Algebraic Geometry J.S. Milne Version 5.21 March 31, 2011 These notes are an introduction to the theory of algebraic varieties. In contrast to most such accounts they study abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory. BibTeX information @misc{milneAG, author={Milne, James S.}, title={Algebraic Geometry (v5.21)}, year={2011}, note={Available at www.jmilne.org/math/}, pages={258} } v2.01 (August 24, 1996). First version on the web. v3.01 (June 13, 1998). v4.00 (October 30, 2003). Fixed errors; many minor revisions; added exercises; added two sections/chapters; 206 pages. v5.00 (February 20, 2005). Heavily revised; most numbering changed; 227 pages. v5.10 (March 19, 2008). Minor fixes; TEXstyle changed, so page numbers changed; 241 pages. v5.20 (September 14, 2009). Minor corrections; revised Chapters 1, 11, 16; 245 pages. v5.21 (March 31, 2011). Minor changes; changed TEXstyle; 258 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of Lake Sylvan, New Zealand. Copyright c 1996, 1998, 2003, 2005, 2008, 2009, 2011 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Contents 3 Notations 6; Prerequisites 6; References 6; Acknowledgements 6 1 Preliminaries 11 Rings and algebras 11; Ideals 11; Noetherian rings 13; Unique factorization 15; Polynomial rings 17; Integrality 18; Direct limits (summary) 21; Rings of fractions 22; Tensor Prod- ucts 25; Categories and functors 28; Algorithms for polynomials 30; Exercises 36 2 Algebraic Sets 37 Definition of an algebraic set 37; The Hilbert basis theorem 38; The Zariski topology 39; The Hilbert Nullstellensatz 40; The correspondence between algebraic sets and ideals 41; Finding the radical of an ideal 44; The Zariski topology on an algebraic set 44; The coor- dinate ring of an algebraic set 45; Irreducible algebraic sets 46; Dimension 48; Exercises 50 3 Affine Algebraic Varieties 53 Ringed spaces 53; The ringed space structure on an algebraic set 55; Morphisms of ringed spaces 58; Affine algebraic varieties 58; The category of affine algebraic varieties 59; Explicit description of morphisms of affine varieties 61; Subvarieties 63; Properties of the regular map defined by specm.˛/ 64; Affine space without coordinates 65; Exercises 66 4 Algebraic Varieties 69 Algebraic prevarieties 69; Regular maps 70; Algebraic varieties 71; Maps from varieties to affine varieties 72; Subvarieties 73; Prevarieties obtained by patching 74; Products of vari- eties 74; The separation axiom revisited 79; Fibred products 82; Dimension 83; Birational equivalence 84; Dominant maps 85; Algebraic varieties as a functors 85; Exercises 87 5 Local Study 89 Tangent spaces to plane curves 89; Tangent cones to plane curves 91; The local ring at a m point on a curve 91; Tangent spaces of subvarieties of A 92; The differential of a regular map 94; Etale maps 95; Intrinsic definition of the tangent space 97; Nonsingular points 100; Nonsingularity and regularity 101; Nonsingularity and normality 102; Etale neigh- bourhoods 103; Smooth maps 105; Dual numbers and derivations 106; Tangent cones 109; Exercises 111 6 Projective Varieties 113 n n n Algebraic subsets of P 113; The Zariski topology on P 116; Closed subsets of A and n n P 117; The hyperplane at infinity 117; P is an algebraic variety 118; The homogeneous 3 n coordinate ring of a subvariety of P 120; Regular functions on a projective variety 121; Morphisms from projective varieties 122; Examples of regular maps of projective vari- eties 123; Projective space without coordinates 128; Grassmann varieties 128; Bezout’s theorem 132; Hilbert polynomials (sketch) 133; Exercises 134 7 Complete varieties 135 Definition and basic properties 135; Projective varieties are complete 137; Elimination theory 138; The rigidity theorem 140; Theorems of Chow 141; Nagata’s Embedding The- orem 142; Exercises 142 8 Finite Maps 143 Definition and basic properties 143; Noether Normalization Theorem 147; Zariski’s main theorem 148; The base change of a finite map 150; Proper maps 150; Exercises 152 9 Dimension Theory 153 Affine varieties 153; Projective varieties 160 10 Regular Maps and Their Fibres 163 Constructible sets 163; Orbits of group actions 166; The fibres of morphisms 168; The fibres of finite maps 170; Flat maps 171; Lines on surfaces 172; Stein factorization 178; Exercises 178 11 Algebraic spaces; geometry over an arbitrary field 179 Preliminaries 179; Affine algebraic spaces 183; Affine algebraic varieties. 184; Algebraic spaces; algebraic varieties. 185; Local study 190; Projective varieties. 191; Complete varieties. 192; Normal varieties; Finite maps. 192; Dimension theory 192; Regular maps and their fibres 192; Algebraic groups 192; Exercises 192 12 Divisors and Intersection Theory 193 Divisors 193; Intersection theory. 194; Exercises 199 13 Coherent Sheaves; Invertible Sheaves 201 Coherent sheaves 201; Invertible sheaves. 