Quotient Categories and Grothendieck's Splitting

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

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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 varieties 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 holomorphic 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 cohomology 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 examples 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 CatégoriesAbélienne [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), (†) 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 modules over a principal ideal domain (PID). The polynomial ring R = k[x, y] is of course not principal, as (x, y) is an ideal which cannot be generated by a single element, but it is still a noetherian domain, which in some senses is the next best thing. We recall this structure theorem for PIDs: Let R be a PID. Then for any ﬁnitely generated R-module M, there exist positive integers r, s and elements fi ∈ R − 0 for i = 1, . . . , r such that

∼ s M = R/(f1) ⊕ · · · ⊕ R/(fr) ⊕ R .

In fact, if R is a graded ring and M is a graded module, we can take fi to be homogeneous (we may also have to shift R by some degrees, but it turns out that for a PID at least, all the shifts of R are isomorphic in the quotient). Then, heuristically, quotienting the category of graded R-modules by the subcategory of

(x)-torsion modules, all the terms R/(fi) would be expected to become 0. In this sense M ∼= Rs, which is formally similar to the statement of (†), so in this sense we may also consider the Splitting Theorem as a generalisation of this structure theorem.

3 1.3 Structure of Thesis

Since the category QCoh(V ) is always an abelian category, we spend a fair bit of time in Chapter 2 setting up this theory in the general setting. The set up of this theory follows that in Chapter 2 of [13]. Some of this terminology is slightly outdated—the current standard is to define additive categories using the notion of enrichment, which while perhaps a more succinct way of defining the notion, it is less intuitive than the definition given here, hence the decision to take the older approach.

Often it is possible to capture the geometry of a space or object V by using the so called coordinate ring2, denoted R = k[V ]. In fact it is suﬃcient to study only the category R -Mod of modules over R, since in suﬃciently well-behaved situations, this is precisely the category QCoh(V ). Unfortunately, in some cases, the coordinate ring may not be available, but often in these cases, the category QCoh(V ) is still available. This is a result of the quotient construction, which will be explored in Chapter 3. Of course it is no longer a category of modules, but it is still abelian, and as such still quite well-behaved. In this chapter we also introduce an important class of quotient categories which correspond to the idea of localisation in geometry.

Chapters 4 and 5 provide a proof of Grothendieck’s Splitting Theorem in the algebraic context. More specifically, chapter 3 introduces some abstract tools from both Homological and Commutative Algebra which are necessary for the proof. This chapter is to be considered a brief summary of relevant results and it should be emphasised that there is no attempt to be compreshensive, with some proofs only sketched and some omitted entirely. Chapter 4 then begins with an introduction to the specific category in which the algebraic formulation of the Splitting Theorem holds—that of graded modules over the polynomial ring in two variables. We then prove a version of the Auslander-Buchsbaum Theorem, which is the key to the final proof, which then ends the chapter.

Finally we come to how all this algebra and category theory actually applies to the geometry of the projective line. In chapter 6 we give a brief introduction to 1 the theory of vector bundles and to quasi-coherent sheaves on Pk, and explain how they are related. The deﬁnitions given here are not the usual ones—they implicitly 1 depend on a particular choice of coordinate chart for Pk. Algebraic geometers are generally interested in a coordinate free approach to treatments such as this, and work has been done to prove that certain ideas are independent of the choice of

2This notation is reminiscent of the notation for a polynomial ring. This is because in the basic case of aﬃne space, this is precisely the coordinate ring. However in general the ring is more complicated.

4 coordinate chart. We ignore this subtle point for the sake of brevity, and instead deﬁne things in terms of the usual coordinate chart. In this chapter we discuss how vector bundles can be thought of as quasi-coherent sheaves in a canonical 1 way, and provide an explicit proof of the equivalence between QCoh(Pk) and the quotient category of chapter 5. Thus via this identiﬁcation of vector bundles as quasi-coherent sheaves, and this equivalence of categories, Grothendieck’s original Splitting Theorem is proved.

Chapter 7 then provides a summary and attempts to give the reader a sense of how generalisable this theorem is, together with some areas of potential further study.

1.4 Assumed Knowledge

The reader is assumed to know basic deﬁnitions and results from module theory, in particular free modules, the noetherian condition, and tensor products.

A familiarity with basic category theory is also assumed. In particular, the deﬁ- nitions and basic results about categories, functors, products, coproducts, kernels, cokernels, adjoints, category equivalences are assumed. Details of these can be found in [2] and [10]. A version of the Yoneda Lemma for abelian categories is also used without proof. A proof of the usual Yoneda Lemma can be found in any intro- ductory category theory textbook, and the abelian version is an easy generalisation.

5 Chapter 2

Abelian Categories

It turns out that the correct categorical framework to discuss the geometry of certain topological spaces is in the theory of abelian categories. This is because according to Grothendieck, the geometry of a space is encoded in a certain category associated with the space, which in general is abelian. In this chapter we build the theory of abelian categories in a piecemeal fashion, starting with pre-additive categories. A good reference for the theory of abelian categories is [13]. The best way to think of abelian categories is as an abstraction of the category of abelian groups and group homomorphisms, Ab.

2.1 Additive Categories

Let A, B, C be abelian groups and let f, f 0 : A → B and g, g0 : B → C. Using f and f 0, we can of course deﬁne a new map, f + f 0 : A → B, by setting (f + f 0)(a) = 0 f(a) + f (a) for all a ∈ A. This operation gives a group structure on HomA(A, B), with inverses given by (−f)(a) = −f(a) and an identity in the zero map, 0(a) = 0. Of course, the commutativity of B makes this group structure abelian.

Furthermore, note that we have g(f +f 0)(a) = g(f(a)+f 0(a)) = g(f(a))+g(f 0(a)) = gf(a) + gf 0(a), so g(f + f 0) = gf + gf 0, and similarly (g + g0)f = gf + g0f, so composition is bilinear with respect to this group operation. This is the ﬁrst property we wish to abstract. Deﬁnition 2.1.1. A category A is called pre-additive if for each A, B ∈ Obj(A),

HomA(A, B) has an abelian group operation with respect to which composition is bilinear.

Remark 2.1.2. The identity of the abelian group HomA(A, B) is denoted 0AB, or simply 0A, if B = A. This is not to be confused with the identity morphism

1A ∈ HomA(A, A).

6 Example 2.1.3. The category R -Mod of modules over a ﬁxed ring R is a pre- additive category, using the group structure induced by the underlying abelian groups of the objects. Example 2.1.4. The category Grp of all groups and group homomorphisms is not pre-additive, since while the Hom-sets have a natural group operation, it is only abelian when the codomain group is abelian. Example 2.1.5. The category Ring of rings and ring homomorphisms is not pread- ditive: since ring homomorphisms are required to send 1 to 1, so there is no zero morphism.

Given two abelian groups A, B, we can form their direct sum

A ⊕ B := {(a, b): a ∈ A, b ∈ B}, with addition given component-wise. This sum comes with canonical injection and projection maps, and satisﬁes both the universal property of a product and also of a coproduct. Of course, it is not always the case that products and coproducts coincide: indeed the coproduct of two groups in Grp is given by the free product, while the product is given by the direct product. However, in a pre-additive category, if a product exists then it is also a coproduct, and vice versa. Proposition 2.1.6. Let A and B be two objects in a pre-additive category A and suppose their product (resp. coproduct) exists. Then it is also a coproduct (resp. product).

Proof. Let A × B denote the product of A and B with projection maps πA and πB.

Let ιA = (1A, 0AB): A → A × B, where 0AB : A → B is the identity in the abelian group HomA(A, B). Similarly, let ιB = (0BA, 1B): B → A × B. Note that

πAιA = 1A and πAιB = 0BA, (∗) and similarly for πB, and so since πA(ιAπA +ιBπB) = πA and πB(ιAπA +ιBπB) = πB, it must be the case that ιAπA + ιBπB = (πA, πB) = 1A×B. Now suppose we have f : A → C and g : B → C. To show that A × B is a coproduct, we must

ﬁnd a unique map [f, g]: A × B → C such that f = [f, g]ιA and g = [f, g]ιB.

Take [f, g] = fπA + gπB. This satisﬁes the conditions, by (∗). To see uniqueness, suppose we have some other k : A × B → C such that f = kιA and g = kιB.

Then [f, g] = [kιA, kιB] = (kιA)πA + (kιB)πB = k. To show that any coproduct is necessarily a product is similar, by duality. 2

7 In general when we have a product A × B which is isomorphic to the coproduct A+B, we denote this common object A⊕B and call it a biproduct 1. The projections and injections of a biproduct necessarily satisfy (in §2 of Chapof [10]) the equations

πAιA = 1A, πAιB = 0BB, πBιB = 1B, πBιA = 0AA, ιAπA + ιBπB = 1A⊕B.

In fact, in §2 of Chapter VIII of [10] it is proved that these equations characterise a biproduct.

We also have a zero object in Ab, given by the identity group 0 = {0}. Indeed, given an abelian group A, we have precisely one map A → 0 which sends everything to the identity, and precisely one map 0 → A which simply injects the indentity into A, so 0 is indeed a zero object. In fact, in much the same way that biproducts are determined by a series of equalities, the zero object in an additive category is characterised by the identity 00 = 10. Indeed, since HomA(0, 0) has only one element, 0 satisﬁes this. On the other hand, if A is some object with 0A = 1A, then the composites 0 → A → 0 and A → 0 → A must be the identities, which shows that A → 0 is an isomorphism. Deﬁnition 2.1.7. A pre-additive category A is called additive if it possesses a zero object, and for any two objects A, B ∈ Obj A, the biproduct A ⊕ B exists. Example 2.1.8. The category R -Mod is also additive: the zero object is given by the zero module, and the direct sum of two modules M and N is given by M ⊕N := {(m, n): m ∈ M, n ∈ N}.

As with any new mathematical object, we should be interested in the structure preserving maps between them. In the case of pre-additive and additive categories, these structure preserving maps are the additive functors. Deﬁnition 2.1.9. Let A and B be two pre-additive categories and let T : A → B be a functor. We say that T is additive if for any objects X,Y ∈ Obj A, the induced map TXY : HomA(X,Y ) → HomB(TX,TY ) is a group homomorphism. It’s clear that this notion preserves the structure of a pre-additive category. What is slightly more surprising (althugh suggested by the name) is that an additive functor preserves all the structure of an additive category. Proposition 2.1.10. Let T : A → B be an additive functor and suppose that for objects A, B ∈ Obj A, the biproduct A ⊕ B exists. Then T (A ⊕ B) ∼= TA ⊕ TB, and T (0) = 0.

1This terminology is not standard and may also be referred to as a direct sum in reference to the direct sum of modules, which is the prototypical example.

8 Proof. The injections and projections πA, πB, ιA and ιB are characterised by the following ﬁve equations:

πAιA = 1A πAιB = 0BB πBιB = 1B πBιA = 0AA ιAπA + ιBπB = 1A⊕B, and each of these equations are preserved by T , making T (A ⊕ B) into a biproduct for TA and TB with projections T (πA), T (πB) and injections T (ιA) and T (ιB). Also, since the zero object of an abelian category is characterised by the property that the map 0 : 0 → 0 and id0 : 0 → 0, T preserves this, since any functor preserves identity maps, and since T is additive, T00 : HomA(0, 0) → HomA(0, 0) is a group homomorphism and so preserves 0. 2

Example 2.1.11. For any object A in an additive category A, the HomA functor

HomA(A, −): A → Ab is additive. Indeed, given f1, f2 : B → C, HomA(A, f1 +f2) maps g ∈ HomA(A, B) to (f1 +f2)g = f1g+f2g in HomA(A, C), and HomA(A, f1)+

HomA(A, f2) maps g to f1g +f2g. This is a result of the bilinearity of addition over composition in additive categories. Similarly HomA(−,B) is an additive functor.

2.2 Abelian Categories

Consider f : A → B in Ab. There are two abelian groups of interest which are associated to this homomorphism. The ﬁrst is the kernel,

ker f := {a ∈ A : f(a) = 0}.

f Take the composite ker f ,→ A −→ B. Clearly this is 0, and in fact if there is some g f other group A0 with the property that A0 −→ A −→ B is zero, then we can conclude that im g ≤ ker f and so we can restrict the codomain of g to get a map A0 → ker f. Of course this just means that ker f is the kernel of f in the categorical sense.

The other group of interest is the quotient B/ im f. In this case, the composite f f g A −→ B → B/ im f is zero, and if we have some other group B0 such that A −→ B −→ B0 is zero, then this tells us that im f ≤ ker g and so g induces a homomorphism B/ im f → B0. Again, this just tells us that B/ im f is the cokernel of f. In other words, since f was arbitrary, Ab has all kernels and cokernels. Deﬁnition 2.2.1. An additive category A is said to be pre-abelian if every morphism f : A → B in A has a kernel and a cokernel. Example 2.2.2. Again, the category R -Mod is pre-abelian, with the kernels and cokernels deﬁned in the same way as for Ab.

9 Example 2.2.3. Let R = C[x1, x2,... ] be the polynomial ring in inﬁnitely many indeterminates. Now let R -mod be the category of ﬁnitely generated R-modules.

Let f : R → R/(x1, x2,... ) be the canonical projection. This is a morphism in

R -mod, however kerf = (x1, x2,... ) which is not ﬁnitely generated and so is not in R -mod. Thus R -mod is not pre-abelian.

It should be noted that for a morphism f : A → B, the kernel of f is a pair (K, k), and the notation ker f is often used to refer to both the object K and also the arrow k. The distinction is usually clear from context. As an example, coker ker f denotes the object which is the cokernel of the morphism ker f.

Let f : A → B be a morphism in a pre-abelian category, and consider the following commutative diagram.

f 0 ker f A B coker f 0 u

coker ker f v ker coker f In this diagram, since the composite ker f → A → B is zero, the map u exists by the universal property of the cokernel. Now, we have u(coker ker f) = f, and (coker f)f = 0, so (coker f)u(coker ker f) = 0. Since coker ker f is an epimorphism, we conclude that (coker f)u = 0, and so the map v exists by the universal property of the kernel, so there is a canonical factorisation of f through v.

Consider this factorisation for a morphism f : A → B in Ab. In this case, coker ker f = A/ ker f, and ker coker f = im f, and v is defined by a+ker f 7→ f(a). Of course, by the First Isomorphism Theorem, this map v is an isomorphism. This is the final property of Ab that we wish to abstract, leading (finally) to the definition of an abelian category. Definition 2.2.4. A pre-abelian category A is abelian if for all objects A, B ∈ Obj A, and all morphisms f : A → B, the canonical morphism v : coker ker f → ker coker f is an isomorphism. Remark 2.2.5. As suggested by the previous paragraph, this is simply an abstraction of the First Isomorphism Theorem.

Once again, for any ring R, the category R -Mod is abelian. Surprisingly though, these are (essentially) the only examples of an abelian category, which is a result of the Mitchell-Freyd Embedding Theorem. Theorem 2.2.6. Let A be a small abelian category. Then there exists a ring R and a fully faithful functor F : A → R -Mod.

10 Proof. See Theorem 11.6 in Chapter 4 of [13]. 2

The upshot of this theorem is that objects of an abelian category can be thought of (to a certain extent) as modules over a ﬁxed ring R, and in fact as we will see, the relations which hold for modules also hold for objects in any abelian category. We also borrow the notation M 0 ≤ M from module theory to denote a subobject of M.

The existence of all kernels and cokernels seems like an innocuous assumption to place on a category, but it actually allows us to deﬁne many notions from module theory in the context of arbitrary abelian categories. Here we will ﬁx an abelian category A and a morphism f : M → N in A.

Firstly, we deﬁne the image of f as im f = ker coker f, and the coimage of f as coim f = coker ker f. In R -Mod, coker f = N/ im f and so this new notion of image coincides with the usual notion.

Let i : M 0 → M be a subobject of M. Then we may deﬁne f(M 0) as the image of composite of f with the injection. That is, f(M 0) := im fi.

Furthermore we can define the quotient M/M 0 := coker i, and this is indeed a quotient object of M/M 0 in the usual sense. We can also define the pre-image of N 0 ≤ N. It is defined to be f −1(N 0) := ker pf, where p : N → N/N 0 is the canonical projection.

0 00 Given two subobjects of M, i1 : M → M and i2 : M → M, we can also deﬁne their intersection and sum. We deﬁne the intersection

M1 ∩ M2 := ker((p1, p2): M → M/M1 ⊕ M/M2), and the sum is

M1 + M2 := im([i1, i2]: M1 ⊕ M2 → M), where ij : Mj → M are the injections and pj = coker ij are the canonical projections pj : M → M/Mj. It’s not too hard to see that in R -Mod, these deﬁnitions all coincide with the usual notions of image, quotient, intersection and sum. Thus by Mitchell’s Embedding Theorem, these objects all have the same relations and theorems as they do in R -Mod, for any abelian category A (if A is large we simply look locally at the abelian category generated by a small set of objects, which is also small). We state the relevant lemmas here without proof. These can also be proven using purely categorical approaches.

