THE RISING SEA Foundations of Algebraic...

THE RISING SEA

Foundations of Algebraic Geometry

math216.wordpress.com

November 18, 2017 draftc 20102017 by Ravi Vakil.

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

remain many issues still to be dealt with in the main part of the notes (including many

of your corrections and suggestions).

Contents

Preface 110.1. For the reader 120.2. For the expert 160.3. 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 471.7. Spectral sequences 57

Chapter 2. Sheaves 712.1. Motivating example: The sheaf of differentiable functions 712.2. Definition of sheaf and presheaf 732.3. Morphisms of presheaves and sheaves 782.4. Properties determined at the level of stalks, and sheafification 822.5. Recovering sheaves from a sheaf on a base 862.6. Sheaves of abelian groups, and OX-modules, form abelian categories 892.7. The inverse image sheaf 92

Part II. Schemes 97

Chapter 3. Toward affine schemes: the underlying set, and topological space 993.1. Toward schemes 993.2. The underlying set of affine schemes 1013.3. Visualizing schemes I: generic points 1133.4. The underlying topological space of an affine scheme 1153.5. A base of the Zariski topology on SpecA: Distinguished open sets 1183.6. Topological (and Noetherian) properties 1193.7. The function I(), taking subsets of SpecA to ideals of A 127

Chapter 4. The structure sheaf, and the definition of schemes in general 1294.1. The structure sheaf of an affine scheme 1294.2. Visualizing schemes II: nilpotents 133

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4.3. Definition of schemes 1364.4. Three examples 1394.5. Projective schemes, and the Proj construction 145

Chapter 5. Some properties of schemes 1535.1. Topological properties 1535.2. Reducedness and integrality 1555.3. Properties of schemes that can be checked affine-locally 1575.4. Normality and factoriality 1615.5. The crucial points of a scheme that control everything: Associated

points and primes 166

Part III. Morphisms 175

Chapter 6. Morphisms of schemes 1776.1. Introduction 1776.2. Morphisms of ringed spaces 1786.3. From locally ringed spaces to morphisms of schemes 1806.4. Maps of graded rings and maps of projective schemes 1866.5. Rational maps from reduced schemes 1886.6. Representable functors and group schemes 1946.7. The Grassmannian (initial construction) 199

Chapter 7. Useful classes of morphisms of schemes 2017.1. An example of a reasonable class of morphisms: Open embeddings 2017.2. Algebraic interlude: Lying Over and Nakayama 2037.3. A gazillion finiteness conditions on morphisms 2077.4. Images of morphisms: Chevalleys Theorem and elimination theory 216

Chapter 8. Closed embeddings and related notions 2258.1. Closed embeddings and closed subschemes 2258.2. More projective geometry 2308.3. The (closed sub)scheme-theoretic image 2368.4. Effective Cartier divisors, regular sequences and regular embeddings240

Chapter 9. Fibered products of schemes, and base change 2479.1. They exist 2479.2. Computing fibered products in practice 2539.3. Interpretations: Pulling back families, and fibers of morphisms 2569.4. Properties preserved by base change 2629.5. Properties not preserved by base change, and how to fix them 2639.6. Products of projective schemes: The Segre embedding 2719.7. Normalization 273

Chapter 10. Separated and proper morphisms, and (finally!) varieties 27910.1. Separated morphisms (and quasiseparatedness done properly) 27910.2. Rational maps to separated schemes 28910.3. Proper morphisms 293

Part IV. Geometric properties: Dimension and smoothness 301

Chapter 11. Dimension 30311.1. Dimension and codimension 30311.2. Dimension, transcendence degree, and Noether normalization 30711.3. Codimension one miracles: Krulls and Hartogss Theorems 31511.4. Dimensions of fibers of morphisms of varieties 32211.5. Proof of Krulls Principal Ideal and Height Theorems 327

Chapter 12. Regularity and smoothness 33112.1. The Zariski tangent space 33112.2. Regularity, and smoothness over a field 33712.3. Examples 34212.4. Bertinis Theorem 34612.5. Another (co)dimension one miracle: Discrete valuation rings 34912.6. Smooth (and etale) morphisms (first definition) 35412.7. Valuative criteria for separatedness and properness 35812.8. More sophisticated facts about regular local rings 36212.9. Filtered rings and modules, and the Artin-Rees Lemma 364

