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MATH 216: FOUNDATIONS OF ALGEBRAIC

GEOMETRY

math216.wordpress.com

June 11, 2013 draft

c© 2010, 2011, 2012, 2013 by Ravi Vakil.

Note to reader: the figures, index, and formatting have yet to be properly dealt with. There

remain a few other issues still to be dealt with in the main part of the notes.

Contents

Preface 110.1. For the reader 120.2. For the expert 150.3. Background and conventions 170.4. ⋆⋆ The goals of this book 18

Part I. Preliminaries 21

Chapter 1. Some category theory 231.1. Motivation 231.2. Categories and functors 251.3. Universal properties determine an object up to unique isomorphism 311.4. Limits and colimits 391.5. Adjoints 431.6. An introduction to abelian categories 461.7. ⋆ Spectral sequences 56

Chapter 2. Sheaves 692.1. Motivating example: The sheaf of differentiable functions. 692.2. Definition of sheaf and presheaf 712.3. Morphisms of presheaves and sheaves 762.4. Properties determined at the level of stalks, and sheafification 802.5. Sheaves of abelian groups, and OX-modules, form abelian categories 842.6. The inverse image sheaf 872.7. Recovering sheaves from a “sheaf on a base” 90

Part II. Schemes 95

Chapter 3. Toward affine schemes: the underlying set, and topological space 973.1. Toward schemes 973.2. The underlying set of affine schemes 993.3. Visualizing schemes I: generic points 1113.4. The underlying topological space of an affine scheme 1123.5. A base of the Zariski topology on SpecA: Distinguished open sets 1153.6. Topological (and Noetherian) properties 1163.7. The function I(·), taking subsets of SpecA to ideals of A 124

Chapter 4. The structure sheaf, and the definition of schemes in general 1274.1. The structure sheaf of an affine scheme 1274.2. Visualizing schemes II: nilpotents 130

3

4.3. Definition of schemes 1334.4. Three examples 1374.5. Projective schemes, and the Proj construction 143

Chapter 5. Some properties of schemes 1515.1. Topological properties 1515.2. Reducedness and integrality 1535.3. Properties of schemes that can be checked “affine-locally” 1555.4. Normality and factoriality 1595.5. Where functions are supported: Associated points of schemes 164

Part III. Morphisms 173

Chapter 6. Morphisms of schemes 1756.1. Introduction 1756.2. Morphisms of ringed spaces 1766.3. From locally ringed spaces to morphisms of schemes 1786.4. Maps of graded rings and maps of projective schemes 1846.5. Rational maps from reduced schemes 1866.6. ⋆ Representable functors and group schemes 1926.7. ⋆⋆ The Grassmannian (initial construction) 197

Chapter 7. Useful classes of morphisms of schemes 1997.1. An example of a reasonable class of morphisms: Open embeddings 1997.2. Algebraic interlude: Lying Over and Nakayama 2007.3. A gazillion finiteness conditions on morphisms 2057.4. Images of morphisms: Chevalley’s theorem and elimination theory 214

Chapter 8. Closed embeddings and related notions 2218.1. Closed embeddings and closed subschemes 2218.2. More projective geometry 2268.3. Smallest closed subschemes such that ... 2328.4. Effective Cartier divisors, regular sequences and regular embeddings236

Chapter 9. Fibered products of schemes, and base change 2419.1. They exist 2419.2. Computing fibered products in practice 2479.3. Interpretations: Pulling back families, and fibers of morphisms 2509.4. Properties preserved by base change 2559.5. ⋆ Properties not preserved by base change, and how to fix them 2579.6. Products of projective schemes: The Segre embedding 2649.7. Normalization 267

Chapter 10. Separated and proper morphisms, and (finally!) varieties 27310.1. Separated morphisms (and quasiseparatedness done properly) 27310.2. Rational maps to separated schemes 28310.3. Proper morphisms 287

Part IV. “Geometric” properties: Dimension and smoothness 293

Chapter 11. Dimension 295

11.1. Dimension and codimension 29511.2. Dimension, transcendence degree, and Noether normalization 29911.3. Codimension one miracles: Krull’s and Hartogs’s Theorems 30711.4. Dimensions of fibers of morphisms of varieties 31311.5. ⋆⋆ Proof of Krull’s Principal Ideal and Height Theorems 318

Chapter 12. Regularity and smoothness 32112.1. The Zariski tangent space 32112.2. Regularity, and smoothness over a field 32612.3. Examples 33212.4. Bertini’s Theorem 33512.5. Discrete valuation rings: Dimension 1 Noetherian regular localrings 33812.6. Smooth (and étale) morphisms (first definition) 34312.7. ⋆ Valuative criteria for separatedness and properness 34712.8. ⋆More sophisticated facts about regular local rings 35112.9. ⋆ Filtered rings and modules, and the Artin-Rees Lemma 352

Part V. Quasicoherent sheaves 355

Chapter 13. Quasicoherent and coherent sheaves 35713.1. Vector bundles and locally free sheaves 35713.2. Quasicoherent sheaves 36313.3. Characterizing quasicoherence using the distinguished affine base 36513.4. Quasicoherent sheaves form an abelian category 36913.5. Module-like constructions 37113.6. Finite type and coherent sheaves 37513.7. Pleasant properties of finite type and coherent sheaves 37713.8. ⋆⋆ Coherent modules over non-Noetherian rings 381

Chapter 14. Line bundles: Invertible sheaves and divisors 38514.1. Some line bundles on projective space 38514.2. Line bundles and Weil divisors 38714.3. ⋆ Effective Cartier divisors “=” invertible ideal sheaves 396

Chapter 15. Quasicoherent sheaves on projective A-schemes 39915.1. The quasicoherent sheaf corresponding to a graded module 39915.2. Invertible sheaves (line bundles) on projective A-schemes 40015.3. Globally generated and base-point-free line bundles 40115.4. Quasicoherent sheaves and graded modules 404

Chapter 16. Pushforwards and pullbacks of quasicoherent sheaves 40916.1. Introduction 40916.2. Pushforwards of quasicoherent sheaves 40916.3. Pullbacks of quasicoherent sheaves 41016.4. Line bundles and maps to projective schemes 41616.5. The Curve-to-Projective Extension Theorem 42316.6. Ample and very ample line bundles 42416.7. ⋆ The Grassmannian as a moduli space 429

Chapter 17. Relative versions of Spec and Proj, and projective morphisms 43517.1. Relative Spec of a (quasicoherent) sheaf of algebras 43517.2. Relative Proj of a sheaf of graded algebras 43817.3. Projective morphisms 44117.4. Applications to curves 447

Chapter 18. Čech cohomology of quasicoherent sheaves 45318.1. (Desired) properties of cohomology 45318.2. Definitions and proofs of key properties 45818.3. Cohomology of line bundles on projective space 46318.4. Riemann-Roch, degrees of coherent sheaves, arithmetic genus, andSerre duality 46518.5. A first glimpse of Serre duality 47318.6. Hilbert functions, Hilbert polynomials, and genus 47618.7. ⋆ Serre’s cohomological characterization of ampleness 48218.8. Higher direct image sheaves 48518.9. ⋆ Chow’s Lemma and Grothendieck’s Coherence Theorem 489

Chapter 19. Application: Curves 49319.1. A criterion for a morphism to be a closed embedding 49319.2. A series of crucial tools 49519.3. Curves of genus 0 49819.4. Classical geometry arising from curves of positive genus 49919.5. Hyperelliptic curves 50119.6. Curves of genus 2 50519.7. Curves of genus 3 50619.8. Curves of genus 4 and 5 50819.9. Curves of genus 1 51119.10. Elliptic curves are group varieties 51819.11. Counterexamples and pathologies using elliptic curves 523

Chapter 20. ⋆ Application: A glimpse of intersection theory 52920.1. Intersecting n line bundles with an n-dimensional variety 52920.2. Intersection theory on a surface 53320.3. The Grothendieck group of coherent sheaves, and an algebraicversion of homology 53920.4. ⋆⋆ The Nakai-Moishezon and Kleiman criteria for ampleness 541

Chapter 21. Differentials 54721.1. Motivation and game plan 54721.2. Definitions and first properties 54821.3. Smoothness of varieties revisited 56121.4. Examples 56421.5. Studying smooth varieties using their cotangent bundles 56921.6. Unramified morphisms 57421.7. The Riemann-Hurwitz Formula 575

Chapter 22. ⋆ Blowing up 58322.1. Motivating example: blowing up the origin in the plane 58322.2. Blowing up, by universal property 585

22.3. The blow-up exists, and is projective 58922.4. Examples and computations 594

Part VI. More 603

Chapter 23. Derived functors 60523.1. The Tor functors 60523.2. Derived functors in general 60923.3. Derived functors and spectral sequences 61323.4. Derived functor cohomology of O-modules 618

23.5. Čech cohomology and derived functor cohomology agree 621

Chapter 24. Flatness 62724.1. Introduction 62724.2. Easier facts 62924.3. Flatness through Tor 63424.4. Ideal-theoretic criteria for flatness 63624.5. Topological aspects of flatness 64324.6. Local criteria for flatness 64724.7. Flatness implies constant Euler characteristic 651

Chapter 25. Smooth, étale, and unramified morphisms revisited 65525.1. Some motivation 65525.2. Different characterizations of smooth and étale morphisms 65725.3. Generic smoothness and the Kleiman-Bertini Theorem 662

Chapter 26. Depth and Cohen-Macaulayness 66726.1. Depth 66726.2. Cohen-Macaulay rings and schemes 67026.3. ⋆⋆ Serre’s R1+ S2 criterion for normality 673