11 Lemma 2.2.7. For any M 0 ≤ M and N 0 ≤ N, the following statements hold in any abelian category A:

1. (First Isomorphism Theorem) For f : M → N, we have

M/ ker f ∼= im f.

This is the isomorphism assumed in the deﬁnition of an abelian category.

2. (Second Isomorphism Theorem) The intersection M 0 ∩ M 00 ≤ M 00 and

M 0 + M 00 M 00 ∼= . M 0 M 0 ∩ M 00

3. (Third Isomorphism Theorem) If further M 0 ≤ M 00, then M 00/M 0 ≤ M/M 00 and M/M 0 ∼= M/M 00. M 00/M 0

4. (Subobjects of Quotients) If N ≤ M/M 0, then N ∼= M 00/M 0, for some M 00 such that M 0 ≤ M 00 ≤ M.

The following proposition is useful in abelian categories for determining the nature of a given morphism using information about its kernel and cokernel. Proposition 2.2.8. Let f : M → N be a morphism in an abelian category.

1. The map f is a monomorphism if and only if ker f = 0

2. The map f is a epimorphism if and only if coker f = 0

3. The map f is an isomorphism if and only if f is both a monomorphism and an epimorphism.

Proof. Since 1 and 2 are dual to each other, we only prove 1 and 3. 1. Suppose f is monomorphism, and suppose h : X → M is such that fh = 0. Then since f is monic, h = 0XM . Then the morphism 0X0 : X → 0 is such that 0XM = h = 00M 0X0, so 0 together with the morphism 00M : 0 → M satisﬁes the universal property of the kernel for f.

Now suppose that ker f = 0, and suppose fh1 = fh2 for h1, h2 : X → M. Then f(h1 − h2) = 0, so by the universal property of the kernel there exists a morphism p : X → 0 such that h1 −h2 = 00M p = 0XM . Thus h1 = h2, so f is a monomorphism

(Of course we necessarily have p = 0X0 here).

12 For 3, ﬁrst suppose f is an isomorphism. Then fh1 = fh2 implies h1 = h2 since we just compose on the left by f −1, so f is monic. Similarly f is epic.

Now suppose f is both monic and epic. Then ker f = 0 = coker f. We also have the canonical factorisation f = ιvπ, where π : M → coker ker f is the canonical projection, ι : ker coker f → N is the canonical injection. Now, since ker f = 0, coker ker f = coker(0 → M). It is quite easy to see that (M, idM ) satisﬁes the universal property for the cokernel in this case, and similarly, we have ker coker f = ker(N → 0) and again, (N, idN ) satisﬁes the universal property here. That is,

π = idM and ι = idN , so f = idB v idA = v, which is an isomorphism. 2

One other important notion we can deﬁne in an arbitrary abelian category is that of the short exact sequence, since we now have a notion of image, kernel and zero. Deﬁnition 2.2.9. Let A be an abelian category and consider a sequence

0 → M 0 → M → M 00 → 0.

This sequence is said the be exact if M 0 → M is a monomorphism, M → M 00 is an epimorphism and im f = ker g.

Exactness will be explored in more detail in Chapter 4, but we show here that short exact sequences hold a lot of information about objects and morphisms in abelian categories, which is a result of the canonical isomorphism between coker ker f and ker coker f for any f. Proposition 2.2.10. Let M 0,M,M 00 be objects in an abelian category A, and consider the short exact sequence

f g 0 → M 0 −→ M −→ M 00 → 0.

Then M 0 ∼= ker g and M 00 ∼= coker f.

Proof. Since the sequence is exact, f is a monomorphism and so ker f = 0. Thus, we have

ker g = im f = ker coker f ∼= coker ker f = coker(0 → M 0) = M 0.

The other assertion is similar. 2

Thus in a short exact sequence as above, M 0 can be identiﬁed with a submodule of M, and M 00 ∼= M/M 0 can be identiﬁed with a quotient of M. This is an idea whose importance will become clearer in the following chapter.

13 Since exact sequences encode information about kernels and cokernels in abelian categories, it would be reasonable to ask a structure preserving functor between abelian categories to preserve exact sequences. Deﬁnition 2.2.11. Let T : A → B be an additive functor between abelian categories and let 0 → M 0 → M → M 00 → 0 be an exact sequence in A. Then we call T an exact functor if

0 → TM 0 → TM → TM 00 → 0 is also exact.

Unfortunately, there are functors between abelian categories which are of interest, but fail to be fully exact, so we also introduce the terminology left and right exact functors. The deﬁnition is what one might expect—a left exact functor preserves the left half of a short exact sequence but not necessarily the rest. That is, with the above notation, T is called left exact if

0 → TM 0 → TM → TM 00 is exact, but the ﬁnal map may not be an epimorphism. The deﬁnition of right exact is similar.

Indeed, exact functors preserve all the structure of an abelian category, since they preserve kernels and cokernels. Proposition 2.2.12. Let T : A → B be an additive functor between abelian categories. Then T is left exact if and only if T preserves kernels. Dually, T is right exact if and only if T preserves cokernels.

Proof. The prove of both assertions is entirely dual, so we just prove the left exact case.

Suppose T is left exact. Then im kerf = ker coker ker f ∼= coker ker ker f = coker 0 = ker f, so we have an exact sequence

0 → ker f → A → B, which T preserves since T is left exact, so Proposition 2.2.10 implies that T (ker f) = ker T f. Now suppose that T preserves kernels. Then if

g 0 → A → B −→ C

14 is an exact sequence, A ∼= ker g, so TA ∼= ker T g, which implies that

T g 0 → TA → TB −→ TC is also exact, so T is left exact. 2

In particular, an additive functor between abelian categories is exact if and only if it preserves kernels and cokernels.

15 Chapter 3

The Quotient of an Abelian Category

As mentioned in the Introduction, the category of modules over a ring R is often not sufficient to capture the geometry of a space, and instead we need some more general notion to find the appropriate category. This is of course the construction of the quotient category. We follow the construction outlined in [14], which in turn follows that presented in Gabriel’s thesis [6]. We also direct the interested reader to [13] for a construction in terms of categories of fractions, which is a different but equivalent way of setting up the theory. We present quotient categories here using a running example which comes from the geometry of affine space. To present this example we need to define the idea of localisation for rings and modules, which comes from commutative algebra.

3.1 Localisation

We now wish to develop some intuition for the construction of quotient categories. To do this we must briefly introduce the idea of localisation in both rings and modules. So to that end we set R to be a commutative ring. A subset S ⊆ R is called multiplicatively closed if 1 ∈ S and s1s2 ∈ S whenever s1, s2 ∈ S. Definition 3.1.1. Let R be a commutative ring and S a multiplicatively closed subset of R. Then for an R-module M, the module of fractions of M at S is −1 defined to be S M := S × M/ ∼, where (s1, m1) ∼ (s2, m2) if and only if there exists non-zero r ∈ S such that r(s2m1 − s1m2) = 0. Again, the equivalence class of (s, m) in S−1M is denoted m/s.

If the module M in this deﬁnition is taken to be R itself, then we call S−1R the ring of fractions of R at S, and this is indeed a ring, with addition given by

r/s + r0/s0 = (rs0 + r0s)/ss0,

16 and multiplication given by

(r/s)(r0/s0) = rr0/ss0.

It is an easy computation to see that these operations are well-deﬁned. Example 3.1.2. If R is a domain and S = R − {0}, then S−1R is simply the ﬁeld of fractions of R. Example 3.1.3. If f ∈ R is non-zero, then for S = {1, f, f 2, . . . , f n,... }, the ring of fractions S−1R is usually denoted R[f −1] and is called the localisation of R at f. We will look at this example in more detail.

One might suspect that the obvious module action of S−1R on S−1M turns S−1M into an S−1R-module, and this is true, which of course justiﬁes the name module of fractions. Proposition 3.1.4. For a commutative ring R, a multiplicatively closed subset S, and an R-module M, the module S−1M is an S−1R-module with addition m/s + m0/s0 = (ms0 + m0s)/ss0 and (r/s)(m/s0) = rm/ss0.

Proof. We omit the proof since all that is needed to show is that the operations are well-deﬁned. 2

For simplicity, we will now only consider the case where S = {f k : k ∈ N}. Con- sider an R-module homomorphism ϕ : M → N. This induces an R[f −1]-module homomorphism ϕ[f −1]: M[f −1] → N[f −1], by sending m/f k 7→ ϕ(m)/f k. This of course suggests the existence of a functor (−)[f −1]: R -Mod → R[f −1]-Mod, deﬁned by M 7→ M[f −1], and ϕ 7→ ϕ[f −1]. Proposition 3.1.5. The functor (−)[f −1]: R -Mod → R[f −1]-Mod is exact.

Proof. First we show that it is additive. Let g1, g2 : M → N be R-module homomorphisms. Firstly,

−1 k k (g1 + g2)[f ](m/f ) = (g1(m) + g2(m))/f k k = g1(m)/f + g2(m)/f −1 k −1 k = g1[f ](m/f ) + g2[f ](m/f ).

−1 −1 −1 −1 Thus (g1 + g2)[f ] = g1[f ] + g2[f ], so (−)[f ] is additive. Now consider a short exact sequence ϕ ψ 0 → M 0 −→ M −→ M 00 → 0. (∗)

17 Applying (−)[f −1] to this gives

ϕ[f −1] ψ[f −1] 0 → M 0[f −1] −−−→ M[f −1] −−−−→ M 00[f −1] → 0. (∗∗)

Suppose that ϕ[f −1](m0/f k) = 0. So ϕ(m0)/f k = 0, and by deﬁnition of the module of fractions we have that f nϕ(m0) = 0 for some n. But then ϕ(f nm0) = 0 and so f nm0 = 0 since ϕ is injective. But this implies that m0/f k = 0, so ϕ[f −1] is injective too.

Now let m00/f k ∈ M 00[f −1]. Since ψ is surjective, there is an m such that ψ(m) = m00. Thus ψ[f −1](m/f k) = ψ(m)/f k = m00/f k, so ψ[f −1] is surjective too.

Now let m0/f k ∈ M 0[f −1] and consider

ψ[f −1](ϕ[f −1](m0/f k)) = ψ(ϕ(m0))/f k = 0/f k = 0, since (∗) is exact at M, so im ϕ[f −1] ⊆ ker ψ[f −1]. Finally, let m/f k ∈ M[f −1] be such that ψ[f −1](m/f k) = 0. Then ψ(m)/f k = 0, so there is some n such that f nψ(m) = 0, so f nm ∈ ker ψ[f −1]. Thus there exists an m0 ∈ M 0 such that ϕ(m0) = f nm since (∗) is exact at M. But then ϕ[f −1](m0/f n+k) = ϕ(m0)/f n+k = f nm/f n+k = m/f k, so we have ker ψ[f −1] ⊆ im ϕ[f −1]. Thus (∗∗) is exact at M[f −1] as well. 2

Of course, we have a map M → M[f −1], sending m 7→ m/1. This map is not always injective, since M may have some torsion. If it does, the localisation must remove it, since if f km = 0, when we localise, f becomes invertible, so we must have m = 0.

Proposition 3.1.6. Given an R-module M, M[f −1] = 0 if and only if for all m ∈ M there is some k ∈ N such that f km = 0.

Proof. Suppose that M is such that all m ∈ M are annihilated by a power of f. Then for any m ∈ M, f km = 0, implying f k(m − 0) = 0, and so immediately we see that m/1 = 0/1 in M[f −1]. Now suppose that M[f −1] = 0. Then m/f k = 0/1 for all m ∈ M and k ∈ N. That is, there exists n such that f n(m − f k0) = 0, in other words f nm = 0. But m was arbitrary, so we are done. 2

The intuition for this localisation idea comes from algebraic geometry, and the speciﬁc case of the polynomial ring R = k[x1, . . . , xn] over some algebraically closed ﬁeld k.

18 Definition 3.1.7. Affine n-space over a field k is defined to be the set of all n-tuples n n of elements of k, and is denoted Ax1,...,xn , or Ak .

n There is a natural correspondence between subsets of Ax1,...,xn and ideals of the polynomial ring R = k[x1, . . . , xn]. Given an ideal I E R, we may deﬁne the set

n V (I): {x ∈ Ax1,...,xn : f(x) = 0 ∀f ∈ I}.

These sets are called algebraic sets and onto these we may deﬁne the coordinate ring, k[V (I)] = {f : V (I) → k : f is given by a polynomial }.

n Thus for aﬃne space, k[Ak ] = k[x1, . . . , xn] = R. For the coordinate ring of an algebraic set V (I), the polynomial ring is a good starting point. But two polynomials f, g ∈ R will determine the same function on V (I) if and only if f − g ∈ I. Thus in this case k[V (I)] = R/I.

Now consider the case where I = (f) for some non-zero, non-constant f ∈ R. n Denoting V (I) by X, we have k[X] = R/(f). Suppose we wanted to study Ak − X instead. This is not an algebraic set since it is not given by the zero set of some set of polynomials, and so we also have to expand our notion of coordinate ring to n account for it. Fortunately there is a natural way to do this. Since Ak − X is the set of points for which f is non-zero, we should be able to invert f on this set in some way. We deﬁne a polynomial p = fz − 1 ∈ k[x1, . . . , xn, z], which gives a set

n Z := V (p) = {x ∈ Ax1,...,xn,z : zf(x) = 1}.

n n n We then project via π : Ax1,...,xn,z → Ax1,...,xn . Note here that π(Z) = Ak − X, so n we deﬁne the coordinate ring of Ak − X to be

n ∼ −1 k[Ak − X] := k[Z] = k[x1, . . . , xn, z]/(zf − 1) = k[x1, . . . , xn, f ], which is precisely the algebraic localisation just discussed. This is also where the n term localisation comes from, since here we are looking at Ax1,...,xn locally in the region where f is non-zero.

−1 It can be shown that k[x1, . . . , xn, f ] is the set of all rational functions which are defined for all x such that f(x) 6= 0. Suppose that we wanted to look locally at the plane (say) with a single point removed, for example the origin. Then even this generalised notion of coordinate ring is not sufficient, since any rational 2 function which is defined on Ax,y − 0 must necessarily be defined at 0 as well, so

19 2 2 k[Ax,y − 0] = k[Ax,y]. So we need some other way of studying the geometry of a space like this.

This is where quotient categories come into the picture. These form the categorical framework in which we can study localisation, and this framework does suﬃciently distinguish between the plane and the punctured plane. Indeed, the coordinate ring captures the geometry of the space, and more abstractly, so too does the category of modules or of quasi-coherent sheaves.

3.2 Quotient Categories

To actually deﬁne a quotient category, we need to know what we can and cannot “divide” by. Unfortunately we can’t just take any subcategory. Indeed, the situation here is akin to that in the category Grp, in which quotients by a subgroup can only be taken if the subgroup is normal. To take a quotient of an abelian category by some subcategory, we need the subcategory to be what is called a Serre subcategory, also referred to in some sources as a thick or dense subcategory. Serre subcategories are in particular closed under subobjects, quotient objects and extensions. Deﬁnition 3.2.1. Let A be an abelian category. A non-empty full subcategory S is called a Serre subcategory if for any short exact sequence

0 → M 0 → M → M 00 → 0, we have M 0,M 00 ∈ S if and only if M ∈ S.

Example 3.2.2. If A = Vectk for some k, then the subcategory of ﬁnite dimensional vector spaces is Serre, which is a consequence of the rank-nullity theorem, which states that in a short exact sequence as above, dim M = dim M 0 + dim M 00. Example 3.2.3. The full subcategory of R -Mod consisting of torsion modules is a Serre subcategory.

This last example in many ways captures the idea behind the deﬁnition. In some sources, the objects in a general Serre subcategory are referred to as torsion objects, because there are many links to the idea of torsion theories in abelian categories, and so the objects of a Serre subcategory can be thought of as in some sense “bad”, and the idea of quotienting can be thought of as removing this.

Indeed, if M ∈ S, then any subobject or quotient object of M can fit into a short exact sequence with M in the middle, and if M 0 and M 00 are in S, then an extension of M 00 by M 0 clearly fits into a short exact sequence. In particular, since 0 ≤ M, and S is defined to be non-empty, 0 ∈ S for any Serre subcategory.

20 In much the same way that a subgroup of some group G is normal if and only if it is the kernel of some group homomorphism, a full subcategory is Serre if and only if it is the kernel of some exact functor. We prove one direction of this claim here and the other later. Proposition 3.2.4. Let F : A → B be an exact functor. Then ker F , the full subcategory of objects A such that FA = 0, is Serre.

Proof. Let M 0,M,M 00 ﬁt into a short exact sequence as above. If M 0,M 00 ∈ ker F , then applying F to said short exact sequence results in

0 → 0 → FM → 0 → 0, since F is exact. Thus FM = 0 as well. Similarly, if M ∈ ker F , then we have

0 → FM 0 → 0 → FM 00 → 0, implying that M 0,M 00 ∈ ker F too. 2

This gives an alternate way of thinking about Serre subcategories, which in some situations might be more useful.