Part V. Quasicoherent sheaves 367

Chapter 13. Quasicoherent and coherent sheaves 36913.1. Vector bundles and locally free sheaves 36913.2. Quasicoherent sheaves 37513.3. Characterizing quasicoherence using the distinguished affine base 37713.4. Quasicoherent sheaves form an abelian category 38113.5. Module-like constructions 38313.6. Finite type and coherent sheaves 38613.7. Pleasant properties of finite type and coherent sheaves 38913.8. Coherent modules over non-Noetherian rings 393

Chapter 14. Line bundles: Invertible sheaves and divisors 39714.1. Some line bundles on projective space 39714.2. Line bundles and Weil divisors 40014.3. Effective Cartier divisors = invertible ideal sheaves 408

Chapter 15. Quasicoherent sheaves and projective A-schemes 41115.1. The quasicoherent sheaf corresponding to a graded module 41115.2. Invertible sheaves (line bundles) on projective A-schemes 41215.3. Globally generated and base-point-free line bundles 41315.4. Quasicoherent sheaves and graded modules 416

Chapter 16. Pushforwards and pullbacks of quasicoherent sheaves 42116.1. Introduction 42116.2. Pushforwards of quasicoherent sheaves 42116.3. Pullbacks of quasicoherent sheaves 42216.4. Line bundles and maps to projective schemes 42816.5. The Curve-to-Projective Extension Theorem 43516.6. Ample and very ample line bundles 43716.7. The Grassmannian as a moduli space 442

Chapter 17. Relative versions of Spec and Proj, and projective morphisms 44717.1. Relative Spec of a (quasicoherent) sheaf of algebras 44717.2. Relative Proj of a sheaf of graded algebras 45017.3. Projective morphisms 45317.4. Applications to curves 459

Chapter 18. Cech cohomology of quasicoherent sheaves 46518.1. (Desired) properties of cohomology 46518.2. Definitions and proofs of key properties 47018.3. Cohomology of line bundles on projective space 47518.4. Riemann-Roch, degrees of coherent sheaves, and arithmetic genus 47718.5. A first glimpse of Serre duality 48518.6. Hilbert functions, Hilbert polynomials, and genus 48818.7. Serres cohomological characterization of ampleness 49418.8. Higher pushforward (or direct image) sheaves 49718.9. From projective to proper hypotheses: Chows Lemma and

Grothendiecks Coherence Theorem 501

Chapter 19. Application: Curves 50519.1. A criterion for a morphism to be a closed embedding 50519.2. A series of crucial tools 50719.3. Curves of genus 0 51019.4. Classical geometry arising from curves of positive genus 51119.5. Hyperelliptic curves 51419.6. Curves of genus 2 51819.7. Curves of genus 3 51919.8. Curves of genus 4 and 5 52119.9. Curves of genus 1 52319.10. Elliptic curves are group varieties 53219.11. Counterexamples and pathologies using elliptic curves 538

Chapter 20. Application: A glimpse of intersection theory 54320.1. Intersecting n line bundles with an n-dimensional variety 54320.2. Intersection theory on a surface 54720.3. The Grothendieck group of coherent sheaves, and an algebraic

version of homology 55420.4. The Nakai-Moishezon and Kleiman criteria for ampleness 556

Chapter 21. Differentials 56121.1. Motivation and game plan 56121.2. Definitions and first properties 56221.3. Smoothness of varieties revisited 57521.4. Examples 57821.5. Understanding smooth varieties using their cotangent bundles 58321.6. Unramified morphisms 58821.7. The Riemann-Hurwitz Formula 589

Chapter 22. Blowing up 59722.1. Motivating example: blowing up the origin in the plane 59722.2. Blowing up, by universal property 599

22.3. The blow-up exists, and is projective 60322.4. Examples and computations 608

Part VI. More 619

Chapter 23. Derived functors 62123.1. The Tor functors 62123.2. Derived functors in general 62523.3. Derived functors and spectral sequences 62923.4. Derived functor cohomology of O-modules 63423.5. Cech cohomology and derived functor cohomology agree 637