Chapter 27. Twenty-seven lines 67927.1. Introduction 67927.2. Preliminary facts 680

27.3. Every smooth cubic surface (over k) has 27 lines 682

27.4. Every smooth cubic surface (over k) is a blown up plane 685

Chapter 28. Cohomology and base change theorems 68928.1. Statements and applications 68928.2. ⋆⋆ Proofs of cohomology and base change theorems 69528.3. Applying cohomology and base change to moduli problems 702

Chapter 29. Power series and the Theorem on Formal Functions 70729.1. Introduction 70729.2. Algebraic preliminaries 70729.3. Defining types of singularities 71129.4. The Theorem on Formal Functions 71329.5. Zariski’s Connectedness Lemma and Stein Factorization 71529.6. Zariski’s Main Theorem 71729.7. Castelnuovo’s criterion for contracting (−1)-curves 721

29.8. ⋆⋆ Proof of the Theorem on Formal Functions 29.4.2 724

Chapter 30. ⋆ Proof of Serre duality 72930.1. Introduction 72930.2. Ext groups and Ext sheaves for O-modules 73430.3. Serre duality for projective k-schemes 73830.4. The adjunction formula for the dualizing sheaf, andωX = KX 742

Bibliography 747

Index 753

June 11, 2013 draft 9

I can illustrate the ... approach with the ... image of a nut to be opened. The firstanalogy that came to my mind is of immersing the nut in some softening liquid, and whynot simply water? From time to time you rub so the liquid penetrates better, and otherwiseyou let time pass. The shell becomes more flexible through weeks and months — when thetime is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

A different image came to me a few weeks ago. The unknown thing to be knownappeared to me as some stretch of earth or hard marl, resisting penetration ... the seaadvances insensibly in silence, nothing seems to happen, nothing moves, the water is sofar off you hardly hear it ... yet finally it surrounds the resistant substance.

— A. Grothendieck [Gr6, p. 552-3], translation by C. McLarty [Mc, p. 1]

Preface

This book is intended to give a serious and reasonably complete introductionto algebraic geometry, not just for (future) experts in the field. The expositionserves a narrow set of goals (see §0.4), and necessarily takes a particular point ofview on the subject.It has now been four decades since David Mumford wrote that algebraic ge-

ometry “seems to have acquired the reputation of being esoteric, exclusive, andvery abstract, with adherents who are secretly plotting to take over all the rest ofmathematics! In one respect this last point is accurate ...” ([Mu4, preface] and[Mu7, p. 227]). The revolution has now fully come to pass, and has fundamentallychanged how we think about many fields of pure mathematics. A remarkablenumber of celebrated advances rely in some way on the insights and ideas force-fully articulated by Grothendieck, Serre, and others.For a number of reasons, algebraic geometry has earned a reputation of being

inaccessible. The power of the subject comes from rather abstract heavy machin-ery, and it is easy to lose sight of the intuitive nature of the objects and methods.Many in nearby fields have only vague ideas of the fundamental ideas of the sub-ject. Algebraic geometry itself has fractured into many parts, and even withinalgebraic geometry, new researchers are often unaware of the basic ideas in sub-fields removed from their own.But there is another more optimistic perspective to be taken. The ideas that al-

low algebraic geometry to connect several parts of mathematics are fundamental,and well-motivated. Many people in nearby fields would find it useful to developa working knowledge of the foundations of the subject, and not just at a super-ficial level. Within algebraic geometry itself, there is a canon (at least for thoseapproaching the subject from this particular direction), that everyone in the fieldcan and should be familiar with. The rough edges of scheme theory have beensanded down over the past half century, although there remains an inescapableneed to understand the subject on its own terms.

0.0.1. The importance of exercises. This book has a lot of exercises. I have foundthat unless I have some problems I can think through, ideas don’t get fixed in mymind. Some exercises are trivial — some experts find this offensive, but I findthis desirable. A very few necessary ones may be hard, but the reader should havebeen given the background to deal with them— they are not just an excuse to pushhard material out of the text. The exercises are interspersed with the exposition,not left to the end. Most have been extensively field-tested. The point of view hereis one I explored with Kedlaya and Poonen in [KPV], a book that was ostensiblyabout problems, but secretly a case for how one should learn and do and thinkabout mathematics. Most people learn by doing, rather than just passively reading.

11

12 Math 216: Foundations of Algebraic Geometry

Judiciously chosen problems can be the best way of guiding the learner towardenlightenment.

0.0.2. Acknowledgments.This one is going to be really hard, so I’ll write this later. (Mike Stay is the

author of Jokes 1.3.11 and 21.5.2.)

0.1 For the reader

This is your last chance. After this, there is no turning back. You take the blue pill,the story ends, you wake up in your bed and believe whatever you want to believe. Youtake the red pill, you stay in Wonderland and I show you how deep the rabbit-hole goes.— Morpheus

The contents of this book are intended to be a collection of communal wisdom,necessarily distilled through an imperfect filter. I wish to say a few words on howyou might use it, although it is not clear to me if you should or will follow thisadvice.Before discussing details, I want to say clearly at the outset: the wonderful

machine of modern algebraic geometry was created to understand basic and naivequestions about geometry (broadly construed). The purpose of this book is to giveyou a thorough foundation in these powerful ideas. Do not be seduced by the lotus-eaters into infatuation with untethered abstraction. Hold tight to the your geometricmotivation as you learn the formal structures which have proved to be so effectivein studying fundamental questions. When introduced to a new idea, always askwhy you should care. Do not expect an answer right away, but demand an answereventually. Try at least to apply any new abstraction to some concrete exampleyou can understand well.Understanding algebraic geometry is often thought to be hard because it con-

sists of large complicated pieces of machinery. In fact the opposite is true; to switchmetaphors, rather than being narrow and deep, algebraic geometry is shallow butextremely broad. It is built out of a large number of very small parts, in keepingwith Grothendieck’s vision of mathematics. It is a challenge to hold the entireorganic structure, with its messy interconnections, in your head.A reasonable place to start is with the idea of “affine complex varieties”: sub-

sets of Cn cut out by some polynomial equations. Your geometric intuition can im-mediately come into play — you may already have some ideas or questions aboutdimension, or smoothness, or solutions over subfields such asR orQ. Wiser headswould counsel spending time understanding complex varieties in some detail be-fore learning about schemes. Instead, I encourage you to learn about schemesimmediately, learning about affine complex varieties as the central (but not exclu-sive) example. This is not ideal, but can save time, and is surprisingly workable.An alternative is to learn about varieties elsewhere, and then come back later.The intuition for schemes can be built on the intuition for affine complex vari-

eties. Allen Knutson and Terry Tao have pointed out that this involves three differ-ent simultaneous generalizations, which can be interpreted as three large themesin mathematics. (i) We allow nilpotents in the ring of functions, which is basicallyanalysis (looking at near-solutions of equations instead of exact solutions). (ii) We


glue these affine schemes together, which is what we do in differential geometry(looking at manifolds instead of coordinate patches). (iii) Instead of working overC (or another algebraically closed field), we work more generally over a ring thatisn’t an algebraically closed field, or even a field at all, which is basically numbertheory (solving equations over number fields, rings of integers, etc.).Because our goal is to be comprehensive, and to understand everything one

should know after a first course, it will necessarily take longer to get to interestingsample applications. You may be misled into thinking that one has to work thishard to get to these applications — it is not true! You should deliberately keep aneye out for examples you would have cared about before. This will take some timeand patience.As you learn algebraic geometry, you should pay attention to crucial stepping

stones. Of course, the steps get bigger the farther you go.

Chapter 1. Category theory is only language, but it is language with an em-bedded logic. Category theory is much easier once you realize that it is designedto formalize and abstract things you already know. The initial chapter on cate-gory theory prepares you to think cleanly. For example, when someone namessomething a “cokernel” or a “product”, you should want to know why it deservesthat name, and what the name really should mean. The conceptual advantages ofthinking this way will gradually become apparent over time. Yoneda’s Lemma —and more generally, the idea of understanding an object through the maps to it —will play an important role.

Chapter 2. The theory of sheaves again abstracts something you already un-derstand well (see the motivating example of §2.1), and what is difficult is under-standing how one best packages and works with the information of a sheaf (stalks,sheafification, sheaves on a base, etc.).

Chapters 1 and 2 are a risky gamble, and they attempt a delicate balance. Attemptsto explain algebraic geometry often leave such background to the reader, refer toother sources the reader won’t read, or punt it to a telegraphic appendix. Instead,this book attempts to explain everything necessary, but as little possible, and triesto get across how you should think about (and work with) these fundamentalideas, and why they are more grounded than you might fear.

Chapters 3–5. Armed with this background, you will be able to think cleanlyabout various sorts of “spaces” studied in different parts of geometry (includ-ing differentiable real manifolds, topological spaces, and complex manifolds), asringed spaces that locally are of a certain form. A scheme is just another kindof “geometric space”, and we are then ready to transport lots of intuition from“classical geometry” to this new setting. (This also will set you up to later thinkabout other geometric kinds of spaces in algebraic geometry, such as complex an-alytic spaces, algebraic spaces, orbifolds, stacks, rigid analytic spaces, and formalschemes.) The ways in which schemes differ from your geometric intuition can beinternalized, and your intuition can be expanded to accomodate them. There aremany properties you will realize you will want, as well as other properties thatwill later prove important. These all deserve names. Take your time becomingfamiliar with them.


Chapters 6–10. Thinking categorically will lead you to ask about morphismsabout schemes (and other spaces in geometry). One of Grothendieck’s funda-mental lessons is that the morphisms are central. Important geometric propertiesshould really be understood as properties of morphisms. There are many classesof morphisms with special names, and in each case you should think through whythat class deserves a name.