In order to deﬁne the quotient of an abelian category, we also need the notion of a direct limit, see [14] or [9] for details. There are three facts about direct limits which we will take as given. Firstly, the category Ab and R -Mod have all direct limits. Secondly, if {Ai, ϕij}i∈I is a direct system over a directed set I, and J ⊆ I is coﬁnal, then lim A ∼ lim A . −→ i = −→ i i∈I i∈J Lastly, the set of equivalence classes

G Ai ∼, i∈I where ∼ identifies xi ∈ Ai and xj ∈ Aj if there is some k such that ϕik(xi) = ϕjk(xj), satisfies the universal property of the direct limit, and since the set I is directed, this set inherits any algebraic structure that the Ai’s all share. Effectively, we take the disjoint union and then identify any element with its image under any of the maps in the system.

21 Let A be an abelian category and S a Serre subcategory of A. For objects M,N ∈ A, deﬁne the set

0 0 0 0 0 0 IM,N := {(M ,N ): M ≤ M,N ≤ N, M/M ,N ∈ S}.

0 0 00 00 We then place a partial order on IM,N by declaring that (M ,N ) ≤ (M ,N ) if and only if M 00 ≤ M 0 and N 0 ≤ N 00.

1 Proposition 3.2.5. For any M,N, the set IM,N is directed.

0 0 00 00 Proof. Let (M ,N ), (M ,N ) ∈ IM,N . Then since S is closed under subobjects and quotient objects, N 0 ∩ N 00 ∈ S and so (N 0 + N 00)/N 0 ∼= N 00/N 0 ∩ N 00 ∈ S. But then since N 0 ∈ S, N 0 +N 00 ∈ S. Also, we have M 0/M 0 ∩M 00 = (M 0 +M 00)/M 00 ≤ M/M 00 and so is in S, and ﬁnally

M/M 0 ∩ M 00 = M/M 0, M 0/M 0 ∩ M 00 which is in S as well, so we conclude that M/M 0 ∩ M 00 ∈ S. So (M 0 ∩ M 00,N 0 + N 00) 0 0 00 00 is in IM,N and is an upper bound of (M ,N ) and (M ,N ). 2

Now if (M 0,N 0) ≤ (M 00,N 00), the inclusions M 00 ≤ M 0 and N 0 ≤ N 00 induce maps 0 0 00 00 0 0 0 0 from HomA(M , N/N ) → HomA(M , N/N ), so {HomA(M , N/N )}(M ,N )∈IM,N is a direct system indexed by IM,N . We are now ready to define the quotient of an abelian category by a Serre subcategory. Definition 3.2.6. Given an abelian category A and a Serre subcategory S of A, we define the quotient category Q := A/S as follows:

• the objects of Q are the same as the objects of A

• for objects M,N ∈ Q, deﬁne

Hom (M,N) = lim Hom (M 0, N/N 0) Q −→ A 0 0 (M ,N )∈IM,N

Of course it’s not immediately clear that this gives a well deﬁned category, since it’s not obvious how to compose morphisms in Q. So take f¯ : M → N andg ¯ : N → S in Q. Then f¯ is the image in the direct limit of some map f : M 0 → N/N 0 andg ¯ is

1 To ensure that IM,N is indeed a set and not a proper class, we actually have to assume that A has a set of generators, the details of which we will not go into here. See [14].

22 the image in the direct limit of some map g : N 00 → S/S0, where M/M 0,N 0, N/N 00 and S0 are all in S.

If we deﬁne S00 := S0 +g(N 0 ∩N 00) and M 00 := f −1((N 0 +N 00)/N 0), then g will induce a map g0 : N 00/N 0 ∩ N 00 → S/S00 and f will induce a map f 0 : M 00 → (N 0 + N 00)/N 0.

Firstly, N 0 ∩ N 00 ≤ N 0, and g(N 0 ∩ N 00) is the image of the composite

g ϕ : N 0 ∩ N 00 → N 00 −→ S/S0.

But then g(N 0 ∩ N 00) ∼= N 0 ∩ N 00/ ker ϕ, which is in S because N 0 ∩ N 00 is.. Also, S0 is in S by assumption, and so S00 is in S.

Next, we have that M 00 ≤ M 0, and

M/M 00 ∼= M/M 0, M 0/M 00 which is in S. But M 00 is the kernel of the composite

0 f N/N ψ : M 0 −→ N/N 0 → = N/(N 0 + N 00), (N 0 + N 00)/N 0 so again M 0/M 00 ∼= im ψ ≤ N/(N 0 + N 00). Finally,

N/N 00 N/(N 0 + N 00) = , (N 0 + N 00)/N 00 so it is a quotient of N/N 00 which is in S by assumption. So M 0/M 00 ∈ S which implies M/M 00 is in S as well, so h is indeed in the direct system.

Then since (N 0 + N 00)/N 0 ∼= N 00/(N 0 ∩ N 00), we can deﬁne h : M 00 → S/S00 to be the composite g0ϕf 0, where ϕ is the canonical isomorphism from N 0 + N 00/N 0 to N 00/N 0 ∩ N 00. Then h is in the direct limit, and so we deﬁneg ¯f¯ := h¯, the image of h in the direct limit.

Now we check that this composite does not depend on the choice of f and g. Before we do this it should be noted that the computation is rather tedious and most sources in the literature omit it.

We begin by varying each variable subobject individually. Suppose f¯ is the image 0 0 ¯ 0 0 00 0 of f1 : M → N/N and also f2 : M → N/N , andg ¯ is the image of g1 : N → S/S 00 ¯0 ¯ 0 0 ¯ 0 ¯0 0 ¯0 and also g2 : N → S/S . Then by replacing M with M ∩ M and S with S + S ¯ 0 0 0 ¯0 if necessary, we may assume that M ≤ M and S ≤ S , and so f2 = f1ι, and ¯ 0 0 0 ¯0 g2 = πg1 where ι : M → M is the canonical inclusion and π : S/S → S/S . Then

23 00 ¯ 00 00 ¯00 deﬁne M , M , S and S as in the deﬁnition of the composite. Then we have h1 and h2 as the rows of the following diagram: N 0 + N 00 N 00 M 00 ∼ S/S00 N 0 N 0 ∩ N 00

N 0 + N 00 N 00 M¯ 00 ∼ S/S¯00. N 0 N 0 ∩ N 00

This diagram is easily seen to commute, and so h1 and h2 have the same image in Hom(M¯ 00, S/S¯00) so they are equal in the direct limit.

¯ 0 0 0 ¯ 0 Now suppose f is the image of f1 : M → N/N and also f2 : M → N/N , and g¯ is the image of g : N 00 → S/S0. Then, as above we assume N 0 is contained in ¯ 0 0 N , so f2 = πf1. Also, since by the above we can make M as small as we like, we 0 0 00 0 ¯ 0 may assume that f1(M ) is contained in (N + N )/N , and similarly that g(N ) is contained in S0. Then we have N 0 + N 00 N 00 M 0 ∼ ` S/S0 N 0 N 0 ∩ N 00 (1) (2) (3) 0 00 00 N¯ + N N ¯ M 0 ∼ ` S/S0. N¯ 0 N 0 ∩ N 00 Here (1) commutes by deﬁnition of the bottom arrow, and (2) commutes canonically. The only square we need to check is (3). The map ` is induced by the projection 00 00 0 00 ¯ π1 : N → N /N ∩ N , so g = `π1, and similarly ` is induced by the projection 00 00 ¯ 0 00 ¯ 00 0 00 π2 : N → N /N ∩ N , so g = `π2. Also, π1 = pπ2, where p : N /N ∩ N → 00 ¯ 0 00 ¯ N /N ∩ N is the canonical projection. So then we have `π1 = `pπ1, so since π1 is an epimorphism, ` = `p¯ , so (3) commutes. ¯ ¯ Thus the rows h1 and h2 are equal and so h1 = h2. Finally we vary N 00. So f¯ is the image of f : M 0 → N/N 0 andg ¯ is the image 00 0 ¯ 00 0 ¯ 00 00 of g1 : N → S/S and g2 : N → S/S , where N ≤ N , so g2 = g1ι, where ι : N¯ 00 → N 00 is the canonical inclusion. Again we assume that M 0 is small enough and S0 is big enough. So we have, N 0 + N 00 N 00 M 0 ∼ ` S/S0 N 0 N 0 ∩ N 00 (1) (2) q (3) 0 00 00 N + N¯ N¯ ¯ M 0 ∼ ` S/S0. N¯ 0 N 0 ∩ N¯ 00

24 (1) and (2) are again easily seen to commute. For (3), the map ` is induced by g1 00 00 0 00 ¯ and π1 : N → N /N ∩ N , so g1 = `π1, and similarly g1ι = g2 = `π2. Also, the map q is induced by π2 and π1ι, so also π1ι = qπ2. Putting this together we have ¯ ¯ ¯ `π1ι = `π2, which implies `qπ2 = `π2, so since π2 is an epimorphism, `q = `, so (3) ¯ ¯ commutes. Thus the maps h1 and h2 are equal so h1 = h2. Since the direct limit of a system of abelian groups is an abelian group, to check that A/S is pre-additive, it suﬃces to show that the group structure is bilinear. Consider (f¯+ f¯0)¯g. Then since composition is independent of the choice of lift we can just take f¯ and f¯0 to be the images of some maps f and f 0 in the same Hom-set in A, and then bilinearity follows from bilinearity in A. Thus A/S is pre-additive and π is an additive functor. Also, since A has all biproducts, so does A/S. It turns out that also A/S is necessarily abelian and π is exact. The following propositions are necessary to prove this. Proposition 3.2.7. If f : M → N is a morphism in A, then πf = 0 in Q if and only if im f ∈ S.

Proof. If im f ∈ S, then f maps to f 0 : M → N/ im f in the direct system, and clearly f 0 = 0, so πf = πf 0 = 0. Now suppose πf = 0. So there exists M 0,N 0 such that M/M 0 and N 0 are in S and f induces a map f 0 : M 0 → N/N 0 which is the 0 map. That is, 0 = im f 0 = (f(M 0) + N 0)/N 0. So f(M 0) ≤ N 0 and so is in S. By the ﬁrst isomorphism theorem we have im f = M/ ker f and f(M 0) = M 0/ ker f ∩ M 0 = (ker f + M 0)/ ker f. So im f/f(M 0) = M/(ker f + M 0), which is a quotient of M/M 0, so is in S. Thus since f(M 0) and im f/f(M 0) are both in S, so too is im f. 2

This Proposition implies that any Serre subcategory is the kernel of its projection functor. This, together with the Proposition 3.2.7 and the following Proposition prove the claim that a subcategory S is Serre if and only if it is the kernel of some exact functor.

The following proof was adapted2 from [6]. Proposition 3.2.8. Let f : M → N be a morphism of A. Then ker(πf) = π(ker f) and coker(πf) = π(coker f). In particular, π is exact.

Proof. Let (K, k) = ker f. Then certainly (πf)(πk) = π(fk) = π(0) = 0, so we should check that (πK, πk) is universal with respect to this property. Suppose there is somep ¯ : P → M in Q such that πfp¯ = 0. We must ﬁnd a morphismq ¯ : P → K such thatp ¯ = (πk)¯q.

2And possibly made clearer.

25 Firstly,p ¯ is the image of some p : P 0 → M/M 0 such that P/P 0 and M 0 are in S. Here consider f 0 : M/M 0 → N/f(M 0) which is induced by f. Then we have ker f 0 = (K +M 0)/M 0 ∼= K/K ∩M 0. Also consider k0 : K/K ∩M 0 → M/M 0 induced by k. Then (K/K ∩ M 0, k0) is the kernel of f 0.

Since π(f 0p) = (πf)¯p = 0, by the previous proposition we have im(f 0p) ∈ S. Deﬁne P 00 := ker(f 0p). Then P 0/P 00 = im(f 0p) ∈ S. So since P 0/P 00 and P/P 0 are both in S, so too is P/P 00.

Let p0 : P 00 → M/M 0, be the map induced by p. Then f 0p0 = (f 0p)i = 0, where i : P 00 → P 0 is the canonical inclusion, so we get a map q : P 00 → K/K ∩ M 0 such that p0 = k0q since K/K ∩ M 0 is the kernel of f 0. Then πq : P → K is a map such thatp ¯ = πp0 = π(k0q) = (πk0)(πq) = (πk)(πq). Soq ¯ = πq is the desired map.

The argument for the preservation of cokernels is similar. Thus Proposition 2.2.12 implies π is exact. 2

The fact that A/S is abelian follows from this proposition without too much more work. Theorem 3.2.9. If A is an abelian category and S is a Serre subcategory, then A/S is an abelian category.

Proof. We have already seen that A/S is additive so it suﬃces to show that A/S has all kernels and cokernels. But this follows from the previous proposition, since given a map f¯ : M → N in A/S, f¯ is the image of some f : M 0 → N/N 0, where M/M 0 and N 0 are in S. So f¯ = πf. But the previous proposition implies that ker f¯ = ker πf = π(ker f), which exists since A is abelian, and similarly coker f¯ = π(coker f). Thus A/S is pre-abelian. But since functors preserve isomorphisms, we have coker ker f¯ = π(coker ker f) ∼= π(ker coker f) = ker coker f¯, so in fact A/S is abelian. 2

It is diﬃcult to see what quotient categories “look like”, simply by using the deﬁni- tion. Often to determine the nature of a quotient category it is necessary to prove some equivalence of categories, which of course involves constructing functors to and from the quotient. Constructing to the quotient is straightforward; construct a functor B → A and then compose with π to obtain a functor B → A/S. Functors out of a quotient are slightly harder, but like most quotient objects, we have a universal property which we can use to construct them. Theorem 3.2.10 (Universal property of Quotient Categories). Let A be an abelian category and S a Serre subcategory of A. If B is any abelian category and F : A → B

26 is an exact functor such that FS = 0 for any S ∈ S, there exists a exact functor F¯ : A/S → B such that F = F¯ π. Furthermore, if S = ker F , then this induced functor is faithful.

Proof. Since πM = M for all M, we deﬁne FM¯ = FM for all M. Now let ¯ ¯ 0 0 0 f ∈ HomA/S (M,N). So f is the image of some f : M → N/N such that M/M and N 0 are in S. Thus by considering the short exact sequence

0 → M 0 → M → M/M 0 → 0, and applying F to it, we see that F i : FM 0 → FM is an isomorphism, and similarly F p : FN → F (N/N 0) is an isomorphism. Thus we may deﬁne F¯f¯ := −1 −1 ¯ (F p) F f(F i) ∈ HomB(FM,FN). In particular, if f = πf for some f, then F f = F¯f¯ = F¯ πf, so F = F¯ π.

Now suppose S = ker F , and suppose some f¯ : M → N in A/S is such that F¯f¯ = 0. So then (F p)−1F f(F i)−1 = 0, so since F p and F i are isomorphisms, F f = 0 and so im(F f) = 0. Therefore since F is exact, F (im f) = 0. So im f ∈ ker F = S, and Proposition 3.2.7 implies f¯ = πf = 0, so F¯ is faithful. 2

In other areas of mathematics, when we take quotients, such as the quotient of a group by a normal subgroup, or the quotient of a ring by an ideal, or the quotient of a topological space by a subspace, we usually deﬁne the quotient object to be the set of equivalence classes under some equivalence relation which (in particular) identiﬁes the subobject by which we are quotienting, with zero. The result is an object which is in some sense smaller than the object we started with.

It is not immediately clear that the quotient category construction described in this section achieves the same goal in the context of categories, but it does. The key diﬀerence is that in category theory, we tend only to think of objects as equal up to isomorphism. By making the Hom-sets bigger in the construction of the quotient category, we are actually adding more isomorphisms to the category, and in the process we make the skeleton of the category smaller, even though the class of objects stays the same. In particular (as in the case of groups, rings, modules etc), any object of S is now isomorphic to the zero object. Lemma 3.2.11. Let A be an abelian category with a Serre subcategory S and let π : A → A/S be the projection functor. Then if M is an object of A, we have πM ∼= 0 if and only if M ∈ S.

¯ ¯ Proof. Suppose M ∈ S and let f ∈ HomA/S (πM, πM). So f is the image of some f : M 0 → M/M 00 where M/M 0 and M 00 are both in S. Then certainly

27 0 00 00 00 ¯ (M ,M ) ≤ (0,M ) in IM,M and f maps to 0 ∈ HomA(0, M/M ), so f = 0. Thus ∼ HomA/S (πM, πM) = 0. So idπM = 0, so πM = 0.

