FOUNDATIONS OF ALGEBRAIC GEOMETRY: Rough notes for course. November 9, c 2005, 2006, 2007 by Ravi Vakil.

Transcription

1 FOUNDATIONS OF ALGEBRAIC GEOMETRY: Rough notes for course November 9, c 2005, 2006, 2007 by Ravi Vakil.

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3 Contents Chapter 1. Strictly edited notes for myself 5 Chapter 2. Introduction 9 Part I. Preliminaries 11 Chapter 3. Some category theory 13 Chapter 4. Sheaves 49 Part II. Schemes and morphisms of schemes 71 Chapter 5. Chapter 6. Toward affine schemes: the underlying set, and the underlying topological space 73 The structure sheaf of an affine scheme, and the definition of schemes in general 95 Chapter 7. Some properties of schemes 107 Chapter 8. Morphisms of schemes 125 Chapter 9. Properties of morphisms of schemes 135 Chapter 10. Fiber products 153 Chapter 11. Projective schemes 165 Part III. Harder properties of schemes 179 Chapter 12. Dimension 181 Chapter 13. Nonsingularity (or regularity or Smoothness ) 199 Part IV. Quasicoherent sheaves 211 Part V. Cohomology 213 Part VI. Properties of morphisms of schemes 215 Part VII. Useful properties and constructions 217 Appendix. Bibliography 219 3

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5 CHAPTER 1 Strictly edited notes for myself Three editing environments: 1.1 Parsimonious list of guidelines forreader [...]. These are comments to ask the (proof-)readers. primordial = primordial ooze forexperts (questions to ask; comments on controversial pedagogical choices) forme (comparison to other results in the literature; how important exercises are; links to elsewhere in the book, e.g. where results are used later; post mortem on how hard things are; things I still need to write or think about writing TEND TO BE IN CAPS or STARRED***) Sectioning: part (last) chapter vsection subsection Theorem Exercise (exercisedone) Definition Remark. Other comment commands: notation reference (also hartshorne) lremind cut (for things I don t ever want to see, but want to keep in the file) 1.2 Tex tips varnothing not emptyset varprojlim lim not lim, and varinjlim lim not lim. hook, dotted, dashed arrows. a hi b c. two arrows pointing to the right A bowed arrows U R n φ ψ B π f 1,...,f n U Boldface in index is index{ notation textbf} (no space) 5

6 6 Foundations of Algebraic Geometry Notation: 1.3 Conventions I will follow Rings are A and B in the French style, not R and S. Scheme morphism is π : X Y, although π may sometimes be f. Symbols that go together: X, A, m. Hence A is a B-algebra. I think we will have maps X Spec B rather than X Spec A, although I m not sure about that. Ideals are I and J. Modules are M and N. Schemes are X, Y, Z, and bases are often S, T. Sheaves are F, G. complexes/indices will be S, where that symbol could be changed, e.g. possibly later to. However, has too many meanings, and is too small. But I may shrink the bullet. For graded rings, we have S 0 = A. S f is (S ) f. Γ(U, F), and later H 0 (U, F). sheaves are underlined, e.g. Hom, not in calligraphic font. Coordinates on affine space are x i. Projective coordinates x 0/1. Duals are not. FF stands for two things: fraction field (and total fraction ring, Eisenbud s total quotient ring), and function field. Defined for reduced and locally Noetherian. Explain in the text why the usual notations of K( ), C( ), k( ) are not good. Blow-up is β : Bl X Y Y with exceptional divisor E X Y. Spellings: Leibniz (following Eisenbud). base-point-free. Unique factorization domain, Krull s Principal Ideal Theorem. Algebraic Hartogs Theorem [better name?]. 1.4 Various stages of notes (long-term planning) serious editing and writing algebraic geometry done right [and comprehensively] teach class and order then a write-through. In each section, have a forme of what the big picture is, and what they should know at the end. go through standard texts and make sure all important results are here public posting Extremely long-term issues: Names: FOAG is my working title. Schemes seems a good one. Kiran suggests Schemes and algebraic geometry, SAG or SAAG. Or Schemes and cohomology in algebraic geometry. Or Foundational/Fundamental Lessons/Lectures in Algebraic Geometry. Possibly: The Rising Sea: Foundations of Algebraic Geometry. Or The Rising Tide. commented out here: Tex for Ravi Vakil, Palo Alto, today Some cover ideas: possibly: the most important diagrams from the text the magic diagram (Exercise 3.3.K) ellipse fact (see below); poncelet quadric surface with 2 rulings, and also 2 lines meeting 4 lines E 6 singularity in the water Joe Harris picture: family of curves Picture of schemes with associated points, maybe with map from Spec of the dual numbers. Serre duality statement RR for curves

7 Ravi Vakil 7 Fun ellipse fact in the style of Poncelet. Given an ellipse in the plane, suppose there is a point such that when you do the following thing a finite number of times, you return to the starting point: take the other point on the ellipse with the same x- coordinate; then the same thing with the y-coordinate. Then show this is true no matter where you start. Idea: degenerate elliptic curve on P 1 P 1. To get concrete examples: translate ellipse so it is centered at the origin, then scale so that it is of the form x 2 + bxy + y 2 = 1. Then (x, y) ( by x, y) ( by x, b( by x) y) = ( by x, bx + (b 2 1)y). Then we get a matrix, whose eigenvalues satisfy λ 2 + λ(2 b 2 ) + 1 = 0. Here b 4, so we get a root of unity. We can get any root we want. A particularly fun case is b = 3. Then we get a 6th root of 1. Fun side fact: if we start over Q, then the period can only a few finite things. Reason: roots of unity are not in quadratic extensions of Q Sources. Hartshorne. Mumford Red book. Mumford Abelian varieties. Complex projective varieties I. Shafarevich. Hindry-Silverman. Debarre; Smith et al. Iitaka. Harris. Eisenbud-Harris. FAC, Dolgachev. Gathmann. Yves Andre. Milne. Fundamentals of Algebraic Geometry (Barbara et al.) Not EGA. ]