Chapters 11–12. You will then be in a good position to think about fundamen-tal geometric properties of schemes: dimension and smoothness. You may be sur-prised that these are subtle ideas, but you should keep in mind that they are subtleeverywhere in mathematics.

Chapters 13–21. Vector bundles are ubiquitous tools in geometry, and algebraicgeometry is no exception. They lead us to the more general notion of quasicoher-ent sheaves, much as free modules over a ring lead us to modules more generally.We study their properties next, including cohomology. Chapter 19, applying theseideas ideas to study curves, may help make clear how useful they are.

Chapters 23–30. With this in hand, you are ready to learn more advanced toolswidely used in the subject. Many examples of what you can do are given, andthe classical story of the 27 lines on a smooth cubic surface (Chapter 27) is a goodopportunity to see many ideas come together.

The rough logical dependencies among the chapters are shown in Figure 0.1.(Caution: this should be taken with a grain of salt. For example, you can avoidusing much of Chapter 19 on curves in later chapters, but it is a crucial source ofexamples, and a great way to consolidate your understanding. And Chapter 29 oncompletions uses Chapters 19, 20 and 22 only in the discussion of Castelnuovo’sCriterion 29.7.1.)In general, I like having as few hypotheses as possible. Certainly a hypothesis

that isn’t necessary to the proof is a red herring. But if a reasonable hypothesis canmake the proof cleaner and more memorable, I am willing to include it.In particular, Noetherian hypotheses are handy when necessary, but are oth-

erwise misleading. Even Noetherian-minded readers (normal human beings) arebetter off having the right hypotheses, as they will make clearer why things aretrue.We often state results particular to varieties, especially when there are tech-

niques unique to this situation that one should know. But restricting to alge-braically closed fields is useful surprisingly rarely. Geometers needn’t be afraidof arithmetic examples or of algebraic examples; a central insight of algebraic ge-ometry is that the same formalism applies without change.Pathological examples are useful to know. On mountain highways, there are

tall sticks on the sides of the road designed for bad weather. In winter, you cannotsee the road clearly, and the sticks serve as warning signs: if you cross this line,you will die! Pathologies and (counter)examples serve a similar goal. They alsoserve as a reality check, when confronting a new statement, theorem, or conjecture,whose veracity you may doubt.When working through a book in algebraic geometry, it is particularly helpful

to have other algebraic geometry books at hand, to see different approaches andto have alternate expositions when things become difficult. This book may serve


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FIGURE 0.1. Important logical dependences among chapters (ormore precisely, a directed graph showing which chapter shouldbe read before which other chapter)

as a good secondary book. If it is your primary source, then two other excellentbooks with what I consider a similar philosophy are [Liu] and [GW]. De Jong’sencyclopedic online reference [Stacks] is peerless. There are many other outstand-ing sources out there, perhaps one for each approach to the subject; you shouldbrowse around and find one you find sympathetic.If you are looking for a correct or complete history of the subject, you have

come to the wrong place. This book is not intended to be a complete guide tothe literature, and many important sources are ignored or left out, due to my ownignorance and laziness.Finally, if you attempt to read this without working through a significant num-

ber of exercises (see §0.0.1), I will come to your house and pummel you with[Gr-EGA] until you beg for mercy. It is important to not just have a vague senseof what is true, but to be able to actually get your hands dirty. As Mark Kisin hassaid, “You can wave your hands all you want, but it still won’t make you fly.”

0.2 For the expert


If you use this book for a course, you should of course adapt it to your ownpoint of view and your own interests. In particular, you should think about anapplication or theorem you want to reach at the end of the course (which maywell not be in this book), and then work toward it. You should feel no compulsionto sprint to the end; I advise instead taking more time, and ending at the rightplace for your students. (Figure 0.1, showing large-scale dependencies among thechapters, may help you map out a course.) I have found that the theory of curves(Chapter 19) and the 27 lines on the cubic surface (Chapter 27) have served thispurpose well at the end of winter and spring quarters. This was true even if someof the needed background was not covered, and had to be taken by students assome sort of black box.Faithfulness to the goals of §0.4 required a brutal triage, and I have made a

number of decisions you may wish to reverse. I will briefly describe some choicesmade that may be controversial.Decisions on how to describe things were made for the sake of the learners.

If there were two approaches, and one was “correct” from an advanced point ofview, and one was direct and natural from a naive point of view, I went with thelatter.On the other hand, the theory of varieties (over an algebraically closed field,

say) was not done first and separately. This choice brought me close to tears, butin the end I am convinced that it can work well, if done in the right spirit.Instead of spending the first part of the course on varieties, I spent the time

in a different way. It is tempting to assume that students will either arrive withgreat comfort and experience with category theory and sheaf theory, or that theyshould pick up these ideas on their own time. I would love to live in that world.I encourage you to not skimp on these foundational issues. I have found thatalthough these first lectures felt painfully slow to me, they were revelatory to anumber of the students, and those with more experience were not bored and didnot waste their time. This investment paid off in spades when I was able to relyon their ability to think cleanly and to use these tools in practice. Furthermore, ifthey left the course with nothing more than hands-on experience with these ideas,the world was still better off for it.For the most part, we will state results in the maximal generality that the proof

justifies, but we will not give a much harder proof where the generality of thestronger result will not be used. There are a few cases where we work harder toprove a somewhat more general result that many readers may not appreciate. Forexample, we prove a number of theorems for proper morphisms, not just projec-tive morphisms. But in such cases, readers are invited or encouraged to ignore thesubtleties required for the greater generality.I consider line bundles (and maps to projective space) more fundamental than

divisors. General Cartier divisors are not discussed (although effective Cartier divi-sors play an essential role).

Cohomology is done first using the Čech approach (as Serre first did), and de-rived functor cohomology is introduced only later. I amwell aware that Grothendieck

thinks of the fact that the agreement of Čech cohomology with derived functor co-homology “should be considered as an accidental phenomenon”, and that “it is

important for technical reasons not to take as definition of cohomology the Čechcohomology”, [Gr4, p. 108]. But I am convinced that this is the right way for most


people to see this kind of cohomology for the first time. (It is certainly true thatmany topics in algebraic geometry are best understood in the language of derivedfunctors. But this is a view from the mountaintop, looking down, and not the bestway to explore the forests. In order to appreciate derived functors appropriately,one must understand the homological algebra behind it, and not just take it as ablack box.)We restrict to the Noetherian case only when it is necessary, or (rarely) when it

really saves effort. In this way, non-Noetherian people will clearly see where theyshould be careful, and Noetherian people will realize that non-Noetherian thingsare not so terrible. Moreover, even if you are interested primarily in Noetherianschemes, it helps to see “Noetherian” in the hypotheses of theorems only whennecessary, as it will help you remember how and when this property gets used.There are some cases where Noetherian readers will suffer a little more than

they would otherwise. As an inflammatory example, instead of using Noetherianhypotheses, the notion of quasiseparated comes up early and often. The cost isthat one extra word has to be remembered, on top of an overwhelming numberof other words. But once that is done, it is not hard to remember that essentiallyevery scheme anyone cares about is quasiseparated. Furthermore, whenever thehypotheses “quasicompact and quasiseparated” turn up, the reader will immedi-ately guess a key idea of the proof. As another example, coherent sheaves andfinite type (quasicoherent) sheaves are the same in the Noetherian situation, butare still worth distinguishing in statements of the theorems and exercises, for thesame reason: to be clearer on what is used in the proof.Many important topics are not discussed. Valuative criteria are not proved

(see §12.7), and their statement is relegated to an optional section. Completelyomitted: devissage, formal schemes, and cohomology with supports. Sorry!

0.3 Background and conventions

“Should you just be an algebraist or a geometer?” is like saying “Would you ratherbe deaf or blind?”—M. Atiyah, [At2, p. 659]

All rings are assumed to be commutative unless explicitly stated otherwise.All rings are assumed to contain a unit, denoted 1. Maps of rings must send 1 to1. We don’t require that 0 6= 1; in other words, the “0-ring” (with one element)is a ring. (There is a ring map from any ring to the 0-ring; the 0-ring only mapsto itself. The 0-ring is the final object in the category of rings.) The definition of“integral domain” includes 1 6= 0, so the 0-ring is not an integral domain. Weaccept the axiom of choice. In particular, any proper ideal in a ring is contained ina maximal ideal. (The axiom of choice also arises in the argument that the categoryof A-modules has enough injectives, see Exercise 23.2.G.)The reader should be familiar with some basic notions in commutative ring

theory, in particular the notion of ideals (including prime andmaximal ideals) andlocalization. Tensor products and exact sequences ofA-modules will be important.We will use the notation (A,m) or (A,m, k) for local rings (rings with a uniquemaximal ideal) — A is the ring, m its maximal ideal, and k = A/m its residue field.


We will use the structure theorem for finitely generated modules over a principalideal domain A: any such module can be written as the direct sum of principalmodules A/(a). Some experience with field theory will be helpful from time totime.

0.3.1. Caution about foundational issues. We will not concern ourselves with subtlefoundational issues (set-theoretic issues, universes, etc.). It is true that some peo-ple should be careful about these issues. But is that really how you want to liveyour life? (If you are one of these rare people, a good start is [KS2, §1.1].)

0.3.2. Further background. It may be helpful to have books on other subjects athand that you can dip into for specific facts, rather than reading them in advance.In commutative algebra, [E] is good for this. Other popular choices are [AtM] and[Mat2]. The book [Al] takes a point of view useful to algebraic geometry. Forhomological algebra, [Wei] is simultaneously detailed and readable.Background from other parts of mathematics (topology, geometry, complex

analysis, number theory, ...) will of course be helpful for intuition and grounding.Some previous exposure to topology is certainly essential.

