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Bundles in Analytic, Zariski, and Étale

Takumi Murayama

Adviser: János Kollár

a senior thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in at Princeton University

May 5, 2014 Acknowledgments

My time here at Princeton University has been quite an adventure, and it is hard to decide who to thank and acknowledge on these pages. First of all, I would like to thank my adviser, János Kollár. It is only through his guidance and patience that I was able to pinpoint a good topic to write about in this thesis, and then complete it to the state where it is at now. No one knows better than he does how confused I can get even with the most basic parts of mathematics, and I cannot stress enough the extent to which I am indebted to his teaching. I know that the lessons and ideas I have learned from him will stay with me for years to come. I would also like to express my sincerest thanks to my second reader for this thesis, Zsolt Patakfalvi. My experience with him in the last year working on has been an enlightening one, and I especially appreciate the effort he put into making sure I understood the foundations of modern algebraic geometry, without losing track of the geometric insight and intuition at its core. My teachers and colleagues in the mathematics department are also very deserving of recognition. I came to Princeton not sure whether or not I wanted to pursue mathematics as my undergraduate career, and without their encouragement and the conversations I had with them, I might not be writing this thesis today. They have shown me that the beauty of mathematics is really unparalleled, and that in many ways mathematics is like art or poetry, full of images and stories that must be discovered and unraveled slowly with a good cup of coffee. In particular, Kevin Tucker and Yu-Han Liu introduced me to algebraic geometry, and it is because of their enthusiasm that I focused my attention on this field in the first place. Lastly, I would like to thank my family and friends for supporting me these last four years. My love of mathematics would never have begun without my father’s explanations of how mathematics is the for the music my mother taught me to appreciate so much. And without my friends, especially those from VTone, I would never have been able to stay up so many nights and work so hard to complete this thesis and all the other work I’ve done these last four years.

This thesis represents my own work in accordance with University regulations.

Takumi Murayama May 5, 2014 Princeton, NJ

i Abstract

We compare the behavior of fiber bundles with structure for the , the projective general linear group , and the general퐺 affine퐺 group . We prove aGL criterion for when a - is in factPGL a -bundle, i.e., a ,GA over the . We alsoGA discuss the behavior ofGL fiber bundles under different topologies; specifically, we compare1 the analytic versus Zariski topologies over the Riemann sphere , ℙ 1 and compare the étale and Zariski topologies on schemes in general. ℙ

ii Contents

Introduction 1

Part 1. Algebraic and holomorphic fiber bundles over the Riemann sphere 3 Chapter 1. Fiber bundles with structure group 4 1. Groups and group actions on varieties 4 2. Definition of fiber bundles 6 3. A comparison between the analytic and Zariski topologies 8 Chapter 2. Fiber bundles on the Riemann sphere 11 1. The case of vector bundles 11 2. The case of projective and affine bundles 15

Part 2. Principal bundles over schemes 20 Chapter 3. Preliminaries 21 1. Group schemes and algebraic groups 21 2. Group actions on schemes 23 Chapter 4. Principal bundles in étale 26 1. Differential forms; unramified and étale 26 2. Principal and fiber bundles in étale topology and a comparison 28 Bibliography 31

iii Introduction

Let be a projective defined over the complex numbers , i.e., is defined as the zero of homogeneous polynomials in the complex 푋 u�+1 ℂ 푋 projective of . can then be givenu� what is called1≤u�≤u� the Zariski topology, u� {푓 ∶ ℂ → ℂ} generated by theℙ openℂ basis of sets푛 defined푋 by the complement in of zero sets of some other homogeneous polynomials . An important feature of algebraic푋 geometry over is the following: if is smooth, i.e.,{푔 foru�} all points , the Jacobian matrixℂ 푋 푥 = (푥0 ∶ ⋯ ∶ 푥u�) ∈ 푋 휕푓1 휕푓1 휕푓1 ⎛ ⋯ ⎞ ⎜휕푥1 휕푥2 휕푥u� ⎟ ⎜ ⎟ ⎜ 휕푓2 휕푓2 휕푓2 ⎟ ⎜ ⎟ ⎜ 1 2 ⋯ u� ⎟ ⎜휕푥 휕푥 휕푥 ⎟ ⎜ ⋮ ⋮ ⋱ ⋮ ⎟ ⎜ ⎟ ⎜ 휕푓u� 휕푓u� 휕푓u� ⎟ ⋯ has rank , where is the dimension⎝휕푥1 휕푥 of2 , then휕푥u�also⎠ has the structure of a complex analytic ,푛 − 푑 i.e.,푑 a complex manifold푋 that is locally푋 biholomorphic to , with a topology given by the from with the standard topology.u� Call this u� ℂ topology the analytic topology. The geometry ofℙ projectiveℂ algebraic varieties over , then, has the benefit of having topological, analytic, and algebraic methods at its disposal;ℂ in particular, projective algebraic varieties of dimension are widely studied as Riemann surfaces. 1 The richness of the structure on a complex , however, also means that there are non-trivial choices to be made. Recall, for example, the notion of a vector bundle: Definition. Let be a projective algebraic variety over .A vector bundle is a space together with a morphism푋 such that for any ℂ there is an containing and an isomorphism such that for any 퐸 푝∶ 퐸 → 푋 u� −1 푥 ∈ 푋 푈 , 푥 , and such that if 푔∶is 푈 another × ℂ → open 푝 (푈) set containing푝 ∘ 푔(푢,with 푒) isomorphism = 푢 u� , then the two′ structures on are the same. 푢′ ∈ 푈′ 푒 ∈u� ℂ −1 ′ 푈 −1 푥 Note푔 ∶ 푈 that× ℂ this→ definition 푝 (푈 ) works in two different settings: 푝 (푥) (1) Consider with the analytic topology, with another complex analytic manifold, and morphisms푋 given by holomorphic functions.퐸 This gives holomorphic vector bundles over . (2) Consider with푋 the Zariski topology, with another algebraic variety over , and morphisms푋 given by regular functions, i.e.,퐸 functions locally given by polynomials.ℂ This gives algebraic vector bundles over . 1 푋 2

At first glance, these notions give very different objects: indeed, while every algebraic vector bundle is clearly holomorphic, there is no reason to believe that the converse is the case. Indeed, the fact that on a projective variety, holomorphic and algebraic vector bundles are the same is a deep result due to Serre [GAGA, Prop. 18]. Moreover, this result is fairly special, in that replacing the vector spaces with some other space gives fiber bundles that are u� no longer always algebraic: sometimesℂ a holomorphic fiber bundle퐹 is no longer an algebraic bundle. Our goal is to investigate the difference between the behavior of fiber bundles in the two topologies above, and also the difference between the behavior when we change the fibers of our bundle. We use Grothendieck’s theorem classifying algebraic vector bundles over 1 [Gro57] as a point of departure. ℙ After proving his result, we extend our methods to some kinds of fiber bundles over , 1 specifically those listed in Example 1.11. In particular, we look at what happens if inℙ the definition above for a vector bundle, we assume that only has the same affine structure when looking at two different neighborhoods on−1 which we have morphisms . In ′푝 (푥) ′ this situation, we prove a new criterion for when푈, the푈 “affine” bundle thus obtained is푔, in 푔 fact a vector bundle. Finally, we discuss fiber bundles in the general context of schemes. For schemes, we can introduce what is called the étale topology, which in some ways provides an analogue for analytic bundles over complex varieties. Finally, we give some comparisons between the behavior of fiber bundles in the Zariski and étale topologies. Part 1

Algebraic and holomorphic fiber bundles over the Riemann sphere CHAPTER 1

Fiber bundles with structure group

We begin the first half of this thesis by talking about complex algebraic varieties and complex analytic , and defining what are fiber bundles on them. Fiber bundles are a generalization of vector bundles. Recall that for a vector bundle, each fiber has the structure of a vector space. For a fiber bundle, though, we just require that each fiber is isomorphic to some fixed space . Unlike in French, we do not have퐹 the luxury of using the word “variété” to refer to both algebraic varieties and complex analytic manifolds; regardless, in the following we use the word “variety” without qualification to emphasize the fact that the definitions are the same in both contexts, by replacing the notion of a in the right way, i.e., with regular maps in the algebraic case, and holomorphic maps in the analytic case. In the algebraic context, we do not have to work over the field ; we will state when something depends on the choice of field . ℂ 푘 1. Groups and group actions on varieties Oftentimes when we have a variety, we would like to analyze the way a group can act on it. The main examples we have in mind are the classical groups, e.g., the general linear group acting on a vector space, and subgroups thereof. In general, we have the following definition: Definition 1.1. Let be a variety. is an algebraic or , depending on the context, if the underlying퐺 set of forms퐺 a group, such that multiplication , and inversion , are morphisms. ′ ′ 퐺 −1 퐺 × 퐺 → 퐺 (푔, 푔Definition) ↦ 푔푔 1.2. Let 퐺be → an 퐺 algebraic푔 ↦ 푔 or Lie group, and a variety. acts (on the left) of if there is a morphism such that and 퐺 푋 퐺 ′ for all , u� . We write u� . If is understood,u� we ′ 푋 푚′ ∶ 퐺 × 푋 → 푋 푚 (1, 푥) = 푥 푚 (푔 푔, 푥) = 푚callu�(푔 ,a 퐹(푔,-variety 푥)) .A morphism푥 ∈ 푋 푔, of 푔 ∈-varieties 퐺 is a morphism푚u�(푔, 푥) = 푔 ⋅ 푥 푚betweenu� -varieties such푋 that퐺 is -invariant, that is,퐺 for all휋∶ 푋 →. 푌 퐺 Example휋 1.3.퐺 Let 휋(푔, the ⋅ 푥)general = 푔 ⋅ 휋(푥) linear group푥 ∈of 푋 degree over . This is given by matrices퐺 = GLu�(푘)with entries in such that . 푟 acts푘 on affine -space with a chosen basisu�u� by the u� 푟u� × 푟 퐴 = (푎 ) 푘 det 퐴 ≠ 0 GL 푟 픸 {푒u�} 푣1 푣1 ⎛ ⎞ ⎛ ⎞ ⎜푣2⎟ ⎜푣2⎟ ⎜ ⎟ ↦ 퐴 ⎜ ⎟ . ⎜ ⋮ ⎟ ⎜ ⋮ ⎟ ⎝푣u� ⎠ 4 ⎝푣u� ⎠ CHAPTER 1. FIBER BUNDLES WITH STRUCTURE GROUP 5

