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Algebraic Geometry Algebraic Geometry J.S. Milne Version 6.01 August 23, 2015 These notes are an introduction to the theory of algebraic varieties emphasizing the simi- larities to the theory of manifolds. 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 (v6.01)}, year={2015}, note={Available at www.jmilne.org/math/}, pages={226} } 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. v5.22 (January 13, 2012). Minor fixes; 260 pages. v6.00 (August 24, 2014). Major revision; 223 pages. v6.01 (August 23, 2015). Minor fixes; 226 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The curves are a tacnode, a ramphoid cusp, and an ordinary triple point. Copyright c 1996, 1998, 2003, 2005, 2008, 2009, 2011, 2012, 2013, 2014 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Contents3 1 Preliminaries from commutative algebra 11 a. Rings and ideals, 11 ; b. Rings of fractions, 16 ; c. Unique factorization, 21 ; d. Integral dependence, 24; e. Tensor Products, 30 ; f. Transcendence bases, 33. 2 Algebraic Sets 35 a. Definition of an algebraic set, 35 ; b. The Hilbert basis theorem, 36; c. The Zariski topology, 37; d. The Hilbert Nullstellensatz, 38; e. The correspondence between algebraic sets and radical ideals, 39 ; f. Finding the radical of an ideal, 43; g. Properties of the Zariski topology, 43; h. Decomposition of an algebraic set into irreducible algebraic sets, 44 ; i. Regular functions; the coordinate ring of an algebraic set, 47; j. Regular maps, 48; k. Hypersurfaces; finite and quasi-finite maps, 48; l. Noether normalization theorem, 50 ; m. Dimension, 52 . 3 Affine Algebraic Varieties 59 a. Sheaves, 59 ; b. Ringed spaces, 60; c. The ringed space structure on an algebraic set, 61 ; d. Morphisms of ringed spaces, 64 ; e. Affine algebraic varieties, 65; f. The category of affine algebraic varieties, 66; g. Explicit description of morphisms of affine varieties, 67 ; h. Subvarieties, 70; i. Properties of the regular map Spm.˛/, 71; j. Affine space without coordinates, 72; k. Birational equivalence, 73; l. Noether Normalization Theorem, 75; m. Dimension, 75 . 4 Local Study 81 a. Tangent spaces to plane curves, 81 ; b. Tangent cones to plane curves, 83 ; c. The local m ring at a point on a curve, 85; d. Tangent spaces to algebraic subsets of A , 86 ; e. The differential of a regular map, 88; f. Tangent spaces to affine algebraic varieties, 89 ; g. Tangent cones, 93; h. Nonsingular points; the singular locus, 94 ; i. Nonsingularity and regularity, 96; j. Examples of tangent spaces, 97. 5 Algebraic Varieties 99 a. Algebraic prevarieties, 99; b. Regular maps, 100; c. Algebraic varieties, 101; d. Maps from varieties to affine varieties, 103; e. Subvarieties, 103 ; f. Prevarieties obtained by patching, 104; g. Products of varieties, 105 ; h. The separation axiom revisited, 110; i. Fibred products, 112 ; j. Dimension, 113; k. Dominant maps, 115; l. Rational maps; birational equivalence, 115; m. Local study, 116; n. Etale´ maps, 117 ; o. Etale´ neighbourhoods, 120 ; p. Smooth maps, 122 ; q. Algebraic varieties as a functors, 123 ; r. Rational and unirational varieties, 126 . 6 Projective Varieties 129 n n a. Algebraic subsets of P , 129; b. The Zariski topology on P , 133; c. Closed subsets of 3 n n n A and P , 134 ; d. The hyperplane at infinity, 135; e. P is an algebraic variety, 135; f. The homogeneous coordinate ring of a projective variety, 137; g. Regular functions on a projective variety, 138; h. Maps from projective varieties, 139; i. Some classical maps of projective varieties, 141 ; j. Maps to projective space, 146; k. Projective space without coordinates, 146; l. The functor defined by projective space, 146; m. Grassmann varieties, 147 ; n. Bezout’s theorem, 150; o. Hilbert polynomials (sketch), 151; p. Dimensions, 153; q. Products, 154 . 