0.3.3. Nonmathematical conventions. “Unimportant” means “unimportant for thecurrent exposition”, not necessarily unimportant in the larger scheme of things.Other words may be used idiosyncratically as well.There are optional starred sections of topics worth knowing on a second or

third (but not first) reading. They are marked with a star: ⋆. Starred sections arenot necessarily harder, merely unimportant. You should not read double-starredsections (⋆⋆) unless you really really want to, but you should be aware of theirexistence. (It may be strange to have parts of a book that should not be read!)Let’s now find out if you are taking my advice about double-starred sections.

0.4 ⋆⋆ The goals of this book

There are a number of possible introductions to the field of algebraic geome-try: Riemann surfaces; complex geometry; the theory of varieties; a nonrigorousexamples-based introduction; algebraic geometry for number theorists; an abstractfunctorial approach; and more. All have their place. Different approaches suit dif-ferent students (and different advisors). This book takes only one route.Our intent is to cover a canon completely and rigorously, with enough exam-

ples and calculations to help develop intuition for the machinery. This is oftenthe content of a second course in algebraic geometry, and in an ideal world, peo-ple would learn this material over many years, after having background coursesin commutative algebra, algebraic topology, differential geometry, complex analy-sis, homological algebra, number theory, and French literature. We do not live inan ideal world. For this reason, the book is written as a first introduction, but achallenging one.This book seeks to do a very few things, but to try to do them well. Our goals

and premises are as follows.


The core of the material should be digestible over a single year. After ayear of blood, sweat, and tears, readers should have a broad familiarity with thefoundations of the subject, and be ready to attend seminars, and learn more ad-vanced material. They should not just have a vague intuitive understanding ofthe ideas of the subject; they should know interesting examples, know why theyare interesting, and be able to work through their details. Readers in other fieldsof mathematics should know enough to understand the algebro-geometric ideasthat arise in their area of interest.This means that this book is not encyclopedic, and even beyond that, hard

choices have to be made. (In particular, analytic aspects are essentially ignored,and are at best dealt with in passing without proof. This is a book about algebraicalgebraic geometry.)This book is usable (and has been used) for a course, but the course should

(as always) take on the personality of the instructor. With a good course, peopleshould be able to leave early and still get something useful from the experience.With this book, it is possible to leave without regret after learning about categorytheory, or about sheaves, or about geometric spaces, having become a better per-son.The book is also usable (and has been used) for learning on your own. But

most mortals cannot learn algebraic geometry fully on their own; ideally youshould read in a group, and even if not, you should have someone you can askquestions to (both stupid and smart questions).There is certainly more than a year’s material here, but I have tried to make

clear which topics are essential, and which are not. Those teaching a class willchoose which “inessential” things are important for the point they wish to getacross, and use them.

There is a canon (at least for this particular approach to algebraic geometry). Ihave been repeatedly surprised at howmuch people in different parts of algebraicgeometry agree on what every civilized algebraic geometer should know after afirst (serious) year. (There are of course different canons for different parts of thesubject, e.g. complex algebraic geometry, combinatorial algebraic geometry, com-putational algebraic geometry, etc.) There are extra bells andwhistles that differentinstructors might add on, to prepare students for their particular part of the fieldor their own point of view, but the core of the subject remains unified, despite thediversity and richness of the subject. There are some serious and painful compro-mises to be made to reconcile this goal with the previous one.

Algebraic geometry is for everyone (with the appropriate definition of “ev-eryone”). Algebraic geometry courses tend to require a lot of background, whichmakes them inaccessible to all but those who know they will go deeply into thesubject. Algebraic geometry is too important for that; it is essential that many ofthose in nearby fields develop some serious familiarity with the foundational ideasand tools of the subject, and not just at a superficial level. (Similarly, algebraic ge-ometers uninterested in any nearby field are necessarily arid, narrow thinkers. Donot be such a person!)For this reason, this book attempts to require as little background as possible.

The background required will, in a technical sense, be surprisingly minimal — ide-ally just some commutative ring theory and point-set topology, and some comfort


with things like prime ideals and localization. This is misleading of course — themore you know, the better. And the less background you have, the harder you willhave to work — this is not a light read. On a related note...

The book is intended to be as self-contained as possible. I have tried tofollow the motto: “if you use it, you must prove it”. I have noticed that moststudents are human beings: if you tell them that some algebraic fact is in some latechapter of a book in commutative algebra, they will not immediately go and readit. Surprisingly often, what we need can be developed quickly from scratch, andeven if people do not read it, they can see what is involved. The cost is that thebook is much denser, and that significant sophistication andmaturity is demandedof the reader. The benefit is that more people can follow it; they are less likely toreach a point where they get thrown. On the other hand, people who already havesome familiarity with algebraic geometry, but want to understand the foundationsmore completely, should not be bored, and will focus on more subtle issues.As just one example, Krull’s Principal Ideal Theorem 11.3.3 is an important

tool. I have included an essentially standard proof (§11.5). I do not want peopleto read it (unless they really really want to), and signal this by a double-star in thetitle: ⋆⋆. Instead, I want people to skim it and realize that they could read it, andthat it is not seriously hard.This is an important goal because it is important not just to know what is true,

but to know why things are true, and what is hard, and what is not hard. Also,this helps the previous goal, by reducing the number of prerequisites.

The book is intended to build intuition for the formidable machinery of al-gebraic geometry. The exercises are central for this (see §0.0.1). Informal languagecan sometimes be helpful. Many examples are given. Learning how to thinkcleanly (and in particular categorically) is essential. The advantages of appropriategenerality should be made clear by example, and not by intimidation. The mo-tivation is more local than global. For example, there is no introductory chapterexplaining why one might be interested in algebraic geometry, and instead thereis an introductory chapter explaining why you should want to think categorically(and how to actually do this).

Balancing the above goals is already impossible. We must thus give up anyhope of achieving any other desiderata. There are no other goals.

Part I

Preliminaries

CHAPTER 1

Some category theory

The introduction of the digit 0 or the group concept was general nonsense too, andmathematics was more or less stagnating for thousands of years because nobody wasaround to take such childish steps...— A. Grothendieck, [BP, p. 4-5]

That which does not kill me, makes me stronger.— F. Nietzsche

1.1 Motivation

Before we get to any interesting geometry, we need to develop a languageto discuss things cleanly and effectively. This is best done in the language ofcategories. There is not much to know about categories to get started; it is justa very useful language. Like all mathematical languages, category theory comeswith an embedded logic, which allows us to abstract intuitions in settings we knowwell to far more general situations.Our motivation is as follows. We will be creating some new mathematical

objects (such as schemes, and certain kinds of sheaves), and we expect them toact like objects we have seen before. We could try to nail down precisely whatwe mean by “act like”, and what minimal set of things we have to check in orderto verify that they act the way we expect. Fortunately, we don’t have to — otherpeople have done this before us, by defining key notions, such as abelian categories,which behave like modules over a ring.Our general approach will be as follows. I will try to tell what you need to

know, and nomore. (This I promise: if I use the word “topoi”, you can shoot me.) Iwill begin by telling you things you already know, and describing what is essentialabout the examples, in a way that we can abstract a more general definition. Wewill then see this definition in less familiar settings, and get comfortable with usingit to solve problems and prove theorems.For example, we will define the notion of product of schemes. We could just

give a definition of product, but then you should want to know why this precisedefinition deserves the name of “product”. As a motivation, we revisit the notionof product in a situation we know well: (the category of) sets. One way to definethe product of sets U and V is as the set of ordered pairs {(u, v) : u ∈ U, v ∈ V}.But someone from a different mathematical culture might reasonably define it as

the set of symbols {uv : u ∈ U, v ∈ V}. These notions are “obviously the same”.

Better: there is “an obvious bijection between the two”.

23


This can be made precise by giving a better definition of product, in terms of auniversal property. Given two setsM and N, a product is a set P, along with mapsµ : P → M and ν : P → N, such that for any set P ′ with maps µ ′ : P ′ → M andν ′ : P ′ → N, these maps must factor uniquely through P:

(1.1.0.1) P ′

∃!