∼ 0 00 Now suppose πM = 0. Then in particular, idπM = 0, so there are some M ,M 0 00 0 00 with M/M and M in S such that idM : M → M maps to 0 : M → M/M . So then (M 0 + M 00)/M 00 = im 0 = 0. That is, M 0 ≤ M 00. But since S is closed under subobjects, M 0 is in S, so since M 0 and M/M 0 are in S, so is M. 2

Now that we have deﬁned the quotient category, we return to the example of

R = k[x1, . . . , xn] and the process of localisation at f. We claimed there that quotient categories are the correct framework to discuss localisation categorically. Here we attempt to provide some intuition as to why this is true. The above lemma together with Proposition 3.1.6 suggests that our Serre subcategory should be the full subcategory of modules M such that for all m ∈ M, f km = 0 for suﬃciently large k. We denote this subcategory f -Tors. Proposition 3.1.6 implies that f -Tors is the kernel of the exact functor (−)[f −1], and so the universal property (Theorem 3.2.10) guarantees us a faithful exact functor (−)[f −1]: R -Mod /f -Tors → R[f −1]-Mod. This notation will become cumbersome, so denote (−)[f −1] by L. Proposition 3.2.12. The functor L : R -Mod /f -Tors → R[f −1]-Mod is an equivalence of categories.

Proof. The functor L is already exact and faithful, so it suﬃces to show that it is essentially surjective and full.

Let N be an R[f −1]-module. Then by simply ignoring the action of f −1 on N, we can consider N as an R-module. Then if n/f k ∈ N, we have f −kn/1 = n/f k, so any any element of N[f −1] can be written as n/1 for some n ∈ N. Then clearly the map N[f −1] → N sending n/1 7→ n is an isomorphism and so N[f −1] ∼= N. Thus LN = N[f −1] ∼= N, so L is essentially surjective.

−1 −1 Now suppose that ϕ ∈ HomR[f −1](M[f ],N[f ]) for some R-modules M and N in R -Mod /f -Tors. We can assume that M and N are f-torsion free, so they embed in their localisations, so we consider them as submodules. Consider ϕ−1(N) ∩ M. Then M/ϕ−1(N) ∩ M is f-torsion, since ϕ(m) = n/f k, so f km ∈ ϕ−1(N). So, ϕ0 : −1 ϕ (N)∩M → N is in the direct limit in the deﬁnition of HomR -Mod /f -Tors(M,N). Also, Lπ(ϕ0) = (ϕ0)[f −1] = ϕ, so L is full. 2

3.3 Localising Subcategories

In this section we expand on the idea that quotient categories capture localisation and further generalise the example of the functor (−)[f −1].

28 Consider again the functor (−)[f −1]: R -Mod → R[f −1]-Mod. There is a natural functor which goes in the other direction, G : R[f −1]-Mod → R -Mod which sends an R[f −1]-module N to the R-module N, where we forget the action of f −1 −1 on N. As seen in the proof of Proposition 3.2.12, the map εN : N[f ] → N mapping n/1 7→ n is an isomorphism. We also have an R-module homomorphism −1 0 −1 ηM : M → M[f ] mapping m 7→ m/1. If ϕ : N → N is an R[f ]-module homomorphism, and ψ : M → M 0 is an R-module homomorphism, then we can consider the squares

η N[f −1] εN N M M M[f −1]

ϕ[f −1] ϕ ψ ψ[f −1] . ε η N 0[f −1] N0 N 0 M 0 M0 M 0[f −1]

−1 Since any fraction in N[f ] can be written as n/1 for some n ∈ N, ϕεN (n/1) = −1 ϕ(n) = εN 0 (ϕ(n)/1) = εN 0 ϕ[f ](n/1), so the left hand square commutes. Also, for −1 −1 m ∈ M, ψ[f ]ηM (m) = ψ[f ](m/1) = ψ(m)/1 = ηM 0 (ψ(m)), so the right hand square commutes too. Thus ηM and εN are the components of the unit and the −1 counit of an adjunction between (−)[f ] and G. Now the maps εN have already been seen to be isomorphisms, but the maps ηM are in general not isomorphisms, so the unit is not a natural isomorphism. So in particular we have (−)[f −1]G ∼= idR[f −1]-Mod. Now since R[f −1]-Mod is equivalent to R -Mod /f -Tors, the projection functor π : R -Mod → R -Mod /f -Tors must have a right adjoint as well, denoted ω := GL. In this case we say that f -Tors is a localising subcategory of R -Mod. Deﬁnition 3.3.1. Let A be an abelian category and S a Serre subcategory. We say that S is a localising subcategory if the projection functor π : A → A/S has a right adjoint ω : A/S → A. This functor is called the localisation functor or the section functor.

In [6], Gabriel provides necessary and suﬃcient conditions for the existence of a section functor. It turns out that if such a functor exists, then for every object M, the set {M 0 ≤ M : M 0 ∈ S} partially ordered by inclusion, has a maximal element, denoted t(M). Proposition 3.3.2. If M is an object of A/S, then t(ωM) = 0.

Proof. Consider HomA(t(ωM), ωM) = HomA/S (πt(ωM),M) = HomA/S (0,M) = 0. In particular, the inclusion i : t(ωM) → ωM is 0, so t(ωM) = 0. 2

29 In this way, we can think of ω as picking out a “nice” (or torsion free) representative of the isomorphism classes of A/S.

In the example of R -Mod and f -Tors above, we saw that the composite functor −1 ∼ (−)[f ]G = idR[f −1]. In fact this is true in general, justifying the name “section functor”. This proof was condensed from the proof of Proposition 14.7 in Chapter 2 of [14]. Proposition 3.3.3. Let A be an abelian category with a localising subcategory S. ∼ Then the composite πω = idA/S , where π : A → A/S is the quotient functor and ω : A/S → A is the section functor.

Proof. We already have a natural transformation ε : πω ⇒ idA/S , the counit of the adjunction, so it suﬃces to show that εN is an isomorphism for all N ∈ A/S. Let πM ∈ A/S and consider the square

HomA(M, ωN) HomA/S (πM, N)

π id

HomA/S (πM,εN ) HomA/S (πM, πωN) HomA/S (πM, N)

This square commutes by virtue of the adjunction between π and ω, and the top arrow is the adjunction isomorphism. Now, since t(ωN) = 0, we have

0 HomA/S (πM, πωN) = lim HomA(M , ωN), −→ where the limit runs over all subobjects M 0 of M such that M/M 0 = 0. Now, since ∼ 0 0 ∼ ∼ πM = πM for all M in the direct limit, we have HomA(M, ωN) = HomA/S (πM, N) = 0 ∼ 0 0 HomA/S (πM ,N) = HomA(M , ωN), so each HomA(M, ωN) → HomA(M , ωN) is an isomorphism, so actually the map π in the diagram is an isomorphism as well.

Thus the map HomA/S (πM, εN ) is an isomorphism, and so since πM was arbitrary, this deﬁnes a natural isomorphism between HomA/S (−, πωN) and HomA/S (−,N) in the additive functor category AdFun(A/S, Ab). But by the Yoneda Lemma, the assignment N 7→ HomA/S (−,N) is fully faithful, so εN : πωN → N is an isomorphism as well, which completes the proof. 2

30 Chapter 4

Homological Algebra

Homological Algebra is quite a technical area of maths, and was born out of the study of the singular homology of topological spaces, although some results were proved prior to the development of algebraic topology. In this chapter we provide a brief introduction to some of the more useful aspects of the subject. It should be emphasised that the deﬁnitions and results in this chapter should be viewed as tools, and as such, proofs are often only sketched or in some cases omitted entirely. At the end of the chapter we prove a speciﬁc case of Hilbert’s famous Syzygies Theorem, which is necessary for the proof of Grothendieck’s Splitting Theorem in the next chapter.

4.1 Chain Complexes and Homology

Consider a sequence of objects Mp in an abelian category, with maps ∂p : Mp →

Mp−1 for all p. Then the chain

∂p+2 ∂p+1 ∂p ∂p−1 M• = ··· −−→ Mp+1 −−→ Mp −→ Mp−1 −−→· · · is called a chain complex if ∂p∂p+1 = 0 for all p. The morphisms ∂p are called the boundary morphisms. If instead the maps are ∂p : M p → M p+1, then we call the chain p−2 p−1 p p+1 M • := ··· −−−→∂ M p−1 −−−→∂ M p −→∂ M p+1 −−→·∂ · · a cochain complex is the corresponding condition ∂p+1∂p = 0 is satisﬁed for all p. Note the convention here is to place the indices at the top for a cochain complex, to highlight the duality with chain complexes. Of course, the only diﬀerence is that the arrows point in the opposite direction, and as such, if one applies a contravariant functor to a chain complex, the result is a cochain complex (at least provided the functor is additive, to preserve the chain condition).

31 In the case of chain complexes, the condition ∂p∂p+1 = 0 implies that im ∂p+1 ≤ ker ∂p, and similarly for cochain complexes, im ∂p+1 ≤ ker ∂p, so it makes sense to take these quotients.

Deﬁnition 4.1.1. If M• is a chain complex, the pth homology is the quotient • object Hp := ker ∂p/ im ∂p+1. If M is a cochain complex, we similarly deﬁne the p-th cohomology to be the quotient Hp := ker ∂p/ im ∂p−1.

We say that a chain complex is exact at Mp if Hp = 0, and simply that it is exact if it is exact at Mp for all p. Note that this notion of exactness coincides with the one given in Chapter 2 for short exact sequences, since Hp = 0 if and only if ker ∂p = im ∂p+1. Of course, there is a corresponding notion of exactness for cochain complexes. So in some sense, the homology measures how far a (co)chain complex is from being exact, and in fact we can use homology to also measure how far a functor is from being exact as well. We demonstrate how to do this with two examples.

While chain complexes are deﬁned for arbitrary abelian categories, the theory is rather more tedious to work through in the general setting, so given the lighter nature of this chapter, we restrict to the case of modules over a ring R. In particular we are mainly interested in two particular functors which (provided R is commutative) map modules to modules.

4.2 The Hom and Tensor Functors

For this section and the next we will consider an exact sequence of modules over a (commutative) ring R: 0 → A → B → C → 0.

We are interested in the exactness (or lack thereof) of the resulting sequence when the functors HomR(M, −) and (−) ⊗R M are applied to the above sequence for a given M.

We summarise a few basic results without proof, since they are fairly easy to see.

1. If R is a commutative ring, then for any R-modules M and N, the abelian

groups HomR(M,N) and M ⊗R N each have a well-deﬁned R-module structure which can be obtained from either the R-module structure of M or that of N. This is not true in the case of a non-commutative ring, so throughout this chapter we will assume R to be commutative, so that the func-

tors HomR(−,N), HomR(M, −) and (−) ⊗R N may considered functors from R -Mod to R -Mod.

32 2. The tensor product functor (−) ⊗R (−): R -Mod ×R -Mod → R -Mod is ∼ symmetric. That is, M ⊗R N = N ⊗R M naturally in M and N. ∼ 3. For any ring R and ideal I E R, there is an isomorphism (R/I)⊗RM = M/IM which is natural in M.

We now show a slightly less trivial result which shows that these two functors are in fact intimately related. Lemma 4.2.1 (Tensor-Hom Adjunction). For any modules M,N,S, we have

∼ HomR(N ⊗R M,S) = HomR(N, HomR(M,S)).

In particular, the functors (−) ⊗R M and HomR(M, −) form an adjoint pair.

Proof. We define a map Φ : HomR(N ⊗R M,S) → HomR(N, Hom(M,S)) and then construct the inverse. Given ϕ : N ⊗R M → S, define Φϕ : N → HomR(M,S) by sending n 7→ (m 7→ ϕ(n ⊗ m)). That this is a homomorphism follows from the linearity of the tensor product. Now given ψ : N → HomR(M,S), define

Ψψ : N ⊗R M → S by sending n ⊗ m 7→ [ψ(n)](m).

Now consider ΨΦϕ : N ⊗R M → S. By deﬁnition,

ΨΦϕ(n ⊗ m) = [Φϕ(n)](m) = ϕ(n ⊗ m),

so ΨΦϕ = ϕ. Also for ψ : N → HomR(M,S), we have

[ΦΨψ(n)](m) = Ψψ(n ⊗ m) = [ψ(n)](m), so ΦΨψ(n) and ψ(n) deﬁne the same map, so ΦΨψ = ψ. Thus Φ and Ψ are mutually inverse, which completes the proof. 2

In particular, the functor HomR(M, −) is left exact and (−) ⊗R M is right exact. Proposition 4.2.2. The sequences

0 → HomR(M,A) → HomR(M,B) → HomR(M,C), and

A ⊗R M → B ⊗R M → C ⊗R M → 0, are exact.

Proof. Right adjoints preserve limits and left adjoints preserve colimits. For the details of this more general statement, see Chapter V, §5 of [10]. 2

33 In general, unfortunately, we can’t say much more than this. For instance, let R = k[x, y], m = (x, y) and denote the quotient module R/m by k. Consider the exact sequence 0 → m → R → k → 0.

Since m and R are both torsion free and k is not, the modules HomR(k, m) and

HomR(k, R) are both 0, but HomR(k, k) is not, so the functor HomR(k, −) is not right exact.

Of course, the map HomR(M,B) → HomR(M,C) may in some cases be surjective. ∼ For example, if we take M = R, then HomR(R,A) = A via the isomorphism

ϕ 7→ ϕ(1), so in this case the HomR(R, −) is right exact as well. To generalise this notion we introduce the notion of a projective module.

Deﬁnition 4.2.3. An R-module M is called projective if the functor HomR(M, −) is exact.

In particular, since HomR(−,N) is additive, any free module is projective. The converse is not true in general, but we will see in the next chapter that for the case where R is a polynomial ring, the converse is true for ﬁnitely generated graded modules.

As indicated in the preceding section, we wish to use homology to measure the exactness of the functors HomR(M, −) and (−) ⊗R M for a given R-module M. To do this we must introduce the notion of a derived functor.

4.3 Ext and Tor Functors

The Ext and Tor functors are the derived functors of Hom and ⊗R, respectively. The idea behind these functors is to extend the sequences

0 → HomR(C,N) → HomR(B,N) → HomR(A, N),

0 → HomR(M,A) → HomR(M,B) → HomR(M,C) and

M ⊗R A → M ⊗R B → M ⊗R C → 0, to obtain long exact sequences. To deﬁne such of functors, we need to use what is known as a projective resolution of a module M.

34 Deﬁnition 4.3.1. Let M be an R-module. A projective resolution of M is an exact chain complex of the form

· · · → Pn → · · · → P2 → P1 → P0 → M → 0, where each Pi is a projective R-module and the boundary maps are module homomorphisms. If furthermore the Pi’s are free R-modules, we call the chain complex a free resolution. If there exists some n, such that for all i > n, Pi = 0, but Pn 6= 0, then the resolution is said to have length n.

For convenience, we introduce the notation

P• := · · · → Pn → · · · → P1 → P0, and write P• → M to indicate that P• is a projective resolution of M. We can use the notion of projective resolution to define the projective dimension of a module, which can measure how far a module is from being projective. We will see later that in the case of the polynomial ring, there is an upper bound on how ‘unprojective’ a module can be. Definition 4.3.2. The projective dimension of an R-module M is defined to be the minimum length of a projective resolution of M. If all projective resolutions of M are infinite, then we say M has infinite projective dimension.

We denote the projective dimension of a module M by pd M.

In particular, the projective R-modules are precisely those with projective dimension 0, since 0 → M → M → 0 is a projective resolution of a projective module, and if M has projective dimension 0, then the exactness of 0 → P0 → M → 0 ∼ implies that M = P0 so is projective.

Let P• → M be a projective resolution. Then we have a cochain complex

HomR(P•,N) := 0 → HomR(P0,N) → HomR(P1,N) → · · · → HomR(Pn,N) → · · · which is in general not exact. Deﬁnition 4.3.3. Let M,N be R-modules for some commutative ring R, and let

P• be a projective resolution of M. We deﬁne the R-module

n n ExtR(M,N) := H (HomR(P•,N)).

35 n Consider the object ExtR(M,N). The R-action on M induces an R-action on the cochain complex HomR(P•,N), and because R is commutative, the action on N induces the same action on this cochain complex. Thus the R-action induced on the cohomology of HomR(P•,N) can be taken from M or N. Thus we have a n functor ExtR(−,N): R -Mod → R -Mod, which is the right derived functor of

HomR(−,N).

n This definition of the ExtR(−,N) functors seems to depend on the choice of projective resolution, but it turns out that the homology groups obtained from different resolutions are isomorphic up to a unique isomorphism, and so up to isomorphism this functor is well-defined (note that the unique choice of isomorphism is necessary to ensure functoriality).

Of course, the purpose of these functors is to obtain a long exact sequence. Given a short exact sequence 0 → A → B → C → 0, a projective resolution of A and a projective resolution of C can be used to provide a projective resolution of B (see Proposition 17.7 of [4]), which forms a short exact sequence of chain complexes. Applying HomR(−,N) to this provides another short exact sequence of chain complexes, and so this induces a long exact sequence of n homology. In terms of the ExtR functors, this sequence is

0 → HomR(C,N) → HomR(B,N) → HomR(A, N) 1 1 1 → ExtR(C,N) → ExtR(B,N) → ExtR(A, N) → · · ·

n We also have a dual theory for the functor ExtR(M, −): R -Mod → R -Mod. Strictly speaking, this is computed using the theory of injectives, which we don’t go into here, which provides the other long exact sequence for the Hom functor. n However, computing ExtR(M,N) via a projective resolution of M or an injective resolution of N results in canonically isomorphic modules (for a proof of this, at 1 least for ExtR(M,N), see [12]), so our theory using projectives is suﬃcient.

n To get a feel for these ExtR functors, instead of going through the details of proving that they satisfy the long exact sequence of homology, we will simply compute some examples.

n For any R-module M, ExtR(R,M) = 0 for n > 0. This is simply because R is ∼ n projective, since HomR(R,M) = M for any M. Then, since ExtR is additive, the same is true for any free module, and more generally any projective module P has n ExtR(P,M) = 0 for all n > 0.