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9 CHAPTER 2 Introduction [forexperts: The title Foundations of Algebraic Geometry was just a quickly chosen working title. What should this be called?] [forme: Brief introduction goes here, and philosophy.] 2.1 Why algebraic geometry? 2.2 Background and conventions 1 All rings are assumed to be commutative unless explicitly stated otherwise. All rings are assumed to have 1. Recall that maps of rings must send 1 to 1. We don t require that 0 1; in other words, the 0-ring (with one element) is assumed to be the ring. (There is a ring map from any ring to the 0-ring; the 0-ring only maps to itself. The 0-ring is the final object in the category of rings.) [forme: 0-ring, zero ring Many reasons for 0-ring, including: final object in rings. For all x A, A x is a ring.] We accept the axiom of choice. In particular, any ideal in a ring is contained in a maximal ideal. 2 AC The reader should be familiar with some basic notions in commutative ring theory, in particular the notion of ideals (including prime and maximal ideals) and localization. For example, the reader should be able to show that if S is a multiplicative set of a ring A, then the primes of S 1 A are in natural bijection with those primes of A not meeting S ( 5.2.E). The notion of tensor products and exact sequences of A-modules will be useful. [forme: Notions from commutative algebra that we use: rings, modules, prime and maximal ideals, ideals and quotients, localization, tensor product, exact sequences.] We will not concern ourselves with set-theoretic issues of sets, classes, universes, small categories, etc. [forme: ADD MAX s issues: About subtle foundational issues (small categories, set-theoretic issues, etc.): It is true that some people should be careful about these isues: but do you really want to be one of those people?] Further background. It may be helpful to have books on other subjects handy that you can dip into for specific facts, rather than reading them in advance. In commutative algebra, Eisenbud [E] is good for this. Other popular choices are Atiyah- Macdonald [AM] and Matsumura [M-CRT]. For homological algebra, Weibel 1 conventions 2 assumeac 9

10 10 Foundations of Algebraic Geometry [W] (homological algebra) has detailed and readable. Another good possibility is Kashiwara-Schapira [KS]. Background from other parts of mathematics (topology, geometry, complex analysis) will be helpful for developing intuition Acknowledgments.

11 Part I Preliminaries

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13 CHAPTER 3 Some category theory [forme: POSSIBLY ADD: Why say epi rather than surjective? because surjective will sometimes have an obvious meaning which may not agree with epi.] That which does not kill me, makes me stronger. Nietzsche [forme: , Twilight of the Idols, 1888, or possibly Thus spake Zarathustra] [forme: Outline of this chapter: Categories. Functors. Universal properties and Yoneda s lemma. Examples: limits. Adjoints. spectral sequences. Here is what they should know from this section: the definition of a category. Functors (covariant and contravariant). Yoneda s lemma. Universal properties, in the important special example of colimits = direct limits. Direct limits and directed systems. We need the adjoint version of Yoneda too, which is currently not in the text below.] 3.1 Motivation 1 Before we get to any interesting geometry, we need to develop the language to discuss things cleanly and effectively. [forme: Counsel patience!] This is best done in the language of categories. There is not much to know about categories to get started; it is just a very useful language. Like all mathematical languages, category theory comes with an embedded logic, which allows us to abstract intuitions in settings we know well to far more general situations. Our motivation is as follows. We will be creating some new mathematical objects (such as schemes, and families of sheaves), and we expect them to act like objects we have seen before. We could try to nail down precisely what we mean by act like, and what minimal set of things we have to check in order to verify that they act the way we expect. Fortunately, we don t have to other people have done this before us, by defining key notions, such as abelian categories, which behave like modules over a ring. For example, we will define the notion of product of the geometric spaces (schemes). We could just give a definition of product, but then you should want to know why this precise definition deserves the name of product. As a motivation, we revisit the notion of product in a situation we know well: (the category of) sets. One way to define the 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 { u v u U, v V }. These notions are obviously the same. Better: there is an obvious bijection between the two. 1 catmot 13

14 14 univ. prop This can be made precise by giving a better definition of product, in terms of a universal property. Given two sets M and N, a product is a set P, along with maps µ : P M and ν : P N, such that for any other set P with maps µ : P M and ν : P N, these maps must factor uniquely through P : 2 (1) P! ν µ P ν N µ M Thus a product is a diagram µ P ν M and not just a set P, although the maps µ and ν are often left implicit. This definition agrees with the usual definition, with one twist: there isn t just a single product; but any two products come with a canonical isomorphism between them. In other words, the product is unique up to unique isomorphism. Here is why: if you have a product ν 1 P 1 N N µ 1 and I have a product M ν 2 P 2 N µ 2 M then by the universal property of my product (letting (P 2, µ 2, ν 2 ) play the role of (P, µ, ν), and (P 1, µ 1, ν 1 ) play the role of (P, µ, ν ) in (1)), there is a unique map f : P 1 P 2 making the appropriate diagram commute (i.e. µ 1 = µ 2 f and ν 1 = ν 2 f). Similarly by the universal property of your product, there is a unique map g : P 2 P 1 making the appropriate diagram commute. Now consider the universal property of my product, this time letting (P 2, µ 2, ν 2 ) play the role of both (P, µ, ν) and (P, µ, ν ) in (1). There is a unique map h : P 2 P 2 such that P 2 h ν 2 µ 2 P 2 ν 2 N µ 2 M 2 setup