Chapter 24. Flatness 64524.1. Introduction 64524.2. Easier facts 64724.3. Flatness through Tor 65224.4. Ideal-theoretic criteria for flatness 65424.5. Topological aspects of flatness 66124.6. Local criteria for flatness 66524.7. Flatness implies constant Euler characteristic 669

Chapter 25. Smooth and etale morphisms, and flatness 67325.1. Some motivation 67325.2. Different characterizations of smooth and etale morphisms 67525.3. Generic smoothness and the Kleiman-Bertini Theorem 681

Chapter 26. Depth and Cohen-Macaulayness 68526.1. Depth 68526.2. Cohen-Macaulay rings and schemes 68826.3. Serres R1+ S2 criterion for normality 691

Chapter 27. Twenty-seven lines 69727.1. Introduction 69727.2. Preliminary facts 69927.3. Every smooth cubic surface (over k) has 27 lines 70027.4. Every smooth cubic surface (over k) is a blown up plane 704

Chapter 28. Cohomology and base change theorems 70728.1. Statements and applications 70728.2. Proofs of cohomology and base change theorems 71328.3. Applying cohomology and base change to moduli problems 720

Chapter 29. Power series and the Theorem on Formal Functions 72529.1. Introduction 72529.2. Algebraic preliminaries 72529.3. Defining types of singularities 72929.4. The Theorem on Formal Functions 73229.5. Zariskis Connectedness Lemma and Stein Factorization 73329.6. Zariskis Main Theorem 73629.7. Castelnuovos Criterion for contracting (1)-curves 740

29.8. Proof of the Theorem on Formal Functions 29.4.2 743

Chapter 30. Proof of Serre duality 74930.1. Introduction 74930.2. Ext groups and Ext sheaves for O-modules 75430.3. Serre duality for projective k-schemes 75830.4. The adjunction formula for the dualizing sheaf, andX = KX 763

Bibliography 769

Index 775

November 18, 2017 draft 9

Je pourrais illustrer la ... approche, en gardant limage de la noix quil sagit douvrir.La premiere parabole qui mest venue a lesprit tantot, cest quon plonge la noix dansun liquide emollient, de leau simplement pourquoi pas, de temps en temps on frotte pourquelle penetre mieux, pour le reste on laisse faire le temps. La coque sassouplit au fil dessemaines et des mois quand le temps est mur, une pression de la main suffit, la coquesouvre comme celle dun avocat mur a point! . . .

Limage qui metait venue il y a quelques semaines etait differente encore, la choseinconnue quil sagit de connatre mapparaissait comme quelque etendue de terre ou demarnes compactes, reticente a se laisser penetrer. ... La mer savance insensiblement etsans bruit, rien ne semble se casser rien ne bouge leau est si loin on lentend a peine...Pourtant elle finit par entourer la substance retive...

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 Alexander Grothendieck, Jean-Pierre Serre, and others.

For a number of reasons, algebraic geometry has earned a reputation of beinginaccessible. 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 a vague sense of the fundamental ideas of thesubject. 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 dont 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 The Rising Sea: Foundations of Algebraic Geometry

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

0.0.2. Structure. You will quickly notice that everything is numbered by chapterand section, and everything is numbered the same way after that (for ease of refer-ence), except exercises are indicated by letters (and are sprinkled throughout thetext, rather than at the end of sections). Individual paragraphs often get numbersfor ease of reference, or to indicate a new topic. Definitions are in bold, and aresometimes given in passing.

0.0.3. Acknowledgments.This one is going to be really hard, so Ill write this later. (Mike Stay is the au-

thor of Jokes 1.3.11 and 21.5.2.) The phrase The Rising Sea is due to Grothendieck[Gr6, p. 552-3], with this particular translation by McLarty [Mc, p. 1], and popu-larized as the title of Daniel Murfets excellent blog [Mur].

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 wonderfulmachine of modern algebraic geometry was created to understand basic and naivequestions about geometry (broadly construed). The purpose of this book is togive you a thorough foundation in these powerful ideas. Do not be seduced by thelotus-eaters into infatuation with untethered abstraction. Hold tight to 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 Grothendiecks 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 as R or Q. Wiser heads


would 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) Weglue 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 thatisnt 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 oneshould 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 steppingstones. 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. Yonedas 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 wont read, or punt it to a telegraphic appendix. Instead,this book attempts to explain everything necessary, but as little as possible, andtries to get across how you should think about (and work with) these fundamentalideas, and why they are more grounded than you might fear.