36 Proposition 4.3.4. Let R = k[x, y] and let m = (x, y), and denote by k the quotient 1 2 module R/m. Then we have ExtR(k, R) = 0 and ExtR(k, R) = k.

Proof. We compute these groups via the so-called Koszul complex, which is a projective resolution of k.

(−y x)T (x y) 0 → R −−−−−→ R ⊕ R −−−→ R −→π k → 0.

It’s not too hard to see that this sequence is exact, and is a free resolution. We then delete the k from the end and apply the functor HomR(−,R) to the resulting ∼ sequence. Using the isomorphism HomR(R,M) = M for all M and the fact that

HomR(−,R) is contravariant, we obtain the cochain complex

(x y)T (−y x) 0 → R −−−−→ R ⊕ R −−−−→ R → 0.

2 1 Then ExtR(k, R) = R/ im(−y x) = R/(R(−y)+Rx) = R/m = k. Also ExtR(k, R) = ker(−y x)/ im(x y)T . Now,

ker(−y x) = {(p, q) ∈ R ⊕ R : qx = py} = R(x, y) = im(x y)T ,

1 so ExtR(k, R) = 0. 2

n It was mentioned before that projective dimension together with the ExtR functors can measure how far a module is from being projective. Since a projective module 1 M has ExtR(M, −) = 0, the following proposition makes this slightly more precise.

Proposition 4.3.5. For all k ≥ n = pd M, Extk+1(M,N) = 0 for all modules N.

Proof. Since pd M = n, there is a projective resolution of M given by

0 → Pn → · · · → P1 → P0 → M → 0.

If we use this to compute the Extk functors, then if we let N be any R-module, we k+1 k+1 k+1 have ExtR (M,N) = Z /B , where

k+1 Z = ker(HomR(Pk+1,N) → HomR(Pk,N)) and k+1 B = im(HomR(Pk+2,N) → HomR(Pk+1,N)),

37 k+1 both of which are submodules of Pk+1. Thus, if k ≥ n, ExtR (M,N) = 0, so pd M ≤ n. 2

d+1 This proposition shows that if d is the smallest integer such that ExtR (M,N) = 0 for all N, then d ≤ pd M. In fact d = pd M, but we will not prove the other inequality here.

We are also interested in deriving the tensor functor. In this case, we use the same projective resolution P• of M as above, and tensor this complex with N, giving the not necessarily exact chain complex:

P• ⊗R N := · · · → Pn ⊗R N → · · · → P1 ⊗R N → P0 ⊗R N → 0.

Deﬁnition 4.3.6. Let M,N be R-modules for a commutative ring R, and let P• be a projective resolution of R. We deﬁne the R-module

R Torn (M,N) := Hn(P• ⊗R N).

R This deﬁnes a functor Torn (M, −): R -Mod → R -Mod. This of course can also be thought of as a functor of two variables, and in this case, much like the tensor product itself, this functor is symmetric. Proposition 4.3.7. For any R-modules M,N, then for all n, the modules

R ∼ R Torn (M,N) = Torn (N,M) naturally in both variables.

R Proof. We merely sketch the proof of this for n = 1. Since Torn (M,N) is computed R via a projective resolution of M, say P•, and Torn (N,M) is computed via a projective resolution of N, say Q•, we need some way of comparing the two resolutions. To do this we use the following commutative diagram:

38 ......

··· Q2 ⊗R P2 Q1 ⊗R P2 Q0 ⊗R P2 N ⊗R P2 0

M id ⊗R∂2

··· Q2 ⊗R P1 Q1 ⊗R P1 Q0 ⊗R P1 N ⊗R P1 0

M (1) id ⊗R∂1

··· Q2 ⊗R P0 Q1 ⊗R P0 Q0 ⊗R P0 N ⊗R P0 0 (2) N N ∂2 ⊗Rid ∂1 ⊗Rid ··· Q2 ⊗R M Q1 ⊗R M Q0 ⊗R M 0

0 0 0

All the rows and columns in this diagram are exact (except for the bottom one and the right most one), since for projective modules P , the functor P ⊗R (−) is exact. R Now, the module Tor1 (M,N) is given by the first homology of the right column, R and Tor1 (N,M) is given by the first homology of the bottom row, so we need to M construct a map between these homology groups. So let x ∈ ker(id ⊗R∂1 ). Since the P1 row of the diagram is exact, there exists an x01 ∈ Q0 ⊗R P1 which maps to x. Let x00 be the image of x01 in Q0 ⊗R P0. Since square (1) commutes, x00 is in the kernel of the horizontal map Q0 ⊗R P0 → N ⊗R P0, so there must be some x10 ∈ Q1 ⊗R P0 that maps to x00. Then we define ϕ(x) to be the image of x10 in

Q1 ⊗R M. Since square (2) commutes and x00 is the image of x01, ϕ(x) is in the N kernel of ∂1 ⊗R id. This deﬁnes a map ϕ : H1(N ⊗R P•) → H1(Q• ⊗R M). We should then check that the image of x under this map doesn’t change if we vary x by some M element of im(id ⊗R∂2 ). This is omitted. Finally we should check that is actually a module homomomorphism, which is not too hard to see so is also omitted.

To see that ϕ is an isomorphism, we just note that the construction of the map is entirely symmetrical and so the deﬁnition of the inverse is clear.

The proof is easily generalisable to other values of n, we just zig-zag down the diagram in the same way. To check naturality, the reader can probably imagine how this gets quite tedious, as the naturality square ends up being three-dimensional, so this is omitted as well. 2

n R Like ExtR(M,N), the R-module structure of Torn (M,N) can be induced by that of either M or N, and applying M ⊗R (−) to a short exact sequence results in a short exact sequence of chain complexes and so Tor satisﬁes the long exact sequence of homology as well.

39 4.4 Hilbert’s Syzygies Theorem

First proved by David Hilbert in 1890, his Syzygies Theorem is extremely useful in controlling the projective dimension of a module over a polynomial ring. It gives a finite upper bound, and perhaps not so surprisingly, this upper bound is precisely the number of indeterminates. We aim to prove this theorem in the restricted case of graded modules over the polynomial ring in two indeterminates. Definition 4.4.1. Let R = L R be a ring, with each R a subgroup of the i∈Z i i underlying abelian group of R. If the condition RiRj ⊆ Ri+j is satisfied, then R is called a graded ring. Let M = L M be an R-module, again with each M i∈Z i i a subgroup of the underlying abelian group of M. If the corresponding condition

RiMj ⊆ Mi+j is satisﬁed, then M is called a graded R-module. The module homomorphisms which we are interested in for graded modules are those which preserve degree. That is, if f : M → N is a homomorphism of graded

R-modules, then f is called graded if f(Mi) ⊆ Ni for all i. Given this notion we can then deﬁne a graded projective resolution of a module M, by insisting that each of the boundary maps be homogeneous. Often though, this is too restrictive, since in some cases the boundary maps shift the degree of a graded module by a constant amount. To allow for this, we deﬁne the degree shift of a module, M(n), by declaring that M(n)i = Mi+n. This notion will be explored in more detail in the next chapter.

Before we arrive at Hilbert’s Theorem, we need a Lemma due to Nakayama. We only prove this Lemma for the case of graded modules over the polynomial ring R = k[x, y], since this is the case we are interested in. The proof is identical for an arbitrary graded ring. The original statement is for arbitrary rings, the proof of which can be found in any commutative algebra textbook. The proofs of Lemma 4.4.2 and Corollary 4.4.3 have been adapted from [5] to the graded case.

In fact, many lemmas take the name of Nakayama’s Lemma, the statements of which are seemingly unrelated, but can all be proven from the following quite simple lemma. Lemma 4.4.2. Let R = k[x, y] let m = (x, y). If M is a ﬁnitely generated graded R-module such that mM = M, then M = 0.

Proof. Since M is ﬁnitely generated and R is graded over N. Let i be the smallest integer for which Mi 6= 0. Note that since M is ﬁnitely generated and R is graded over N, i 6= −∞. Then since m is positively graded, Mi cannot be a subset of mM, which is a contradiction, so no such i exists. That is, M = 0. 2

40 If we have some graded module M, then of course M/mM is a k-vector space. Furthermore, if M is finitely generated, then M/mM is finite dimensional. The following corollary of Nakayama’s lemma allows us to use this fact to find a minimal generating set of M. Corollary 4.4.3. With R as above, if M is a finitely generated graded R-module, and {m1, . . . , mr} is a set of homogeneous elements such that their images in the quotient form a basis for M/mM over k, then {m1, . . . , mr} is a minimal generating set for M.

Proof. Let N = Rm1 + ··· + Rmr be the submodule generated by {m1, . . . , mr}, so N is a graded submodule. Then certainly M = N + mM, and so m(M/N) = (mM + N)/N = M/N, so by Nakayama’s Lemma, M/N = 0, that is, M = N. Furthermore, this generating set is necessarily minimal, since the residues form a basis of M/mM. 2

This allows us to construct minimal free resolutions of a ﬁnitely generated module

M over R. We choose a set of homogeneous elements {mi} which project to a basis Lr for M/mM. Then we deﬁne F0 = i=1 R(ni), where mi ∈ M−ni . This gives a projection π : F0 → M such thatπ ¯ : F0/mF0 → M/mM is an isomorphism. We then perform the same construction on ker π, and continue in this way until we have a free resolution ∂3 ∂2 ∂1 π · · · → F3 −→ F2 −→ F1 −→ F0 −→ M → 0.

Then, by construction, the induced maps Fi/mFi → Fi−1/mFi−1 are all 0 (except for the one induced by π), since for each i we have an exact sequence

∂i Fi −→ Fi−1 → ker ∂i−2 → 0, where the map Fi−1/mFi−1 → ker ∂i−2/m ker ∂i−2 is an isomorphism, and so the induced map Fi/mFi → Fi−1/mFi−1 maps to the kernel of the following isomorphism, which is 0. This characterisation of minimal resolutions is only possible in the graded case. Theorem 4.4.4 (Hilbert’s Syzygies Theorem). Let R = k[x, y] and let M be a graded R-module. Then any graded free resolution has length at most 2.

41 Proof. We let P• → M be a minimal graded free resolution of M. Tensoring this ∼ resolution with k and using the natural isomorphism k ⊗R (−) = (−)/m(−) we see that k ⊗ P• is isomorphic as chain complexes to the chain complex

· · · → Pn/mPn → · · · → P2/mP2 → P1/mP1 → P0/mP0 → 0.

But since the resolution was chosen to be minimal, all these maps are 0, so the R R R Tori (M, k) are precisely Pi/mPi. But Tori (M, k) = Tori (k, M), so these modules can also be computed via a projective resolution of k, for example the (graded) Koszul complex,

(−y x)T (x y) 0 → R(−2) −−−−−→ R(−1) ⊕ R(−1) −−−→ R → k → 0.

R R So in particular, for i > 2, we have 0 = Tori (k, M) = Tori (M, k) = Pi/mPi.

So Pi = mPi, and by Nakayama’s lemma we conclude that Pi = 0 for all i > 2, completing the proof. 2

One of the many upshots of this theorem is the following corollary, which follows from the fact that the projective dimension of M is equal to the smallest d such d+1 that ExtR (M,N) = 0 for all N. Corollary 4.4.5. For any modules M, N over the polynomial ring k[x, y], we have n ExtR(M,N) = 0 for all n ≥ 3.

42 Chapter 5

Grothendieck’s Splitting Theorem

We come now to the proof of the Splitting Theorem in the context of a particular quotient of an abelian category. Since Grothendieck’s Splitting Theorem pertains to vector bundles on the projective line, we must set up the appropriate algebraic context to discuss the geometry of the projective line. How this context applies to the geometry will be explored in more detail in the next chapter, but the general idea is that we construct the projective line as the set of 1-dimensional subspaces 2 × of Ak modulo the action of non-zero scaling. In other words, we consider the k 2 orbits of the scaling action on Ak − 0. As brieﬂy explained in Chapter 3, localising 2 to the open subset Ak − 0 can’t be done using ring localisation, and so quotient categories must be appealed to. It turns out that to capture the group action of k×, it’s appropriate to consider the category of graded modules over the coordinate ring of the plane.

5.1 The Category of Graded k[x, y]-modules

In the last chapter we introduced the notion of a graded module and a graded homomorphism. Here we treat this idea in more detail, and consider some examples.

The classic example of a graded ring is the polynomial ring R = k[x1, . . . , xn]. In degree i, Ri is the subgroup generated over k by degree i monomials. We are mainly interested in the case of the polynomial ring in two variables, R = k[x, y]. Example 5.1.1. R = k[x, y] is of course itself a graded R-module. Example 5.1.2. If M is a graded module over R, and S is a multiplicatively closed subset of S consisting of homogeneous elements, then the localisation S−1M is a graded module. If m ∈ M has degree i and s ∈ S has degree j, then m/s has degree i − j.

Example 5.1.3. Any ring R can be given the trivial grading R0 = R, and Ri = 0 for i 6= 0. In this case any R-module can be graded with the same trivial grading.

43 When graded modules were introduced in the context of graded resolutions, we required a homomorphism to preserve the grading of the module. In fact we can relax this condition slightly, by appealing to the notion of the degree shift, deﬁned in the last chapter. Deﬁnition 5.1.4. Let M,N be graded modules over a graded ring R. If f : M → N is an R-module homomorphism such that f(Mi) ⊆ Ni+d for all i, then f is said to be homogeneous of degree d.

The graded morphisms which were deﬁned earlier are precisely the homogeneous homomorphisms of degree 0. Let f : M → N be homogeneous of degree d. Then f : M → N(d) is graded. Indeed, for any i, we have f(Mi) ⊆ Ni+d = N(d)i. Given a graded R-module M, and a submodule N ≤ M, we say that N is a graded submodule of M if N = L M ∩ N, and in this case, the quotient module M/N i∈Z i has a natural grading, since M/N = L M /M ∩ N. i∈Z i i This discussion suggests that for a graded ring R, we have a category, R -Gr of graded R-modules together with the degree 0 homogeneous homomorphisms. Proposition 5.1.5. The category R -Gr is abelian.

Proof. Since any morphism in R -Gr is in particular a ring homomorphism, and R -Mod is abelian, it suﬃces to show that for a morphism f : M → N in R -Gr, the kernel and cokernel of f both have a natural grading induced by the grading of M and N and the map f. Clearly f(M) = L f(M ), so f(M) is a graded submodule i∈Z i of N, and by the above, coker f = N/f(M) is graded and hence in R -Gr. We must show that ker f = L M ∩ ker f. Let a ∈ ker f. In particular a ∈ M so we can i∈Z i write a = a1 + ··· + an for ai ∈ Mi. Then 0 = f(a) = f(a1) + f(a2) + ··· + f(an).

But since f is graded, f(ai) ∈ Ni for all i, and since the sum is direct, we conclude that each f(a ) = 0 and so a ∈ M ∩ ker f. Thus a ∈ L M ∩ ker f. The i i i i∈Z i reverse inclusion is obvious, so we have equality. Furthermore, the isomorphism M/ ker f → im f is induced by f and so is certainly graded. 2

One important aside to mention here is that if a graded R-module M is finitely generated, then each Mi is necessarily finite dimensional over k. This is not hard to see, since Ri is finite dimensional over k for all i, it would not be possible to generate an infinite k-basis for Mi using a finite generating set. Since we wish to use homological algebra to study graded modules, we should be careful about the distinction between the functors HomR(M, −) and HomR -Gr(M, −). If M is finitely generated there is at least a nice connection between the two.