15 15 commutes. However, I can name two such maps: the identity map id P2, and g f. Thus g f = id P2. Similarly, f g = id P1. Thus the maps f and g arising from the universal property are bijections. In short, there is a unique bijection between P 1 and P 2 preserving the product structure (the maps to M and N). This gives us the right to name any such product M N, since any two such products are canonically identified. This definition has the advantage that it works in many circumstances, and once we define category, we will soon see that the above argument applies verbatim in any category to show that products, if they exist, are unique up to unique isomorphism. Even if you haven t seen the definition of category before, you can verify that this agrees with your notion of product in some category that you have seen before (such as the category of vector spaces, where the maps are taken to be linear maps; or the category of real manifolds, where the maps are taken to be submersions). This is handy even in cases that you understand. For example, one way of defining the product of two manifolds M and N is to cut them both up into charts, then take products of charts, then glue them together. But if I cut up the manifolds in one way, and you cut them up in another, how do we know our resulting manifolds are the same? We could wave our hands, or make an annoying argument about refining covers, but instead, we should just show that they are indeed products, and hence the same (i.e. isomorphic). We will shortly formalize this argument in Section 3.3. Another set of notions we will abstract are categories that behave like modules. We will want to define kernels and cokernels for new notions, and we should make sure that these notions behave the way we expect them to. This leads us to the definition of abelian categories, first defined by Grothendieck in his Tôhoku paper [Gr]. In this chapter, we ll give an informal introduction to these and related notions, in the hope of giving just enough familiarity to comfortably use them in practice. [forme: The reader can also instead read [W, App. A].] abelian category 3.2 Categories and functors We begin with an informal definition of categories and functors Categories. category A category consists of a collection of objects, and for each pair of objects, a set of maps, or morphisms (or arrows), between them. The collection of objects of obj, mor, Mor, arrows a category C are often denoted obj(c), but we will usually denote the collection C also by C. If A, B C, then the morphisms from A to B are denoted Mor(A, B). A morphism is often written f : A B, and A is said to be the source of f, and B the target of f. Morphisms compose as expected: there is a composition Mor(A, B) Mor(B, C) Mor(A, C), and if f Mor(A, B) and g Mor(B, C), then their composition is denoted g f. Composition is associative: if f Mor(A, B), g Mor(B, C), and h Mor(C, D), then h (g f) = (h g) f. For each object A C, there is always an identity morphism id A : A A, such that when you (left- or

16 16 iso, aut right-)compose a morphism with the identity, you get the same morphism. More precisely, if f : A B is a morphism, then f id A = f = id B f. If we have a category, then we have a notion of isomorphism between two objects (if we have two morphisms f : A B and g : B A, both of whose compositions are the identity on the appropriate object), and a notion of automorphism of an object (an isomorphism of the object with itself) 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. (Because Russell s paradox shows that there is no set of all sets, we did not say earlier that there is a set of all objects. But as stated in 2.2, we are deliberately omitting all set-theoretic issues.) Example. Another good example is the category Vec k of vector spaces over a given field k. The objects are k-vector spaces, and the morphisms are linear transformations. 4 groupoid Aut 3.2.A. Unimportant exercise. A category in which each morphism is an isomorphism is called a groupoid. (This notion is not important in this book. The point of this exercise is to give you some practice with categories, by relating them to an object you know well.) (a) A perverse definition of a group is: a groupoid with one element. Make sense of this. (b) Describe a groupoid that is not a group. (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 from x y are paths from x to y, up to homotopy. Then the automorphism group of x 0 is the (pointed) fundamental group π 1 (X, x 0 ). In the case where X is connected, and the π 1 (X) is not abelian, this illustrates the fact that for a connected groupoid whose definition you can guess the automorphism groups of the objects are all isomorphic, but not canonically isomorphic.) [forme: Cut this exercise most likely.] 3.2.B. Exercise. If A is an object in a category C, show that the Mor(A, A) forms a group (called the automorphism group of A, denoted Aut(A)). What are the automorphism groups of the objects in Examples and 3.2.3? Show that two isomorphic objects have isomorphic automorphism groups Example: abelian groups. The abelian groups, along with group homomorphisms, form a category Ab Example: modules over a ring. If A is a ring, then the A-modules form a category Mod A. (This category has additional structure; it will be the prototypical example of an abelian category, see Sect. 3.6.) Taking A = k, we obtain Example 3.2.3; taking A = Z, we obtain Example Sets 4 Veck 5 Ab