It is an algebraic/Lie group by taking the open subset of 2 defined by those 2 u� u� such that . 픸 (푎u�u�) ∈ 픸 u�u� We note,det(푎 however,) ≠ 0 that the automorphisms defined by matrices in are not all of the u� automorphisms of ; for example, a translation for fixed픸 gives an automorphism of ,u� but is not given by a in since is not fixedu� by this 픸u� 푣 ↦ 푣 + 푤 u� 푤 ∈ 픸 automorphism. By픸 embedding in , even, thereGL areu� already0 ∈ more 픸 automorphisms of . To describe these, we first describeu� theu� analogue of for projective -space: u� 픸 ℙ u� 픸 Example 1.4. Let , theGLprojective general푟 linear group × of degree over . This퐺 is = given PGLu� by(푘) = GLu�+1(푘)/푘 matrices in modulo scalar multiplication by . acts on projective -space by the map u�+1 푟 푘 × (푟 + 1) × (푟 + 1) u� 퐴 GL 푘 PGLu� 푟 ℙ 푣0 푣0 ⎡ ⎤ ⎡ ⎤ ⎢푣1⎥ ⎢푣1⎥ ⎢ ⎥ ↦ 퐴 ⎢ ⎥ . ⎢ ⋮ ⎥ ⎢ ⋮ ⎥ ⎣푣u� ⎦ ⎣푣u� ⎦ It is an algebraic/Lie group by taking the open subset of 2 defined by those 2 u� u� such that . ℙ [푎u�u�] ∈ ℙ u�u� Unlikedet[푎 for ] ≠acting 0 on , however, we know that is in fact the entire group of u� automorphisms ofu� : u� GL u� 픸 PGL Proposition 1.5.ℙ Every automorphism of is linear, i.e., is induced by an automor- phism of . u� u�+1 PGL u�+1 Proof퐴 ∈ ([Har95, GL p.픸 229]). Let and be a hyperplane and a line in , respec- u� u� tively, meeting at one point. By Bézout’s퐻 theorem퐿 ⊂ ℙ [Har95, Thm. 18.3], since an automorphismℙ of must preserve intersection numbers, we have that maps to a hyperplane and u� 휑to a line,ℙ such that they still meet at one point. This implies퐻 that hyperplanes must map to퐿 hyperplanes under the automorphism . If on an affine chart is given by 휑 휑 푓u�(푧1, … , 푧u�) (푧1, … , 푧u�) ↦ (푤1, … , 푤u�), 푤u� = the requirement that maps hyperplanes to hyperplanes implies푔u�(푧1, …that , 푧 au�) linear relation between the implies one between휑 the , hence the degrees of the are all at most , with all scalar푤u� multiples of each other. 푧u� 푓u�, 푔u� 1 푔u� Note that Bézout’s theorem only works if our field is algebraically closed; we will give a different proof later that works for non-algebraically closed fields as well as finitely generated UFD’s over . 푘 This example gives us a slightly larger group of automorphisms of : 푘 u� u� Example 1.6. PGLLet be the subgroup that that fixes the픸 hyperplane in . This is the generalu� affine groupu� of degree over , and is given by u� 퐺 = GA (푘) ⊂ PGL (푘) matrices0 in modulo scalar multiplication by such that that {푥 = 0} ℙ 푟 × 푘 (푟+1)×(푟+1) (푎u�u�) GLu�+1 푘 푎01 = 푎02 = ⋯ = 2. DEFINITION OF FIBER BUNDLES 6

. Note then that , hence normalizing by gives that every of −1 푎has0u� = a representative 0 of the푎00 form≠ 0 푎00 GAu�

1 0 ⋯ 0 ⎛ ⎞ ⎜ ∗ ⎟ ⎜ ⎟ ⎜ ⋮ 퐴 ⎟ ∗ where . is an algebraic/Lie⎝ group since⎠ the requirement 퐴defines ∈ GLu� a(푘) closedGAu� subset of or that is also a subgroup. Note푎01 that= 푎02 by= taking ⋯ = the0u� hyperplane out of the u�+1that u�acts on, we get an automorphism of . 푎 = 0 GLu� PGL u� {푥0 = 0} ℙ GAu� 픸 This way of representing elements in gives rise to the following diagram relating our three examples above: GAu�

u�+1↠ GL ← (푘) (1.7) → ↩ GLu�(푘) ↩ → GAu�(푘) ↩ → PGLu�(푘) Note this relationship is on the level of algebraic/Lie groups; there is no reason why we should expect morphisms to lift from to , for example. This is the main topic of Chapter 2. PGLu� GLu�

2. Definition of fiber bundles We can now define our main object of study, that is, fiber bundles with structure group. These generalize vector bundles, and provide a good framework in which to talk about vector bundles and related constructions with different fibers. For example, vector bundles have fibers that are isomorphic to some vector space , but we can always stipulate that the fibers are isomorphic to some other variety or scheme푉 . In the examples we are interested in, moreover, there is some group acting on these fibers퐹 , just like how acts on the fibers of a vector bundle. Let be an퐺 algebraic or Lie group.퐹 For more discussion,GL see [Gro58a]. 퐺 Definition 1.8. A fiber system with structure group and fiber is a -variety together with a morphism such that 퐺 for some퐹 fixed 퐺-variety 퐸,

and for all .A morphism−1 of two fiber systems and

−1 −1 휋∶ 퐸 → 푋 휋 (푥) ≅ 퐹 퐺 퐹 is a morphism of -varieties ← , such that it makes the diagram

′ 퐺′ ⋅ 휋 (푥) ⊂ 휋 (푥) 푥 ∈ 푋 ′ 휋∶ 퐸 → 푋

휋 ∶ 퐸 → 푋 퐺 푓 ∶ 퐸 →→ 퐸 ← u� ′ 퐸 → → 퐸 ← ′ u� u� 푋 commute. If is a morphism, we define the fiber system as the -variety ′ together with the morphism in the bottom fiber∗ square ′ 푓 ∶ 푋′ → 푋 ′ ′ ′ 푓 퐸 퐺 퐸 = 푋 ×u� 퐸 휋 ∶ 퐸 → 푋

CHAPTER 1. FIBER BUNDLES WITH STRUCTURE GROUP 7 below, and the -action is given by the top square:← 퐺

′ ← ← ← → 퐺 × 퐸 → 퐺→ × 퐸 u�u�′ u�u�

′ ← ← ← → 퐸 → 퐸→ ′ u� u�

′ u� In the special case , we call 푋 the→restriction푋 of the fiber system to . ′ ′ ∗ ′ u� It is hard to automatically푋 ⊂ 푋 see which푓 | ≔ fiber 푓 퐸 systems are isomorphic, that is, when there푋 are two morphisms whose composition in both directions is the identity, so we have the −1 following criterion:푓 , 푓 Lemma 1.9. Let and be two fiber systems and let ′ ′ ′ be a morphism between휋∶ 퐸them. → 푋 is an휋 isomorphism∶ 퐸 → 푋 if and only if for all 푓, ∶there 퐸 → is 퐸 a neighborhood containing such푓 that induces an isomorphism 푥 ∈ 푋 . ∼ ′ 푈u� 푥 푓 퐸|u�u� → 퐸 |u�u� Proof. This follows since morphisms of -varieties are defined locally.  We would moreover like our fiber systems퐺 to have a “local triviality condition”: Definition 1.10. The trivial fiber system is that defined by the .A fiber bundle with structure group and fiber or simply a -bundle with fiber푋 × 퐹is → a 푋 fiber

system over together with an open cover of and isomorphisms 퐺 ← 퐹 퐺 퐹

−1 ∼

, 푋 {푈u�} 푋 휑u� ∶ 휋 (푈u�) →

← 푈u� × 퐹 → u� −1 u� u� u� 휋 (푈 ) ∼ → 푈 × 퐹 u� → ←pr1 u� commutes, and such that the maps 푈

−1 are given by 휑u� ∘ 휑u� ∶for (푈 someu� ∩ 푈u� morphisms) × 퐹 → (푈u� ∩ 푈u�) × 퐹 , which satisfy the cocycle condition(푥, 휉) ↦ (푥, 푡u�u�(푥)휉). Morphisms of fiber bundles푡u�u� ∶ are 푈u� morphisms∩ 푈u� → 퐺 as fiber systems. u�u� u�u� u�u� Examples 1.11.푡 In= 푡this푡 thesis, we focus on the following examples: (1) Let acting on (1.3). is then a vector space by choosing the u� origin to be theu� stabilizer of the -action, and the -bundles with fiber are 퐺 = GL 퐹 = 픸 퐹 u� called vector bundles on . 퐺 GLu� 픸 (2) Let acting on (1.4). The -bundles with fiber are called 푋 u� u� projective퐺 = PGL bundlesu� on . 퐹 = ℙ PGLu� ℙ (3) Let acting on (1.6). The -bundles with fiber are called 푋 u� u� affine퐺 bundles = GAu� on . Note퐹 = they 픸 are a specialGA caseu� of projective bundles,픸 and that vector bundles are푋 a special case of affine bundles. 3. A COMPARISON BETWEEN THE ANALYTIC AND ZARISKI TOPOLOGIES 8

These fiber bundles are related in the same way as the groups are as illustrated in (1.7). The main goal of the first half of this thesis is to describeGL, PGL, the relationship GA between these different fiber bundles over the base space . 1 Remark 1.12. Note that for vector bundles, the transition푋 = data ℙ can be viewed as matrices with entries that are analytic functions on in the푡 analyticu�u� case, or regular푟 × 푟 functions in in the algebraic case. We푈 alsou� ∩ 푈 remarku� here about how different choices of can giveu� isomorphicu� vector bundles. Suppose is given by matrices and 풪(푈 ∩ 푈 ) ′ by . Then,u�u� suppose can be written for some a matrix of analytic (resp.u�u� regular) ′ 푡 ′ ′ 퐸 푡 퐸 functionsu�u� on . Theu�u� morphism u�u� thenu� u�u� defined byu� gluing together the morphisms 푡 푡 푡 =′ 푓 푡 푓 isu� an isomorphism, hence as vector bundles. ′−1 푈 퐸 → 퐸 ′ 휑u� A∘ key 푓u� ∘ question휑u� we ask is whether or퐸 not ≅ all 퐸 projective bundles arise from projectivizing vector bundles, i.e., by taking the quotient by in each fiber. If this were the case, then we × could always lift our transition matrices in 푘 to and work with transition matrices only. This is the basis for why our methods in ChapterPGL 2GL work, where we can compute everything in terms of matrices. Remark 1.13. For a general fiber bundle (without a structure group), we can define a fiber bundle as a fiber space that is locally isomorphic to a trivial bundle, i.e., such that for all there is a neighborhood퐸 → 푋 such that is isomorphic to the trivial system on . In this푥 ∈ case, 푋 the isomorphisms 푈u� as퐸| fiberu�u� systems can be identified with maps 퐸 [Gro58a, Prop. 1.4.1].푋 ×퐹 →We 푋 would × 퐹 like a similar statement for fiber systems with푓 ∶ 푋 a → structure Aut(퐹) group that are locally isomorphic to a trivial bundle in the above sense, so that we can identify isomorphisms with maps . This is an issue, however, since while we do get a map , the fact that푋 → 퐺may not act freely and transitively on means that we cannot푓 ∶ 푋 →assume Autu�(퐹) that this map becomes퐺 a morphism , for there might퐹 not be an element such that , or even if there is one,푋 → it may퐺 not be unique. For example, suppose푔 ∈ 퐺 form푓 (푥) = an 푔 open ⋅ − cover of such that . We can have the group 푈1, 푈2 ⊊ 푋act on by 푋 , where푈1 ∩ 푈2 ≠ ∅ . Then gluing together via the map 1 for gives a fiber system 퐺 = ℤ/21 ⊕ ℤ/2 픸 (푖, 푗) ⋅ 푣 ↦ 푖 ⋅ 푣 × 푖, 푗 ∈ {+1, −1} with structure group푈u� × 픸, but this is not(푥, a 푣) fiber ↦ bundle(푥, 푐푣) for푐 the ∈ transition푘 − {±1} is not in . Moreover, even if 퐺 , , so there is ambiguity as to which element in 퐺 to choose to define푐 our = 1 fiber(1, bundle. ±1) ⋅ 푣= This 푣 is why it is often more convenient to work with퐺 principal bundles, which are fiber bundles with fibers isomorphic to , since in this case we are guaranteed that acts freely and transitively on fibers. This is퐺 the viewpoint we will take in Chapter 4. 퐺 3. A comparison between the analytic and Zariski topologies We will see later that for , the choice of analytic versus Zariski topology on 1 does not matter when it comes푋 to = what ℙ fiber bundles we can have, at least for our examples in푋 Example 1.11. On other base spaces , though, it can make a difference. This is illustrated by the following example: 푋 Example 1.14. Let , the affine plane minus the two lines and , and define the fiber system2 by having fibers projecting onto 푋 = 픸 − {푠푡 = 0} 2 2 2 2 {푠 = 0} {푡 = 0} 퐸 {푥 + 푠푦 = 푡푧 } ⊂ ℙ CHAPTER 1. FIBER BUNDLES WITH STRUCTURE GROUP 9