7 Complete Varieties 157 a. Definition and basic properties, 157 ; b. Proper maps, 159; c. Projective varieties are complete, 160 ; d. Elimination theory, 161 ; e. The rigidity theorem; abelian varieties, 165; f. Chow’s Lemma, 167 ; g. Analytic spaces; Chow’s theorem, 169; h. Nagata’s Embedding Theorem, 170 . 8 Normal Varieties; (Quasi-)finite maps; Zariski’s Main Theorem 173 a. Normal varieties, 173 ; b. Regular functions on normal varieties, 177 ; c. Finite and quasi-finite maps, 179 ; d. The fibres of finite maps, 184; e. Zariski’s main theorem, 186 ; f. Stein factorization, 191; g. Blow-ups, 192 ; h. Resolution of singularities, 192 . 9 Regular Maps and Their Fibres 195 a. The constructibility theorem, 195; b. The fibres of morphisms, 198; c. Flat maps and their fibres, 201 ; d. Lines on surfaces, 208; e. Bertini’s theorem, 213; f. Birational classification, 214. Index 223 4 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. We sometimes write S for the number of elements in a finite set S. 7! W ! j j Throughout, k is an algebraically closed field. Unadorned tensor products are over k. For a k-algebra R and k-module M , we often write MR for R M . The dual Hom .E;k/ ˝ k-linear of a finite-dimensional k-vector space E is denoted by E_. All rings will be commutative with 1, and homomorphisms of rings are required to map 1 to 1. We use Gothic (fraktur) letters for ideals: abcmnpqABCMNPQ abcmnpqABCMNPQ Finally 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). ' A reference “Section 3m” is to Section m in Chapter 3; a reference “(3.45)” is to this item in chapter 3; a reference “(67)” is to (displayed) equation 67 (usually given with a page reference unless it is nearby). Prerequisites The reader is assumed to be familiar with the basic objects of algebra, namely, rings, modules, fields, and so on. References Atiyah and MacDonald 1969: Introduction to Commutative Algebra, Addison-Wesley. CA: Milne, J.S., Commutative Algebra, v4.01, 2014. FT: Milne, J.S., Fields and Galois Theory, v4.50, 2014. Hartshorne 1977: Algebraic Geometry, Springer. Shafarevich 1994: Basic Algebraic Geometry, Springer. A reference monnnn (resp. sxnnnn) is to question nnnn on mathoverflow.net (resp. math.stackexchange.com). We sometimes refer to the computer algebra programs CoCoA (Computations in Commutative Algebra) http://cocoa.dima.unige.it/. Macaulay 2 (Grayson and Stillman) http://www.math.uiuc.edu/Macaulay2/. 5 Acknowledgements I thank the following for providing corrections and comments on earlier versions of these notes: Jorge Nicolas´ Caro Montoya, Sandeep Chellapilla, Rankeya Datta, Umesh V. Dubey, Shalom Feigelstock, Tony Feng, B.J. Franklin, Sergei Gelfand, Daniel Gerig, Darij Grinberg, Lucio Guerberoff, Isac Heden,´ Guido Helmers, Florian Herzig, Christian Hirsch, Cheuk-Man Hwang, Jasper Loy Jiabao, Dan Karliner, Lars Kindler, John Miller, Joaquin Rodrigues, Sean Rostami, David Rufino, Hossein Sabzrou, Jyoti Prakash Saha, Tom Savage, Nguyen Quoc Thang, Bhupendra Nath Tiwari, Israel Vainsencher, Soli Vishkautsan, Dennis Bouke Westra, Felipe Zaldivar, and others. 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 equations is. This would cost years of study to Plato. QUESTION: It should be nice to have a little faith that after two thousand years every good high school graduate can understand what an affine scheme is . From the notes of a lecture series that Grothendieck gave at SUNY at Buffalo in the summer of 1973 (in 167 pages, Grothendieck manages to cover very little). 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 polynomial equations, fi .X1;:::;Xn/ 0; i 1;:::;m; fi kŒX1;:::;Xn: D D 2 One immediate difference between linear equations and polynomial equations is that theo- rems 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.
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