Chapters 35. 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 kind


of geometric space, and we are then ready to transport lots of intuition fromclassical 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 610. Thinking categorically will lead you to ask about morphismsof schemes (and other spaces in geometry). One of Grothendiecks fundamentallessons is that the morphisms are central. Important geometric properties shouldreally be understood as properties of morphisms. There are many classes of mor-phisms with special names, and in each case you should think through why thatclass deserves a name.

Chapters 1112. 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 1321. 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 to study curves, may help make clear how useful they are.

Chapters 2330. 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 CastelnuovosCriterion 29.7.1.)

In general, I like having as few hypotheses as possible. Certainly a hypothesisthat isnt 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 neednt be afraidof arithmetic examples or of algebraic examples; a central insight of algebraic ge-ometry is that the same formalism applies without change.


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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)

Pathological examples are useful to know. On mountain highways, there aretall 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 helpfulto have other algebraic geometry books at hand, to see different approaches andto have alternate expositions when things become difficult. This book may serveas 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 Jongsencyclopedic 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 havecome 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 sense ofwhat is true, but to be able to actually get your hands dirty. To quote Mark Kisin:You can wave your hands all you want, but it still wont make you fly. Note: Thehints may help you, but sometimes they may not.

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 anumber 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 timein 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 proofjustifies, but we will not give a much harder proof if the generality of the strongerresult will not be used. There are a few cases where we work harder to provea somewhat more general result that many readers may not appreciate. For ex-ample, we prove a number of theorems for proper morphisms, not just projective


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

Cohomology is done first using the Cech approach (as Serre first did), and de-rived functor cohomology is introduced only later. I am well aware that Grothendieckthinks of the fact that the agreement of Cech cohomology with derived functor co-homology should be considered as an accidental phenomenon, and that it isimportant for technical reasons not to take as definition of cohomology the Cechcohomology, [Gr4, p. 108]. But I am convinced that this is the right way for mostpeople 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 itreally 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 thanthey would otherwise. As an inflammatory example, instead of using Noetherianhypotheses, the notion of quasiseparatedness comes up early and often. The costis that 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 dont require that 0 = 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 ofintegral domain includes 1 = 0, so the 0-ring is not an integral domain. Weaccept the Axiom of Choice. In particular, any proper ideal in a ring is containedin a maximal ideal. (The Axiom of Choice also arises in the argument that thecategory of A-modules has enough injectives, see Exercise 23.2.G.)

The reader should be familiar with some basic notions in commutative ringtheory, in particular the notion of ideals (including prime and maximal 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, [Weib] is simultaneously detailed and readable.

Background from other parts of mathematics (topology, geometry, complexanalysis, 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 orthird (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!)

Lets 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 goalsand 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, hardchoices 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. Butmost 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 years material here, but I have tried to makeclear 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 how much 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 and whistles 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 comfortwith 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 and maturity 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 can focus on more subtle issues.

As just one example, Krulls Principal Ideal Theorem 11.3.3 is an importanttool. 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. 45]

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 mathematicalobjects (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 dont 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 you what you need toknow, and no more. (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 justgive 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 asthe 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 termsof a universal property. Given two sets M and N, a product is a set P, along withmaps : P M and : P N, such that for any set P with maps : P M and : P N, these maps must factor uniquely through P:(1.1.0.1) P

!

((PPPPP

PPPPPP

PPPP

00000000000000

P

//

N

M

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

P //

N

M

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

isnt 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

P11 //

1

N

M

and I have a product

P22 //

2

N

M

then by the universal property of my product (letting (P2, 2, 2) play the role of(P, , ), and (P1, 1, 1) play the role of (P , , ) in (1.1.0.1)), there is a uniquemap f : P1 P2 making the appropriate diagram commute (i.e., 1 = 2 f and1 = 2 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, 2, 2) play the role ofboth (P, , ) and (P , , ) in (1.1.0.1). There is a unique map h : P2 P2 such


thatP2

hAA

A

AAA

2

''PPPPP

PPPPPP

PPPP

2

00000000000000

P2 2//

2

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 to M 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, andonce 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 havent 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 tobe linear maps; or the category of differentiable manifolds, where the maps aretaken to be submersions, i.e., differentiable maps whose differential is everywheresurjective).

