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Algebraic Geometry Algebraic Geometry J.S. Milne Version 5.20 September 14, 2009 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 (v5.20)}, year={2009}, note={Available at www.jmilne.org/math/}, pages={239+vi} } 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; TeX style changed, so page numbers changed; 241 pages. v5.20 (September 14, 2009). Minor corrections; revised Chapters 1, 11, 16; 245 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 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Contents iii Notations vi; Prerequisites vi; References vi; Acknowledgements vi Introduction 1 1 Preliminaries 4 Rings and algebras 4; Ideals 4; Noetherian rings 6; Unique factorization 8; Polynomial rings 10; Integrality 11; Direct limits (summary) 14; Rings of fractions 15; Tensor Prod- ucts 18; Categories and functors 21; Algorithms for polynomials 23; Exercises 29 2 Algebraic Sets 30 Definition of an algebraic set 30; The Hilbert basis theorem 31; The Zariski topology 32; The Hilbert Nullstellensatz 33; The correspondence between algebraic sets and ideals 34; Finding the radical of an ideal 37; The Zariski topology on an algebraic set 37; The coor- dinate ring of an algebraic set 38; Irreducible algebraic sets 39; Dimension 41; Exercises 44 3 Affine Algebraic Varieties 45 Ringed spaces 45; The ringed space structure on an algebraic set 47; Morphisms of ringed spaces 50; Affine algebraic varieties 50; The category of affine algebraic varieties 51; Explicit description of morphisms of affine varieties 53; Subvarieties 55; Properties of the regular map defined by specm(˛) 56; Affine space without coordinates 57; Exercises 58 4 Algebraic Varieties 60 Algebraic prevarieties 60; Regular maps 61; Algebraic varieties 62; Maps from varieties to affine varieties 63; Subvarieties 64; Prevarieties obtained by patching 65; Products of vari- eties 66; The separation axiom revisited 70; Fibred products 73; Dimension 74; Birational equivalence 75; Dominant maps 76; Algebraic varieties as a functors 76; Exercises 78 5 Local Study 80 Tangent spaces to plane curves 80; Tangent cones to plane curves 82; The local ring at a m point on a curve 82; Tangent spaces of subvarieties of A 83; The differential of a regular map 85; Etale maps 86; Intrinsic definition of the tangent space 88; Nonsingular points 91; Nonsingularity and regularity 92; Nonsingularity and normality 93; Etale neighbourhoods 94; Smooth maps 96; Dual numbers and derivations 97; Tangent cones 100; Exercises 102 6 Projective Varieties 103 n n n Algebraic subsets of P 103; The Zariski topology on P 106; Closed subsets of A and iii n n P 107; The hyperplane at infinity 107; P is an algebraic variety 108; The homogeneous n coordinate ring of a subvariety of P 110; Regular functions on a projective variety 111; Morphisms from projective varieties 112; Examples of regular maps of projective vari- eties 113; Projective space without coordinates 118; Grassmann varieties 118; Bezout’s theorem 122; Hilbert polynomials (sketch) 123; Exercises 124 7 Complete varieties 125 Definition and basic properties 125; Projective varieties are complete 127; Elimination theory 128; The rigidity theorem 130; Theorems of Chow 131; Nagata’s Embedding The- orem 132; Exercises 132 8 Finite Maps 133 Definition and basic properties 133; Noether Normalization Theorem 137; Zariski’s main theorem 138; The base change of a finite map 140; Proper maps 140; Exercises 141 9 Dimension Theory 143 Affine varieties 143; Projective varieties 150 10 Regular Maps and Their Fibres 152 Constructible sets 152; Orbits of group actions 155; The fibres of morphisms 157; The fibres of finite maps 159; Flat maps 160; Lines on surfaces 161; Stein factorization 167; Exercises 167 11 Algebraic spaces; geometry over an arbitrary field 168 Preliminaries 168; Affine algebraic spaces 172; Affine algebraic varieties. 173; Algebraic spaces; algebraic varieties. 174; Local study 179; Projective varieties. 180; Complete varieties. 181; Normal varieties; Finite maps. 181; Dimension theory 181; Regular maps and their fibres 181; Algebraic groups 181; Exercises 182 12 Divisors and Intersection Theory 183 Divisors 183; Intersection theory. 184; Exercises 189 13 Coherent Sheaves; Invertible Sheaves 190 Coherent sheaves 190; Invertible sheaves. 192; Invertible sheaves and divisors. 193; Direct images and inverse images of coherent sheaves. 195; Principal bundles 195 14 Differentials (Outline) 196 15 Algebraic Varieties over the Complex Numbers 198 16 Descent Theory 201 Models 201; Fixed fields 201; Descending subspaces of vector spaces 202; Descending subvarieties and morphisms 204; Galois descent of vector spaces 205; Descent data 206; Galois descent of varieties 210; Weil restriction 211; Generic fibres and specialization 212; Rigid descent 212; Weil’s descent theorems 215; Restatement in terms of group actions 216; Faithfully flat descent 218 17 Lefschetz Pencils (Outline) 222 Definition 222 iv 18 Algebraic Schemes and Algebraic Spaces 225 A Solutions to the exercises 226 B Annotated Bibliography 233 Index 236 v Notations We use the standard (Bourbaki) notations: N 0; 1; 2; : : : , Z ring of integers, R D f g D D field of real numbers, C field of complex numbers, Fp Z=pZ field of p elements, p D D D a 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, Section 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, v4.20, 2008 (www.jmilne.org/math/). CA: Milne, J.S., Commutative Algebra, v2.10. 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, Umesh V. Dubey, Shalom Feigelstock, B.J. Franklin, Sergei Gelfand, Daniel Gerig, Darij Grinberg, Lucio Guerberoff, Guido Helmers, Christian Hirsch, Jasper Loy Jiabao, Lars Kindler, Sean Rostami, David Rufino, Tom Savage, Nguyen Quoc Thang, Soli Vishkautsan, Dennis Bouke Westra, and others. vi 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. Of course, the polynomial functions form a much less rich class than the others, but by restricting our study to polynomials we are able to do calculus over any field: we simply define d X i X i 1 ai X iai X : dX D 1 For example, suppose that the system (1) has coefficients aij k and that K is a field containing k.
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