17 Example: rings. There is a category Rings, where the objects are rings, and the morphisms are morphisms of rings (which I ll assume send 1 to 1). [forme: Chad remarks: the category Mod, (A, M, φ : A End(M)).] Example: topological spaces. The topological spaces, along with continuous maps, form a category Top. The isomorphisms are homeomorphisms Example: partially ordered sets. 6 A partially ordered set, or poset, is a set (S, ) along with a binary relation satisfying: (i) x x, (ii) x y and y z imply x z (transitivity), and (iii) if x y and y x then x = y. A partially ordered set (S, ) can be interpreted as a category whose objects are the elements of S, and with a single morphism from x to y if and only if x y (and no morphism otherwise). A trivial example is (S, ) where x y if and only if x = y. Another example is 7 (2) Here there are three objects. The identity morphisms are omitted for convenience, and the three non-identity morphisms are depicted. A third example is (3) Here the obvious morphisms are again omitted: the identity morphisms, and the morphism from the upper left to the lower right. Similarly, depicts a partially ordered set, where again, only the generating morphisms are depicted Example: the category of subsets of a set, and the category of open sets in a topological space. If X is a set, then the subsets form a partially ordered set, where the order is given by inclusion. Similarly, if X is a topological space, then the open sets form a partially ordered set, where the order is given by inclusion. 8 (What is the initial object? What is the final object?) Example. A subcategory A of a category B has as its objects some of the objects of B, and some of the morphisms, such that the morphisms of A include the identity morphisms of the objects of A, and are closed under composition. (For example, (2) is in an obvious way a subcategory of (3).) Functors. 6 d:poset 7 three3 8 categoryofopensets

18 18 covariant, functor forgetful functor 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 a 1, a 2 A a morphism m : a 1 a 2, F (m) is a morphism from F (A 1 ) F (A 2 ) in B. F preserves identity morphisms: for A A, F (id A ) = id F (A). F preserves composition: F (m 1 m 2 ) = F (m 1 ) F (m 2 ). If F : A B and G : B C, then we may define a functor G F : A C in the obvious way. Composition of functors is associative Example: a forgetful functor. Consider the functor from the category of complex vector spaces Vec k to Sets, that associates to each vector space its underlying set. The functor sends a linear transformation to its underlying map of sets. This is an example of a forgetful functor, where some additional structure is forgotten. Another example of a forgetful functor is Mod A Ab from A-modules to abelian groups, remembering only the abelian group structure of the A-module Topological examples. Examples of covariant functors include the fundamental group functor π 1, which sends a topological space with X choice of a point x 0 X to a group π 1 (X, x 0 ), and the ith homology functor Top Ab, which sends a topological space X to its ith homology group H i (X, Z). The covariance corresponds to the fact that a (continuous) morphism of pointed topological spaces f : X Y with f(x 0 ) = y 0 induces a map of fundamental groups π 1 (X, x 0 ) π 1 (Y, y 0 ), and similarly for homology groups Example. Suppose A is an element of a category C. Then there is a functor h A : C Sets sending B C to Mor(A, B), and sending f : B 1 B 2 to Mor(A, B 1 ) Mor(A, B 2 ) described by 10 [g : A B 1 ] [f g : A B 1 B 2 ] Definition. A contravariant functor is defined in the same way as a covariant functor, except the arrows switch directions: in the above language, F (A 1 A 2 ) is now an arrow from F (A 2 ) to F (A 1 ). It is wise to always state whether a functor is covariant or contravariant. If it is not stated, the functor is often assumed to be covariant Topological example (cf. Example ). The the ith cohomology functor H i (, Z) : Top Ab is a contravariant functor Example. If A is the category of complex vector spaces, then taking duals gives a contravariant functor : A A. Indeed, to each linear transformation f : V W, we have a dual transformation f : W V, and (f g) = g f Example. There is a contravariant functor Top Rings taking a topological space X to the continuous functions on X. A morphism of topological spaces X Y (a continuous map) induces the pullback map from functions on Y to maps on X Example (cf ). Suppose A is an element of a category C. Then there is a contravariant functor h A : C Sets sending B C to Mor(B, A), and 9 homologyexample, referred to in coho e.g. 10 ha, referred to in contra e.g.

19 19 sending f : B 1 B 2 to Mor(B 2, A) Mor(B 1, A) described by 11 [g : B 2 A] [g f : B 2 B 1 A]. This example initially looks weird and different, but the previous two examples are just special cases of this; do you see how? What is A in each case? Full and faithful functors. A covariant functor F : A B is faithful if for all A, A A, the map Mor A (A, A ) Mor B (F (A), F (A )) is injective, and full if it is surjective. A functor that is full and faithful is fully faithful. A subcategory i : A B is a full subcategory if i is full. [forme: Where will we use these notions? full, faithful functors Full Yoneda, and Freyd-Mitchell] 3.3 Universal properties determine an object up to unique isomorphism GET RID OF YONEDA!!! 12 [forme: [H, p. 241] refers to the new edition of EGA I, Ch. 0, 1] Given some category that we come up with, we often will have ways of producing new objects from old. In good circumstances, such a definition can be made using the notion of a universal property. Informally, we wish that there is an object with some property. We first show that if it exists, then it is essentially unique, or more precisely, is unique up to unique isomorphism. Then we go about constructing an example of such an object to show existence. With a little practice, universal properties are useful in proving things quickly slickly. However, explicit constructions are often intuitively easier to work with, and sometimes also lead to short proofs. We have seen one important example of a universal property argument already in Section 3.1: products. You should go back and verify that our discussion there gives a notion of product in category, and shows that products, if they exist, are unique up to canonical isomorphism Another good example of a universal property construction is the notion of a tensor product of A-modules product tensor product A : obj(mod A ) obj(mod A ) obj(mod A ) M N M A N The subscript A is often suppressed when it is clear from context. Tensor product is often defined as follows. Suppose you have two A-modules M and N. Then elements of the tensor product M A N are of the form m n (m M, n N), subject to relations (m 1 +m 2 ) n = m 1 n+m 2 n, m (n 1 +n 2 ) = m n 1 +m n 2, a(m n) = (am) n = m (an) (where a A). If A is a field k, we get the tensor product of vector spaces. 11 haha 12 s:yoneda 13 tpexists

