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My Way to Algebraic Geometry Marco Lo Giudice

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My Way to Algebraic Geometry Marco Lo Giudice My way to Algebraic Geometry Varieties and Schemes from the point of view of a PhD student Marco Lo Giudice July 29, 2005 Introduction I began writing these notes during the last weeks of year 2002, collecting some work I had already written in my studentship. This is intended to be my personal approach to Algebraic Geometry, in particular to Hartshorne (1977), and originally it was meant to be a collection of solved exercises. But as my understanding grew stronger I began to add comments of any sort, so as to reach the level of exposition you will find. I am very pleased to hear that there is somebody out there who thinks those comments are interesting, and I really wish them to be of any help. To those students approaching the subject, wishing to learn something about the abstract concepts that arise in Algebraic Geometry and finding themselves lost into a sea of difficult to grasp material, I have to say that unfortunately there is no easy way, they will have to work a lot and pretty much alone. A word of caution: don’t take these notes for granted! I am a lone student, I’m writing as I’m learning the subject and I’m not a good nor an experienced mathematician. Chances are that some of my proofs are wrong, and be ad- vised that also some of the statements are mine and may be wrong. For this reason when I state a result I always try to give a reference for it, in particular most of the times I will deal with statements that are actually exercises in some book and the reader may want to check their proofs carefully if not to try to work them out on his own. We wish to learn the formalism of schemes, therefore throughout these notes our main reference will be Hartshorne (1977) which is the most cele- brated book where to learn the machinery. The reader will be assumed famil- iar with the notations and definitions in there, however there are quite a few other books one can look at, such as Eisenbud and Harris (2000), Shafarevich (1994b) or Mumford (1999). The original work by Grothendieck is also still a good reference, although very abstract, you can obtain all the Elem´entsde´ g´eom´etriealg´ebrique from <http://www.numdam.org/> (rigorously in French), and many other articles, included the S´eminaire de g´eom´etriealg´ebrique from <http://www.grothendieck-circle.org/>. Unfortunately the concepts we are go- ing to introduce cannot make any sense if you haven’t at least a basic knowl- 3 Marco Lo Giudice My Way to Algebraic Geometry edge of Commutative Algebra. I used to study this subject in Atiyah and Macdonald (1969) and this is the main reference for these notes, but you may want to read Eisenbud (1995) or the classic textbooks by Matsumura (1989), Zariski and Samuel (1958, 1960), or Bourbaki (1998). The word “ring” will always mean “commutative ring with unit” and a morphism of rings will be always assumed to respect this structure, that is will send the unit element to the unit element. As Eisenbud and Harris say in the introduction to their book, “the basic definitions of scheme theory appear as natural and necessary ways of dealing with a range of ordinary geometric phenomena, and the constructions in the theory take on an intuitive geometric content which makes them much easier to learn and work with.” For this reason, in line with the authors, I think there is no point in learning all the machinery first, and then work out the geometry as a consequence of it. But I still think that any student has to think a little through the basic definitions before going straight to geometry, and in this re- spect I disagree with Eisenbud and Harris. Chapter one is my attempt to com- promise, with all the basic abstract concepts introduced in a systematic way before to arrive at the very definition of algebraic variety. The reader famil- iar with the second chapter of Hartshorne (1977) will surely recognise where the discussion is going, and will find some useful comments helping him to familiarise with abstract definitions otherwise totally detached by the geomet- ric content they are supposed to carry over. In fact, this approach is essentially different from any other I am aware of. Classic references as Mumford (1999), or Shafarevich (1994a,b), develop the theory quite extensively before getting into schemes, while more recent accounts such as Liu (2002) or Ueno (1999, 2001, 2003) concentrate more on the abstract machinery