ν ′

((PPPPP

PPPPPP

PPPP

µ ′

��000

0000

0000

000

Pν

//

µ

��

N

M

(The symbol ∃means “there exists”, and the symbol ! here means “unique”.) Thusa product is a diagram

Pν //

µ

��

N

M

and not just a set P, although the maps µ and ν are often left implicit.This definition agrees with the traditional definition, with one twist: there

isn’t just a single product; but any two products come with a unique isomorphismbetween them. In other words, the product is unique up to unique isomorphism.Here is why: if you have a product

P1ν1 //

µ1

��

N

M

and I have a product

P2ν2 //

µ2

��

N

M

then by the universal property of my product (letting (P2, µ2, ν2) play the role of(P, µ, ν), and (P1, µ1, ν1) play the role of (P

′, µ ′, ν ′) in (1.1.0.1)), there is a uniquemap f : P1 → P2 making the appropriate diagram commute (i.e., µ1 = µ2 ◦ f andν1 = ν2 ◦ f). Similarly by the universal property of your product, there is a uniquemap g : P2 → P1 making the appropriate diagram commute. Now consider theuniversal property of my product, this time letting (P2, µ2, ν2) play the role of both


(P, µ, ν) and (P ′, µ ′, ν ′) in (1.1.0.1). There is a unique map h : P2 → P2 such that

P2

h

AAA

AAA

ν2

''PPPPP

PPPPPP

PPPP

µ2

��00000000000000

P2 ν2//

µ2

��

N

M

commutes. However, I can name two such maps: the identity map idP2 , and f ◦ g.Thus f ◦ g = idP2 . Similarly, g ◦ f = idP1 . Thus the maps f and g arising fromthe universal property are bijections. In short, there is a unique bijection betweenP1 and P2 preserving the “product structure” (the maps toM and N). This givesus the right to name any such product M × N, since any two such products areuniquely identified.This definition has the advantage that it works in many circumstances, and

once we define categories, we will soon see that the above argument applies ver-batim in any category to show that products, if they exist, are unique up to uniqueisomorphism. Even if you haven’t seen the definition of category before, you canverify that this agrees with your notion of product in some category that you haveseen before (such as the category of vector spaces, where the maps are taken to belinear maps; or the category of manifolds, where the maps are taken to be submer-sions, i.e., differentiable maps whose differential is everywhere surjective).This is handy even in cases that you understand. For example, one way of

defining the product of two manifoldsM andN is to cut them both up into charts,then take products of charts, then glue them together. But if I cut up the manifoldsin one way, and you cut them up in another, how do we know our resulting mani-folds are the “same”? We could wave our hands, or make an annoying argumentabout refining covers, but instead, we should just show that they are “categoricalproducts” and hence canonically the “same” (i.e., isomorphic). We will formalizethis argument in §1.3.Another set of notions we will abstract are categories that “behave like mod-

ules”. We will want to define kernels and cokernels for new notions, and weshould make sure that these notions behave the way we expect them to. Thisleads us to the definition of abelian categories, first defined by Grothendieck in hisTôhoku paper [Gr1].In this chapter, we will give an informal introduction to these and related no-

tions, in the hope of giving just enough familiarity to comfortably use them inpractice.

1.2 Categories and functors

Before functoriality, people lived in caves.— B. Conrad

We begin with an informal definition of categories and functors.

1.2.1. Categories.


A category consists of a collection of objects, and for each pair of objects, aset of morphisms (or arrows) between them. (For experts: technically, this is thedefinition of a locally small category. In the correct definition, the morphisms needonly form a class, not necessarily a set, but see Caution 0.3.1.) Morphisms are ofteninformally called maps. The collection of objects of a category C is often denotedobj(C ), but we will usually denote the collection also by C . If A,B ∈ C , then theset of morphisms from A to B is denoted Mor(A,B). A morphism is often writtenf : A → B, and A is said to be the source of f, and B the target of f. (Of course,Mor(A,B) is taken to be disjoint from Mor(A ′, B ′) unless A = A ′ and B = B ′.)Morphisms compose as expected: there is a compositionMor(B,C)×Mor(A,B)→

Mor(A,C), and if f ∈ Mor(A,B) and g ∈ Mor(B,C), then their composition is de-noted g ◦ f. Composition is associative: if f ∈ Mor(A,B), g ∈ Mor(B,C), andh ∈Mor(C,D), then h ◦ (g ◦ f) = (h ◦ g) ◦ f. For each object A ∈ C , there is alwaysan identity morphism idA : A→ A, such that when you (left- or right-)compose amorphism with the identity, you get the same morphism. More precisely, for anymorphisms f : A → B and g : B → C, idB ◦f = f and g ◦ idB = g. (If you wish,you may check that “identity morphisms are unique”: there is only one morphismdeserving the name idA.) This ends the definition of a category.We have a notion of isomorphism between two objects of a category (a mor-

phism f : A → B such that there exists some — necessarily unique — morphismg : B → A, where f ◦ g and g ◦ f are the identity on B and A respectively), and anotion of automorphism of an object (an isomorphism of the object with itself).

1.2.2. Example. The prototypical example to keep in mind is the category of sets,denoted Sets. The objects are sets, and the morphisms are maps of sets. (BecauseRussell’s paradox shows that there is no set of all sets, we did not say earlier thatthere is a set of all objects. But as stated in §0.3, we are deliberately omitting allset-theoretic issues.)

1.2.3. Example. Another good example is the category Veck of vector spaces overa given field k. The objects are k-vector spaces, and the morphisms are lineartransformations. (What are the isomorphisms?)

1.2.A. UNIMPORTANT EXERCISE. A category in which each morphism is an iso-morphism is called a groupoid. (This notion is not important in what we willdiscuss. The point of this exercise is to give you some practice with categories, byrelating them to an object you know well.)(a) A perverse definition of a group is: a groupoid with one object. Make sense ofthis.(b) Describe a groupoid that is not a group.

1.2.B. EXERCISE. If A is an object in a category C , show that the invertible ele-ments of Mor(A,A) form a group (called the automorphism group of A, denotedAut(A)). What are the automorphism groups of the objects in Examples 1.2.2and 1.2.3? Show that two isomorphic objects have isomorphic automorphismgroups. (For readers with a topological background: if X is a topological space,then the fundamental groupoid is the category where the objects are points of X,and the morphisms x → y are paths from x to y, up to homotopy. Then the auto-morphism group of x0 is the (pointed) fundamental group π1(X, x0). In the case


where X is connected, and π1(X) is not abelian, this illustrates the fact that fora connected groupoid — whose definition you can guess — the automorphismgroups of the objects are all isomorphic, but not canonically isomorphic.)

1.2.4. Example: abelian groups. The abelian groups, along with group homomor-phisms, form a category Ab.

1.2.5. Important example: modules over a ring. IfA is a ring, then theA-modules forma categoryModA. (This category has additional structure; it will be the prototypi-cal example of an abelian category, see §1.6.) TakingA = k, we obtain Example 1.2.3;taking A = Z, we obtain Example 1.2.4.

1.2.6. Example: rings. There is a category Rings, where the objects are rings, andthe morphisms are maps of rings in the usual sense (maps of sets which respectaddition and multiplication, and which send 1 to 1 by our conventions, §0.3).

1.2.7. Example: topological spaces. The topological spaces, along with continuousmaps, form a category Top. The isomorphisms are homeomorphisms.

In all of the above examples, the objects of the categories were in obviousways sets with additional structure (a concrete category, although we won’t usethis terminology). This needn’t be the case, as the next example shows.

1.2.8. Example: partially ordered sets. A partially ordered set, or poset, is a set Salong with a binary relation ≥ on S satisfying:

(i) x ≥ x (reflexivity),(ii) x ≥ y and y ≥ z imply x ≥ z (transitivity), and(iii) if x ≥ y and y ≥ x then x = y (antisymmetry).

A partially ordered set (S,≥) can be interpreted as a category whose objects arethe elements of S, and with a single morphism from x to y if and only if x ≥ y (andno morphism otherwise).A trivial example is (S,≥) where x ≥ y if and only if x = y. Another example

is

(1.2.8.1) •

��• // •Here there are three objects. The identity morphisms are omitted for convenience,and the two non-identity morphisms are depicted. A third example is

(1.2.8.2) •

��

// •

��• // •Here the “obvious” morphisms are again omitted: the identity morphisms, andthe morphism from the upper left to the lower right. Similarly,

· · · // • // • // •depicts a partially ordered set, where again, only the “generating morphisms” aredepicted.


1.2.9. Example: the category of subsets of a set, and the category of open sets in a topo-logical space. If X is a set, then the subsets form a partially ordered set, where theorder is given by inclusion. Informally, if U ⊂ V , then we have exactly one mor-phism U→ V in the category (and otherwise none). Similarly, if X is a topologicalspace, then the open sets form a partially ordered set, where the order is given byinclusion.

1.2.10. Definition. A subcategory A of a categoryB has as its objects some of theobjects of B, and some of the morphisms, such that the morphisms of A includethe identity morphisms of the objects of A , and are closed under composition.(For example, (1.2.8.1) is in an obvious way a subcategory of (1.2.8.2). Also, wehave an obvious “inclusion functor” i : A → B.)

1.2.11. Functors.A covariant functor F from a categoryA to a categoryB, denoted F : A → B,

is the following data. It is a map of objects F : obj(A ) → obj(B), and for eachA1, A2 ∈ A , and morphism m : A1 → A2, a morphism F(m) : F(A1) → F(A2) inB. We require that F preserves identity morphisms (for A ∈ A , F(idA) = idF(A)),and that F preserves composition (F(m2 ◦m1) = F(m2) ◦ F(m1)). (You may wishto verify that covariant functors send isomorphisms to isomorphisms.) A trivialexample is the identity functor id : A → A , whose definition you can guess.Here are some less trivial examples.

1.2.12. Example: a forgetful functor. Consider the functor from the category ofvector spaces (over a field k) Veck to Sets, that associates to each vector space itsunderlying set. The functor sends a linear transformation to its underlying map ofsets. This is an example of a forgetful functor, where some additional structure isforgotten. Another example of a forgetful functor isModA → Ab from A-modulesto abelian groups, remembering only the abelian group structure of the A-module.

1.2.13. Topological examples. Examples of covariant functors include the funda-mental group functor π1, which sends a topological space Xwith choice of a pointx0 ∈ X to a group π1(X, x0) (what are the objects and morphisms of the source cat-egory?), and the ith homology functor Top→ Ab, which sends a topological spaceX to its ith homology groupHi(X,Z). The covariance corresponds to the fact that a(continuous) morphism of pointed topological spaces φ : X → Y with φ(x0) = y0induces a map of fundamental groups π1(X, x0) → π1(Y, y0), and similarly forhomology groups.

1.2.14. Example. Suppose A is an object in a category C . Then there is a func-tor hA : C → Sets sending B ∈ C to Mor(A,B), and sending f : B1 → B2 toMor(A,B1)→Mor(A,B2) described by

[g : A→ B1] 7→ [f ◦ g : A→ B1 → B2].

