QUOTIENT CATEGORIES AND GROTHENDIECK'S SPLITTING THEOREM Zachary Murphy Supervisor: Associate Professor Daniel Chan School of Mathematics and Statistics UNSW Sydney October 2017 Submitted in partial fulfillment of the requirements of the degree of Bachelor of Science (Advanced Mathematics) 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 When I finished high school and decided to study pure mathematics at university, I thought I knew a lot of maths. Over the four years of my degree, I have gradually come to the realisation that I know very little, and nothing has made that more apparent than researching this project under my supervisor, Associate Professor Daniel Chan. I have learnt so much maths from him this year that I never knew existed, and for that he has my most sincere and profound thanks. His unwavering support, patience and enthusiasm made this honours year a joy. I would also like to thank my high school maths teacher Bruce Smith, who was the first person to pique my interest in mathematics and without whom I would never have considered studying it at the tertiary level. I would also like to thank my friends for their friendship, support and banter throughout this trying year. Finally, sincere thanks goes to my family, in particular my parents, who have put up with me for the longest. Without their support, encouragement and proof-reading, I would never have made it this far. Zachary Murphy, 27 October 2017. ii Abstract At first glance, quotient categories might seem like a strange notion to define, but in actuality they turn out to be quite useful constructions. They arise naturally in geometry as a way of studying spaces which may not be distinguishable using more elementary algebraic methods. Specifically, quotient categories provide a way to construct the category of quasi-coherent sheaves on a space. This category encodes much of the geometry of the case, and in particular contains as a subcategory the category of vector bundles, which have been important objects of study in algebraic geometry since the start of the 20th century. In 1957, Grothendieck proved an important structure theorem for vector bundles on the projective line. This thesis is intended firstly as an expository work on the theory of quotient categories and how they interact with geometry, and secondly we offer an alternative approach to Grothendieck's Splitting Theorem. iii Contents Chapter 1 Introduction 1 1.1 Preface . .1 1.2 Introduction . .1 1.3 Structure of Thesis . .4 1.4 Assumed Knowledge . .5 Chapter 2 Abelian Categories 6 2.1 Additive Categories . .6 2.2 Abelian Categories . .9 Chapter 3 The Quotient of an Abelian Category 16 3.1 Localisation . 16 3.2 Quotient Categories . 20 3.3 Localising Subcategories . 28 Chapter 4 Homological Algebra 31 4.1 Chain Complexes and Homology . 31 4.2 The Hom and Tensor Functors . 32 4.3 Ext and Tor Functors . 34 4.4 Hilbert's Syzygies Theorem . 40 Chapter 5 Grothendieck's Splitting Theorem 43 5.1 The Category of Graded k[x; y]-modules . 43 5.2 Torsion in Graded Modules . 45 5.3 The Splitting Theorem . 48 Chapter 6 Applications to Geometry 56 6.1 The Projective Line . 56 6.2 Quasi-Coherent Sheaves . 57 6.3 Vector Bundles . 63 v Chapter 7 Conclusion 67 7.1 Summary . 67 7.2 Generalisations and Further Work . 67 References 69 vi Chapter 1 Introduction 1.1 Preface Category theory is often referred to (endearingly or otherwise) as \abstract non- sense", and the theory of quotient categories is unsurprisingly, no exception. How- ever, the main goal of this thesis is to motivate them and to give the reader a sense of how these are actually natural and useful constructions. Category theory is an essential tool in studying geometry, so while this is first and foremost an exposition of the theory of quotient categories, the motivation comes from the geometry of both affine and projective space, and as such some parts of this thesis have a rather geometric flavour. This thesis attempts to motivate the theory of quotient categories by presenting a proof of Grothendieck's famous Splitting Theorem for vector bundles. This is a theorem which is purely geometric in statement and nature, but we attempt to reinterpret it and consequently prove it in a purely algebraic way. The pieces which fit together to form this proof draw from aspects of category theory of course, but also from homological algebra. As far as we can tell, the proof of the Splitting Theorem presented in this essay has not appeared in print before. However it is something that experts in the field are most likely aware of|a \folklore result", so to speak. As such, much of the information I have included in this thesis, particularly in the final two chapters, I have gleaned from conversations with my supervisor, Associate Professor Daniel Chan, which is why the reference list may not be as long as one might expect. 1.2 Introduction Vector bundles are very important objects of study in geometry, because they turn out to capture quite a lot of the geometry and invariants of certain algebraic vari- eties and manifolds. In some ways the correspondence between vector bundles and 1 varieties or manifolds can be likened to that between modules and rings, and this analogy becomes even clearer when we expand the category of vector bundles to the category of quasi-coherent sheaves. Rougly speaking, a vector bundle on a space is a way of assigning a vector space to each point, in a consistent manner. Two basic examples of 1-dimensional vector bundles on the circle S1 are the cylinder and the M¨obiusband. If the base space has some extra structure, the assignation of the vector spaces should of course be compatible with that structure: for example, for complex manifolds, we would like the function to be holomorphic. In 1957, Alexander Grothendieck [7], proved that any holomorphic vector bundle on the complex projective line is isomorphic to a direct sum of 1-dimensional holo- morphic vector bundles, or line bundles. Hazewinkel and Martin [8] then proved in 1982 that the same result holds for algebraic vector bundles on the projective line over any field k. The statement of this theorem is thus: If E is an algebraic vector 1 bundle on Pk, then ∼ E = O(n1) ⊕ · · · ⊕ O(nr); where O(n) is the nth twisted line bundle. In the case of the real projective line, 1 is homeommorphic to S1, and it can PR be shown that the cylinder and the M¨obiusband are the only two examples of line bundles (up to vector bundle isomorphism), so in this case the classification is rather simple1. Classically, Grothendieck's Splitting Theorem is proved geometrically, using coho- mology of line bundles. Our approach in this paper is to reinterpret the statement of the theorem using quotient categories. Via a canonical identification, vector bundles on a space V can be considered ex- amples of quasi-coherent sheaves on V . These are algebraic objects which form an abelian category. This category, denoted QCoh(V ), encodes all the geometric information about the space X, and so in Grothendieck's philosophy, is considered the quintessential object of study in algebraic geometry. The reason we embed the vector bundles into this larger category is precisely so that we obtain the properties of an abelian category. This is the incentive for developing quotient categories, since this construction gives a way of determining the category QCoh(V ) for a certain class of varieties V . Quotients of abelian categories were first treated in detail by Gabriel in his famous masters thesis Des Cat´egoriesAb´elienne [6]. This notion of quotient achieves the 1Of course, simple in the sense that there are few examples of vector bundles up to isomorphism. The proof is still non-trivial. 2 same goal as quotients in other areas of mathematics|it makes the category in question \smaller" in some way. The way it achieves this however, seems counter productive in the sense that the resulting category has more things in it. What happens is that instead of identifying objects in the way that one identifies elements in a quotient group, isomorphisms are added to the category, which has the ultimate effect of reducing the number of isomorphism classes, even though the class of objects remains the same. This is consistent with the usual philosophy of category theory that things should only be considered unique up to isomorphism. 1 As it turns out, the category QCoh(Pk) can be obtained by taking the category of graded modules over a polynomial ring in two variables and quotienting out by the subcategory of m = (x; y)-torsion modules. In this case, torsion free noetherian objects satisfy a structure theorem: ∼ M = R(n1) ⊕ · · · ⊕ R(nr); (y) where R(n) denotes the nth degree shift of R. This version of Grothendieck's Splitting Theorem is of algebraic importance as well|as a generalisation of the usual structure theorem for finitely generated mod- ules over a principal ideal domain (PID).