. This is a fiber system with structure group since for fixed , 2 2 2 (푠,is a 푡) smooth ∈ 푋 conic, hence is isomorphic to . PGL1 푠, 푡 {푥 +푠푦 = 푡푧 } Now note . Giving the analytic1 topology, denote to be minus the × × ℙ part of the 푋axis, = ℂ and× ℂ let 푋 ℂ± ℂ ± ℝ

On each of푈 these1 = ℂ open+ × ℂ sets,+, we 푈2 have= ℂ+ the× ℂ trivialization−, 푈3 = ℂ given− × ℂ by+, 푈4 = ℂ− × ℂ−.

1 푈u� × ℙ → 퐸|u�u� 2 2 2 2 2푢푣 푢 + 푣 (푠, 푡) × (푢 ∶ 푣) ↦ (푢 − 푣 ∶ 1/2 ∶ 1/2 ) where are the positive or negative square roots푠 depending푡 on the . The transition 1/2 1/2 data is푠 then, 푡 given by 푈u�

(1.15) 1 1 1 푡13 = 푡24 = [ ] , 푡12 = 푡34 = [ ] , 푡14 = 푡23 = [ ] and so defines a holomorphic−1 projective−1 bundle. 1 On퐸 the → other 푋 hand, in the Zariski topology, this cannot be locally trivial since the fiber at the is the , which has no rational points. 2 2 2 For suppose it had a point {푥 + 푠푦where= 푡푧 } → Spec ℂ(푠, 푡) . Clearing denominators we can assume 0 . Let0 0 0 0 0 be the terms with lowest total degree (푥 , 푦 , 푧 )u�1 u�1 u�2푥 u�,2 푦 ,u�3 푧 u�3∈ ℂ(푠, 푡) in respectively.0 0 0 Then, the term with lowest total degree in is one of 푥 , 푦 , 푧 ∈ ℂ[푠, 푡] 푠 푡 , 푠 푡 , 푠 푡 2 2 2 0 0 0 , but these clearly do not cancel. 2u�푥1 2u�, 푦1 , 푧2u�2+1 2u�2 2u�3 2u�3+1 푥 + 푠푦 − 푡푧 푠 Note,푡 , 푠 however,푡 , 푠 that푡 the matrices in (1.15) do define a in the Zariski topology, which shows that there can exist fiber systems that are isomorphic by holomorphic maps, but not algebraic maps. We cannot hope to construct a similar example with plane conics as fibers with a base space because of the following: Theoremℂ 1.16 (Tsen [Tse33]). Let , algebraically closed. Then any homoge- neous polynomial in of degree퐾 = 푘(푠) 푘 has a non-trivial zero. 1 u� Proof. Let 푓 (푡 , … , 푡 ) 퐾 푑 푁푑 + 훿 + 1 So if we had a family of non-degenerate푁 푑 < 푛 conics over a Zariski-open subset of , our 1 argument above no longer퐸 works as the generic fiber is has a point by Tsen’s theorem,픸 hence is rational. 3. A COMPARISON BETWEEN THE ANALYTIC AND ZARISKI TOPOLOGIES 10

Remark 1.17. At first glance, the matrices in (1.15) look like they might not commute if lifted up to . We will return to this example in Remark 2.17; we remark for now that surprisingly,GL the2 bundle defined by these matrices does lift to a vector bundle, even if algebraically, the fibration of conics that we defined does not. CHAPTER 2

Fiber bundles on the Riemann sphere

Analyzing the examples of fiber bundles on the Riemann sphere we are interested in 1 is convenient because, as we shall see, we can do so using matrices andℙ linear algebra. 1. The case of vector bundles In this , we classify vector bundles on . Recall that is defined as fol- lows: let and , 1 1 and ℙ −1ℙ 0 . 1 is then obtained01 by gluing together 0 along the 푈 =−1 Spec 푘[푠] 푈 1= Spec 푘[푡] 푈 = Spec 푘[푠, 푠 ] = 푈 − {0} isomorphism10 identifying 1 and by the isomorphism induced by 0 1 푈 = Spec 푘[푡, 푡 ] = 푈 − {0} ℙ 푈 −1, 푈 −1 defined by . Note that01 each 10is a copy of . Similarly, we can get as∼ a complex −1 푈 푈 1 푘[푠, 푠 1 ] → 푘[푡, 푡 ] analytic manifold푠 ↦ 푡 by doing the same.푈u� Note that the픸 proofs in this section doℙ not depend on our field . Our goal푘 is to prove the following result due to Grothendieck: Theorem 2.1 ([Gro57, Thm. 2.1]). Let be a rank vector bundle over . Then is 1 isomorphic to a of line bundles 퐸 푟 ℙ 퐸

and the are퐸 ≅ uniquely 풪(휅1) ⊕ determined ⋯ ⊕ 풪(휅 byu�), the 휅 isomorphism1 ≥ ⋯ ≥ 휅u�, class 휅u� ∈ of ℤ,. 푖 = 1, … , 푟, where we휅u� note that each is defined by the transition matrix퐸 , and that the direct −u�u� sum above corresponds to풪(휅 takingu�) direct sums of transition matrices.(푠 ) Our strategy is as follows: we show that any vector bundle is trivial over and . Then, the question reduces to how two trivial bundles on above can glue푈0 together푈1 on their intersection. The gluing operation corresponds to one푈0, transition 푈1 matrix by Remark 1.12. Finally, we can obtain a diagonal transition matrix realizing our vector bundle as a direct sum of line bundles via an elementary linear algebra argument from [HM82]. We pause here for a moment to discuss the history of this result. Our approach here shows that Grothendieck’s result boils down to an algebraic statement about transition matrices. However, this result on matrices was apparently known to Dedekind and Weber; see [Sch05] for a discussion. In addition, Grothendieck’s theorem above in the analytic setting is also the consequence of a theorem due to G.D. Birkhoff on matrices of analytic functions [Bir13] as pointed out by Seshadri [Ses57]. To show that any vector bundle over or is trivial, it suffices to show Proposition 2.2. Every vector bundle푈0 on 푈1 is trivial. 1 In the algebraic setting, this result is actually픸 a rather easy consequence of the following linear algebraic result: 11 1. THE CASE OF VECTOR BUNDLES 12

Theorem 2.3 (Smith normal form [Jac85, Thm. 3.8]). Let be a PID, and an matrix in . Then, there exist such that 퐷 퐴 푟 × 푟 퐷 푃, 푄 ∈ GLu�(퐷) 푑1 ⎜⎛ 2 ⎟⎞ ⎜ 푑 0 ⎟ ⎜ ⋱ ⎟ 푃퐴푄 = ⎜ ⎟ ⎜ 푑u� ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ where and if . ⎝ ⋱⎠ We푑 canu� ≠ 0 now prove푑u� ∣ 푑 ouru� 푖 proposition. ≤ 푗 Proof of Proposition 2.2. Suppose a vector bundle is trivial on 1 and . We claim that it is isomorphic to the trivial bundle휋∶ 퐸 → on 픸 . There퐷(푓1) is a transition2 matrix that1 defines the2 gluing 퐷(푓 ) −1 −1 퐷(푓 ) ∪ 퐷(푓 ) of these trivial bundles12 by definition.u� Multiplying1 2 u� by 1 2 , we have that is 푡 ∈ GL (풪(퐷(푓 푓 ))) = GL (푘[푠, 푓 , 푓 u� ]) u� a matrix in , hence has a Smith normal form; the푡12 matrices(푓1푓2) above correspond(푓1푓2) 푡12 to automorphisms푘[푠] of the trivial bundles on respectively푃, 푄 by Remark 1.12. Note that since is invertible in 퐷(푓1,), its 퐷(푓 Smith2) normal form is of the form 푡12 GLu�(풪(퐷(푓1푓2))) u�1 u�1 1 2 푓 푓 2 2 ⎛ u� u� ⎞ u� ⎜ 푓1 푓2 ⎟ (푓1푓2) ⋅ 푃푡12푄 = ⎜ ⎟ ⎜ ⋱ u�u� u�u� ⎟ for and ⎝ . Then, 푓1 푓2 ⎠ 0 ≤ 푚1 ≤ ⋯ ≤ 푚u� 0 ≤ 푛1 ≤ ⋯ ≤ 푛u� u�−u�1 u�−u�1 1 2 푓 2 푓 2 ⎛ u�−u� ⎞ ⎛ u�−u� ⎞ ⎜ 푓1 ⎟ ⎜ 푓2 ⎟ ⎜ ⎟ 푃푡12푄 ⎜ ⎟ = Idu�, ⎜ ⋱ u�−u�u� ⎟ ⎜ ⋱ u�−u�u� ⎟ and so is trivial⎝ on 푓1 . Thus,⎠ any⎝ vector bundle is trivial푓2 on⎠ , by choosing a 1 finite open cover of on1 which2 the bundle is trivial repeating the process above.  휋 퐷(푓1 ) ∪ 퐷(푓 ) 픸 We can now prove픸 Grothendieck’s theorem 2.1. In the algebraic case, this follows by a straightforward linear algebraic argument as in [HM82]. On , any vector bundle , by Proposition 2.2, is trivial on , hence 1 1 is the result of gluing together and along an isomorphism 0 1 ℙ 휋∶ 퐸 →u� ℙ u� 푈 , 푈 u� 퐸 which is given by0 1 , where is0 a matrix∼ u� 푈 × 픸 푈−1× 픸 −1 −1 푈 − {0} × 픸 → with푈1 − entries {0} × 픸 that are regular functions.(푠, 푣) ↦ (푠 , 퐴(푠, 푠 )푣) 퐴(푠, 푠 ) 푟 × 푟 Let be a rank vector bundle. The matrix mentioned above is invertible for all 1 since it is invertible on , hence−1 휋∶ 푃 → ℙ −1 푟 퐴(푠, 푠 ) 푠 ≠ 0, 푠 ≠ 0 푈0 ∩ 푈1 (2.4) −1 u� × By Remark 1.12, we notedet that 퐴(푠, depending 푠 ) = 푐 on ⋅ 푠 our, 푛choice ∈ ℤ, of 푐 ∈isomorphisms 푘 . −1 , we can have different matrices that give isomorphic vectoru� bundles.u� But∼ u� −1 휑 ∶ 휋 (푈 ) → isomorphismsu� of are of the form for , so 푈 × 픸 u� 퐴(푠, 푠 ) 푈0 × 픸 (푠, 푣) ↦ (푠, 푈(푠)푣) 푈(푠) ∈ GL(푟, 푘[푠]) CHAPTER 2. FIBER BUNDLES ON THE RIEMANN SPHERE 13