This seemingly silly functor ends up surprisingly being an important concept.

1.2.15. Definitions. If F : A → B and G : B → C are covariant functors, then wedefine a functor G ◦ F : A → C (the composition of G and F) in the obvious way.Composition of functors is associative in an evident sense.


A covariant functor F : A → B is faithful if for all A,A ′ ∈ A , the mapMorA (A,A

′) → MorB(F(A), F(A ′)) is injective, and full if it is surjective. A func-tor that is full and faithful is fully faithful. A subcategory i : A → B is a fullsubcategory if i is full. Thus a subcategory A ′ of A is full if and only if for allA,B ∈ obj(A ′), MorA ′(A,B) = MorA (A,B). For example, the forgetful functorVeck → Sets is faithful, but not full; and if A is a ring, the category of finitelygenerated A-modules is a full subcategory of the categoryModA of A-modules.

1.2.16. Definition. A contravariant functor is defined in the same way as a covari-ant functor, except the arrows switch directions: in the above language, F(A1 →A2) is now an arrow from F(A2) to F(A1). (Thus F(m2 ◦m1) = F(m1) ◦ F(m2), notF(m2) ◦ F(m1).)It is wise to state whether a functor is covariant or contravariant, unless the

context makes it very clear. If it is not stated (and the context does not make itclear), the functor is often assumed to be covariant.(Sometimes people describe a contravariant functor C → D as a covariant

functor C opp → D , where C opp is the same category as C except that the arrowsgo in the opposite direction. Here C opp is said to be the opposite category to C .)One can define fullness, etc. for contravariant functors, and you should do so.

1.2.17. Linear algebra example. If Veck is the category of k-vector spaces (introducedin Example 1.2.3), then taking duals gives a contravariant functor (·)∨ : Veck →Veck. Indeed, to each linear transformation f : V →W, we have a dual transforma-tion f∨ : W∨ → V∨, and (f ◦ g)∨ = g∨ ◦ f∨.

1.2.18. Topological example (cf. Example 1.2.13) for those who have seen cohomology. Theith cohomology functor Hi(·,Z) : Top→ Ab is a contravariant functor.

1.2.19. Example. There is a contravariant functor Top→ Rings taking a topologicalspace X to the ring of real-valued continuous functions on X. A morphism oftopological spaces X → Y (a continuous map) induces the pullback map fromfunctions on Y to functions on X.

1.2.20. Example (the functor of points, cf. Example 1.2.14). Suppose A is an objectof a category C . Then there is a contravariant functor hA : C → Sets sendingB ∈ C to Mor(B,A), and sending the morphism f : B1 → B2 to the morphismMor(B2, A)→Mor(B1, A) via

[g : B2 → A] 7→ [g ◦ f : B1 → B2 → A].

This example initially looks weird and different, but Examples 1.2.17 and 1.2.19may be interpreted as special cases; do you see how? What is A in each case? Thisfunctor might reasonably be called the functor of maps (toA), but is actually knownas the functor of points. We will meet this functor again in §1.3.10 and (in thecategory of schemes) in Definition 6.3.7.

1.2.21. ⋆ Natural transformations (and natural isomorphisms) of covariant func-tors, and equivalences of categories.(This notion won’t come up in an essential way until at least Chapter 6, so you

shouldn’t read this section until then.) Suppose F andG are two covariant functorsfrom A to B. A natural transformation of covariant functors F → G is the data


of a morphism mA : F(A) → G(A) for each A ∈ A such that for each f : A → A ′in A , the diagram

F(A)F(f) //

mA

��

F(A ′)

mA ′

��G(A)

G(f)// G(A ′)

commutes. A natural isomorphism of functors is a natural transformation suchthat each mA is an isomorphism. (We make analogous definitions when F and Gare both contravariant.)The data of functors F : A → B and F ′ : B → A such that F ◦ F ′ is naturally

isomorphic to the identity functor idB on B and F ′ ◦ F is naturally isomorphic toidA is said to be an equivalence of categories. “Equivalence of categories” is anequivalence relation on categories. The right notion of when two categories are“essentially the same” is not isomorphism (a functor giving bijections of objects andmorphisms) but equivalence. Exercises 1.2.C and 1.2.D might give you some vaguesense of this. Later exercises (for example, that “rings” and “affine schemes” areessentially the same, once arrows are reversed, Exercise 6.3.D) may help too.Two examples might make this strange concept more comprehensible. The

double dual of a finite-dimensional vector space V is not V , but we learn early tosay that it is canonically isomorphic to V . We can make that precise as follows. Letf.d.Vec

kbe the category of finite-dimensional vector spaces over k. Note that this

category contains oodles of vector spaces of each dimension.

1.2.C. EXERCISE. Let (·)∨∨ : f.d.Veck→ f.d.Vec

kbe the double dual functor from

the category of finite-dimensional vector spaces over k to itself. Show that (·)∨∨is naturally isomorphic to the identity functor on f.d.Vec

k. (Without the finite-

dimensionality hypothesis, we only get a natural transformation of functors fromid to (·)∨∨.)Let V be the category whose objects are the k-vector spaces kn for each n ≥ 0

(there is one vector space for each n), and whose morphisms are linear transfor-mations. The objects of V can be thought of as vector spaces with bases, and themorphisms as matrices. There is an obvious functor V → f.d.Vec

k, as each kn is a

finite-dimensional vector space.

1.2.D. EXERCISE. Show that V → f.d.Veckgives an equivalence of categories,

by describing an “inverse” functor. (Recall that we are being cavalier about set-theoretic assumptions, see Caution 0.3.1, so feel free to simultaneously choosebases for each vector space in f.d.Vec

k. To make this precise, you will need to use

Gödel-Bernays set theory or else replace f.d.Veckwith a very similar small category,

but we won’t worry about this.)

1.2.22. ⋆⋆ Aside for experts. Your argument for Exercise 1.2.D will show that (mod-ulo set-theoretic issues) this definition of equivalence of categories is the same asanother one commonly given: a covariant functor F : A → B is an equivalenceof categories if it is fully faithful and every object of B is isomorphic to an objectof the form F(A) for some A ∈ A (F is essentially surjective). Indeed, one can showthat such a functor has a quasiinverse, i.e., a functor G : B → A (necessarily also


an equivalence and unique up to unique isomorphism) for which G ◦ F ∼= idA andF ◦G ∼= idB, and conversely, any functor that has a quasiinverse is an equivalence.

1.3 Universal properties determine an object up to uniqueisomorphism

Given some category that we come up with, we often will have ways of pro-ducing new objects from old. In good circumstances, such a definition can bemade using the notion of a universal property. Informally, we wish that there werean object with some property. We first show that if it exists, then it is essentiallyunique, or more precisely, is unique up to unique isomorphism. Then we go aboutconstructing an example of such an object to show existence.Explicit constructions are sometimes easier to work with than universal prop-

erties, but with a little practice, universal properties are useful in proving thingsquickly and slickly. Indeed, when learning the subject, people often find explicitconstructions more appealing, and use them more often in proofs, but as they be-comemore experienced, they find universal property arguments more elegant andinsightful.

1.3.1. Products were defined by a universal property. We have seen one im-portant example of a universal property argument already in §1.1: products. Youshould go back and verify that our discussion there gives a notion of product inany category, and shows that products, if they exist, are unique up to unique iso-morphism.

1.3.2. Initial, final, and zero objects. Here are some simple but useful conceptsthat will give you practice with universal property arguments. An object of acategory C is an initial object if it has precisely one map to every object. It is afinal object if it has precisely one map from every object. It is a zero object if it isboth an initial object and a final object.

1.3.A. EXERCISE. Show that any two initial objects are uniquely isomorphic. Showthat any two final objects are uniquely isomorphic.

In other words, if an initial object exists, it is unique up to unique isomorphism,and similarly for final objects. This (partially) justifies the phrase “the initial object”rather than “an initial object”, and similarly for “the final object” and “the zeroobject”. (Convention: we often say “the”, not “a”, for anything defined up tounique isomorphism.)

1.3.B. EXERCISE. What are the initial and final objects in Sets, Rings, and Top (ifthey exist)? How about in the two examples of §1.2.9?

1.3.3. Localization of rings and modules. Another important example of a defi-nition by universal property is the notion of localization of a ring. We first review aconstructive definition, and then reinterpret the notion in terms of universal prop-erty. A multiplicative subset S of a ring A is a subset closed under multiplicationcontaining 1. We define a ring S−1A. The elements of S−1A are of the form a/s


where a ∈ A and s ∈ S, and where a1/s1 = a2/s2 if (and only if) for some s ∈ S,s(s2a1 − s1a2) = 0. We define (a1/s1) + (a2/s2) = (s2a1 + s1a2)/(s1s2), and(a1/s1) × (a2/s2) = (a1a2)/(s1s2). (If you wish, you may check that this equal-ity of fractions really is an equivalence relation and the two binary operations onfractions are well-defined on equivalence classes and make S−1A into a ring.) Wehave a canonical ring map

(1.3.3.1) A→ S−1A

given by a 7→ a/1. Note that if 0 ∈ S, S−1A is the 0-ring.There are two particularly important flavors of multiplicative subsets. The

first is {1, f, f2, . . . }, where f ∈ A. This localization is denoted Af. The second isA− p, where p is a prime ideal. This localization S−1A is denoted Ap. (Notationalwarning: If p is a prime ideal, thenAp means you’re allowed to divide by elementsnot in p. However, if f ∈ A, Af means you’re allowed to divide by f. This can beconfusing. For example, if (f) is a prime ideal, then Af 6= A(f).)Warning: sometimes localization is first introduced in the special case whereA

is an integral domain and 0 /∈ S. In that case, A →֒ S−1A, but this isn’t always true,as shown by the following exercise. (But we will see that noninjective localizationsneedn’t be pathological, and we can sometimes understand them geometrically,see Exercise 3.2.L.)