This is handy even in cases that you understand. For example, one way ofdefining 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 hisTohoku 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 composition Mor(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. (BecauseRussells 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 1(X, x0). In the case


where X is connected, and 1(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. If A is a ring, then the A-modulesform a category ModA. (This category has additional structure; it will be the pro-totypical example of an abelian category, see 1.6.) Taking A = k, we obtain Exam-ple 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 wont usethis terminology). This neednt 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 exampleis

(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 subsets of 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 category B 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 category A to a category B, denoted F : A B,is the following data. It is a map of objects F : obj(A ) obj(B), and for each A1,A2 A , and morphismm : A1 A2, a morphism F(m) : F(A1) F(A2) in B. Werequire that F preserves identity morphisms (forA A , F(idA) = idF(A)), and thatF preserves composition (F(m2 m1) = F(m2) F(m1)). (You may wish to verifythat covariant functors send isomorphisms to isomorphisms.) A trivial example isthe identity functor id : A A , whose definition you can guess. Here are someless 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 is ModA Ab from A-modulesto abelian groups, remembering only the abelian group structure of theA-module.

1.2.13. Topological examples. Examples of covariant functors include the funda-mental group functor 1, which sends a topological space Xwith choice of a pointx0 X to a group 1(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 group Hi(X,Z). The covariance corresponds to the fact thata (continuous) morphism of pointed topological spaces : X Y with (x0) = y0induces a map of fundamental groups 1(X, x0) 1(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. (Inclusions are always faithful, so there is no need for thephrase faithful subcategory.) Thus a subcategory A of A is full if and only if forall A,B obj(A ), MorA (A,B) = MorA (A,B). For example, the forgetful func-tor Veck Sets is faithful, but not full; and if A is a ring, the category of finitelygenerated A-modules is a full subcategory of the category ModA 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 thecontext 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 covariantfunctor 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 6.3.9.

1.2.21. Natural transformations (and natural isomorphisms) of covariant func-tors, and equivalences of categories.

(This notion wont come up in an essential way until at least Chapter 6, so youshouldnt read this section until then.) Suppose F andG are two covariant functors


from A to B. A natural transformation of covariant functors F G is the dataof a morphism mA : F(A) G(A) for each A A such that for each f : A A inA , 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 naturallyisomorphic 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 areessentially 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. Thedouble 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.Veck be the category of finite-dimensional vector spaces over k. Note that thiscategory contains oodles of vector spaces of each dimension.

1.2.C. EXERCISE. Let () : f.d.Veck f.d.Veck be the double dual functor fromthe category of finite-dimensional vector spaces over k to itself. Show that ()is naturally isomorphic to the identity functor on f.d.Veck. (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.Veck, as each kn is afinite-dimensional vector space.

1.2.D. EXERCISE. Show that V f.d.Veck gives 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.Veck. To make this precise, you will need to useGodel-Bernays set theory or else replace f.d.Veck with a very similar small category,but we wont 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, a term we will not


need). Indeed, one can show that such a functor has a quasiinverse (another termwe will not use later), i.e., a functor G : B A (necessarily also an equivalenceand unique up to unique isomorphism) for which G F = idA and F 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-come more 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 objectrather 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 multiplication


containing 1. We define a ring S1A. The elements of S1A are of the form a/swhere 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 S1A into a ring.) Wehave a canonical ring map

(1.3.3.1) A S1Agiven by a 7 a/1. Note that if 0 S, S1A is the 0-ring.

There are two particularly important flavors of multiplicative subsets. Thefirst is {1, f, f2, . . . }, where f A. This localization is denoted Af. The second isA p, where p is a prime ideal. This localization S1A is denoted Ap. (Notationalwarning: If p is a prime ideal, thenAp means youre allowed to divide by elementsnot in p. However, if f A, Af means youre allowed to divide by f. This can beconfusing. For example, if (f) is a prime ideal, then Af = A(f).)