; similarly, isomorphisms of correspond to , × u� −1 −1 so . This gives us the following:1 det 푈(푠) ∈−1 푘 × 푈 × 픸 푉(푠 ) ∈ GL(푟, 푘[푠 ]) detProposition 푉(푠 ) ∈ 푘 2.5. Isomorphism classes of rank vector bundles over are in one-to-one 1 correspondence to the set 푟 ℙ

−1 −1 ′ −1 ′ −1 −1 −1 where{GL(푟, 푘[푠, 푠 ])/퐴(푠,, 푠 ) ∼ 퐴 (푠, 푠 ) ⟺ 퐴. (푠, 푠 ) = 푉(푠 )퐴(푠, 푠 )푈(푠)} −1 −1 In particular,푈(푠) ∈ this GL(푟, means 푘[푠]) that푉(푠 we can) ∈ assume GL(푟, 푘[푠 that in]) (2.4), the constant . We now find a canonical form for matrices in this set 푐 = 1 to find all isomorphism classes of rank vector bundles over . −1 1 GL(푟, 푘[푠, 푠 ])/ ∼ Proposition 2.6. Let 푟 ℙ with for . Then, there exist −1 , −1 such that−1 u� 퐴(푠, 푠 ) ∈ GL(푟,−1 푘[푠, 푠 ]) −1 det 퐴(푠, 푠 ) = 푠 푛 ∈ ℤ 푈(푠) ∈ GL(푟, 푘[푠]) 푉(푠 ) ∈ GL(푟, 푘[푠 ]) u�1 푠 u�2 0 (2.7) −1 −1 ⎜⎛ ⎟⎞ ⎜ 푠 ⎟ 푉(푠 )퐴(푠, 푠 )푈(푠) = ⎜ ⎟ ⎜ ⋱ u�u� ⎟ with , . The are uniquely⎝ 0 determined 푠 by⎠ . −1 Proof.푑1 ≥ 푑2We≥ ⋯first ≥ prove 푑u� 푑u� uniqueness.∈ ℤ 푑 Writeu� for the matrix퐴(푠, on푠 the) right side of (2.7). If there are two matrices and1 u� equivalent to , then 퐷(푑 , … ,′ 푑 ) ′ −1 there are , 1 u� 1 such thatu� 퐷(푑−1 , … , 푑 ) 퐷(푑−1 , … , 푑 ) 퐴(푠, 푠 ) 푈(푠) ∈ GL(푟, 푘[푠]) 푉(푠 ) ∈ GL(푟, 푘[푠 ]) (2.8) −1 ′ ′ We now recall the Cauchy-Binet퐶 = 푉(푠 )퐷(푑 formula1, … [Gan59, , 푑u�) = 퐷(푑 I, §2.5].1, … , If 푑u�)푈(푠).is an matrix and is an matrix, denoting 퐴 푚 × 푛 퐵 푛 × 푞 푖1 푖2 ⋯ 푖u� 퐴 ( ) to be the minor of obtained by taking푗1 the푗2 ⋯ 푗u� of after removing all rows with index in 퐴 and removing all columns with퐴 index in , then {1, … , 푟}−{푖1, … , 푖u�} {1, … , 푟}−{푗1, … , 푗u�}

푖1 푖2 ⋯ 푖u� 푖1 푖2 ⋯ 푖u� ℓ1 ℓ2 ⋯ ℓu� (퐴퐵) ( ) = ∑ 퐴 ( ) 퐵 ( ). We want to apply푗1 푗 this2 ⋯ to (2.8). 푗u� Weℓ1<ℓ first2<⋯<ℓ noteu� ℓ1 ℓ2 ⋯ ℓu� 푗1 푗2 ⋯ 푗u�

ℓ1 ℓ2 ⋯ ℓu� 퐷(푑1, … , 푑u�)( ) ≠ 0 ⟺ ℓu� = 푗u�∀푖. Thus, we have 푗1 푗2 ⋯ 푗u�

1 2 ⋯ 푘 −1 1 2 ⋯ 푘 u�u�1+u�u�2+⋯+u�u�u� 퐶 ( ) = 푉(푠 )( ) 푠 푖1 푖2 ⋯ 푖u� 푖1 푖2 ⋯ 푖u� ′ ′ ′ u�1+u�2+⋯+u�u� 1 2 ⋯ 푘 = 푠 푈(푠) ( ) 푖1 푖2 ⋯ 푖u� 1. THE CASE OF VECTOR BUNDLES 14

for all and sequences . Since , for all , there exists at least one sequence푘 푖1 < 푖2

u�1 푠 0u�2 ⋯ 0 (2.9) ⎜⎛ 2 ⎟⎞ 1 0−1 1 0 ⎜푐 푠 0 ⎟ 퐶(푠) = ( 2 ) 퐵 ( 2 ) = ⎜ ⎟ 0 푉 (푠 ) 0 푈 (푠) ⎜ ⋮ ⋱ u�u� ⎟ u� for some and . By⎝ multiplying푐 0 푠 by⎠ suitable −1 −1 on the left,푘1, i.e., … , by 푘u� performing∈ ℤ≥0 푐 elementary2, … , 푐u� ∈ 푘[푠, row 푠 operations,] we can assume퐶(푠) that 푉(푠for) all . 푐u� ∈ 푘[푠] 푖Now consider all matrices equivalent to of the form (2.9). There is one representative with maximal, for implies 퐵(푠) . We claim that for all 1 . So suppose2 u� . Subtracting1 a suitable -multiple of the first1 row,u� (2.9) 푘 푘 , … , 푘 ≥ 0 푘 ≤ deg(det 퐵(푠))−1 푘 ≥ 푘 then has for1 someu� . Interchanging the first and -th rows, we get 푖 = 2, … , 푟 u�1+1 ′ 푘 < 푘 ′ 푘[푠 ] a polynomial푐u� = matrix푠 푐 (푠) such that푐 (푠) the ∈ greatest 푘[푠] common divisor of the first푖 row is ′ with . But applying′ to the same process as above for , we get a u�1of the ′ 퐵 (푠) ′ ′ 푠 form1 (2.9)1 with , contradicting maximality of . 푘 ≥ 푘 + 1 ′ 퐵 (푠) 퐵(푠) 퐶 (푠) We can therefore푘1 > 푘1 assume in (2.9) that and푘1 for . Now subtracting suitable -multiples of the th푘1 column≥ 푘u� for 푐u� ∈ 푘[푠] from푖 = the 2, first … , 푟 column (i.e., multiplying by suitable푘[푠] on the right)푖 we get a matrix푖 = 2,(2.9) … , 푟with for all . Then, , and so subtracting suitable -multiples of the first rowu� fromu� the 푈(푠) −1 deg 푐 ≤ 푘 th row (i.e., multiplyingu� 1 by suitable on the left), we get in 푖 deg 푐 < 푘 −1 푘[푠 ] (2.9)푖 . This shows that there are 푉(푠 ) , 푐2and= 푐3 = ⋯ = 푐u� = 0 , such that1 u� ≥0 1 2 u� −1 −1 푘 , … , 푘 ∈ ℤ 푘 ≥ 푘 ≥ ⋯ ≥ 푘 푈(푠) ∈ GL(푟, 푘[푠]) 푉(푠 ) ∈ GL(푟, 푘[푠 ]) −1 u� −1 −1 1 u� Multiplying by 푉(푠gives)푠 퐴(푠, 푠 )푈(푠) = 푉(푠 )퐵(푠)푈(푠) = 퐷(푘 , … , 푘 ). −u� 푠  −1 −1 푉(푠 )퐴(푠, 푠 )푈(푠) = 퐷(푑1, … , 푑u�), 푑u� = 푘u� − 푛. CHAPTER 2. FIBER BUNDLES ON THE RIEMANN SPHERE 15

Finally, let , be the over defined by the gluing matrix . Then, the bundle defined by the gluing matrix1 −1 −u�풪(푑) 푑 ∈ ℤ ℙ −1 퐴(푠,is equal 푠 ) to = the (푠 direct) sum , for the direct sum퐴(푠, of 푠 two)= vector 퐷(푑1 bundles, … , 푑u�) defined by gluing matrices and 1 on the sameu� cover is defined as the vector bundle defined 풪(−푑 ′), … , 풪(−푑 ) by gluing matrices .u�u� This givesu�u� us Theorem 2.1. ′ 푡 푡 We pause here to푡u�u� give⊕ 푡u�u� some references as to what happens in the holomorphic case. In terms of steps, the proof is the same. The fact that vector bundles over the complex plane 1 are trivial follows from the fact that vector bundles on non-compact Riemann surfaces areℂ holomorphically trivial [For91, §30], or the fact that is a Stein manifold [Gri74, VIII]. Then, the question again becomes how trivial bundles onu� each copy of glue together. The ℂ 1 critical step is the following lemma due to Birkhoff: ℂ Lemma 2.10 ([Bir13; Ses57, Lem. 3.31]). Let be a matrix of functions on 1 holomorphic for for some , such that (푙u�u�(푧)) for . Then, there existsℂ a matrix of entire functions|푧| ≥ 푅 푅with det(푙u�u�(푧))such ≠ 0 that|푧| ≥ 푅 (휖u�u�(푧)) det(휖u�u�(푧)) ≠ 0 u�u� u�u� u�u� u�u� u�u� where the are , (푙 (푧))(휖is a matrix(푧)) = of (푎 functions(푧))(훿 holomorphic푧 ) in a neighborhood of and u� is a diagonalu�u� matrix with diagonal entries . 푘 u�u� (푎 (푧)) u�u� u�u� ∞ The(훿 proof푧 ) then proceeds as follows: if is the affine푧 chart containing , and the matrix corresponds to our transition matrix푈0 for the vector bundle , then∞ for some small neighborhood(푙u�u�(푧)) containing there are matrices 퐸and that correspond to automorphisms푉 ⊂ 푈0 ∩ of 푈1 restricted to∞ and , respectively.(휖u�u�(푧)) The diagonal(푎u�u�(푧)) matrix is exactly that which we obtained algebraically1 above. The main difficulty of the proof u�u� 퐸 푈 푉 훿ofu�u� Lemma푧 2.10 is that unlike in the algebraic case, the isolated singularities of the entries in the matrix can be essential singularities; this is the main difficulty in Birkhoff’s proof of this lemma.푙u�u�(푧) After reducing to the case of poles by using a Fredholm integral equation, the rest of his proof follows analgously to our own proof of Proposition 2.6.