20 A. Exercise (if you haven t seen tensor products before). Calculate Z/10 Z Z/12. (This exercise is intended to give some hands-on practice with tensor products.) 14 [forme: Have them generalize to Z/m Z Z/n?] 3.3.B. Exercise: right-exactness of A N. Show that A N gives a covariant functor Mod A Mod A. Show that A N is a right-exact functor, i.e. if M M M 0 is an exact sequence of A-modules, then the induced sequence M A N M A N M A N 0 right exact ex.done tensor and def is also exact. [forme: Tony Licata gave a universal property proof, at least partially. By construction it is easy to check surjectivity, and that the composition is 0. So we show that (M A N/M A N) satisfies the universal property for M A N. Well, it clearly has a map there, and then we win.] (You will be reminded of the definition of right-exactness in ) 15 This is a weird definition, and really the wrong definition. To motivate a better one: notice that there is a natural A-bilinear map M N M A N. Any A-bilinear map M N C factors through the tensor product uniquely: M N M A N C. (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 any other such map factors through t that given any other t : M N T, there is a unique map f : T T such that t = f t. M N t t T!f T 3.3.C. 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 such pairs. Then follow the analogous argument for the product. [forme: In class: explain this. If I could create something satisfying this property, (M A N), and you were to create something else (M A N), then by my universal property for C = (M A N), there would be a unique map (M A N) (M A N) interpolating M N (M A N), and similarly by your universal property there would be a unique universal map (M A N) (M A N). The composition of these two maps in one order (M A N) (M A N) (M A N) has to be the identity, by the universal property for C = (M A N), and similarly for the other composition. Thus we have shown that these two maps are inverses, and our two spaces are isomorphic. In short: our two definitions may not be the same, but there is a canonical isomorphism between them. Then the usual construction works, but someone else may have another construction which works just as well.] ex.done 14 tensorbaby 15 tensorrightexact

21 21 In short: 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 f : M A N T such that t = f t. Note that this argument shows uniqueness assuming existence. We need to still show the existence of such a tensor product. This forces us to do something constructive. 3.3.D. Exercise. Show that the construction of satisfies the universal property of tensor product. The uniqueness of tensor product is our second example of the proof of uniqueness (up to unique isomorphism) by a universal property. If you have never seen this sort of argument before, then you might think you get it, but you don t, so you should think over it some more. We will be using such arguments repeatedly in the future. [forme: teaching: It s hard, but easy. It is black, but white. This takes us to a powerful fact, that is very zen: it is very deep, but also very shallow.] Some stray remarks about tensor products that you should be familiar with: if S is a multiplicative set of A and M is an A-module, then there is a natural isomorphism (S 1 A) A B = S 1 M. If further A B is a morphism of rings, then B A M naturally has the structure of a B-module. If further A C is a morphism of rings, then B A C is a ring. Here is another exercise involving a universal property Definition. An object of a category C is an initial object if it has precisely one map to every other object. It is a final object if it has precisely one map from every other object. It is a zero-object if it is both an initial object and a final object. 16 ex.done initial, final, zero object 3.3.E. Exercise. Show that any two initial objects are canonically isomorphic. Show that any two final objects are canonically isomorphic. This (partially) justifies the phrase the initial object rather than an initial object, and similarly for the final object and the zero object. 3.3.F. Exercise. State what the initial and final objects are in Sets, Rings, and Top (if they exist) Example: Fibered products. (This notion of fibered product will be important for us later.) Suppose we have morphisms X, Y Z (in any category). Then the fibered product is an object X Z Y along with morphisms to X and Y, where the two compositions X Z Y Z agree, such that given any other object W with maps to X and Y (whose compositions to Z agree), these maps factor 16 def0object

22 22 through some unique W X Z Y : W! X Z Y π Y Y π X f X Z 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 use of 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 what maps to it are: they are precisely maps to X and maps to Y that agree on maps to Z. The right way to interpret this is first to think about what it means in the category of sets. g 3.3.G. Exercise. Show that in Sets, X Z Y = {(x X, y Y ) f(x) = g(y)}. More precisely, describe a natural isomorphism between the left and right sides. (This will help you build intuition for fibered products.) 3.3.H. Exercise. If X is a topological space, show that fibered products always exist in the category of open sets of X, by describing what a fibered product is. (Hint: it has a one-word description.) 3.3.I. Exercise. If Z is the final object in a category C, and X, Y C, then X Z Y = X Y : the fibered product over Z is canonically isomorphic to the product. (This is an exercise about unwinding the definition.) 3.3.J. Useful Exercise: towers of fiber diagrams are fiber diagrams. If the two squares in the following commutative diagram are fiber diagrams, show that the outside rectangle (involving U, V, Y, and Z) is also a fiber diagram. 17 U V W Y X Z 3.3.K. Useful exercise: My favorite fiber diagram. Suppose we are given morphisms W, X Y and Y Z. Describe the natural morphism W Y X 17 towerofpower