leaving the geometry slightly aside. Finally in Iitaka (1982) commutative algebra is developed to- gether with algebraic geometry, which makes it the most self-contained book. Once again let me say that these notes are not to be trusted that much, but if you are going to read them for whatever reason then let me ask you something in return. If you find any mistake, misprint or anything wrong just let me know, you can contact me by e-mail on [email protected] and you can find out something more about me on my homepage at the University of Bath <http://www.maths.bath.ac.uk/»mapmlg/>, any comments or suggestions are also more than welcome. Marco Lo Giudice University Of Bath November 2004 4 Contents Introduction 3 1 Affine Varieties 7 1.1 The Spectrum of a Ring . 7 1.2 Topological Properties . 14 1.3 The Structure Sheaf . 21 1.4 Functoriality . 26 2 General Properties of Schemes 35 2.1 Schemes . 35 2.2 The Projective Spectrum of a Graded Ring . 43 2.3 Gluing Constructions . 50 2.4 Reduced, Integral and Noetherian Schemes . 61 3 Attributes of Morphisms 67 3.1 Schemes of Finite Type . 67 3.2 Product of Schemes . 73 3.3 Separated and Proper Morphisms . 81 3.4 Finite Morphisms . 89 4 Basic Algebraic Geometry 95 4.1 Algebraic Varieties . 95 4.2 Hilbert’s Nullstellensatz . 101 4.3 Rational Maps . 103 4.4 Dimension . 107 5 Sheaves over Schemes 115 5.1 Sheaves of Modules . 115 5.2 Locally Free Sheaves and the Picard Group . 120 5.3 Coherent Sheaves . 124 5.4 Vector Bundles on Schemes . 134 5 Marco Lo Giudice My Way to Algebraic Geometry 6 Projective Geometry 137 6.1 Projective Schemes . 137 6.2 Projective Varieties . 143 6.3 Very Ample Invertible Sheaves . 149 6.4 Global Proj and Blow-ups . 157 A Algebra 167 A.1 Rings and Modules of Finite Length . 167 B Sheaf Theory 177 B.1 Basic properties . 177 B.2 Functorial constructions . 181 B.3 Topological attributes . 187 Bibliography 189 6 Chapter 1 Affine Varieties Our goal in this chapter is to define affine varieties from an abstract point of view. The main effort required to the reader is getting accustomed to switch- ing frequently between algebra and geometry, for what we will do is giving geometric names to purely algebraic objects. We will follow the guidelines of xI.1 in Eisenbud and Harris (2000), but we will try to keep as close as possible to standard introductions such as Chapter II in Reid (1988). In fact the author has been highly inspired by Milne (2005), where a wonderfully clear account of the basic material is accompanied with a precise description of some of the abstract concepts. It must be said that originally this was simply a collection of solved exer- cises from Atiyah and Macdonald (1969), therefore some background knowl- edge of commutative algebra is required throughout (nothing more than a standard undergraduate course, see for example Reid, 1995). The reader will also be assumed familiar with sheaf theory, at least with what Hartshorne (1977) reads about it, but occasionally we may refer to Tennison (1975) for more advanced material. 1.1 The Spectrum of a Ring 1.1.1 Zeros of an Ideal The main object of study in Algebraic Geometry are systems of algebraic equations and their sets of solutions. Let k be a field and let R be the polynomial ring k[x1, ... , xn], a system of algebraic equations S is a collection of equations 8 <>F1(x1, ... , xn) = 0 S = ... :> Fr(x1, ... , xn) = 0 7 Marco Lo Giudice 1. Affine Varieties where Fi is a polynomial in R for every i, and a solution of S in k is an n- n tuple (l1, ... , ln) 2 k satisfying every equation. If we denote by a the ideal generated by F1, ... , Fr, then solutions of S are in one-to-one correspondence with n-tuples in kn annihilating any polynomial in a, in particular any other set of generators for a will define a system of algebraic equations with exactly the same set of solutions as S. We encode this information in our language by saying that (l1, ... , ln) is a zero of a. Proposition. Let k be any field. There is a natural injective correspondence from n-tuples in kn to maximal ideals of R, given by l = (l1, ... , ln) 7! ml = (x1 ¡ l1, ... , xn ¡ ln) If a ⊆ R is any ideal, then l is a zero of a if and only if the maximal ideal ml contains a. Proof. The ideal ml is maximal because it is the kernel of the surjective ho- momorphism j : k[x1, ... , xn] ! k defined by evaluation in l. Indeed it is enough to show that the kernel is contained in ml: we can apply the division algorithm to any polynomial F 2 ker j, to obtain F = G + a where G 2 ml and a is a constant.
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