Warning: sometimes localization is first introduced in the special case whereAis an integral domain and 0 / S. In that case,A S1A, but this isnt always true,as shown by the following exercise. (But we will see that noninjective localizationsneednt be pathological, and we can sometimes understand them geometrically,see Exercise 3.2.L.)

1.3.C. EXERCISE. Show that A S1A 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. More generally, ifM is anA-module, thena A is a zerodivisor for M if there is a nonzero m M with am = 0. The otherelements of A are called non-zerodivisors forM.)

IfA is an integral domain and S = A{0}, then S1A 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 S1A satisfies the following universal property:S1A 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 Bare the same.) Translation: any map A B where every element of S is sent to aninvertible element must factor uniquely through A S1A. Another translation:a ring map out of S1A is the same thing as a ring map from A that sends everyelement of S to an invertible element. Furthermore, an S1A-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 S1A by the universal property, and toshow that it exists, and to use the universal property to check various propertiesS1A has. Lets get some practice with this by defining localizations of modulesby universal property. Suppose M is an A-module. We define the A-module map : M S1M as being initial amongA-module mapsM N such that elements


of S are invertible in N (s : N N is an isomorphism for all s S). Moreprecisely, any such map : M N factors uniquely through :

M //

##F

FFFF

FFFF

S1M

!N

(Translation: M S1M is universal (initial) among A-module maps from M tomodules that are actually S1A-modules. Can you make this precise by definingclearly the objects and morphisms in this category?)

Notice: (i) this determines : M S1M up to unique isomorphism (youshould think through what this means); (ii) we are defining not only S1M, butalso the map at the same time; and (iii) essentially by definition the A-modulestructure on S1M extends to an S1A-module structure.

1.3.E. EXERCISE. Show that : M S1M exists, by constructing somethingsatisfying the universal property. Hint: define elements of S1M to be of theform m/s where m M and s S, and m1/s1 = m2/s2 if and only if for somes S, s(s2m1 s1m2) = 0. Define the additive structure by (m1/s1) + (m2/s2) =(s2m1 + s1m2)/(s1s2), and the S1A-module structure (and hence the A-modulestructure) 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, if M1, . . . , Mn are A-modules, describe an iso-morphism (ofA-modules, and of S1A-modules) S1(M1 Mn) S1M1 S1Mn.(b) Show that localization commutes with arbitrary direct sums.(c) Show that localization does not necessarily commute with infinite products:the obvious map S1(

iMi)i S1Mi induced by the universal property of

localization is not always an isomorphism. (Hint: (1, 1/2, 1/3, 1/4, . . . ) QQ .)

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) //MA 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 productMAN are finiteA-linear combinations of symbolsm n (m M, n N), subject to relations (m1 +m2) n = m1 n +m2 n,m (n1+n2) = mn1+mn2, a(mn) = (am)n = m (an) (where a A,


m1,m2 M, n1, n2 N). More formally,MAN is the free A-module generatedby M 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 HAVENT 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 MM 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 NMA NM A N 0is also exact. This exercise is repeated in Exercise 1.6.F, but you may 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. (If M,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 mapMN P factors throughthe tensor product uniquely: MNMA N P. (Think this through!)

We can take this as the definition of the tensor product as follows. It is anA-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.

MN t //

t ##GGG

GGGG

GGT

!fT

1.3.I. EXERCISE. Show that (T, t : MN 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: given M 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 uniquenessassuming 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 BAM 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 BA 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 and M is an A-module, describe a natural isomorphism (S1A)AM = S1M (as S1A-modulesand as A-modules).

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

M (iINi) // iI (MNi) .

1.3.6. Essential Example: Fibered products. Suppose we have morphisms : X Z and : Y Z (in any category). Then the fibered product (or fi-bred product) is an object X Z Y along with morphisms prX : X Z Y X andprY : XZ Y Y, where the two compositions prX, prY : XZ Y Z agree,such that given any object W with maps to X and Y (whose compositions to Zagree), these maps factor through some uniqueW XZ Y:

W

!##

555

5555

5555

5555

5

))SSSSSSS

SSSSSSSS

SSSS

XZ Y

prX

prY// Y

X

// Z

(Warning: the definition of the fibered product depends on and , even thoughthey are omitted from the notation XZ Y.)