1.3.C. EXERCISE. Show that A → S−1A is injective if and only if S contains nozerodivisors. (A zerodivisor of a ringA is an element a such that there is a nonzeroelement b with ab = 0. The other elements of A are called non-zerodivisors. Forexample, an invertible element is never a zerodivisor. Counter-intuitively, 0 is azerodivisor in every ring but the 0-ring.)

IfA is an integral domain and S = A−{0}, then S−1A is called the fraction fieldofA, which we denote K(A). The previous exercise shows thatA is a subring of itsfraction field K(A). We now return to the case where A is a general (commutative)ring.

1.3.D. EXERCISE. Verify thatA→ S−1A satisfies the following universal property:S−1A is initial among A-algebras B where every element of S is sent to an invert-ible element in B. (Recall: the data of “an A-algebra B” and “a ring map A → B”are the same.) Translation: any map A→ B where every element of S is sent to aninvertible element must factor uniquely through A → S−1A. Another translation:a ring map out of S−1A is the same thing as a ring map from A that sends everyelement of S to an invertible element. Furthermore, an S−1A-module is the samething as an A-module for which s × · : M → M is an A-module isomorphism forall s ∈ S.In fact, it is cleaner to define A → S−1A by the universal property, and to

show that it exists, and to use the universal property to check various propertiesS−1A has. Let’s get some practice with this by defining localizations of modulesby universal property. SupposeM is an A-module. We define the A-module mapφ : M→ S−1M as being initial amongA-module mapsM→ N such that elementsof S are invertible in N (s × · : N → N is an isomorphism for all s ∈ S). More


precisely, any such map α : M→ N factors uniquely through φ:

Mφ //

α##F

FFFF

FFFF

S−1M

∃!

��N

(Translation: M → S−1M is universal (initial) among A-module maps fromM tomodules that are actually S−1A-modules. Can you make this precise by definingclearly the objects and morphisms in this category?)Notice: (i) this determines φ : M → S−1M up to unique isomorphism (you

should think through what this means); (ii) we are defining not only S−1M, butalso the map φ at the same time; and (iii) essentially by definition the A-modulestructure on S−1M extends to an S−1A-module structure.

1.3.E. EXERCISE. Show that φ : M → S−1M exists, by constructing somethingsatisfying the universal property. Hint: define elements of S−1M to be of the formm/s where m ∈ M and s ∈ S, and m1/s1 = m2/s2 if and only if for some s ∈ S,s(s2m1−s1m2) = 0. Define the additive structure by (m1/s1)+(m2/s2) = (s2m1+s1m2)/(s1s2), and the S

−1A-module structure (and hence theA-module structure)is given by (a1/s1) · (m2/s2) = (a1m2)/(s1s2).

1.3.F. EXERCISE.(a) Show that localization commutes with finite products, or equivalently, withfinite direct sums. In other words, ifM1, . . . ,Mn are A-modules, describe an iso-morphism (ofA-modules, and of S−1A-modules) S−1(M1×· · ·×Mn)→ S−1M1×· · · × S−1Mn.(b) Show that localization commutes with arbitrary direct sums.(c) Show that “localization does not necessarily commute with infinite products”:the obvious map S−1(

∏iMi)→

∏i S

−1Mi induced by the universal property oflocalization is not always an isomorphism. (Hint: (1, 1/2, 1/3, 1/4, . . . ) ∈ Q × Q ×· · · .)

1.3.4. Remark. Localization does not always commute with Hom, see Exam-ple 1.6.8. But Exercise 1.6.G will show that in good situations (if the first argumentof Hom is finitely presented), localization does commute with Hom.

1.3.5. Tensor products. Another important example of a universal property con-struction is the notion of a tensor product of A-modules

⊗A : obj(ModA) × obj(ModA) // obj(ModA)

(M,N)� //M⊗A N

The subscriptA is often suppressed when it is clear from context. The tensor prod-uct is often defined as follows. Suppose you have two A-modulesM andN. Thenelements of the tensor productM⊗AN are finiteA-linear combinations of symbolsm ⊗ n (m ∈ M, n ∈ N), subject to relations (m1 +m2) ⊗ n = m1 ⊗ n +m2 ⊗ n,m⊗ (n1+n2) = m⊗n1+m⊗n2, a(m⊗n) = (am)⊗n = m⊗ (an) (where a ∈ A,m1,m2 ∈M, n1, n2 ∈ N). More formally,M⊗AN is the free A-module generated


byM ×N, quotiented by the submodule generated by (m1 +m2, n) − (m1, n) −(m2, n), (m,n1+n2)−(m,n1)−(m,n2), a(m,n)−(am,n), and a(m,n)−(m,an)for a ∈ A, m,m1,m2 ∈ M, n,n1, n2 ∈ N. The image of (m,n) in this quotient ism⊗ n.If A is a field k, we recover the tensor product of vector spaces.

1.3.G. EXERCISE (IF YOU HAVEN’T SEEN TENSOR PRODUCTS BEFORE). Show thatZ/(10) ⊗Z Z/(12) ∼= Z/(2). (This exercise is intended to give some hands-on prac-tice with tensor products.)

1.3.H. IMPORTANT EXERCISE: RIGHT-EXACTNESS OF (·)⊗AN. Show that (·)⊗ANgives a covariant functor ModA → ModA. Show that (·) ⊗A N is a right-exactfunctor, i.e., if

M ′ →M→M ′′ → 0is an exact sequence of A-modules (which means f : M → M ′′ is surjective, andM ′ surjects onto the kernel of f; see §1.6), then the induced sequence

M ′ ⊗A N→M⊗A N→M ′′ ⊗A N→ 0is also exact. This exercise is repeated in Exercise 1.6.F, but youmay get a lot out ofdoing it now. (You will be reminded of the definition of right-exactness in §1.6.5.)

In contrast, you can quickly check that tensor product is not left-exact: tensorthe exact sequence of Z-modules

0 // Z×2 // Z // Z/(2) // 0

with Z/(2).The constructive definition ⊗ is a weird definition, and really the “wrong”

definition. To motivate a better one: notice that there is a natural A-bilinear mapM × N → M ⊗A N. (IfM,N, P ∈ ModA, a map f : M × N → P is A-bilinear iff(m1 + m2, n) = f(m1, n) + f(m2, n), f(m,n1 + n2) = f(m,n1) + f(m,n2), andf(am,n) = f(m,an) = af(m,n).) AnyA-bilinear mapM×N→ P factors throughthe tensor product uniquely:M×N→M⊗A N→ P. (Think this through!)We can take this as the definition of the tensor product as follows. It is an A-

module T along with an A-bilinear map t : M × N → T , such that given anyA-bilinear map t ′ : M × N → T ′, there is a unique A-linear map f : T → T ′ suchthat t ′ = f ◦ t.

M×N t //

t ′ ##GGG

GGGG

GGT

∃!fT ′

1.3.I. EXERCISE. Show that (T, t : M×N→ T) is unique up to unique isomorphism.Hint: first figure out what “unique up to unique isomorphism” means for suchpairs, using a category of pairs (T, t). Then follow the analogous argument for theproduct.

In short: givenM and N, there is an A-bilinear map t : M × N → M ⊗A N,unique up to unique isomorphism, defined by the following universal property:


for any A-bilinear map t ′ : M ×N → T ′ there is a unique A-linear map f : M ⊗AN→ T ′ such that t ′ = f ◦ t.As with all universal property arguments, this argument shows uniqueness

assuming existence. To show existence, we need an explicit construction.

1.3.J. EXERCISE. Show that the construction of §1.3.5 satisfies the universal prop-erty of tensor product.

The three exercises below are useful facts about tensor products with whichyou should be familiar.

1.3.K. IMPORTANT EXERCISE.(a) IfM is an A-module and A→ B is a morphism of rings, give B⊗AM the struc-ture of a B-module (this is part of the exercise). Show that this describes a functorModA →ModB.(b) If further A→ C is another morphism of rings, show that B⊗A C has a naturalstructure of a ring. Hint: multiplication will be given by (b1 ⊗ c1)(b2 ⊗ c2) =(b1b2)⊗ (c1c2). (Exercise 1.3.U will interpret this construction as a fibered coprod-uct.)

1.3.L. IMPORTANT EXERCISE. If S is a multiplicative subset of A andM is an A-module, describe a natural isomorphism (S−1A)⊗AM ∼= S−1M (as S−1A-modulesand as A-modules).

1.3.M. EXERCISE (⊗ COMMUTES WITH ⊕). Show that tensor products commutewith arbitrary direct sums: ifM and {Ni}i∈I are allA-modules, describe an isomor-phism

M⊗ (⊕i∈INi) ∼ // ⊕i∈I (M⊗Ni) .

1.3.6. Essential Example: Fibered products. Suppose we have morphisms α :X→ Z and β : Y → Z (in any category). Then the fibered product (or fibred product)is an object X×Z Y along with morphisms prX : X×Z Y → X and prY : X×Z Y → Y,where the two compositions α ◦ prX, β ◦ prY : X ×Z Y → Z agree, such that givenany objectW with maps to X and Y (whose compositions to Z agree), these mapsfactor through some uniqueW → X×Z Y:

W

∃!