44 Lemma 5.1.6. If M is a ﬁnitely generated graded R-module, then

M HomR(M,N) = HomR -Gr(M,N(i)). i∈Z

Proof. It is clear that L Hom (M,N(i)) ⊆ Hom (M,N), so we prove the i∈Z R -Gr R other inclusion. Suppose m1, . . . , mr is a homogeneous generating set for M of degrees deg mi = di, and let f ∈ HomR(M,N). Then each f(mi) can be written Pki as a ﬁnite sum f(mi) = j=1 nij, where deg nij = hij. Then deﬁne fij : M → P N(hij − di) by fij(mi) = nij and fij(m`) = 0 for ` 6= i. Then f = i,j fij ∈ L Hom (M,N(i)), since even though some f ’s may have the same degree, i∈Z R -Gr ij the sum of two degree d homomorphisms is a degree d homomorphism. Thus we have equality. 2

n If we denote Ext(i)(−,N) to be the right derived functor of HomR -Gr(−,N(i)), we n n have a similar relation between ExtR(M,N) and Ext(i)(M,N). Proposition 5.1.7. If M is a ﬁnitely generated graded R-module, then for all n,

n M n ExtR(M,N) = Ext(i)(M,N). i∈Z

n Proof. Take a projective resolution P• of M. Then ExtR(M,N) is given by the n nth homology of HomR(P•,N), while Ext(i)(M,N) is given by the nth homology of HomR -Gr(P•,N(i)). If we denote the boundary maps of HomR(P•,N) by ∂• and (i) (i) those of HomR -Gr(P•,N(i)) by ∂• , we note that ker ∂n = ker ∂n ∩ HomR(Pn,N), so the result follows from the previous proposition. 2

In particular, for ﬁnitely generated M, Hom(M,N) and Extn(M,N) are both graded R-modules, and hence objects in R -Gr.

5.2 Torsion in Graded Modules

Recall that given an element m of an R-module M, we let the annihilator of m be the ideal ann(m) = {r ∈ R : rm = 0}.

Of course, 0 ∈ ann(m) for any m. We say that an R-module M is a torsion module if ann(m) 6= 0 for all m ∈ M.

In the case of graded modules, another notion of torsion is of interest. As usual, we let R = k[x, y] and m = (x, y). An R-module M is called graded torsion or m

45 -torsion if for all m ∈ M, there exists an n > 0 such that mn ⊆ ann(m). In other words, M is a m-torsion module if for all m ∈ M, there exists n such that mnm = 0.

This is of course a diﬀerent notion to the usual deﬁnition of torsion, and care should be taken from here, since both notions are used in this chapter. For the sake of clarity, the usual notion of torsion will be termed R-torsion, and the graded notion will be termed m-torsion. Of course, m-torsion implies R-torsion, but not conversely, for example M = k[x, y]/(y) is R-torsion since everything is annihilated by y, but not m-torsion since nothing is annihilated by x.

We denote the full subcategory of R -Gr of m-torsion modules by m -Tors. This is a Serre subcategory of R -Gr. Indeed, it is non-empty since 0 is certainly m-torsion. Suppose f g 0 → M 0 −→ M −→ M 00 → 0 is a short exact sequence and ﬁrstly assume M ∈ m -Tors. Let m ∈ M 0. Then f(m) ∈ M is m-torsion, so there exists n such that mnf(m) = 0. Thus f(mnm) = 0, so since f is injective, mnm = 0, and M 0 ∈ m -Tors. Now let m ∈ M 00. So there exists x ∈ M such that g(x) = m. So there is also some n0 such that mn0 x = 0, so since mn0 m = mng(x) = g(mnx) = g(0) = 0, we conclude that M 00 ∈ m -Tors.

Now suppose M 0,M 00 ∈ m -Tors. Since M 00 ∼= M/M 0, and M 00 is m -Tors, for any m ∈ M, there is some n such that mnm ⊆ M 0, and so since M 0 ∈ m -Tors, there is some n0 such that mn0 mnm = 0, so M ∈ m -Tors.

So we can deﬁne the quotient category Q = R -Gr /m -Tors. In fact, m -Tors is a localising subcategory. Here we brute force the proof of this by taking a module M and showing that the only object that ωM could be equal to, actually exists. Certainly, any module has a maximal m-torsion submodule, so by replacing M with M/t(M), we may assume that M is m-torsion free. Then we have

ωM = HomR(R, ωM) M = HomR -Gr(R, ωM(d)) d∈Z M = HomQ(πR, M(d)) d∈Z M n = lim HomR -Gr(m ,M(d)) n→∞ d∈Z M n = lim HomR -Gr(m ,M(d)) n→∞ d∈Z n = lim HomR(m ,M), n→∞

46 where we have used the fact that R and mn are both ﬁnitely generated over R, the adjoint property, and the fact that arbitrary colimits commute (the direct limit and n ∞ the direct sum). We have also used the fact that the set {(m ,M)}n=0 is coﬁnal in

IR,M . To see this, suppose that J ≤ R is such that R/J ∈ m -Tors. Then since R/J is generated by 1 + J, and there must be some N such that mN (1 + J) = 0, that N N is, m ≤ J, so (J, M) ≤ (m ,M) in IR,M , and since M is m-torsion free, the only torsion submodule of M is 0. Thus, since R -Gr has all direct limits, we conclude that ωM is well-deﬁned.

In fact, we can say more since M can be taken m-torsion free. Let ϕ : mn → M be such that ϕ(r) = 0 for all r ∈ mn+1. Then if s ∈ mn, sϕ(x) = ϕ(sx) = 0, so ϕ(x) must be m-torsion. But M is m-torsion free, so ϕ(x) = 0, and similarly n n+1 ϕ(y) = 0, so ϕ = 0. Thus all the maps HomR(m ,M) → HomR(m ,M) are injections, so the disjoint union/equivalence formulation of the direct limit implies S∞ n ωM = n=0 HomR(m ,M). The objects of R -Gr can often be thought of as graded R-modules, but of course, it is not that simple. As an example, consider the following construction. Given a graded R-module, M, we denote by M>n the graded module such that (M>n)i = Mi for i > n and (M>n)i = 0 for i ≤ n. This is called the n-th tail of M. Lemma 5.2.1. If M is a ﬁnitely generated graded R-module, then M ∈ m -Tors if and only if M>n = 0 for some large enough n.

Proof. If M >n = 0 for some n, then clearly mn+1M = 0, so M ∈ m -Tors. In fact this direction holds even if M is not ﬁnitely generated.

Now suppose that M ∈ m -Tors. Let {m1, . . . , mr} be a set of homogeneous gener- N ators for M with degrees ni respectively. Let N be large enough so that m mi = 0 for all i ∈ {1, . . . , r}. Let d = maxi ni, and let m ∈ Mk where k ≥ Nd. Then Pr m = i=1 aimi. Each aimi has degree Nd, which means that ai must be homogeneous of degree at least N, but then aimi = 0 and so m = 0. Thus M>Nd−1 = 0, as required. 2

This Lemma has an interesting consequence. Firstly note that for any module M, and for n > 0, the exact sequence

0 → M>n → M → M/M>n → 0

∼ ∼ implies that M>n = M in Q, since M/M>n is clearly m-torsion. Now, if M>n = N>n for some n, then clealy M ∼= N in Q. Conversely, for ﬁnitely generated M and N, if M ∼= N in Q, then M/N ∼= 0 in Q, so since M/N is ﬁnitely generated, the Lemma

47 implies (M/N)>n = 0 for some n. But then M>n = N>n, so the image of a ﬁnitely generated module in Q is completely determined by its tail.

Another way in which Q diﬀers from R -Gr is that projective modules may no longer be projective in the quotient. This is a consequence (of course) of the deﬁnition of the Hom-sets in Q. If p : M → N is an epimorphism in Q, and we have some map q : R → N, it is not always possible to lift q through p to get a map from R to M, because although R is projective in R -Gr, m is not, and q might actually be the image of some map out of m instead of R. So it is certainly not immediate that R is projective in Q, and it turns out that it is indeed not—and as a consequence of Lemma 5.3.3 in the next section, this implies that all the projective modules in R -Gr become unprojective in Q.

5.3 The Splitting Theorem

As usual, in this section R = k[x, y], m = (x, y), and Q = R -Gr /m -Tors. Theorem 5.3.1 (Splitting Theorem). Let M ∈ Q be a noetherian object. If M is ∼ torsion free, then M = R(n1) ⊕ R(n2) ⊕ · · · ⊕ R(nr). To get a sense of the diﬃculty of a theorem like this, consider the module

x2y − xy2  x10 + 3x8y2   x   3   3 7 8 2 4 6   M = R  y  + R x y − x y + x y  + R  y  ≤ R ⊕ R ⊕ R. x3 + 2y3 x5y5 x + y

Being a submodule of R3, it is certainly torsion free, and it is ﬁnitely generated so is noetherian, so it must be isomorphic to a direct sum of shifts of R. But this is not at all obvious. For a given module M, the main question that arises here is how to determine the correct graded free module. We will show that the correct module ∼ to choose is ωM, where ω is the section functor associated to Q. Since πω = idQ (Proposition 3.3.3), if ωM ∼= ωN in R -Gr, then we may conclude that M ∼= N in Q.

The general idea of the proof is to split up into cases based on the projective dimension of M, considered as an object of R -Gr. Hilbert’s Syzygies Theorem, 4.4.4 implies that we only have three cases to check. First though, we need to know what the noetherian objects of Q look like. The proof that π preserves noetherian objects has been adapted from [14]. Proposition 5.3.2. The full subcategory of Q of noetherian objects, which we will denote QN , is equivalent to the quotient R -gr /m -tors, where R -gr is the full subcategory of R -Gr of graded noetherian modules.

48 Proof. Let M ∈ R -gr, and suppose that

N1 ,→ N2 ,→ N3 ,→ · · · be an ascending chain of subobjects of πM, in Q. Then since ω is a left adjoint, it is left exact and so

ωN1 ,→ ωN2 ,→ ωN3 ,→ · · · is a chain of subobjects of ωπM, and so intersecting with M we have

ωN1 ∩ M,→ ωN2 ∩ M,→ ωN3 ∩ M,→ · · · is a chain of submodules of M, so since M is noetherian, there exists an i such that ∼ ωNi ∩ M = ωNi+1 ∩ M = ··· . Then since πω = idQ, we have Ni = πωNi ∩ πM =

π(ωNi ∩M) = π(ωNi+1 ∩M) = πωNi+1 ∩πM = Ni+1. So πM is also noetherian, so we have a functor R -gr → QN . The kernel of this map is clearly m -tors, the full subcategory of R -gr of noetherian m-torsion modules. So by the universal property we get a faithful exact functor H : R -gr /m -tors → QN , so we just need to check essential surjectivity and fullness.

Suppose M ∈ Q is noetherian. Take m1, . . . , mi ∈ M and consider Rm1 + Rm2 + ∼ ··· + Rmi ⊆ M in Q. If M/(Rm1 + Rm2 + ··· + Rmi) = 0 in Q, then M =

H(Rm1 + Rm2 + ··· + Rmi). Otherwise, there exists mi+1 whose residue is not m-torsion in M/(Rm1 + Rm2 + ··· + Rmi). We then repeat the process with

Rm1 + Rm2 + ··· + Rmi + Rmi+1. This gives an ascending chain of subobjects of

M in Q, so it must terminate at some r, that is M/(Rm1 + ··· + Rmr) = 0 in Q, so H(Rm1 + ··· + Rmr) = M. Thus, H is essentially surjective. For M ∈ R -gr, HM = M, so we just need to check that

HomR -gr /m -tors(M,N) = HomQN (M,N) = HomQ(M,N).

But this is trivial since any subobject of a noetherian object is necessarily noetherian, so the direct limits in each case are taken over the same directed set. 2

We now come to the beginning of the series of Lemmas and Propositions which will come together to form the proof of the Splitting Theorem. We begin by showing that for graded modules over k[x, y], projective and free are equivalent notions. So actually Hilbert’s Syzygies Theorem can be restated in terms of projective resolutions instead of free ones, since the notions are actually equivalent.

49 Lemma 5.3.3. Let M be a ﬁnitely generated graded R-module with projective dimension 0. Then M is graded free, that is,

∼ M = R(n1) ⊕ R(n2) ⊕ · · · ⊕ R(nr).

Proof. Since M is ﬁnitely generated, we can ﬁnd some minimal projection onto M as follows: ϕ 0 → ker ϕ → P −→ M → 0.

But since M is projective, there exists a homomorphism ψ : M → P such that

ϕψ = idM . In other words, the short exact sequence above splits, so P = M ⊕ker ϕ. But the mapϕ ¯ : P/mP → M/mM induced by ϕ is an isomorphism, so 0 = kerϕ ¯ = (ker ϕ + mP )/mP , so ker ϕ ⊆ mP . So we have,

P = ker ϕ + M ⊆ mP + M.

But mP + M ⊆ P , so in fact m(P/M) = (mP + M)/M = P/M, so Nakayama’s Lemma implies that M = P and we are done. 2

Note here that this isomorphism holds in R -Gr, so in the projective case, this lemma is actually stronger than the statement of Theorem 5.3.1. This lemma is however the key to proving the theorem. We will show that ωM necessarily has projective dimension 0, regardless of the projective dimension of M, which will be sufficient by Proposition 3.3.3. To proceed here we need more information about the projective dimension of a graded k[x, y]-module, and to that end we prove the Auslander-Buchsbaum Theorem in the specific case of R = k[x, y]. Definition 5.3.4. The depth d of an R-module M is the smallest integer such that Extd(k, M) 6= 0. Theorem 5.3.5 (Auslander-Buchsbaum). If M is a finitely generated graded R- module with projective dimension p and depth d, then p + d = 2.

This is a very speciﬁc restatement of the Auslander-Buchsbaum theorem. Most sources state this theorem for modules over local rings, but it actually holds for graded modules over an arbitrary graded rings. We prove it speciﬁcally for the case R = k[x, y] by looking at each dimension individually. The techniques here can be generalised to the polynomial ring in more than 2 indeterminates, but in that case a more general proof may be preferred. Ideas for the proofs of the following Lemmas were taken from [1].

50 Proposition 4.3.4 implies that the depth of R is 2, so the previous proposition takes n care of the pd M = 0 case by the additivity of ExtR(k, −). The following two Lemmas prove the case pd 1 and pd 2 respectively. Lemma 5.3.6. Let M be a ﬁnitely generated R-module with projective dimension 1 1. Then HomR(k, M) = 0 and ExtR(k, M) 6= 0.

Proof. Since pd M = 1, we have a minimal graded projective resolution

∂1 ∂0 0 → P1 −→ P0 −→ M → 0.

Applying the functor HomR(k, −) to this results in the following long exact sequence:

0 → HomR(k, P1) → HomR(k, P0) → HomR(k, M) → 1 1 1 → ExtR(k, P1) → ExtR(k, P0) → ExtR(k, M) → 2 2 2 → ExtR(k, P1) → ExtR(k, P0) → ExtR(k, M) → 0, terminating at n = 2 by Corollary 4.4.5. Now, since P0 and P1 are projective and hence by Lemma 5.3.3 are isomorphic to a direct sum of shifts of R, by Proposition 1 4.3.4 and the additivity of Hom and Ext, HomR(k, Pi) = 0 = ExtR(k, Pi). So we have

0 → HomR(k, M) → 0, so HomR(k, M) = 0. We also have

1 2 f 2 2 0 → ExtR(k, M) → ExtR(k, P1) −→ ExtR(k, P0) → ExtR(k, M) → 0.

Since P1 and P0 are both direct sums of shifts of R, ∂1 is a matrix with entries in R. ¯ But by the minimality of the resolution, ∂1 : P1/mP1 → P0/mP0 is zero, so in fact 1 ∂1 is a matrix with entries in m. So, since f is induced by ∂1 and ExtR is additive, f is given by multiplication by this same matrix, so since the module structure on 1 1 ExtR(k, Pi) is induced by the module structure on k, ExtR(k, Pi) is annihilated by 1 2 m, so f is zero. Thus, ExtR(k, M) = ker f = ExtR(k, P1), which is non-zero by Proposition 4.3.4. 2

51 Lemma 5.3.7. Let M be a ﬁnitely generated R-module with projective dimension 0 2. Then ExtR(k, M) = HomR(k, M) 6= 0.

Proof. By taking a minimal graded projection onto M, we have

∂0 0 → C → P0 −→ M → 0,

∼ where P0 is projective and P0/mP0 = M/mM, and C is the kernel of the projection onto M. Now, since C = ker ∂0, if

· · · → Pn → · · · → P2 → P1 → C → 0, is a minimal graded projective resolution of C, then clearly

· · · → Pn → · · · → P2 → P1 → P0 → M → 0 is a minimal graded projective resolution of M. So, since the projective dimension of a module is the length of its minimal projective resolution, in this case we have 1 pd C = pd M − 1 = 1. Thus ExtR(k, C) 6= 0 by Lemma 5.3.6. We now apply

HomR(k, −) to the above sequence to obtain:

0 → HomR(k, C) → HomR(k, P0) → HomR(k, M) → 1 1 1 → ExtR(k, C) → ExtR(k, P0) → ExtR(k, M) → · · ·

1 But we know that HomR(k, P0) = 0 = ExtR(k, P0), so we have

1 0 → HomR(k, M) → ExtR(k, C) → 0.

∼ 1 Thus, HomR(k, M) = ExtR(k, C) 6= 0. 2

This completes the proof of the Auslander-Buchsbaum Theorem for R = k[x, y]. Note in the previous proof we used the fact that pd C = pd M − 1. Provided that pd M > 0, this argument holds for any dimension, so it is easy to see how this argument can generalise to an induction argument on the dimension of M.

n Now let M be an object in Q. Since ωM = lim HomR(m ,M), and since we can n→∞ assume that M is m-torsion free (by replacing M with M/t(M)), we have a sequence

2 M,→ HomR(m,M) ,→ HomR(m ,M) ,→ · · · ,→ ωM.

The following lemma highlights the relationship between each of these Hom-sets.