203; Invertible sheaves and divisors. 204; Direct images and inverse images of coherent sheaves. 206; Principal bundles 207 14 Differentials (Outline) 209 15 Algebraic Varieties over the Complex Numbers 213 16 Descent Theory 217 Models 217; Fixed fields 217; Descending subspaces of vector spaces 218; Descending subvarieties and morphisms 220; Galois descent of vector spaces 221; Descent data 222; Galois descent of varieties 226; Weil restriction 227; Generic fibres and specialization 227; Rigid descent 228; Weil’s descent theorems 230; Restatement in terms of group actions 232; Faithfully flat descent 234 17 Lefschetz Pencils (Outline) 237 Definition 237 18 Algebraic Schemes and Algebraic Spaces 241 4 A Solutions to the exercises 243 B Annotated Bibliography 251 Index 255 QUESTION: If we try to explain to a layman what algebraic geometry is, it seems to me that the title of the old book of Enriques is still adequate: Geometrical Theory of Equations.... GROTHENDIECK: Yes! but your “layman” should know what a system of algebraic equa- tions is. This would cost years of study to Plato. 5 Notations We use the standard (Bourbaki) notations: N 0;1;2;::: , Z ring of integers, R field D f g D D of real numbers, C field of complex numbers, Fp Z=pZ field of p elements, p a D D D prime number. Given an equivalence relation, Œ denotes the equivalence class containing . A family of elements of a set A indexed by a second set I , denoted .ai /i I , is a function 2 i ai I A. 7! W ! A field k is said to be separably closed if it has no finite separable extensions of degree > 1. We use ksep and kal to denote separable and algebraic closures of k respectively. All rings will be commutative with 1, and homomorphisms of rings are required to map 1 to 1. For a ring A, A is the group of units in A: A a A there exists a b A such that ab 1 : D f 2 j 2 D g We use Gothic (fraktur) letters for ideals: abcmnpqABCMNPQ abcmnpqABCMNPQ X def YX is defined to be Y , or equals Y by definition; D X YX is a subset of Y (not necessarily proper, i.e., X may equal Y ); X YX and Y are isomorphic; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism). ' Prerequisites The reader is assumed to be familiar with the basic objects of algebra, namely, rings, mod- ules, fields, and so on, and with transcendental extensions of fields (FT, Chapter 8). References Atiyah and MacDonald 1969: Introduction to Commutative Algebra, Addison-Wesley. Cox et al. 1992: Varieties, and Algorithms, Springer. FT: Milne, J.S., Fields and Galois Theory, v4.22, 2011. CA: Milne, J.S., Commutative Algebra, v2.22, 2011. Hartshorne 1977: Algebraic Geometry, Springer. Mumford 1999: The Red Book of Varieties and Schemes, Springer. Shafarevich 1994: Basic Algebraic Geometry, Springer. For other references, see the annotated bibliography at the end. Acknowledgements I thank the following for providing corrections and comments on earlier versions of these notes: Sandeep Chellapilla, Rankeya Datta, Umesh V. Dubey, Shalom Feigelstock, B.J. Franklin, Sergei Gelfand, Daniel Gerig, Darij Grinberg, Lucio Guerberoff, Guido Helmers, Christian Hirsch, Cheuk-Man Hwang, Jasper Loy Jiabao, Lars Kindler, Joaquin Rodrigues, Sean Rostami, David Rufino, Hossein Sabzrou, Jyoti Prakash Saha, Tom Savage, Nguyen Quoc Thang, Soli Vishkautsan, Dennis Bouke Westra, and others. 6 Introduction There is almost nothing left to discover in geometry. Descartes, March 26, 1619 Just as the starting point of linear algebra is the study of the solutions of systems of linear equations, n X aij Xj bi ; i 1;:::;m; (1) D D j 1 D the starting point for algebraic geometry is the study of the solutions of systems of polyno- mial equations, fi .X1;:::;Xn/ 0; i 1;:::;m; fi kŒX1;:::;Xn: D D 2 Note immediately one difference between linear equations and polynomial equations: the- orems for linear equations don’t depend on which field k you are working over,1 but those for polynomial equations depend on whether or not k is algebraically closed and (to a lesser extent) whether k has characteristic zero. A better description of algebraic geometry is that it is the study of polynomial functions and the spaces on which they are defined (algebraic varieties), just as topology is the study of continuous functions and the spaces on which they are defined (topological spaces), differential topology the study of infinitely differentiable functions and the spaces on which they are defined (differentiable manifolds), and so on: algebraic geometry regular (polynomial) functions algebraic varieties topology continuous functions topological spaces differential topology differentiable functions differentiable manifolds complex analysis analytic (power series) functions complex manifolds. The approach adopted in this course makes plain the similarities between these different areas of mathematics.