2. The case of projective and affine bundles We now want to extend some our results for §1 to the case of projective and affine bundles. We also want to know when an affine bundle is in fact a vector bundle; we prove such a criterion below.

2.1. Projective bundles. We first want to have an analogue of Proposition 2.2 for pro- jective bundles. At first glance, this seems rather easy since we could just try lifting our transition matrix on some open set from to . The issue is that while the matrices themselves in lift to푈 PGLasu� illustratedGLu�+1 in (1.7), this does not ensure that our maps from toPGLu�(푘) andGLu�+1(푘) necessarily lift as we mentioned before. So note that after푈 shrinkingPGLu�+1(푘)if necessary,GLu�+1(푘) for some localization of the polynomial . Then, the푈 map 푈 =corresponds Spec 푅 to an automorphism푅 of the bundle ; this is therefore an automorphismu�+1 of over , that is, an automorphism u� 푘[푠] 푈 → PGL u� of -dimensional푈 ×ℙu� over the ring .ℙ Weu� firstSpec show 푅 the following: 푟 푅 2. THE CASE OF PROJECTIVE AND AFFINE BUNDLES 16

Proposition 2.11. If is a finitely-generated -algebra that is also a UFD, then all automorphisms of are induced by a matrix in . u� 푅 푘 Proof. First considerℙu� ; this is Proposition 1.5GL butu�+1 we(푅) provide another proof as promised u� that works over any . Letℙu� be an automorphism. This induces an isomorphism on Picard groups, hence 푘 푓 or . But a section of gives a section of , hence ∗ . Now this gives an isomorphism of vector spaces ∗ ∗ 푓 풪(1) ≅ 풪(−1) 풪(1) 풪(1) u� 푓 풪(1) , which corresponds to an invertible matrix in . u� u� 푓 ∗풪(1) ≅ 풪(1) Γ(ℙ , 풪(1)) ≅ Γ(ℙTheu�, 푓 case풪(1)) when is a UFD that is finitely-generated as a GL-algebrau�+1(푘) follows similarly by [Har77, II, Exc. 6.1], since then , and also since 푅 u� u� 푘 by [Har77, II, Prop. 6.2]. We thereforeu� have an isomorphism ofu� free modules Pic ℙ ≅ Pic Spec 푅 × Pic ℙ Pic Specu� 푅 = 1 , which is also given by an invertible matrix in . u�  u� ∗ Γ(ℙ , 풪(1)) ≅ Γ(ℙRemarku�, 푓 풪(1)) 2.12. It is important that here is a UFD, since ifGL weu�+1 replace(푅) with any connected scheme (e.g., if were just푅 a domain and not necessarily aSpec UFD), 푅 then the group of automorphisms of is isomorphic to the group of invertible sheaves on 푆 푅u� plus matrices u� of sections of such that as a section of ℙ ℒ u�+1푆 never(푟 vanishes; + 1) × (푟 cf. + [GIT, 1) Ch. 0,(푎 §5;u�u�) EGAII, §4.2]. Thisℒ recoversdet(푎 our resultu�u�) since ℒ for a UFD which is finitely-generated as a -algebra has only one (isomorphism푆 = class Spec of) 푅 invertible푅 sheaves, namely, the structure ,푘 and so the matrix of sections given above is exactly a matrix in . We now want toPGL proveu�(푅) the analogue of Proposition 2.2. We first prove an easy commuta- tive algebraic result: Lemma 2.13. If is a noetherian UFD, then the localization is also a UFD if −1 . 푅 푆 푅 0 ∉Proof. 푆 is a noetherian domain, and so by [Mat70, Thm. 47], is a UFD if and only if every height−1 one prime is principal. But if is a height one prime−1 in , then there 푆 푅 푆 푅 −1 is a prime such that . But localization푃 does not affect푆 height,푅 hence has height one, hence is principal since−1 is a UFD, so let . Then , 푄 ⊂ 푅 푃 = 푆 푄 −1 u�푄 so is principal, and is a UFD. 1 −1 푅 푄 = (푎) 푃 = 푆 (푎) = ( ) 푃We can then prove푆 푅 Proposition 2.14. Every projective bundle on is trivial. 1 Proof. Suppose our projective bundle is trivial on픸 and ; we first show it is triv- ial on . By Proposition 2.11, we have a transition1 matrix2 in , 퐷(푓 ) 퐷(푓 ) −1 −1 since the퐷(푓 localization1)∪퐷(푓2) of a UFD is a UFD by Lemma 2.13. Since there isPGL onlyu�(푘[푠, one 푓 transition1 , 푓2 ]) matrix, we can lift this matrix to to define a vector bundle on . By Proposition 2.2, this is the trivial vectorGLu�+1 bundle, and the automorphisms of퐷(푓 the1) vector ∪ 퐷(푓2 bundle) restricted to and descend to automorphisms of the projective bundle restricted to and퐷(푓1) . 퐷(푓2)  퐷(푓A1 similar) 퐷(푓 argument2) also works for the proof of Proposition 2.6, i.e., our transition matrix in can be lifted to one in , and then we proceed as before. We −1 −1 moreoverPGL(푘[푠, have 푠 the]) following: GL(푘[푠, 푠 ]) CHAPTER 2. FIBER BUNDLES ON THE RIEMANN SPHERE 17

Corollary 2.15. Every projective bundle of rank on is of the form , i.e., is 1 obtained by projectivizing each fiber in some vector bundle푟 ℙ. 푃(퐸) Proof. Let be our transition퐸 matrix; we can simply lift it to −1 −1 to define a vectoru� bundle such that is our original projective bundle. −1 퐴(푠, 푠 ) ∈ PGL (푘[푠, 푠 ]) Theu�+1 reason this works for is that there is only one transition matrix we have to worry GL (푘[푠, 푠 ]) 1 퐸 푃(퐸) about; for other spaces, weℙ need to make sure that the representatives for each transition matrix still satisfy the cocycle condition.  Remark 2.16. One thing we would like to note is that even though it seems like we can always normalize to use a representative matrix of determinant one, this is not always the case. For example,

푠 0 ( ) defines a vector bundle and hence a projective0 1 bundle, but is not equivalent in to a matrix of determinant one. PGL1 Remark 2.17. The matrices from (1.15) seem to provide a good example of a set of matrices in that do not lift to while still verifying the cocycle condition. However, defining PGL1 GL2

1 1 1 푡13 = −푡24 = ( ) , 푡12 = 푡34 = ( ) , 푡14 = 푡23 = ( ) in fact verifies the cocycle−1 condition. This is actually−1 somewhat surprising,1 given that the matrices define a projective representation of the Klein -group, and this representation푡12 does, 푡13 not, 푡23 lift to a linear representation [Kar87, Ex. 2.3.2]. The4 reason that we can still consistently lift these matrices, then, stems from the fact that unlike for the projective representation of the Klein -group, we do not have to consider all possible products of these matrices, only the ones4 that arise as a cocycle condition. Thus, the class of projective bundles that do not lift to vector bundles is related, but not the same, as trying to find groups with a nontrivial Schur multiplier . 2 × 퐺 2.2. Affine bundles. For affine bundles,퐻 (퐺, ℂ proving) analogous statements is just a bit more involved, since while they are special cases of projective bundles, all isomorphisms of affine bundles must be isomorphisms of projective bundles that also fix the hyperplane . Recall that an affine bundle is obtained as a projective bundle whose transition maps{푧 always0 = 0} fix a hyperplane . So, to prove our analogue of Proposition 2.2, we lift our transition matrix in {푧to0 one= 0} in by first embedding it in and then lifting to as in (1.7); noteu�+1 that again, thisu�+1 is only possible because we areu�+1 working over , on whichu�+1 GA GL PGL 1 GL all projective bundles lift to vector bundles as proven in the previous section.ℙ The fact that this lifted matrix must fix , though, gives that the matrix is of the form {푧0 = 0} ∗ 0 ⋯ 0 ⎛ ⎞ (2.18) ⎜ ∗ ⎟ ⎜ ⎟ ⎜ ⋮ 퐴 ⎟ where . ⎝ ∗ ⎠ 퐴 ∈ GLu� 2. THE CASE OF PROJECTIVE AND AFFINE BUNDLES 18

Proposition 2.19. Every affine bundle on is trivial. 1 Proof. As in Proposition 2.2, assume our픸 affine bundle is trivial on . The same argument from before lets us reduce to the case where the matrix (2.18)퐷(푓1), has 퐷(푓 the2) form

u� u� 푓1 푓2 0 ⋯ 0 ⎛ ⎞ ⎜ 푣1 1 0 ⎟ ⎜ ⎟ ⎜ ⋮ ⋱ ⎟ where , for some ⎝. Then푣u� 0 1 ⎠ 푣u� ∈ 푘[푠] 푚, 푛 ∈ ℤ −u� u� u� −u� 1 1 2 2 푓 −u� −u� 0 ⋯ 0 푓 푓 0 ⋯ 0 푓 0 ⋯ 0 ⎜⎛ 1 1 2 ⎟⎞ ⎜⎛ 1 ⎟⎞ ⎜⎛ ⎟⎞ ⎜ −푣 푓 푓 1 0 ⎟ ⎜ 푣 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = Id, ⎜ −u�⋮ −u� ⋱ ⎟ ⎜ ⋮ ⋱ ⎟ ⎜ ⋮ ⋱ ⎟ u� 1 2 u� and then⎝ we−푣 proceed푓 푓 by0 gluing step1 ⎠ ⎝ by step푣 as0 before 1 ⎠ ⎝ 0 0 1 ⎠  Now we would like to prove an analogue of Theorem 2.1 for affine bundles. Of course, we do not have a notion of direct sum of affine bundles, but we can still ask when our transition can be diagonalized, i.e., when an affine bundle is actually a vector bundle. This produces the following result: Theorem 2.20. Suppose we have an affine bundle, which we view as a projective bundle that has transition matrix holding the hyperplane fixed. Let be its lift. Let 퐵be the line bundle associated to the coordinate . Then,{푧0 =is 0} a vector bundle퐵̃ if and only퐿 if . 푧0 퐵 ̃ ̃ 퐵 ≅Proof. 퐿 ⊕ 퐵/퐿 has transition matrix of the form 퐵̃ u� 푠 0 0 ⋯ 0 ⎜⎛ 1 11 12 1u� ⎟⎞ (2.21) ⎜ 푣 푎 푎 ⋯ 푎 ⎟ ⎜ 푣2 푎21 푎22 ⋯ 푎2u� ⎟ ⎜ ⎟ ⎜ ⋮ ⋮ ⋮ ⋱ ⋮ ⎟ with determinant a unit in ⎝ ,푣 byu� Corollary푎u�1 푎u�2 2.15⋯ and푎u�u� the⎠ discussion at the beginning of −1 §3. Then, by applying the푘[푠, method 푠 ] of Proposition 2.6, is isomorphic to the vector bundle with matrix of the form 퐵̃ u� 푠 0u�1 0 ⋯ 0 ⎜⎛ 1 ⎟⎞ ⎜ 푣 푠 2 ⎟ ⎜ u� ⎟ ⎜ 푣2 푠 ⎟ ⎜ ⎟ ⎜ ⋮ ⋱ u�u� ⎟ By attempting to use the inductive⎝ 푣u� step in the proof푠 of Proposition⎠ 2.6, we can moreover assume that each , since that inductive step keeps the hyperplane u�u�−1 u� fixed. Finally, we see푣u� = that ∑u�=1−u� by Theorem푏u�u�푠 2.1 and the definition of a quotient bundle, {푧0 = 0} and . 퐿 ≅ 풪(−푛) 퐵̃ ≅ 풪(−푑1) ⊕ ⋯ ⊕ 풪(−푑u�) CHAPTER 2. FIBER BUNDLES ON THE RIEMANN SPHERE 19