23 23 W Z X. Show that the following diagram is a commutative diagram. W Y X Y W Z X Y Z Y The following diagram is surprisingly incredibly useful so useful that we will call it the magic diagram. 18 We make a definition to set up an exercise Definition. A morphism f : X Y is a monomorphism if any two monomorphism morphisms g 1, g 2 : Z X such that f g 1 = f g 2 must satisfy g 1 = g 2. This a generalization of of an injection of sets. In other words, there is a unique way of filling in the dotted arrow so that the following diagram commutes. 1 Z X f Y. Intuitively, it is the categorical version of an injective map, and indeed this notion generalizes the familiar notion of injective maps of sets. 19 The notion of an epimorphism is dual to this diagramatic definition, where all the arrows are reversed. This concept will not be central for us, although it is necessary for the definition of an abelian category. Intuitively, it is the categorical version of a surjective map. 20 epimorphism 3.3.L. Exercise. Prove a morphism is a monomorphism if and only if the natural morphism X X Y X is an isomorphism. We may then take this as the definition of monomorphism. (Monomorphisms aren t very central to future discussions, although they will come up again. This exercise is just good practice.) 3.3.M. Exercise. Suppose X Y is a monomorphism, and W, Z X are two morphisms. Show that W X Z and W Y Z are canonically isomorphic. We will use this later when talking about fibered products. (Hint: for any object V, give a natural bijection between maps from V to the first and maps from V to the second. It is also possible to use the magic diagram, Exercise 3.3.K) 3.3.N. Exercise. Given X Y Z, show that there is a natural morphism X Y X X Z X, assuming that both fibered products exist. (This is trivial once you figure out what it is saying. The point of this exercise is to see why it is trivial.) 3.3.O. Unimportant exercise. Define coproduct in a category by reversing all the arrows in the definition of product. Show that coproduct for Sets is disjoint union. [forme: This definition is here because of the next exercise.] coproduct 18 magicdiagram 19 monodef 20 epidef

24 P. Exercise. Suppose C A, B are two ring morphisms, so in particular A and B are C-modules. Define a ring structure A C B with multiplication given by (a 1 b 1 )(a 2 b 2 ) = (a 1 a 2 ) (b 1 b 2 ). There is a natural morphism A A C B given by a (a, 1). (This is not necessarily an inclusion, see Exercise 3.3.A.) Similarly, there is a natural morphism B A C B. Show that this gives a coproduct on rings, i.e. that A C B B satisfies the universal property of coproduct. A C [forme: Not too hard.] Q. Important Exercise for later. We continue the notation of the previous exercise. Let I be an ideal of A. Let I e be the extension of I to A C B. (These are the elements j i j b j where i j I, b j B.) Show that there is a natural isomorphism (A/I) C B = (A C B)/I e. (Hint: consider I A A/I 0, and use the right-exactness of C B.) 22 Hence the natural morphism B B C (A/I) is a surjection. As an application, we can compute tensor products of finitely generated k algebras over k. For example, we have a canonical isomorphism k[x 1, x 2 ]/(x 2 1 x 2) k k[y 1, y 2 ]/(y y3 2 ) = k[x 1, x 2, y 1, y 2 ]/(x 2 1 x 2, y y3 2 ). 3.4 Limits and colimits Limits and colimits provide two important examples defined by universal properties. They generalize a number of familiar constructions. I ll give the definition first, and then show you why it is familiar. [forme: Ben s analogy: kernel/cokernel, 23 product/coproduct, limit/colimit.] index category index cat Limits. We say that a category is an small category if the objects form a set. (This is a technical condition intended only for experts.) Suppose I is any small category, and C is any category. Then a functor F : I C (i.e. with an object A i 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 an index category. Our index categories will all be partially ordered sets (Example 3.2.8), in which in particular there is at most one morphism between any two objects. (However, other examples are sometimes useful.) [forme: e.g. equalizer] For example, if is the category 21 e:ringcoproduct 22 tensorclosed 23 d:limit

25 25 and A is a category, then a functor A is precisely the data of a commuting square in A. Then the limit is an object lim A i of C along with morphisms f i : lim A i such I I that if m : i j is a morphism in I, then lim A i I f j inverse limit, limit, proj lim f i A i F (m) commutes, and this object and maps to each A i is universal (final) respect to this property. [forme: Fix wording.] (The limit is sometimes called the inverse limit or projective limit.) By the usual universal property argument, if the limit exists, it is unique up to unique isomorphism Examples: products. For example, if I is the partially ordered set we obtain the fibered product. If I is A j we obtain the product. If I has an initial object e, then A e is the limit, and in particular the limit always exists. If I is a set (i.e. the only morphisms are the identity maps), then the limit is called the product of the A i, and is denoted i A i. The special case where I has two elements is the example of the previous paragraph Example: the p-adics. The p-adic numbers, Z p, are often described informally (and somewhat unnaturally) as being of the form Z p =?+?p+?p 2 +?p 3 +. They are an example of a limit in the category of rings: Z p Z/p 3 Z/p 2 Z/p Limits do not always exist. For example, there is no limit of Z/p 3 Z/p 2 Z/p 0 in the category of finite rings. However, you can often easily check that limits exist if the elements of your category can be described as sets with additional structure, and arbitrary products exist (respecting the set structure). 3.4.A. Exercise. Show that in the category Sets, {(a i ) i I A i : F (m)(a i ) = a j for all [m : i j] Mor(I)}, i along with the projection maps to each A i, is the limit lim A i. I

