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 check- ing). 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: i 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. ii Abstract A classical theorem of Morita theory states that if an abelian category has a pro- generator 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 cate- gories 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 iv 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 cate- gory 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 2 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 2 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.