Algebraic Geometry J.S. Milne Version 5.22 January 13, 2012 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 (v5.22)}, year={2012}, note={Available at www.jmilne.org/math/}, pages={260} } 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. v5.22 (January 13, 2012). Minor fixes; 260 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, 2012 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 16; Polynomial rings 18; Integrality 19; Direct limits (summary) 21; Rings of fractions 22; Tensor Prod- ucts 25; Categories and functors 28; Algorithms for polynomials 31; Exercises 37 2 Algebraic Sets 39 Definition of an algebraic set 39; The Hilbert basis theorem 40; The Zariski topology 41; The Hilbert Nullstellensatz 42; The correspondence between algebraic sets and ideals 43; Finding the radical of an ideal 46; The Zariski topology on an algebraic set 46; The coor- dinate ring of an algebraic set 47; Irreducible algebraic sets 48; Dimension 50; Exercises 52 3 Affine Algebraic Varieties 55 Ringed spaces 55; The ringed space structure on an algebraic set 57; Morphisms of ringed spaces 60; Affine algebraic varieties 60; The category of affine algebraic varieties 61; Explicit description of morphisms of affine varieties 63; Subvarieties 65; Properties of the regular map defined by specm.˛/ 66; Affine space without coordinates 67; Exercises 68 4 Algebraic Varieties 71 Algebraic prevarieties 71; Regular maps 72; Algebraic varieties 73; Maps from varieties to affine varieties 74; Subvarieties 75; Prevarieties obtained by patching 76; Products of vari- eties 76; The separation axiom revisited 81; Fibred products 84; Dimension 85; Birational equivalence 86; Dominant maps 87; Algebraic varieties as a functors 87; Exercises 89 5 Local Study 91 Tangent spaces to plane curves 91; Tangent cones to plane curves 93; The local ring at a m point on a curve 93; Tangent spaces of subvarieties of A 94; The differential of a regular map 96; Etale maps 97; Intrinsic definition of the tangent space 99; Nonsingular points 102; Nonsingularity and regularity 103; Nonsingularity and normality 104; Etale neigh- bourhoods 105; Smooth maps 107; Dual numbers and derivations 108; Tangent cones 111; Exercises 113 6 Projective Varieties 115 n n n Algebraic subsets of P 115; The Zariski topology on P 118; Closed subsets of A and n n P 119; The hyperplane at infinity 119; P is an algebraic variety 120; The homogeneous 3 n coordinate ring of a subvariety of P 122; Regular functions on a projective variety 123; Morphisms from projective varieties 124; Examples of regular maps of projective vari- eties 125; Projective space without coordinates 130; Grassmann varieties 130; Bezout’s theorem 134; Hilbert polynomials (sketch) 135; Exercises 136 7 Complete varieties 137 Definition and basic properties 137; Projective varieties are complete 139; Elimination theory 140; The rigidity theorem 142; Theorems of Chow 143; Nagata’s Embedding The- orem 144; Exercises 144 8 Finite Maps 145 Definition and basic properties 145; Noether Normalization Theorem 149; Zariski’s main theorem 150; The base change of a finite map 152; Proper maps 152; Exercises 154 9 Dimension Theory 155 Affine varieties 155; Projective varieties 162 10 Regular Maps and Their Fibres 165 Constructible sets 165; Orbits of group actions 168; The fibres of morphisms 170; The fibres of finite maps 172; Flat maps 173; Lines on surfaces 174; Stein factorization 180; Exercises 180 11 Algebraic spaces; geometry over an arbitrary field 181 Preliminaries 181; Affine algebraic spaces 185; Affine algebraic varieties. 186; Algebraic spaces; algebraic varieties. 187; Local study 192; Projective varieties. 193; Complete varieties. 194; Normal varieties; Finite maps. 194; Dimension theory 194; Regular maps and their fibres 194; Algebraic groups 194; Exercises 194 12 Divisors and Intersection Theory 195 Divisors 195; Intersection theory. 196; Exercises 201 13 Coherent Sheaves; Invertible Sheaves 203 Coherent sheaves 203; Invertible sheaves. 205; Invertible sheaves and divisors. 206; Direct images and inverse images of coherent sheaves. 208; Principal bundles 209 14 Differentials (Outline) 211 15 Algebraic Varieties over the Complex Numbers 215 16 Descent Theory 219 Models 219; Fixed fields 219; Descending subspaces of vector spaces 220; Descending subvarieties and morphisms 222; Galois descent of vector spaces 223; Descent data 224; Galois descent of varieties 228; Weil restriction 229; Generic fibres and specialization 229; Rigid descent 230; Weil’s descent theorems 232; Restatement in terms of group actions 234; Faithfully flat descent 236 17 Lefschetz Pencils (Outline) 239 Definition 239 18 Algebraic Schemes and Algebraic Spaces 243 4 A Solutions to the exercises 245 B Annotated Bibliography 253 Index 257 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, Tony Feng, B.J. Franklin, Sergei Gelfand, Daniel Gerig, Darij Grinberg, Lucio Guerberoff, Guido Helmers, Florian Herzig, Christian Hirsch, Cheuk-Man Hwang, Jasper Loy Jiabao, Lars Kindler, John Miller, Joaquin Rodrigues, Sean Rostami, David Rufino, Hossein Sabzrou, Jyoti Prakash Saha, Tom Savage, Nguyen Quoc Thang, Bhupendra Nath Tiwari, Soli Vishkaut- san, 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.