26 26 This clearly also works in the category Mod A of A-modules, and its specializations such as Vec k and Ab. From this point of view, 2 + 3p + 2p 2 + Z p can be understood as the sequence (2, 2 + 3p, 2 + 3p + 2p 2,... ). direct limit sum Colimits. 24 More immediately relevant for us will be the dual of the notion of limit (or inverse limit). We just flip all the arrows in that definition, and get the notion of a colimit. Again, if it exists, it is unique up to unique isomorphism. (The colimit is sometimes called the direct limit or injective limit.) A limit maps to all the objects in the big commutative diagram indexed by I. A colimit has a map from all the objects. Even though we have just flipped the arrows, somehow colimits behave quite differently from limits Example. The ring 5 Z of rational 25 numbers whose denominators are powers of 5 is a colimit lim 5 i Z. More precisely, 5 Z is the colimit of Z 5 1 Z 5 2 Z The colimit over an index set I is called the coproduct, denoted i A i, and is the dual notion to the product B. Exercise. (a) Interpret the statement Q = lim nz. (b) Interpret the union of some subsets of a given set as a colimit. (Dually, the intersection can be interpreted as a limit.) Colimits always exist in the category of sets: 3.4.C. Exercise. Consider the set {(i I, a i A i )} modulo the equivalence generated by: if m : i j is an arrow in I, then (i, a i ) (j, F (m)(a i )). Show that this set, along with the obvious maps from each A i, is the colimit. Thus in Example 3.4.5, each element of the direct limit is an element of something upstairs, but you can t say in advance what it is an element of. For example, 17/125 is an element of the 5 3 Z (or 5 4 Z, or later ones), but not 5 2 Z Example: inverse limits of A-modules. A variant of this construction works in a number of categories that can be interpreted as sets with additional structure (such as abelian groups, A-modules, groups, etc.). While in the case of sets, the direct limit is a quotient object of the direct sum (= disjoint union) of the A i, in the case of A-modules (for example), the direct limit is a quotient object of the direct sum. Thus the direct limit is A i modulo a j F (m)(a i ) for every m : i j in I D. Exercise. Verify that the A-module described above is indeed the colimit Summary. One useful thing to informally keep in mind is the following. In a category where the objects are set-like, an element of a colimit can be thought of ( has a representative that is ) an element of a single object in the diagram. And an element of a limit can be thought of as an element in each object in the 24 directlimit 25 five 26 alsoused, used in definition of a stalk

27 27 diagram, that are compatible. Swap the order. Even though the definitions of limit and colimit are the same, just with arrows reversed, these interpretations are quite different. [forme: I should check out [AM, p. 32, Exercise 14].] 3.5 Adjoints [forme: Say why we care?] Here is another example of a construction closely related to universal properties. 27 Just as a universal property essentially (up to unique isomorphism) determines an object in a category (assuming it exists), adjoints essentially determine a functor (again, assuming it exists). [forme: Weibel Defn 2.3.9] Two covariant functors F : A B and G : B A are adjoint if there is a natural bijection for all A A and B B (4) τ AB : Mor B (F (A), B) Mor A (A, G(B)). We say that (F, G) form an adjoint pair, and that F is left-adjoint to G (and G is right-adjoint to F ). By natural we mean the following. [forme: Make sure to comment on the word natural the first time I use it.] For all f : A A in A, we require 28 adjoint functors (5) Mor B (F (A ), B) F f Mor B (F (A), B) τ A B Mor A (A, G(B)) f τ AB Mor A (A, G(B)) to commute, and for all g : B B in B we want a similar commutative diagram to commute. (Here f is the map induced by f : A A, and F f is the map induced by F f : L(A) L(A ).) 3.5.A. Exercise. Write down what this diagram should be. (Hint: do it by extending diagram (5) above.) [forme: We could figure out what this should mean if the functors were both contravariant. I haven t tried to see if this could make sense. Remark: unique up to unique isomorphism, by universal properties.] 3.5.B. Exercise. Show that the map τ AB (4) is given as follows. For each A there is a map A GF A so that for any g : F (a) B, the corresponding f : A G(B) is given by the composition A GF A Gg GB. Similarly, there is a map B F GB for each B so that for any f : A G(B), the corresponding map g : F (A) B is given by the composition F (A) F f F (G(B)) B. 27 adjoint 28 adjointdiagram

28 28 [forexperts: Is there an easy way to check adjointness using these two maps? Here is what I ve figured out. In order to get naturality in the first term, we need the commutativity of A A GF A GF A The naturality in the second term is automatic. I think in order to make these constructions opposite of each other, we need to have that (Gt) B t GB = id GB and t F! F t A = id F A. I think it will suffice to have Gt B and t GB to be inverse isomorphisms, and simlarly for F and A. This is not necessary. ] Here is an example. 3.5.C. Exercise. Suppose M, N, and P are A-modules. Describe a natural bijection Mor A (M A N, P ) = Mor A (M, Mor A (N, P )). (Hint: try to use the universal property.) If you want, you could check that A N and Mor A (N, ) are adjoint functors. (Checking adjointness is never any fun!) [forme: We may later see why this implies left-exactness of tensor product; refer back to that, and forward.] [forme: A good example is Frobenius reciprocity, because it is so useful!. Mor G (V, W G ) = Mor H (V H, W ), and Mor G (W G, V ) = Mor H (W, V H ). H and H are both left and right exact.] Example: groupification. 29 Here is another motivating example: getting an abelian group from an abelian semigroup. An abelian semigroup is just like a group, except you don t require an inverse. One example is the non-negative integers 0, 1, 2,... under addition. Another is the positive integers under multiplication 1, 2,.... From an abelian semigroup, you can create an abelian group, and this could be called groupification. Here is a formalization of that notion. If S is a semigroup, then its groupification is a map of semigroups π : S G such that G is a group, and any other map of semigroups from S to a group G factors uniquely through G. S G π 3.5.D. Exercise. Define groupification H from the category of abelian semigroups to the category of abelian groups. (One possibility of a construction: given an abelian semigroup S, the elements of its groupification H(S) are (a, b), which you may think of as a b, with the equivalence that (a, b) (c, d) if a + d = b + c. Describe addition in this group, and show that it satisfies the properties of an abelian group. Describe the semigroup map S H(S).) Let F be the forgetful morphism from the category of abelian groups Ab to the category of abelian semigroups. Show that H is left-adjoint to F. (Here is the general idea for experts: We have a full subcategory of a category. We want to project from the category to the subcategory. We have! G 29 groupification