##

��555

5555

5555

5555

5

))SSSSSS

SSSSSS

SSSSSS

S

X×Z YprX

��

prY// Y

β

��X

α // Z

(Warning: the definition of the fibered product depends on α and β, even thoughthey are omitted from the notation X×Z Y.)By the usual universal property argument, if it exists, it is unique up to unique

isomorphism. (You should think this through until it is clear to you.) Thus the useof the phrase “the fibered product” (rather than “a fibered product”) is reasonable,and we should reasonably be allowed to give it the name X×Z Y. We know whatmaps to it are: they are precisely maps to X and maps to Y that agree as maps to Z.


Depending on your religion, the diagram

X×Z YprX

��

prY// Y

β

��X

α // Z

is called a fibered/pullback/Cartesian diagram/square (six possibilities — evenmore are possible if you prefer “fibred” to “fibered”).The right way to interpret the notion of fibered product is first to think about

what it means in the category of sets.

1.3.N. EXERCISE. Show that in Sets,

X×Z Y = {(x, y) ∈ X× Y : α(x) = β(y)}.

More precisely, show that the right side, equipped with its evident maps to X andY, satisfies the universal property of the fibered product. (This will help you buildintuition for fibered products.)

1.3.O. EXERCISE. If X is a topological space, show that fibered products alwaysexist in the category of open sets of X, by describing what a fibered product is.(Hint: it has a one-word description.)

1.3.P. EXERCISE. If Z is the final object in a category C , and X, Y ∈ C , show that“X ×Z Y = X × Y”: “the” fibered product over Z is uniquely isomorphic to “the”product. Assume all relevant (fibered) products exist. (This is an exercise aboutunwinding the definition.)

1.3.Q. USEFUL EXERCISE: TOWERS OF FIBER DIAGRAMS ARE FIBER DIAGRAMS. Ifthe two squares in the following commutative diagram are fiber diagrams, showthat the “outside rectangle” (involving U, V , Y, and Z) is also a fiber diagram.

U //

��

V

��W //

��

X

��Y // Z

1.3.R. EXERCISE. Given morphisms X1 → Y, X2 → Y, and Y → Z, show that thereis a natural morphism X1×Y X2 → X1×Z X2, assuming that both fibered productsexist. (This is trivial once you figure out what it is saying. The point of this exerciseis to see why it is trivial.)

1.3.S. USEFUL EXERCISE: THE MAGIC DIAGRAM. Suppose we are given mor-phisms X1, X2 → Y and Y → Z. Show that the following diagram is a fibered


square.

X1 ×Y X2 //

��

X1 ×Z X2

��Y // Y ×Z Y

Assume all relevant (fibered) products exist. This diagram is surprisingly useful— so useful that we will call it themagic diagram.

1.3.7. Coproducts. Define coproduct in a category by reversing all the arrows inthe definition of product. Define fibered coproduct in a category by reversing allthe arrows in the definition of fibered product.

1.3.T. EXERCISE. Show that coproduct for Sets is disjoint union. This is why weuse the notation

∐for disjoint union.

1.3.U. EXERCISE. Suppose A → B and A → C are two ring morphisms, so inparticular B and C are A-modules. Recall (Exercise 1.3.K) that B ⊗A C has a ringstructure. Show that there is a natural morphism B→ B⊗A C given by b 7→ b⊗ 1.(This is not necessarily an inclusion; see Exercise 1.3.G.) Similarly, there is a naturalmorphism C→ B⊗AC. Show that this gives a fibered coproduct on rings, i.e., that

B⊗A C Coo

B

OO

Aoo

OO

satisfies the universal property of fibered coproduct.

1.3.8. Monomorphisms and epimorphisms.

1.3.9. Definition. A morphism π : X → Y is a monomorphism if any two mor-phisms µ1 : Z→ X and µ2 : Z→ X such that π ◦ µ1 = π ◦ µ2 must satisfy µ1 = µ2.In other words, there is at most one way of filling in the dotted arrow so that thediagram

Z

≤1

�� ???

????

?

Xπ

// Y

commutes — for any object Z, the natural map Mor(Z,X)→ Mor(Z, Y) is an injec-tion. Intuitively, it is the categorical version of an injective map, and indeed thisnotion generalizes the familiar notion of injective maps of sets. (The reason wedon’t use the word “injective” is that in some contexts, “injective” will have anintuitive meaning which may not agree with “monomorphism”. One example: inthe category of divisible groups, the map Q → Q/Z is a monomorphism but notinjective. This is also the case with “epimorphism” vs. “surjective”.)

1.3.V. EXERCISE. Show that the composition of twomonomorphisms is amonomor-phism.


1.3.W. EXERCISE. Prove that a morphism π : X → Y is a monomorphism if andonly if the fibered product X×Y X exists, and the induced morphism X→ X×Y Xis an isomorphism. We may then take this as the definition of monomorphism.(Monomorphisms aren’t central to future discussions, although they will come upagain. This exercise is just good practice.)

1.3.X. EASY EXERCISE. We use the notation of Exercise 1.3.R. Show that if Y → Zis a monomorphism, then the morphism X1 ×Y X2 → X1 ×Z X2 you described inExercise 1.3.R is an isomorphism. (Hint: for any object V , give a natural bijectionbetweenmaps from V to the first andmaps from V to the second. It is also possibleto use the magic diagram, Exercise 1.3.S.)

The notion of an epimorphism is “dual” to the definition of monomorphism,where all the arrows are reversed. This concept will not be central for us, althoughit turns up in the definition of an abelian category. Intuitively, it is the categori-cal version of a surjective map. (But be careful when working with categories ofobjects that are sets with additional structure, as epimorphisms need not be surjec-tive. Example: in the category Rings, Z→ Q is an epimorphism, but obviously notsurjective.)

1.3.10. Representable functors and Yoneda’s lemma. Much of our discussionabout universal properties can be cleanly expressed in terms of representable func-tors, under the rubric of “Yoneda’s Lemma”. Yoneda’s lemma is an easy fact statedin a complicated way. Informally speaking, you can essentially recover an objectin a category by knowing the maps into it. For example, we have seen that thedata of maps to X × Y are naturally (canonically) the data of maps to X and to Y.Indeed, we have now taken this as the definition of X× Y.Recall Example 1.2.20. Suppose A is an object of category C . For any object

C ∈ C , we have a set of morphisms Mor(C,A). If we have a morphism f : B→ C,we get a map of sets

(1.3.10.1) Mor(C,A)→Mor(B,A),

by composition: given a map from C to A, we get a map from B to A by precom-posing with f : B → C. Hence this gives a contravariant functor hA : C → Sets.Yoneda’s Lemma states that the functor hA determines A up to unique isomor-phism. More precisely:

1.3.Y. IMPORTANT EXERCISE THAT YOU SHOULD DO ONCE IN YOUR LIFE (YONEDA’SLEMMA).(a) Suppose you have two objects A and A ′ in a category C , and morphisms

(1.3.10.2) iC :Mor(C,A)→Mor(C,A ′)

that commute with the maps (1.3.10.1). Show that the iC (as C ranges over theobjects of C ) are induced from a unique morphism g : A → A ′. More precisely,show that there is a unique morphism g : A → A ′ such that for all C ∈ C , iC isu 7→ g ◦ u.(b) If furthermore the iC are all bijections, show that the resulting g is an isomor-phism. (Hint for both: This is much easier than it looks. This statement is sogeneral that there are really only a couple of things that you could possibly try.For example, if you’re hoping to find a morphism A → A ′, where will you find


it? Well, you are looking for an element Mor(A,A ′). So just plug in C = A to(1.3.10.2), and see where the identity goes.)

There is an analogous statement with the arrows reversed, where instead ofmaps into A, you think of maps from A. The role of the contravariant functor hAof Example 1.2.20 is played by the covariant functor hA of Example 1.2.14. Becausethe proof is the same (with the arrows reversed), you needn’t think it through.The phrase “Yoneda’s lemma” properly refers to a more general statement.

Although it looks more complicated, it is no harder to prove.

1.3.Z. ⋆ EXERCISE.(a) Suppose A and B are objects in a category C . Give a bijection between the nat-ural transformations hA → hB of covariant functors C → Sets (see Example 1.2.14for the definition) and the morphisms B→ A.(b) State and prove the corresponding fact for contravariant functors hA (see Ex-ample 1.2.20). Remark: A contravariant functor F from C to Sets is said to berepresentable if there is a natural isomorphism

ξ : F∼ // hA .

Thus the representing object A is determined up to unique isomorphism by thepair (F, ξ). There is a similar definition for covariant functors. (We will revisitthis in §6.6, and this problem will appear again as Exercise 6.6.C. The elementξ−1(idA) ∈ F(A) is often called the “universal object”; do you see why?)(c) Yoneda’s lemma. Suppose F is a covariant functor C → Sets, and A ∈ C .Give a bijection between the natural transformations hA → F and F(A). (Thecorresponding fact for contravariant functors is essentially Exercise 9.1.C.)

In fancy terms, Yoneda’s lemma states the following. Given a category C , wecan produce a new category, called the functor category of C , where the objects arecontravariant functors C → Sets, and the morphisms are natural transformationsof such functors. We have a functor (which we can usefully call h) from C to itsfunctor category, which sends A to hA. Yoneda’s Lemma states that this is a fullyfaithful functor, called the Yoneda embedding. (Fully faithful functors were definedin §1.2.15.)

1.3.11. Joke. The Yoda embedding, contravariant it is.

1.4 Limits and colimits

Limits and colimits are two important definitions determined by universalproperties. They generalize a number of familiar constructions. I will give the def-inition first, and then show you why it is familiar. For example, fractions will bemotivating examples of colimits (Exercise 1.4.B(a)), a

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