52 Lemma 5.3.8. For any M ∈ Q we have

(i) i (i−1) M := HomR(m ,M) = {η ∈ ωM : mη ⊆ M } for all i ∈ N.

Proof. As usual, we replace M with M/t(M) if necessary and thus assume that M is m-torsion free.

First we must show that for any i,

i ∼ HomR(m ,M) = HomR(m, HomR(m, HomR(m, ··· HomR(m,M)) ··· ), where the right hand side here means we apply the functor HomR(m, −) i times to i M. For brevity, denote this instead by HomR(m,M). We prove this by induction n ∼ n on i. The case i = 1 is clear. So suppose that HomR(m ,M) = HomR(m,M) for all i+1 i ∼ i n ≤ i. Then, HomR (m,M) = HomR(m, HomR(m,M)) = HomR(m, HomR(m ,M)) by induction, so it suﬃces to show that

i ∼ i+1 HomR(m, HomR(m ,M)) = HomR(m ,M).

By the Tensor-Hom adjunction, we have

k ∼ k HomR(m, HomR(m ,M)) = HomR(m ⊗R m ,M),

i ∼ i+1 so it suﬃces to show that HomR(m ⊗R m ,M) = HomR(m ,M). This isn’t true in general, however it is true in this case since M is m-torsion free. Consider the sequence 0 → m → R → k → 0, and tensor this with mi. Using the isomorphism R ⊗ mi ∼= mi, and the fact that

R ⊗R (−) is exact, we have the sequence

R i i c i i 0 → Tor1 (k, m ) → m ⊗ m −→ m → k ⊗R m → 0.

The map c is given by r ⊗ r0 7→ rr0, so im c = mi+1, so we have a surjective map i i+1 R i i ∼ i+1 c : m ⊗ m → m with ker c = Tor1 (k, m ). So since m ⊗ m / ker c = m , there is i+1 i an isomorphism between HomR(m ,M) and the submodule of HomR(m ⊗ m ,M) R i consisting of all maps which send ker c to 0. But since ker c = Tor1 (k, m ) is i annihilated by m, and M is m-torsion free, any map in HomR(m ⊗ m ,M) must send ker c to 0, so this isomorphism is in fact the desired one.

53 (i) (i−1) (i−1) In particular, M = HomR(m,M ) for all i, and since each M is m-torsion (i−1) free, if we apply HomR(−,M ) to the exact sequence

0 → m → R → k → 0, we get an exact sequence

(i−1) (i) g 1 (i−1) 0 → M → M −→ ExtR(k, M ) → 0.

Now we are ﬁnally in a position to prove the result. Let η ∈ M (i), so η : mi → M. 1 (i−1) Then in the above short exact sequence, g(η) ∈ ExtR(k, M ), so mg(η) = 0. Thus mη ⊆ ker g = M (i−1).

Now suppose η ∈ ωM is such that mη ⊆ M (i−1). Then η deﬁnes a map η : m → (i−1) (i−1) (i−1) (i) M by m 7→ mη ∈ M . So η ∈ HomR(m,M ) = M , as required. 2

The following proposition is the key to the proof of the splitting theorem. Proposition 5.3.9. If M is R-torsion free with projective dimension 1, then ωM has projective dimension 0.

i Proof. Firstly note that if M is R-torsion free, then HomR(m ,M) is R-torsion free S∞ n for all i. Consequently, ωM = n=0 HomR(m ,M) is R-torsion free. We have a sequence

M,→ M (1) ,→ M (2) ,→ · · · ,→ ωM,

(i) i (i−1) 1 (i) where M = HomR(m ,M) = {η ∈ ωM : mη ⊆ M }. Since ExtR(k, M ) is annihilated by m, it is finite dimensional, so M (i)/M (i−1) is finite dimensional. (i) (i−1) Let di denote the maximum degree of any homogeneous element m ∈ M /M . (i) (i−1) If M /M 6= 0, then di < di−1. To see this, suppose not, so there is some (i) (i−1) (i−1) (i−2) m ∈ M − M with deg m ≥ di−1. Then mm ⊆ M . If mm 6⊆ M , then 0 (i−2) 0 there is some m ∈ mm − M , but deg m > deg m ≥ di−1, so the residue of 0 (i−1) (i−2) m in M /M is non-zero of degree larger than di−1, which is impossible. So mm ⊆ M (i−2), but by definition this implies that m ∈ M (i−1), a contradiction. So di < di−1. Now suppose that M (i) has projective dimension 1 for all i. Then M (i+1)/M (i) = 1 (i) Ext (k, M ) 6= 0. So 0 6= ki+1 < ki for all i. Let x ∈ N. Then there is some (n) n with kn < −x, so there exists z ∈ M ⊆ ωM with −d = deg z ≤ −x. But ∼ since ωM is torsion free, Rd+nz = Rd+n which has dimension d + n + 1 over k. But

54 Rd+nz ⊆ (ωM)n, which is the same as Mn, provided n > k1. But M is finitely generated and so Mn must be finite dimensional, so by taking x and thus d large enough, we find a contradiction. So there exists i such that M (i) does not have projective dimension 1. But M (i) ⊆ ωM and so is torsion free, implying M (i) must 1 (i) have projective dimension 0. Thus since ExtR(k, M ) = 0, (Lemma 5.3.3) the exact sequence (i) (i+1) 1 (i) 0 → M → M → ExtR(k, M ) → 0, implies that M (i) ∼= M (i+1) ∼= M (i+2) ∼= ··· ∼= ωM. So ωM has projective dimension 0. 2

We can now ﬁnally prove the theorem.

Proof of Theorem 5.3.1 Since M is noetherian, we can assume that M is ﬁnitely generated. Then Hilbert’s Syzygies Theorem implies that pd M ≤ 2. Thus, if ∼ • pd M = 0, then Lemma 5.3.3 implies M = R(n1) ⊕ · · · ⊕ R(nr) in R -Gr, and thus also in Q.

• pd M = 1, then Proposition 3.3.3 together with Proposition 5.3.9 and Lemma ∼ ∼ 5.3.3 imply that M = ωM = R(n1) ⊕ · · · ⊕ R(nr) in Q. • pd M = 2, then Lemma 5.3.7 implies that if pd M = 2, then M must have torsion, but we assume M is torsion free. 2

Thus we conclude the proof of our algebraic reformulation of Grothendieck’s Split- ting Theorem. As suggested before we proved the Auslander-Buchsbaum Theorem, many of the techniques in this section can be generalised to the polynomial ring in more than two indeterminates—however, there are more cases to check here, since for R = k[x1, . . . , xn], the projective dimension of a module is only bounded above by n, and so it is slightly too optimistic to suspect that in this case ωM would have the same form for each projective dimension. In fact it doesn’t, since higher dimensional projective space does not have a corresponding structure theorem for vector bundles. In any case, study of the quotient category for the case R = k[x1, . . . , xn] n would certainly give insight into Pk . This leads nicely into the next chapter, where we brieﬂy discuss this insight in the case n = 2.

55 Chapter 6

Applications to Geometry

In this chapter, we provide a brief introduction to the geometry of the projective line and how the category explored in the last chapter applies to this. In particular we 1 define vector bundles and quasi-coherent sheaves on Pk. The definitions presented here are not the usual ones. Firstly, they are specific to the projective line, but they are presented in such a way which should make the generalisation to other spaces clear. Secondly, they depend on a choice of coordinate chart, which is generally undesirable, but it simplifies the treatment. We also state many facts without proof, since this chapter is intended to be nothing more than a brief overview of the 1 geometry. We do however prove in detail the equivalence between Q and QCoh(Pk).

6.1 The Projective Line

Intuitively, the real projective line is the real line together with a so called ideal point at infinity. There are multiple ways of defining rigorously what the projective line actually is, and depending on the context, one definition may be more useful than another. We will give two definitions, and use them both at different times as it suits.

1 The most common way of defining the projective line Pk is as the set of 1-dimensional 2 subspaces of AX,Y . In this way, the lines with non-zero gradient can be identified 1 with the usual points of Ak by, say, identifying them with the point of intersection with some fixed line, usually Y = 1. The point at infinity then corresponds to the line Y = 0 which of course is the only 1-dimensional subspace which doesn’t intersect Y = 1.

1 Deﬁnition 6.1.1. The projective line Pk is deﬁned to be the set of all 1-dimensional 2 1 2 × 2 subspaces of Ak, that is Pk = (Ak −0)/k , the set of all non-zero points of Ak modulo the action of non-zero scalars in k.

56 1/X

1 1 We may define a map ϕX : AX → Pk by sending X to the subspace defined by 1 1 (X, 1). This map is clearly injective, so AX may be identified with a subset of Pk, but it is not surjective since the subspace Y = 0 is not in the image.

1 1 In a similar way we deﬁne a map ϕY : AY → Pk by sending Y to the subspace deﬁned by (1,Y ). Once again this is injective but not surjective; in this case we 1 miss the line X = 0. However, these injections suggest that we can view Pk as the 1 1 1 union AX ∪ AY . Formally these two lines are disjoint, but their images in Pk are not, so we should describe how we glue them together.

2 Let p ∈ im ϕX ∩ im ϕY . So p is a 1-dimensional subspace of Ak which intersects both the line Y = 1 and the line X = 1. Suppose it intersects Y = 1 in the point (X, 1). Then from the diagram above it is not diﬃcult to see that it intersects the line X = 1 in the point (1, 1/X). This gives the second deﬁnition.

1 Deﬁnition 6.1.2. The projective line Pk is the one-dimensional manifold given by 1 1 −1 1 1 1 charts {AX , AY } with transition map X 7→ X . In other words Pk = AX ∪ AX−1 .

6.2 Quasi-Coherent Sheaves

What follows is a very loose introduction to the theory of sheaves on the projective line. The deﬁnitions given are not the usual ones, as they depend on a particular coordinate chart, but they will be suﬃcient for expository purposes.

n Deﬁnition 6.2.1. A quasi-coherent sheaf on Ax1,...,xn is simply a k[x1, . . . , xn]- module.

1 1 Since Pk can be covered by aﬃne lines, we may deﬁne a quasi-coherent sheaf on Pk 1 in terms of these charts. That is, we put a k[X]-module on the part covered by AX −1 1 and a k[X ]-module on the part covered by AX−1 , adding a condition that these modules agree somehow on the intersection of the covers. More precisely, we have:

57 1 0 ∞ Deﬁnition 6.2.2. A quasi-coherent sheaf on Pk is a triple (M ,M , θ), where M 0 is a k[X]-module, M ∞ is a k[X−1]-module, and θ : M 0[X−1] → M ∞[X] is a k[X,X−1]-module isomorphism.

This of course makes sense, since localising M 0 at X−1 turns it into a k[X,X−1]- module, and likewise M ∞[X] is the localisation at (X−1)−1, which is also a k[X,X−1]- module. The isomorphism θ is called the gluing map or transition function, and this is where a lot of the information in the sheaf is stored since we can have two sheaves with the same modules but diﬀerent gluing maps be non-isomorphic (as sheaves). Example 6.2.3. Let M 0 = k[X] and M ∞ = k[X−1]. Then M 0[X−1] = k[X,X−1] and M ∞[X] = k[X,X−1], so θ can be given by multiplication by an invertible element of k[X,X−1]. Take θ to be multiplication by X−i, for any i ∈ Z. This is the line bundle O(i).

1 0 ∞ 0 ∞ Given two quasi-coherent sheaves on Pk,(M ,M , θM ) and (N ,N , θN ) a mor- 0 ∞ 0 ∞ phism ϕ :(M ,M , θM ) → (N ,N , θN ) consists of a k[X]-module homomorphism 0 0 −1 ∞ ∞ ϕ0 : M → N , and a k[X ]-module homomorphism ϕ∞ : M → N such that the following diagram commutes:

M 0[X−1] θM M ∞[X]

−1 ϕ0[X ] ϕ∞[X]

N 0[X−1] θN N ∞[X]

0 ∞ 0 ∞ 0 ∞ 0 ∞ If ϕ :(M ,M , θM ) → (N ,N , θN ) and ψ :(N ,N , θN ) → (S ,S , θS) are two sheaf morphisms, it is straightforward to see that the composition ψϕ given by (ψϕ)0 = ψ0ϕ0 and (ψϕ)∞ = ψ∞ϕ∞ is a sheaf morphism, since the composition of module homormorphisms is again a module homomorphism, and the diagram below commutes since each square commutes by assumption.

M 0[X−1] θM M ∞[x]

−1 ϕ0[X ] ϕ∞[X]

N 0[X−1] θN N ∞[x]

−1 ψ0[X ] ψ∞[X]

S0[X−1] θS S∞[X]

58 1 Deﬁnition 6.2.4. The category QCoh(Pk) has objects the quasi-coherent sheaves 1 on Pk and morphisms the sheaf morphisms as deﬁned above.

1 Since the objects of QCoh(Pk) are defined in terms of modules and gluing isomorphisms, it would be reasonable to expect this to be an abelian category, and indeed 1 it is. We will not prove this directly—instead we will prove that QCoh(Pk) is exactly equivalent to a category known to be abelian. Theorem 6.2.5. The two functors Φ and Ψ just defined are inverse equivalences, 1 and so the categories QCoh(Pk) and Q are equivalent. This theorem is one of two main results of this chapter and as such is quite long and technical so we have split it up into three sections. We first define the equivalence Φ and its inverse Ψ, and then prove first that ΦΨ(V ) ∼= V naturally in V = (M 0,M ∞, θ), and finally that ΨΦ(M) ∼= M naturally in M ∈ Q.

1 The functor Φ: Q → QCoh(Pk). Let M ∈ k[x, y]-Gr. Then M[x−1] is a graded k[x, y, x−1]-module (with the natural grading, ie, x−1 has degree −1), and M[y−1] is a k[x, y, y−1]-module. Define M 0 := −1 −1 0 M[y ]0 to be the degree 0 part of M[y ]. Then M is a k[x/y]-module, since ∞ −1 x/y has degree 0. Similarly, defining M := M[x ]0 gives a k[y/x]-module. Then replacing x/y with X, we have M 0 a k[X]-module and M ∞ a k[X−1]-module. We now just need to find a transition function θ : M 0[X−1] → M ∞[X]. But 0 −1 0 −1 −1 −1 ∞ ∞ then M [X ] = M [y/x] = M[y ]0[y/x] = M[y , x ]0 and M [X]M [x/y] = −1 −1 −1 M[x ]0[x/y] = M[x , y ]0, so we can just take θ = id. Let ϕ : M → N be a graded homomorphism. Then since ϕ is graded it induces a map M 0 → N 0 and M ∞ → N ∞ which obviously commutes with the transition functions since we take them to be identities. Moreover, since localisation is an exact functor, this 1 defines an exact functor F : k[x, y]-Gr → QCoh(Pk). Also, ker F = m -Tors, so 1 the universal property induces an exact and faithful functor Φ : Q → QCoh(Pk).

1 The functor Ψ : QCoh(Pk) → Q. Let (M 0,M ∞, θ) be a quasi-coherent sheaf on 1. Deﬁne M 0[y, y−1] = L M 0yd. Pk d∈Z This is a k[X, y, y−1]-module, which, by considering the identity yX = x, is actually a k[x, y, y−1]-module. Similarly, we deﬁne M ∞[x, x−1] = L M ∞xd, which is a d∈Z k[x, y, x−1]-module. Now consider the map

(−θ, id) : M 0[y, y−1] ⊕ M ∞[x, x−1] → M ∞[x, x−1, y, y−1].

Then strangely enough we deﬁne S(M 0,M ∞, θ) := K = ker(−θ, id). The reason for this is that we are attempting to somehow take an intersection of these two

59 very diﬀerent objects, so the kernel here actually does a nice job. Here K inherits 0 ∞ its grading from the direct sum. Then given a morphism ϕ :(M ,M , θM ) → 0 ∞ (N ,N , θN ), consider the diagram

kϕ KM KN

(ϕ0,ϕ∞)

M 0[y, y−1] ⊕ M ∞[x, x−1] N 0[y, y−1] ⊕ N ∞[x, x−1]

(−θM ,id) (−θN ,id)

M ∞[x, x−1, y, y−1] N ∞[x, x−1, y, y−1].

0 ∞ The map kϕ = (ϕ , ϕ )|KM here is induced by the universal property of the kernel 0 −1 ∞ −1 ∞ −1 −1 since the composite KM → N [y, y ] ⊕ N [x, x ] → N [x, x , y, y ] is 0, since 1 ϕ commutes with θ. So deﬁne S(ϕ) = kϕ. This deﬁnes a functor S : QCoh(Pk) → 1 k[x, y]-Gr, which we compose with π to give a functor Ψ := πS : QCoh(Pk) → Q. Lemma 6.2.6. For any quasi-coherent sheaf V = (M 0,M ∞, θ), ΦΨ(V ) ∼= V naturally in V .

