equivalent to is something we have already seen: cohproj A! Theorem 2.4 of [12], which our main result is a generalisation of, says that if an abelian category A has an ample Z-indexed sequence of objects E, then

A' cohproj A(E).

The category coh P1 has an ample sequence E = (O(n)), where O(n) is the sheaf corresponding to the shifted algebra A[n]. Then

M 1 ∼ M A(E) = Homcoh(O(−j), O(−i)) = HomGr(A[−j],A[−i]) = A, i,j∈Z i,j∈Z

1 as HomGr(A[−j],A[−i]) = Aj−i. So we see coh P ' cohproj A. Calling back to Example 5.1.2, we see that if Aq = kq[x, y] then cohproj Aq has an ample Z-indexed ∼ sequence (eiAq). Recall that Aq = A, so we see that

1 cohproj Aq ' cohproj A ' coh P .

1 In this way we can view Aq as a ‘noncommutative P ’, in the sense that the quotient is equivalent to the category of coherent sheaves on P1. Now we give a similar example where our more general result is of use. We consider the category coh(P1 × P1) of coherent sheaves on the biprojective space P1 × P1. We want to determine if this category is a quotient of the category of coherent modules over an indexed algebra. As was the case with P1, the there is a natural ample sequence in coh(P1 × P1) given by the sheaves corresponding to shifts of the coordinate algebra of P1 × P1. However, in this case the coordinate 2 algebra is the Z -graded algebra B = k[x1, y1, x2, y2], where deg x1 = deg y1 = (1, 0) ∼ and deg x2 = deg y2 = (0, 1). That is B = A ⊗k A. So in this way we see that the ample sequence in coh(P1 × P1) is the Z2-indexed sequence (O(m, n)). Thus Theorem 5.3.5 can be applied to show that

1 1 coh(P × P ) ' cohproj B.

1 The convention from algebraic geometry is to consider Homcoh(O(−j), O(−i)) rather than Homcoh(O(j), O(i)), as was our convention. This change only aﬀects the indexing on the algebra, as it would now be negatively indexed.

51 References

[1] M Artin and JJ Zhang. Non-commutative projective schemes. Adv. Math, 109:228–287, 1994. [2] Aleksei Igorevich Bondal and Michel Van den Bergh. Generators and repre- sentability of functors in commutative and noncommutative geometry. Moscow Mathematical Journal, 3(1):1–36, 2003. [3] Daniel Chan. Twisted multi-homogeneous coordinate rings. Journal of Algebra, 223(2):438–456, 2000. [4] Harm Derksen and Jerzy Weyman. An introduction to quiver representations, volume 184. American Mathematical Soc., 2017. [5] David Eisenbud. Commutative Algebra: with a view toward algebraic geometry, volume 150. Springer Science & Business Media, 2013. [6] Peter J Freyd. Abelian categories, volume 1964. Harper & Row New York, 1964. [7] Pierre Gabriel. Des catégoriesabéliennes. Bulletin de la SociétéMathématique de France, 90:323–448, 1962. [8] Robin Hartshorne. Algebraic geometry, volume 52. Springer Science & Business Media, 2013. [9] I Mori and A Nyman. Local duality for connected z-algebras. arXiv preprint arXiv:1910.06301, 2019. [10] Zachary Murphy. Quotient categories and Grothendieck’s splitting theorem. Bachelor’s thesis, University of New South Wales, 2017. [11] Adam Nyman. An abstract characterization of noncommutative projective lines. arXiv preprint arXiv:1704.04544, 2017. [12] A Polishchuk. Noncommutative proj and coherent algebras. arXiv preprint math/0212182, 2002. [13] Emily Riehl. Category theory in context. Courier Dover Publications, 2017. [14] Jean-Pierre Serre. Faisceaux algébriquescohérents. Annals of Mathematics, pages 197–278, 1955. [15] S Paul Smith. Noncommutative algebraic geometry, 2000. [16] Mitsuhiro Takeuchi. A simple proof of gabriel and popesco’s theorem. J. Algebra, 18(1):112–113, 1971.