Now is a vector bundle if and only if in (2.21). But by the definition퐵 of quotient bundle and direct sum,푣 this1 = is 푣2 equivalent= ⋯ = 푣 tou� = 0 .  We hope that something like this criterion holds for other base퐵̃ spaces ≅ 퐿 ⊕ 퐵/퐿,̃ and that there is a cohomological interpretation of this statement. 푋 Part 2

Principal bundles over schemes CHAPTER 3

Preliminaries

In this second half of this thesis, we move to the setting of schemes, and principal bundles over them. We will see later that this is a generalization of the previous setting of varieties and fiber bundles. For completeness, we gather here some general definitions and facts concerning group schemes and group actions on schemes. In the following, by a scheme we mean a scheme with a morphism of schemes. If is affine, we say푋/푆 . refers to the 푋identity morphism 푋 → 푆 , and if 푆 = Spec, 푅 are -morphisms푋/푅 1u� of schemes, then 푋 → 푋 is their푓 ∶ 푋 product,1 → 푋2 and푔∶ 푌 if1 → 푌2 푆 , is their푓 pullback.× 푔∶ 푋1 ×u� If푌1 → 푋2 ×u� 푌2are affine, then refers푋1 = to 푌1 the= corresponding 푍 (푓 , 푔)∶ 푍 → pushout 푋2 ×u� 푌2 of ∘ ∘ algebra homomorphisms.푍, 푋2, 푌2, Varieties 푆 will refer(푓 to, 푔reduced,) separated schemes of finite over a field . 푘 1. Group schemes and algebraic groups In this section we define group schemes and algebraic groups. For more discussion, see [GIT, Ch. 0; Bor91, Ch. I]. Definition 3.1. A group scheme is a morphism with -morphisms

(the group law), 퐺/푆 (the inverse morphism휋∶ 퐺), → 푆 푆 (the identity

← ←

휇∶morphism 퐺 ×u� 퐺) → such 퐺← that the following훽∶ diagrams 퐺 → 퐺 commute: 푒∶ 푆 → 퐺

← ←

u�×1u� (u�,1u�) (u�∘u�,1u�)

← ←

← ← ×u� ×u� ← ×u� ×u� ×u�

퐺 퐺← 퐺 → 퐺 ← 퐺 퐺 → 퐺 ← 퐺 퐺 → 퐺 ← 퐺 → → → → → 1u� → u� u�∘u� u� 1u�×u� u� (1 ,u�) u� (1 ,u�∘u�) u� u� u� → u� → 퐺 ×u� 퐺(Associativity)→ 퐺 퐺 ×u� 퐺 (Inverse)→ 퐺 퐺 ×u� 퐺 (Identity)→ 퐺

Let be another group scheme with morphisms .A morphism of group schemes

′ ′ ′

← ← over퐻/푆is an -morphism← of schemes such휇 , that 훽 , 푒the diagrams below commute: 푆 푆 푓 ∶ 퐻 → 퐺

u� ×u� u� → u�

←

← ←

← ×u� ← ×u� 퐻 ← 퐻 → 퐺 ← 퐺 퐻 → 퐺 퐻 → 퐺 (3.2) → → → → ′ → ′ u� u� u� ′ u� u� ← u� u� u� An algebraic group퐻 over→ a퐺 field is a group퐻 scheme→ 퐺 which is푆 a reduced, separated scheme of finite type퐺 over , i.e., a푘 group scheme that is퐺/푘 also a variety over . 푘 21 푘 1. GROUP SCHEMES AND ALGEBRAIC GROUPS 22

Affine group schemes, that is, group schemes that are affine over some ring are what we are most interested in, especially is퐺/푅 a field and is in fact a variety,푅 i.e.,

when it is even an . For affine푅 = group 푘 schemes over퐺/푅 a (commutative) ring ,

→ →

the diagrams above→ correspond to the diagrams 푅

→ →

∘ ∘ ∘ ∘ u� ⊗1u� (u� ,1u�) (u� ∘u� ,1u�)

→ →

→ → 퐴 ⊗u� 퐴 ⊗u� 퐴 → ← 퐴 ⊗u� 퐴 퐴 ← 퐴 ⊗u� 퐴 퐴 ← 퐴 ⊗u� 퐴 → → → →

← ← ← ∘ ∘ ← ← ← ∘ ∘ ∘ 1u� ∘ ∘ u� u� ∘u� ∘ u� ∘ 1u�⊗u� u� (1 ,u� ) u� (1 ,u� ∘u� ) u� ∘ ∘ ∘ u� u� ← u� ← 퐴 ⊗u� 퐴(Associativity← )퐴 퐴 ⊗u� 퐴 (Inverse )← 퐴 퐴 ⊗u� 퐴(Identity ←) 퐴 ′ ′ ′ in the of -algebras, where morphisms are -algebra homomorphisms. Remark 3.3. 푅-algebras with morphisms 푅 as above are called Hopf algebras, ∘ ∘ ∘ and these form a category푅 opposite퐴 to that of affine휇 , 휋 group, 푒 schemes over with morphisms those that satisfy diagrams dual to (3.2). The study of affine group schemes푅 is a complicated one, in that there are multiple points of view being used simultaneously: an algebraic group can be thought of as a variety with group actions or dually as a Hopf algebra. The last point of view, that of looking at the these schemes represents, is for example the main viewpoint of [DG70]. See [Mil12, Preface] for a discussion. The idea is that group schemes are the scheme-theoretic version of algebraic groups, and so at least for varieties, we can think of the axioms above as defining operations on -valued points to give us the same intuition as for Lie groups. 푅 We redefine our primary examples from Chapter 1, §1 in this setting. Example 3.4. Our first example is , the general linear group of degree over a ring . If is understood, we abbreviateGL thisu�(푅) as . This is 푟 푅 푅 GLu� 푅[푎11, 푎12, … , 푎u�u�, 푡] 휋∶ Spec u�u� → Spec 푅, with the morphism induced by the inclusiondet(푎 )푡 − 1 . ∘ 11 12 u�u� u�u� is then the open subset of 2 where 휋 ∶ 푅 → 푅[푎. In this, 푎 case,, … , 푎 defining, 푡]/(det(푎 )푡 − 1) u� × GLu� 픸u� det(푎u�u�) ∈ 푅 (3.5) ∘ ∘ ∘ u�+u� −1 푒 (푎u�u�) = 훿u�u�, 휇 (푎u�u�) = ∑ 푎u�u� ⊗ 푎ℎu�, 훽 (푎u�u�) = (−1) det(푎u�u�) det(푎u�u�)u�≠u�,u�≠u� and extending linearly turnsℎ into a group scheme. We can think of the -valued points in as the set of GLu� invertible matrices over , in which case the 푅-algebra homomorphismsGLu�(푅) given above correspond푟 × 푟 to the usual operations푅 on as a group.푅 u� Example 3.6. Similarly, we can define , the projectiveGL general linear group

of degree over . It is formed by taking the openu� subset of 2 where , PGL (푅) u� +2u� where we푟 note the푅 determinant is a homogeneous polynomialℙ in the . Notedet(푎 thenu�u�) ≠ that 0 this is an affine scheme , where is the zeroeth푎 degreeu�u� of the ring , and the groupSpec morphisms푅[푎u�u�](det) defined푅[푎 similarlyu�u�](det) to (3.5). 푅[푎u�u�][1/ det] CHAPTER 3. PRELIMINARIES 23

Definition 3.7. Let be a morphism of group schemes over . is a closed subgroup scheme of if 푖is ∶ 퐻 a closed → 퐺 . is moreover a closed algebraic푆 퐻 subgroup of if both are퐺 algebraic푖 groups. By definition,퐻 a closed algebraic subset of is a closed퐺 algebraic퐻, 퐺 subgroup if and only if it is a subgroup of . 퐻 퐺 Example 3.8. The closed algebraic subgroups of 퐺 are called linear algebraic groups. Many classical groups are examples of these. ForGL example,u�(푘) is the subgroup of defined by the ideal . SLu� u� u�u� GL Example 3.9. The lastdet(푇 example) = 1 of a closed algebraic subgroup we give is , the general affine group of degree . It is formed by taking the closed subgroupGA of u�(푅) such that 푟 . PGLu�(푅) 12 13 1u� The simplest푇 = 푇 examples= ⋯ = 푇 of groups,= 0 that is, finite groups, also have a scheme-theoretic analogue: Example 3.10 (Constant group scheme). Let be a finite group; it can be turned into a group scheme over a ring that has only as idempotents퐺 by taking the disjoint union of copies푅 of . 0,then 1 is equal to 퐺u� =. The∐|u�| morphismSpec 푅 |퐺| Spec 푅 퐺u� Spec 푅 × ⋯ × 푅 = Spec 푅[푒u�]u�∈u�

∘ 휇 ∶ 푅 → 푅[푒u�]u�∈u� ⊗u� 푅[푒u�]u�∈u� 푒u� ↦ ∑ (푒u� ⊗ 푒u�) u�=u�u� gives a group law. Inverse is defined by and identity by ∘ −1 훽 ∶ 푒u� ↦ 푒u�

u� u�∈u� u� u�1 i.e., if , otherwise.푒∶ 푅[푒 ] See [Tat97,→ 푅 푒 2.10]↦ 훿 for, details. 푒u� ↦ 1 푔 = 1 ∈ 퐺 0 2. Group actions on schemes We can now talk about group actions on schemes, keeping in mind that we want to generalize our notion of fiber bundles from before. Definition 3.11. A group scheme acts (on the left) of a scheme if an -morphism is given, such that 퐺/푆 푋/푆 푆

u� 푚 ∶ 퐺(a)×u� The푋 → diagram 푋 below commutes, where← is the group law for : 휇 퐺

1u�×u�u� ×u� ×u� ← ×u�

퐺 퐺← 푋 → 퐺 ← 푋 → → u�×1u� u�u� u�u�

퐺 ×u� 푋 → 푋

← ← (b) The composition below← is the identity , where is the identity morphism for : 1u� 푒 퐺 u�×1u� u�u� 푋 ∼→ 푆 ×u� 푋 → 퐺 ×u� 푋 → 푋 2. GROUP ACTIONS ON SCHEMES 24