0 ∞ −1 −1 Proof. First consider ΦΨ(M ,M , θ) = (K[y ]0,K[x ]0, id), where K is the kernel of the map (−θ, id), as above. We wish to show that (M 0,M ∞, θ) and −1 −1 (K[y ]0,K[x ]0, id) are isomorphic as sheaves.

0 −1 ∞ −1 Consider the maps g0 : M → K[y ]0 and g∞ : M → K[x ]0 which send a 7→ (a, θ(a))/1 and b 7→ (θ−1(b), b)/1 respectively. These are both well-deﬁned since a and θ(a) both have degree 0 in K[y−1], and similarly for b and θ−1(b). These maps are both certainly injective, since K ∈ Q and can thus be taken to be k −1 k 0 m-torsion free. Let (c, d)/y ∈ K[y ]0. So d = θ(c), and c = ay for some a ∈ M , k since the degree must be 0. But then (a, θ(a))/1 = (c, θ(c))/y , so ϕ0 is surjective, and a similar argument shows that ϕ∞ is too. Now consider the diagram

−1 id −1 K[y ]0[y/x] K[x ]0[x/y]

g0 g∞

M 0[y/x] θ M ∞[x/y].

This trivially commutes, so g0 and g∞ deﬁne an isomorphism of sheaves.

60 To see that this isomorphism is natural, we let

0 ∞ 0 ∞ 0 ∞ ϕ = (ϕ , ϕ ):(M ,M , θM ) → (N ,N , θN ) be a morphism of sheaves. We must show that the following square commutes.

0 ∞ 0 ∞ (M ,M , θM ) ΦΨ(M ,M , θM )

ϕ ΦΨ(ϕ)

0 ∞ 0 ∞ (N ,N , θN ) ΦΨ(N ,N , θN ).

0 −1 ∞ −1 0 To see this, we note that ΦΨ(ϕ) = Φ(kϕ) = (ϕ [y ], ϕ [x ])|KM . So for a ∈ M we have

0 0 0 ∞ [Φ(kϕ)] gM (a) = (ϕ (a), ϕ (θM (a)))/1 0 0 = (ϕ (a), θN (ϕ (a)))/1 0 0 = gN (ϕ (a)), and similarly for b ∈ M ∞. So the isomorphism is indeed natural. 2

Lemma 6.2.7. For any module M ∈ Q, we have ΨΦ(M) ∼= M naturally in M.

Proof. Now consider ΨΦ(M) for some M ∈ Q. We need to show ﬁrstly that the kernel K of the map

−1 −1 −1 −1 −1 −1 −1 (− id, id) : M[y ]0[y, y ] ⊕ M[x ]0[x, x ] → M[x ]0[x, x , y, y ],

−1 −1 −1 is isomorphic to M in Q. We note here that M[y ]0[y, y ] = M[y ], and similarly −1 −1 −1 −1 −1 −1 −1 −1 M[x ]0[x, x ] = M[x ] and M[x ]0[x, x , y, y ] = M[x , y ]. We have a map ν : M → M[y−1] ⊕ M[x−1] which sends m 7→ (m/1, m/1) which clearly composes with (− id, id) to give 0, so there is a map z : M → K induced by the kernel. To show that M and K are isomorphic in Q, it suﬃces to show that T = ker z and C = coker z are both m-torsion.

Firstly note that T = ker ν as well. To see this, note that for some p : P → M, we have νp = 0 if and only if zp = 0, since ν = ιz, where ι : K → M[y−1] ⊕ M[x−1] is the canonical monomorphism.

Let t ∈ T be homogeneous. Then (t/1, t/1) = 0 in M[y−1] ⊕ M[x−1], so ymt = 0 for some m, in order for t/1 to be 0 in the ﬁrst coordinate, and similarly xnt = 0 for

61 some n. But then mm+nt = 0, since any element of mm+n is of the form xkym+n−k, and if k < n, then m + n − k ≥ m. So T is m-torsion.

Since C = K/z(M), it suﬃces to show that for any homogeneous (a/ym, b/xn) ∈ K, there exists an N such that mN (a/ym, b/xn) ⊆ z(M). Now since (a/ym, b/xn) ∈ K, we have a/ym = b/xn in M[x−1, y−1], which means there exist m0, n0 ∈ N such that xn0 ym0 (xna − ymb) = 0. Take N = n0 + m0 + n + m. Then we should show that for all i = 0,...,N, xiyN−i(a/ym, b/xn) = (c/1, c/1) for some c ∈ M. First suppose i ≥ n + n0. Then xiyN−i(a/ym, b/xn) = (ayN−i−mxi, bxi−nyN−i). In this case we take c = bxi−nyN−i ∈ M and show that byN−ixi−n/1 = ayN−i−mxi in M[y−1]. So we need some k such that yk(byN−ixi−n − ayN−i−mxi) = 0. So we have ykyN−i−mxi−n(bym − axn) = 0, provided k + N − i − m ≥ m0, since i − n ≥ n0. Such a k certainly exists, so we are done.

Now suppose i < n + n0, in which case N − i ≥ m + m0. The details of this are similar to the previous case so we don’t go into full detail, but the proof amounts to the fact that the power of y in xiyN−i is large enough to clear the denominators of y in (a/ym, b/xn), and so in this case we take c = ayN−i−mxi and show that ayN−i−mxi/1 = bxi−nyN−i in M[x−1]. In any case, we have mN (a/ym, b/xn) ⊆ z(M), so C is m-torsion, and thus in Q, the mapz ¯ : M → K is an isomorphism.

To show naturality, we let f¯ : M → N be a morphism in Q and we should show that the following square commutes.

z¯M M KΦ(M)

f¯ ΨΦ(f¯) (∗)

z¯N N KΦ(N)

This is diﬃcult though since f¯ is the image of some f : M 0 → N/N 0 in R -Gr where M/M 0 and N 0 are m-torsion. So consider this diagram in Q.

0 z¯M0 M KΦ(M 0)

z¯M M KΦ(M)

πf f¯ ΨΦ(f¯) ΨΦ(πf)

z¯N N KΦ(N)

z¯ 0 0 N/N N/N KΦ(N/N 0)

62 0 0 Since the arrows M → M, N → N/N , KΦ(M 0) → KΦ(M) and KΦ(N) → KΦ(N/N 0) are all isomorphisms in Q, the inner square commutes if and only if the outer square commutes, and if the outer square commutes in R -Gr, then it commutes in Q. So, in fact it suﬃces to show naturality in the case where f¯ is the image of some map f : M → N. So consider the square in (∗) where all the maps are in R -Gr. Then the map on the right is induced by (f[y−1], f[x−1]), so we have −1 −1 (f[y ], f[x ])(zM (m)) = (f(m)/1, f(m)/1) = zN (f(m)), so the square commutes and thus the isomorphism is natural. 2

As suggested by the two different statements of Grothendieck’s Splitting Theo- rem, the line bundles defined earlier correspond exactly to the shifts of R. To see this, consider the line bundle O(i) = (k[X], k[X−1],X−i). Since k[x/y][y, y−1] = k[x, y, y−1], k[y/x][x, x−1] = k[x, y, x−1] and k[y/x][x, x−1, y, y−1] = k[x, x−1, y, y−1], we have that Ψ(O(i)) is defined to be the kernel K of the map

(−(y/x)i, id) : k[x, y, y−1] ⊕ k[x, y, x−1] → k[x, x−1, y, y−1].

−i −i −i −i Clearly, (y , x ) ∈ K−i. We claim that (y , x ) generates K over k[x, y]. Indeed, ` d−` j d−j if (a, b) ∈ Kd, then a = x y and b = x y , so since this is in the kernel, we have x`−iyd−`+i = xjyd−j, from which we conclude that ` − i = j. So then a = y−i(x`yd−`+i) and b = x−i(xj+iyd−j). So (a, b) = x`yd−`+i(y−i, x−i), and thus

Ψ(O(i)) = K = k[x, y](y−i, x−i) ∼= R(i), the ith shift of R.

6.3 Vector Bundles

We now come to our brief treatment of the original statement of Grothendieck’s Splitting Theorem. Originally stated in terms of holomorphic vector bundles on the complex projective line, the result also holds for algebraic vector bundles on the projective line over any ﬁeld k, and that is the setting in which we discuss the theorem. A direct proof of the theorem will not be given, rather we will explain how to extract the result from the equivalence of categories which was proved in the previous section.

1 Deﬁnition 6.3.1. An algebraic vector bundle of rank r over Pk is an algebraic 1 −1 variety V over k, plus an algebraic function f : V → Pk such that f (p) is an 1 −1 1 ∼ 1 r r-dimensional vector space over k for all p ∈ Pk, and f (AX ) = AX × k and −1 1 ∼ 1 r f (AX−1 ) = AX−1 × k in such a way that the following diagrams commute.

63 −1 1 ∼ 1 r −1 1 ∼ 1 r f (AX ) AX × k f (AX−1 ) AX−1 × k

f pr1 f pr1 1 1 AX AX−1

Here pri denotes the ith projection map.

1 1 Note that the definition given here is specific to the coordinate chart {AX , AX−1 }. The usual definition of a vector bundle does not make reference to a particular coordinate chart. A consequence of our definition is that any vector bundle can be constructed as 1 r 1 r V = [AX × k ∪ AX−1 × k ]/ ∼, (†) where ∼ is the equivalence relation that identifies (z, v) ∼ (z, γ(z)v) for some γ : 1 1 AX ∩ AX−1 → GLr(k). In fact, this is true even using a more general definition of 1 vector bundle, since any vector bundle on Ak is necessarily trivial.

1 1 1 Suppose we have two vector bundles f1 : V1 → Pk and f2 : V2 → Pk on Pk. Then a morphism of vector bundles is an algebraic function ϕ : V1 → V2 such that f1 = f2ϕ.

1 Given a vector bundle V → Pk, we can form the sheaf of sections of V , by deﬁning 0 1 1 −1 1 ∞ M to be Γ(AX ,V ) := {g : AX → f (AX ): fg = id} and likewise for M . 1 Because the ﬁbre of any point p ∈ Pk is a vector space, these sections have a natural k[X] and k[X−1] module structure.

1 1 1 There is a natural restriction map rX : Γ(AX ,V ) → Γ(AX ∩ AX−1 ,V ), which, when −1 1 −1 1 1 we localise at X to obtain rX [X ] : Γ(AX ,V )[X ] → Γ(AX ∩ AX−1 ,V ), becomes 1 an isomorphism. Of course we also have a restriction map rX−1 : Γ(AX−1 ,V ) → 1 1 −1 Γ(AX−1 ∩AX ,V ), which becomes an isomorphism when we localise at X . So then

−1 1 −1 rX [X ] 1 1 Γ(AX ,V )[X ] Γ(AX ∩ AX−1 ,V )

θ id

r [X] 1 X−1 1 1 Γ(AX−1 ,V )[X] Γ(AX−1 ∩ AX ,V ), so the isomorphism θ is canonically deﬁned.

Now, given a morphism of vector bundles ϕ : V1 → V2, this induces a map of sheaves 1 between the respective sheaves of sections. Indeed, given a section g : AX → V1, 1 composition with ϕ gives a section ϕg : AX → V2, since f2(ϕg) = f1g = id. The following square also commutes, since θ1 and θ2 are each (essentially) given by identities.

64 1 −1 ϕ◦(−) 1 −1 Γ(AX ,V1)[X ] Γ(AX ,V2)[X ]

θ1 θ2

1 ϕ◦(−) 1 Γ(AX−1 ,V1)[X] Γ(AX−1 ,V2)[X]

1 Thus the category of vector bundles on Pk can be identiﬁed with a subcategory of 1 1 QCoh(Pk) .

1 Let f : V → Pk be a rank r vector bundle, and choose trivialisation isomorphisms −1 1 1 r −1 1 r 1 tX : f (AX ) → AX × k and tX−1 : f (AX−1 ) → AX−1 × k . If g ∈ Γ(AX ,V ) 1 1 r is a section, then tX g : AX → AX × k must be a section of the projection p : 1 r 1 1 r AX × k → AX . So tX g(x) = (x, g2(x)), for some algebraic function g2 : AX → k . 1 But g2 must then be given by an r-tuple of algebraic functions from AX → k. 1 r These are precisely the elements of the coordinate ring of AX , so g2 ∈ k[X] . 1 r This gives an isomorphism Γ(AX ,V ) → k[X] , and a similar argument gives an 1 −1 r isomorphism Γ(AX−1 ,V ) → k[X ] , and of course we also have an isomorphism 1 1 −1 r from Γ(AX ∩ AX−1 ,V ) → k[X,X ] . Thus we have the following square:

1 1 id 1 1 Γ(AX ∩ AX−1 ,V ) Γ(AX−1 ∩ AX ,V )

tX ◦(−) tX−1 ◦(−)

k[X,X−1]r θ k[X,X−1]r, where the vertical arrow of course take g to the second component of tX g. We −1 define θ to be the obvious choice, induced by tX−1 tX , so the square commutes by −1 1 r 1 1 r definition. Also, tX−1 tX : AX ∩ AX−1 × k → AX−1 ∩ AX × k , is precisely the map which defines the equivalence relation in (†), so in fact, for p ∈ k[X,X−1]r, θ(p) = γp, so the transition function for the vector bundle is precisely the transition function for the corresponding sheaf.

These sheaves which correspond to vector bundles are also referred to as vector 1 bundles in the category QCoh(Pk). The above discussion implies that any vector 1 r −1 r bundle in QCoh(Pk) is isomorphic to a sheaf of the form (k[X] , k[X ] , θ). Finally, suppose we have some noetherian object M ∈ Q which is torsion free. Then −1 of course M[y ]0 will be torsion free too, and so by the structure theorem for ﬁnitely −1 r1 generated modules over a PID, there is an isomorphism ϕ : M[y ]0 → k[X] for −1 −1 r2 some r1, and similarly there is an isomorphism ψ : M[x ]0 → k[X ] for some −1 −1 r2. But since localisation preserves isomorphisms and M[y ]0[y/x] = M[x ]0[x/y],

1We should really check that this identiﬁcation is fully faithful, but that is ommitted from this thesis.

65 −1 −1 r1 −1 r2 then θ := ψ[x/y]ϕ[y/x] : k[X,X ] → k[X ,X] is an isomorphism, so r1 = ∼ r −1 r r2 = r. Thus, Φ(M) = (k[X] , k[X ] , θ). It is also easy to check that for any r, Ψ(k[X]r, k[X−1]r, θ) is a torsion free noetherian module in Q, so we see that 1 under the embedding of the category of vector bundles into QCoh(Pk), and the 1 equivalence between Q and QCoh(Pk), the torsion free noetherian modules in Q 1 correspond exactly to the vector bundles in QCoh(Pk). Here we come now to the other main result of this chapter, the structure theorem 1 for vector bundles in QCoh(Pk). Luckily, the proof of this theorem follows from theory we have already built and so is quite short.

1 Theorem 6.3.2. If E is a vector bundle over Pk, then E = O(n1) ⊕ · · · ⊕ O(nr), where O(n) is the nth line bundle as deﬁned above.

Proof. Under the equivalence of Theorem 6.2.5, the result follows from Theorem 5.3.1. 2

66 Chapter 7

Conclusion

7.1 Summary

We began in the abstract setting, by developing the theory of abelian categories in general, and then used it to set up a useful construction in abelian category theory—the quotient of an abelian category by a Serre subcategory, motivated by the process of localisation in geometry.

This then provided the correct framework which allowed us to discuss an algebraic version of Grothendieck’s original splitting theorem, which used techniques from category theory and from homological algebra, particularly Hilbert’s Syzygies The- orem and the Auslander-Buchsbaum Theorem, both of which were proved for our speciﬁc case. We then provided a purely algebraic proof of this theorem, in the context of graded modules in a quotient category.

Finally we took this algebraic construction and showed how this can be applied to the geometry of the projective line in order to extract the classical statement of the theorem from the categorical setting.

7.2 Generalisations and Further Work

Grothendieck’s Splitting Theorem is one which lends itself to many generalisations, in both the geometric and algebraic settings. One such geometric generalisation is to the case of vector valued modular forms, presented in [11].

The algebraic formulation may also be generalised to the noncommutative setting. Noncommutative algebraic geometry is a very active area of research at present, because until recently, under the duality between algebra and geometry, most known geometric objects correspond to aspects of commutative algebra. Of course, the noncommutative case must also correspond to something geometrically. In fact,

67 one such generalisation of the splitting theorem to the noncommutative setting was presented this year by my supervisor A/Prof. Daniel Chan, in [3].

If one takes the so called skew polynomial ring, Sq = khx, yi/(yx − qxy), for some × q ∈ k , and perform the same quotient construction Sq -Gr /m -Tors, it turns out that the resulting quotient category is equivalent to Q presented in Chapter 5 of this paper. Of course, q = 1 is precisely our case. While I unfortunately was unable this year to delve into the details of why this might be true, the construction of this quotient for skew commutative or other noncommutative graded rings may well provide insight into the theory of noncommutative projective varieties.

68 References

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