By the usual universal property argument, if it exists, it is unique up to uniqueisomorphism. (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 XZ 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

XZ Y

prX

prY// Y

X

// 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 aboutwhat it means in the category of sets.

1.3.N. EXERCISE (FIBERED PRODUCTS OF SETS). Show that in Sets,

XZ Y = {(x, y) X Y : (x) = (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 thatX Z Y = X Y: the fibered product over Z is uniquely isomorphic to theproduct. Assume all relevant (fibered) products exist. (This is an exercise aboutunwinding the definition.)

1.3.Q. USEFUL EXERCISE: TOWERS OF CARTESIAN DIAGRAMS ARE CARTESIAN DI-AGRAMS. If the two squares in the following commutative diagram are Cartesiandiagrams, show that the outside rectangle (involving U, V , Y, and Z) is also aCartesian 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 X1Y X2 X1Z 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 Cartesian


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 the magic 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 BA C given by b 7 b 1.(This is not necessarily an inclusion; see Exercise 1.3.G.) Similarly, there is a naturalmorphism C BAC. Show that this gives a fibered coproduct on rings, i.e., that

BA 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 : X Y is a monomorphism if any two mor-phisms 1 : Z X and 2 : Z X such that 1 = 2 must satisfy 1 = 2.In other words, there is at most one way of filling in the dotted arrow so that thediagram

Z

1 ?

????

???

X

// Y

commutes for any object Z, the natural map Mor(Z,X) Mor(Z, Y) is an in-jection. Intuitively, it is the categorical version of an injective map, and indeedthis notion generalizes the familiar notion of injective maps of sets. (The reasonwe dont use the word injective is that in some contexts, injective will havean intuitive meaning which may not agree with monomorphism. One example:in the category of divisible groups, the map Q Q/Z is a monomorphism butnot injective. This is also the case with epimorphism (to be defined shortly) vs.surjective.)

1.3.V. EXERCISE. Show that the composition of two monomorphisms is a monomor-phism.


1.3.W. EXERCISE. Prove that a morphism : X Y is a monomorphism if andonly if the fibered product XY X exists, and the induced morphism X XY Xis an isomorphism. We may then take this as the definition of monomorphism.(Monomorphisms arent 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 bijectionbetween maps from V to the first and maps 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 Yonedas Lemma. Much of our discussionabout universal properties can be cleanly expressed in terms of representable func-tors, under the rubric of Yonedas Lemma. Yonedas 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 objectC 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.Yonedas 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 (YONEDASLEMMA).(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 the ob-jects of C ) are induced from a unique morphism g : A A . More precisely, showthat there is a unique morphism g : A A such that for all C C , iC is u 7 gu.(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 youre 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 neednt think it through.

The phrase Yonedas 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

: F // hA .

Thus the representing object A is determined up to unique isomorphism by thepair (F, ). 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 element1(idA) F(A) is often called the universal object; do you see why?)(c) Yonedas 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, Yonedas 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. Yonedas 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.C(a)), and the p-adic integers (Exam-ple 1.4.3) will be motivating examples of limits.

1.4.1. Limits. We say that a category is a small category if the objects and themorphisms are sets. (This is a technical condition intended only for experts.) Sup-pose I is any small category, and C is any category. Then a functor F : I C(i.e., with an object Ai C for each element i I , and appropriate commuting


morphisms dictated by I ) is said to be a diagram indexed by I . We call I anindex category. Our index categories will usually be partially ordered sets (Ex-ample 1.2.8), in which in particular there is at most one morphism between anytwo objects. (But other examples are sometimes useful.) For example, if is thecategory

//

//

and A is a category, then a functor A is precisely the data of a commutingsquare in A .

Then the limit of the diagram is an object limI

Ai of C along with morphisms

fj : limI

Ai Aj for each j I , such that ifm : j k is a morphism in I , then(1.4.1.1) lim

I

Ai

fj

fk

!!CCC

CCCC

AjF(m) // Ak

commutes, and this object and maps to eachAi are universal (final) with respect tothis property. More precisely, given any other objectW along with maps gi : W Ai commuting with the F(m) (ifm : j k is a morphism in I , then gk = F(m)gj),then there is a unique map

g : W limI

Ai

so that gi = fi g for all i. (In some cases, the limit is sometimes called the inverselimit or projective limit. We wont use this l

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