Similarly, we can define the notion of that acts on the right of . If the action is understood, we call a -scheme. 퐺/푆 푋/푆 Let be another푋 퐺 -scheme with action , and let be a morphism푌/푆 over . is then퐺 a morphism of -schemes푚u� ∶ 퐺or×u� 푌 a →-equivariant 푌 morphism푓 ∶ 푋 → 푌if the

diagram below commutes:푆 푓 퐺← 퐺

(1u�,u� ) ×u� ← ×u�

퐺 ← 푋 → 퐺 ← 푌 → → u�u� u�u� u� We return to our favorite examples:푋 → 푌 Example 3.12 ( acting on ). We want to define an action . u� u� u� Let GLu�;(푅) we can define픸 thisu� action as the morphism on schemesGLu�(푅) corresponding×u� 픸u� → 픸u� u� to the픸u� -algebra= 푅[푥1, … homomorphism , 푥u�] 푅 푅[푎u�u�][푡] 푅[푥1, … , 푥u�] → ⊗ 푅[푥1, … , 푥u�] det(푎u�u�)푡 − 1 u� 푥u� ↦ ∑ 푎u�u� ⊗ 푥u� u� Example 3.13 ( acting on ). We want to define an action u� u� . This is a bit trickier,PGLu�(푅) but recall fromℙu� [Har77, II, Thm. 7.1] that the휎∶ morphism PGLu�(푅) ×u�canℙu� be→ u� givenu� by finding an on that is generated by global sections ℙ u� 휎 , and then by the rule PGL. Byu�(푅) [EGAII,×u� ℙu� §4.2], ∗ ∗ u�2+2u� ∗ u� 푠is0, such … , 푠 anu� invertible sheaf, and휎 we(푥u� can) = define 푠u� via 푝1[풪ℙ (1)] ⊗ 푝2[풪ℙ (1)] 휎

∗ u� ∗ u�2+2u� ∗ u� 휎 (풪ℙ (1)) ≅ 푝1[풪ℙ (1)] ⊗ 푝2[풪ℙ (1)] u� ∗ ∗ ∗ 휎 (푥u�) = ∑ 푝1(푎u�u�) ⊗ 푝2(푥u�). u�=0 The idea of group actions on schemes suggests that we might be able to take their quotients by such an action. It is easy to write down a definition: Definition 3.14. The quotient scheme of by is a scheme and a morphism of schemes such that 푋 퐺 푌 푝∶ 푋 → 푌 (1) is invariant under , i.e., for all , and (2) 푝 have the following퐺 universal푝 ∘ 푚u�(푔, property: −) = 푝 for every푔 ∈ 퐺 scheme , and every (푌,-invariant 푝) morphism , there is a unique morphism 푍/푆 that makes

퐺the diagram 푓 ∶ 푋 → 푍 ℎ∶ 푌 → 푍 ← ← ← 푋→ u� u� → ℎ commute. 푌 → 푍 CHAPTER 3. PRELIMINARIES 25

However, there is no reason to expect that such an object exists; indeed, this is the main topic of [GIT]. On the other hand, if is a constant group scheme associated to a finite group as in Example 3.10, we have퐺 theu� following nice criterion due to Chevalley: Theorem퐺 3.15 ([SGA1, Exp. V, Prop. 1.8]). Let be a scheme on which a constant group scheme acts on the right. The quotient of by푋 exists if and only is the union of affine open sets퐺 that are invariant under the action푋 of 퐺, or in other words,푋 every orbit of in is contained in an affine open set. 퐺 퐺 푋On a -invariant affine open subset of , since acts on the right of , we see that acts on the퐺 left of the coordinate ring . Thus,푋 the statement퐺 above really boils푋 down to 퐺 Proposition 3.16 ([Bou64, Ch. V,퐴 §1, no 9, Prop. 22; §2, no 2, Thm. 2]). Let be a ring on which a finite group acts on the left, the subring of invariants of , u� 퐴 and , 퐺 the morphism퐵 =associated 퐴 to the inclusion, invariant퐴 푋 = under Spec 퐴. Then,푌 = Spec 퐵 푝∶ 푋 → 푌 퐺 (1) is integral over , i.e., is an integral morphism; (2) 퐴 is surjective, its퐵 fibers are푝 the orbits of , and the topology on is the quotient 푝topology induced by ; 퐺 푌 (3) Letting , 푋 , the stabilizer of , then is a normal (“quasi- Galois”)푥 extension ∈ 푋 푦 = of 푝(푥) 퐺andu� the morphism 푥 푘(푥) is surjective; (4) is the quotient scheme푘(푦) of by . 퐺 → Gal(푘(푥)/푘(푦)) The idea(푌, 푝)is then to take these affine schemes푋 퐺 and glue them together using the uniqueness of the pair ; see [SGA1, Exp. V] for details. Of course(푌, there 푝) are examples where a finite group acts on a scheme, but there does not exist a quotient scheme. The canonical example where this occurs is the following: Example 3.17 ([Hir62]). Let be complex projective three-space, and u� and two conics that intersect normally푉0 = ℙ inℂ exactly two points and . For 훾1, construct훾2 by first , and then in the result. Let푃1 be the푃2 open set푖 = in 1, 2 lying over푉 u� ; by gluing훾u� together 훾along3−u� along , we get푉u� a scheme . Letting푉 u� (푉0 be− 푃 the3−u�) projective transformation푉 u� interchanging푉u� and , and푈 , 휎induces0 ∶ 푉0 → an 푉 automorphism0 of order . Hironaka in푃1 [Hir62]푃2 showed훾1 that훾2 this휎0 scheme under the action 휎∶ 푈 →has 푈 no quotient2 in the sense above, but it does have the structure of what is called퐺 an =algebraic {1, 휎} space [Knu71, p. 15–16]. CHAPTER 4

Principal bundles in étale topology

Our proof of Theorem 2.1 relied on the fact that holomorphic vectors bundles on the variety are isomorphic to algebraic ones. But this is not always the case for fiber bundles 1 with differentℙ structure groups, as we saw in Example 1.14. But having to talk about holomorphic bundles separately from algebraic ones is inconvenient given that there is then no unified way to talk about holomorphic bundles and algebraic bundles on algebraic varieties defined over , and also we would like to have an analogue of holomorphic bundles for fields that are notℂ . The solution to this is to use what is called the étale topology; even though this is notℂ strictly a topology, it provides a framework in which to talk about holomorphic bundles algebraically. We discuss first what is the étale topology, and use it to describe that for structure group being or , fiber systems are locally trivial in étale topology if and only if they퐺 are in locallyGLu� trivialGAu� in Zariski topology, but not for . 퐺 = PGLu� 1. Differential forms; unramified and étale morphisms The idea of an étale morphism is that we want an analogue of a local isomorphism of complex analytic manifolds, which in particular induces an bijection of tangent spaces, and so it becomes important to talk about tangent bundles. In the algebraic world, though, it is more convenient to talk about its dual, the sheaf of differentials. So we first review the algebraic theory of differentials here, following [Har77, II.8]. Let be a ring, an -algebra, and an -. An -derivation of into is an additive map such that , and for all . 퐴 퐵 퐴 ′ 푀 퐵 ′ ′ 퐴 퐷 퐵 푀 Definition 4.1.퐵 →The 푀 module of푑(푏푏 relative) = differential푏푑(푏 ) + 푏푑(푏 forms) of 푑(푎)over = 0to be a 푎-module ∈ 퐴 , together with an -derivation , which satisfies퐵 the퐴 following퐵 universal Ωproperty:u�/u� for any -module퐴 , and푑 foru�/u� any∶ 퐵 →-derivation Ωu�/u� , there exists a unique -module homomorphism such that ′ . 퐵 푀 퐴 ′ 푑 ∶ 퐵 → 푀 u�/u� u�/u� 퐵 To construct such a module,푓 ∶ Ω consider→ 푀 the “diagonal”푑 = morphism푓 ∘ 푑 defined by , and let be the of this morphism. Consider u� as a -module by ′ ′ 퐵 ⊗ 퐵 → 퐵 multiplication on the left. Then has the structure of a -module;u� denote 푏 ⊗ 푏 ↦ 푏푏 퐼 2 퐵 ⊗ 퐵 퐵 to be the -module obtained by restriction of scalars. Define theu� map u�/u�by 퐼/퐼 퐵 ⊗ 퐵 Ω2 (modulo ). u�/u� 퐵 2 푑 ∶ 퐵 → 퐼/퐼 푑u�/u�We(푏) have = 1 ⊗ the 푏 following − 푏 ⊗ 1 facts about퐼 : u�/u� Proposition 4.2 ([Mat70, p. 186]).ΩIf and are -algebras, and , then ′ ′ ′ . 퐴 퐵 퐴 퐵 = 퐵 ⊗u� 퐴 ′ ′ ′ Ωu� /u� ≅ Ωu�/u� ⊗u� 퐵 26 CHAPTER 4. PRINCIPAL BUNDLES IN ÉTALE TOPOLOGY 27

Proposition 4.3 ([Mat70, Thm. 57]). Let u� u� be rings and homomorphisms. Then there is a natural of -modules퐴 → 퐵 → 퐶 퐶 u� u� u�/u� u� u�/u� u�/u� We can then globalize thisΩ definition⊗ 퐶 → to define Ω → sheaves Ω over→ 0. our space. Let be a scheme over , and the diagonal morphism . This is an immersion푋 (i.e., the composition푌 Δ ofu�/u� a closed= Δ then open immersion),푋 and → so 푋 ×u� 푋 is a closed subscheme of an open subscheme of . Δ(푋) 푉 u� Definition 4.4.푋Let × 푋 be the ideal sheaf corresponding to the closed subscheme . isu� then a quasi- on , which we denote ; this ∗ 2 풥 Δ(푋)is the sheaf ⊂ 푉 ofΔ relative(풥u�/풥 differentialu�) forms of over . 푋 Ωu�/u� If and , then 푋 푌 , and in the general case, these glue together푌 =toSpec give 퐴 .푋 This = Spec means 퐵 that eachΩu�/u� of the≅ Ω̃ commutativeu�/u� algebraic results above have sheaf-theoretic analoguesΩu�/u� which we use below. In the following, we assume that all our schemes are locally noetherian. The definitions here are from [SGA1, I]. Definition 4.5. Let be a morphism locally of finite type, and let , . is unramified 푓if ∶ 푋 → 푌 for all . 푥 ∈ 푋 푦 = 푓 (푥)is étale푓 if it is flat, i.e.,(Ωu�/u� )u� = 0 is flat푥 ∈ for 푋 all , and unramified. 푓A finite family of étale morphisms풪u�,u� (u�) → 풪u�,u� where푥 ∈ 푋are affine is an étale cover if the of the cover . {푓u� ∶ 푈u� → 푌} 푈u� u� Unramified and푓 étale푌 morphisms work nicely under composition and base change:

Proposition 4.6. The composition of unramified (resp. étale) morphisms u� u� is unramified (resp. étale); the base change of an unramified (resp. étale) morphism푋 → 푌 → is푍 unramified (resp. étale). Proof. This follows by Propositions 4.2 and 4.3 for unramified maps; for flatness, this is [Mat70, 3.B and 3.C].  We also note that we have an alternate description of unramified morphisms: Theorem 4.7 ([EGAIV , Thm. 17.4.1]). Let be a morphism locally of finite type. Then, is unramified if4 and only if it is quasi-finite푓 ∶ 푋 → and 푌 the ring is a field for all 푓 , , and is a finite separable extension of the residue풪u�,u� field/픪u�풪u�,u� . Definition푥 ∈ 푋 푦 = 4.8. 푓 (푥)We call the degree of the field extension 푘(푦) the degree of our unramified morphism. It is locally constant, hence푘(푦) constant → 풪u�,u� on/픪 a connectedu�풪u�,u� component of . Now an étale푌 cover is galois if it is of the form for a finite group of automor- o phisms of . This happens if and only if acts freely푋 → on 푋/픤 [Ser58a, n 1.4].픤 Note that by Theorem 4.7,푋 we can think of this as the scheme-theoretic픤 푋 version of a galois extension of fields. Now let be an étale cover; we claim there is an associated galois cover: {푓u� ∶ 푈u� → 푌} 2. PRINCIPAL AND FIBER BUNDLES IN ÉTALE TOPOLOGY AND A COMPARISON 28

Proposition 4.9 ([Ser58a, no 1.4]). Let be connected, and a surjective étale morphism, i.e., an étale cover of , of degree푌 . Then, there exists푓 ∶ a 푋 scheme → 푌 such that is an galois cover, and factors푌 through 푛. 휋∶ 푍 → 푌 Proof푓 Sketch. Let be étale; then푓 is a surjective étale cover

×u� ×u� ×u� of degree by Proposition 4.6. There is then← a cartesian diagram u� 푓 ∶ 푋 → 푌 푓 ∶ 푋 → 푌 푛

×u� Δ̃ ×u� u� ← ← ← 푋→ → 푋→ ×u� u� ̃ u�

Δ ×u� is then a degree étale cover of , and푌 acts→ on푌 . Let be the inverse image through u� u� ×u� 푓 ̃ of the open subset푛 of consisting푌 푆 of -tuples푋u� 푍 where each is distinct. ×u� o Δ̃ is then an étale cover푋 of degree , and푛 acts(푦 freely1, … , on푦u�) ; see [Ser58a,푦u� n 1.4] for o 푍details. → 푌 The factorization comes from [Ser58a,푛! 푆 nu� 1.3(f)]. 푍 

2. Principal and fiber bundles in étale topology and a comparison Étale morphisms allow us to talk about a new “topology,” namely, instead of thinking of open sets in , we allow “open covers” which are actually just étale covers. This generalizes the notion of푋 an open covering, and gives us a new class of fiber bundles which are not necessarily algebraic, i.e., locally trivial in Zariski topology. Recalling our definitions from Chapter 1, we have the following analogue of fiber bundles in this new topology: Definition 4.10. A fiber system is locally trivial in étale topology if there exists an étale cover of such that are isomorphic to the trivial bundle. ′ 퐸 → 푋 ∗ u� u� u� u� u�u� We recall Example{푓 ∶ 푈 → 1.14, 푈 } which푋 gives an푓 example퐸| of a -bundle that is locally trivial in Zariski topology but not in the étale topology. PGL Example 4.11 (See Example 1.14). Let

−1 −1 1/2 −1/2 1/2 −1/2 Then as rings.퐴 = This ℂ[푠, is 푠 a, flat 푡, 푡 ring], homomorphism, 퐵 = ℂ[푠 , 푠 for, 푡 ,is 푡 the]. free module over generated by . We have , hence . Thus, the 퐴 ↪ 퐵 1/2 1/2 1/2 1/2 퐵 퐴 map 1, 푠 , 푡 on, 푠 spectra푡 is an étaleker cover 퐵 ⊗ ofu� 퐵 = 퐼 =, and0 the mapΩu�/u� defined= 0 in (1.14) is thenSpec a local 퐵 → trivialization Spec 퐴 of the bundle in this étaleSpec cover. 퐴 Hence Example 1.14 is locally trivial in étale topology. In contrast, however, vector bundles that are locally trivial in étale topology are actually locally trivial in étale topology. To prove this, we change our point of view slightly to principal bundles. Definition 4.12. Let be a fiber system with structure group as we had before. is a principal -bundle locally푃 → 푋trivial in Zariski (resp. étale topology) if퐺 there is an open cover푃 of (resp. an étale cover of ) such that (resp. ). 퐺 ∗ 푈u� 푋 {푓u� ∶ 푈u� → 푋} 푋 푃|u�u� ≅ 푈u� × 퐺 푓u� 푃 ≅ 푈u� × 퐺 CHAPTER 4. PRINCIPAL BUNDLES IN ÉTALE TOPOLOGY 29

Note this is a special case of a fiber system such that our fibers are isomorphic to itself. If is a fixed -scheme, then principal -bundles and fiber퐹 bundles with structure퐺 group 퐹and fiber are퐺 in one-to-one correspondence,퐺 since we can just take our transition functions퐺 and use them퐹 to patch together a principal -bundle by having them act on . We remind the reader of our discussion in Remark 1.13:퐺 these objects are nicer to deal퐺 with because there are canonical ways to choose morphisms that represent automorphisms of the trivial bundle. 푋 → 퐺 Moreover, we have the nice characterization for when the following criterion for principal bundles makes them even better because there is a principal bundles are locally trivial: Proposition 4.13. Let be connected. A principal -bundle is locally trivial if and only if there exists퐺 a Zariski local (resp. étale local)퐺 section휋∶ at 푃 each → 푋 . Note an étale local section is an étale morphism with in its image and a푥 morphism∈ 푋 such that . ′ 푓 ∶ 푈 → 푋 푥 휎∶ 푈Proof. → 푈 ×If 푃 is an open푓 ∘ set 휎 (resp.= 휋 is an étale morphism) on which is trivial, then clearly we푈 are done. In the other푓 direction, ∶ 푈 → 푋 if is a Zariski local section on휋 (resp. is an étale local section) then 휎 defines an isomorphism 푈 휎 (resp. ), where the inverse is as defined in [Ser58a, Prop. 2]. Note in ∗ Ψ(푥, 푔) = 휎(푥) ⋅ 푔 푈 × 퐺 → 푃 particular푈 × that 퐺 this → 푓 morphism푈푃 is bijective by the fact that acts effectively on itself since it’s connected. 퐺  Remark 4.14. We note that this is a feature specific to principal bundles that does not apply to fiber bundles. For example, every vector bundle has the section defined by choosing the origin in each fiber. Proposition 4.15 ([Ser58a, Prop. 1]). The isomorphism classes of principal -bundles over that are trivial after base change through a Galois cover with associated group ′ 퐺 of automorphisms푋 is in bijection with the (non-abelian) group푋 → 푋 , where is the group of morphisms , on which acts by 1 ′ .

픤 ′ ′ 퐻 (픤,′ Γ(푋 ′, 퐺)) Proof.Γ(푋 ,We 퐺) have the commutative diagram푋 →← 퐺 픤 (휎푓 )(푥 ) = 푓 (푥 ⋅ 휎)

′ ← ← ← → 푋 × 퐺 → 푃→ ′ u�

′ is identified with quotiented푋 out by→ 푋by the above. The action of must be compatible with the projection′ ′ , and so we have 푃 푃 = 푋 × 푃 ′ 픤 픤 휋 ′ ′ ′ for some , and(푥 so, 푔) ⋅ 휎is = a (푥-cochain⋅ 휎, 휑u�(푥 of) ⋅ 푔), 휎 ∈ 픤. The associativity of the ′ ′ action of 휑u� ∶ 푋 → 퐺 휑u� 1 픤 → Γ(푋 , 퐺) 픤 ′ ′ implies the identity (푥 , 푔) ⋅ 휎휏 = ((푥 , 푔) ⋅ 휎) ⋅ 휏, 휎, 휏 ∈ 픤

′ ′ ′ 휑u�u�(푥 ) = 휑u�(푥 ⋅ 휎) ⋅ 휑u�(푥 ), 2. PRINCIPAL AND FIBER BUNDLES IN ÉTALE TOPOLOGY AND A COMPARISON 30 which is exactly the cocycle condition. Conversely, such a cocycle gives a well defined operation of on , hence defines . Finally, this implies that two cocycles correspond to the same bundle픤 푋 if × and퐺 only if they are푃 cohomological.  We can now prove Theorem 4.16 ([Ser58a, Thm. 2], [Ser58b, no 15]). A with group is locally trivial in étale topology if and only if it is locally trivial푃 in→ Zariski 푋 topology.GLu� Proof. By restriction, we can assume that is étale-trivial on , and so let be a galois cover with group , such that is trivial. Let be the semilocal ring for ∗ 푃 푋 휋∶ 푈 → 푋 in . The group of germs of morphisms u� can be identified −1 픤 휋 푃 퐴−1 with . By the Propositionu� u� above, defines an element u� ; it 휋 (푥) 푈 Γ (푈, GL ) 휋 → GL 1 suffices tou� showu� that this is trivial, i.e., . u� u� u� GL (퐴 ) 1 푃 푝 ∈ 퐻 (픤, GL (퐴 )) Now for , let . Choose anu� u� matrix in that is the identity on −1 퐻 (픤, GL (퐴 )) = 0 but zero on other points1 of . If , thenu� put 푥 ∈ 푋 푦 ∈ 휋 −1(푥) ℎ 1 푛 × 푛 퐴 푦1 휋 (푥) 휑u� ∈ 퐻 (픤, GLu�(퐴u�)) 푎 = ∑ 휏(ℎ) ⋅ 휑u�. is then invertible on every point in u�∈픤 . Moreover, −1 푎 휋 (푥) 휎(푎)휑u� = ∑ 휎휏(ℎ) ⋅ 휎(휑u�)휑u� = ∑ 휎휏(ℎ) ⋅ 휑u�u� = 푎, and so is a coboundary elementu�∈픤 as desired. u�∈픤  The휑 proofu� above constructs a section of our principal bundle, by proving that the section defined by is in fact invariant under the action of , hence descends ∗ 푋 → 푃 to a morphism푈 → 휋 푃 by the property푎 of quotient schemes. We can do픤 the same thing with an affine bundle;푋 → 푃 indeed, the same construction above gives a section by using instead. u� 1 휑 ∈ 퐻 (픤,Remark GAu�(퐴 4.17.u�)) Groups that satisfy the property in Theorem 4.16 are called special. For a full classification of groups of this type, see [Gro58b]. Note that specialness is preserved under group extensions [Ser58a, Lem. 6; Gro58a, Prop. 5.3.1], giving another way to prove that is special. WeGA recallu� that the analogue for Theorem 4.16 does not hold for projective bundles. Tsen’s theorem 1.16 from before can be used to prove that for projective bundles over curves, though, an analogue does hold. For projective spaces, this is also true, but the proof gets much more involved. This is the subject of étale cohomology and Brauer groups. Bibliography

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