29 29 Mor category (S, H) = Mor subcategory (G, H) automatically; thus we are describing the left adjoint to the forgetful functor. How the argument worked: we constructed something which was in the small category, which automatically satisfies the universal property.) 3.5.E. Exercise. Show that if a semigroup is already a group then groupification is the identity morphism, by the universal property. 3.5.F. Exercise. The purpose of this exercise is to give you some practice with adjoints of forgetful functors, the means by which we get groups from semigroups, and sheaves from presheaves. Suppose A is a ring, and S is a multiplicative subset. Then S 1 A-modules are a fully faithful subcategory of the category of A-modules (meaning: the objects of the first category are a subset of the objects of the second; and the morphisms between any two objects of the second that are secretly objects of the first are just the morphisms from the first). Then M S 1 M satisfies a universal property. Figure out what the universal property is, and check that it holds. In other words, describe the universal property enjoyed by M S 1 M, and prove that it holds. (Here is the larger story. Let S 1 A-Mod be the category of S 1 A-modules, and A-Mod be the category of A-modules. Every S 1 A-module is an A-module, and this is an injective map, so we have a (covariant) forgetful functor F : S 1 A-Mod A-Mod. In fact this is a fully faithful functor: it is injective on objects, and the morphisms between any two S 1 A-modules as A-modules are just the same when they are considered as S 1 A-modules. Then there is a functor G : A-Mod S 1 A-Mod, which might reasonably be called localization with respect to S, which is left-adjoint to the forgetful functor. Translation: If M is an A-module, and N is an S 1 A-module, then Mor(GM, N) (morphisms as S 1 A-modules, which is incidentally the same as morphisms as A-modules) are in natural bijection with Mor(M, F N) (morphisms as A-modules).) Here is a table of adjoints that will come up for us. situation category category left-adjoint right-adjoint A B F : A B G : B A A-modules A N Hom A (N, ) (pre)sheaves on a presheaves sheaves on X sheafification forgetful topological space X on X (semi)groups semigroups groups groupification forgetful sheaves, f : X Y sheaves on Y sheaves on X f 1 f quasicoherent sheaves, quasicoherent quasicoherent f f f : X Y sheaves on Y sheaves on X Useful comment for experts. Here is one last useful comment intended only for people who have seen adjoints before. If (F, G) is an adjoint pair of functors, then F commutes with colimits, and G commutes with limits. We ll prove this in when we ll extend this result. 30 Also, limits commute with limits and colimits commute with colimits. (This isn t hard once you figure out what it means.) [forme: Main idea: product index category I I.] 30 preadjointfun

30 Kernels, cokernels, and exact sequences: A brief introduction to abelian categories Since learning linear algebra, the reader has been familiar with the notions and behaviors of kernels, cokernels, etc. Later in your life you saw them in the category of abelian groups, and later still in the category of A-modules. Each of these notions generalizes the previous one. The notion of abelian category formalizes kernels etc. [forme: THINK THROUGH PRECISELY WHAT I WANT TO HAVE HERE. HOW LITTLE CAN I INCLUDE? Why do I need to discuss abelian categories? We want to define notions which behave like modules over a ring. Our only examples are: Ab, Mod A, sheaves of abelian groups, O X -modules, qcoh/ft/coherent sheaves. Motivate definitions of: 0-object, monomorphism (subobject) and epimorphism, kernel and cokernel, image (not coimage). The reader should understand that they behave the way he expects. This could be done by defining a good subcategory C of an abelian category C (contains 0; given any morphism f in the subcategory, its kernel and cokernel and image [in C] are actually in C ; then they are actually cokernels and images in the small subcategory; etc.), and in our cases they are all good subcategories of some Mod A. derived functors Definitions of abelian catogories from various sources (2 from wikipedia, [KS], Gelfand-Manin, [W], Lang) are in the primordial bit below. ] [forme: Weibel appendix. According to [MilneLET] p. 2, abelian categories were defined by Buchsbaum (and he extended the Cartan-Eilenberg theory of derived functors to such categories). The name is due to Grothendieck.] 31 We now briefly introduce a few notions about abelian categories. We will soon define some new categories (certain sheaves) that will have familiar-looking behavior, reminiscent of that of modules over a ring. The notions of kernels, cokernels, images, and more will make sense, and they will behave the way we expect from our experience with modules. This can be made precise through the notion of an abelian category. We will see enough to motivate the definitions that we will see in general: monomorphism (and subobject), epimorphism, kernel, cokernel, and image. But we will avoid having to show that they behave the way we expect in a general abelian category because the examples we will see will be directly interpretable in terms of modules over rings. Abelian categories are the right general setting in which one can do homological algebra, in which notions of kernel, cokernel, and so on are used, and one can work with complexes and exact sequences. 32 Two key examples of an abelian category are the category Ab of abelian groups, and the category Mod A of A-modules. The first is a special case of the second (just take A = Z). As we give the definitions, you should verify that Mod A is an abelian zero object category. additive category We first define the notion of additive category. We will use it only as a stepping stone to the notion of an abelian category. 31 s:abcat 32 aac