Algebraic Geometry J.S. Milne Taiaroa Publishing Erehwon Version 5.00 February 20, 2005 Abstract 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. 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; 206 pages. v5.00 (February 20, 2005). Heavily revised; most numbering changed; 227 pages. Please send comments and lists of corrections to me at [email protected] Available at http://www.jmilne.org/math/ Copyright c 1996, 1998, 2003, 2005. J.S. Milne. This work is licensed under a Creative Commons Licence (Attribution-NonCommercial-NoDerivs 2.0) http://creativecommons.org/licenses/by-nc-nd/2.0/ Contents Introduction 3 1 Preliminaries 4 Algebras 4; Ideals 4; Noetherian rings 6; Unique factorization 8; Polynomial rings 10; Integrality 11; Direct limits (summary) 13; Rings of fractions 14; Tensor Products 17; Categories and functors 20; Algorithms for polynomials 22; Exercises 28 2 Algebraic Sets 29 Definition of an algebraic set 29; The Hilbert basis theorem 30; The Zariski topology 31; The Hilbert Nullstellensatz 31; The correspondence between algebraic sets and ideals 32; Finding the radical of an ideal 35; The Zariski topology on an algebraic set 36; The coor- dinate ring of an algebraic set 36; Irreducible algebraic sets 37; Dimension 40; Exercises 42 3 Affine Algebraic Varieties 43 Ringed spaces 43; The ringed space structure on an algebraic set 44; Morphisms of ringed spaces 47; Affine algebraic varieties 48; The category of affine algebraic varieties 49; Explicit description of morphisms of affine varieties 50; Subvarieties 53; Properties of the regular map defined by specm(α) 54; Affine space without coordinates 54; Exercises 56 4 Algebraic Varieties 57 Algebraic prevarieties 57; Regular maps 58; Algebraic varieties 59; Maps from varieties to affine varieties 60; Subvarieties 60; Prevarieties obtained by patching 61; Products of vari- eties 62; The separation axiom revisited 67; Fibred products 69; Dimension 70; Birational equivalence 71; Dominating maps 72; Algebraic varieties as a functors 72; Exercises 74 5 Local Study 75 Tangent spaces to plane curves 75; Tangent cones to plane curves 76; The local ring at a m point on a curve 77; Tangent spaces of subvarieties of A 78; The differential of a regular map 79; Etale maps 81; Intrinsic definition of the tangent space 83; Nonsingular points 85; Nonsingularity and regularity 87; Nonsingularity and normality 88; Etale neighbourhoods 88; Smooth maps 90; Dual numbers and derivations 91; Tangent cones 94; Exercises 95 6 Projective Varieties 97 n n n Algebraic subsets of P 97; The Zariski topology on P 100; Closed subsets of A and n n P 100; The hyperplane at infinity 101; P is an algebraic variety 102; The homogeneous n coordinate ring of a subvariety of P 103; Regular functions on a projective variety 104; Morphisms from projective varieties 105; Examples of regular maps of projective vari- eties 107; Projective space without coordinates 111; Grassmann varieties 111; Bezout’s theorem 115; Hilbert polynomials (sketch) 116; Exercises 117 7 Complete varieties 118 Definition and basic properties 118; Projective varieties are complete 119; Elimination theory 121; The rigidity theorem 123; Theorems of Chow 124; Nagata’s Embedding Prob- lem 124; Exercises 125 8 Finite Maps 126 Definition and basic properties 126; Noether Normalization Theorem 130; Zariski’s main theorem 131; The base change of a finite map 133; Proper maps 133; Exercises 134 9 Dimension Theory 135 Affine varieties 135; Projective varieties 141 10 Regular Maps and Their Fibres 144 Constructible sets 144; Orbits of group actions 147; The fibres of morphisms 148; The fibres of finite maps 150; Flat maps 152; Lines on surfaces 153; Stein factorization 158; Exercises 158 11 Algebraic spaces; geometry over an arbitrary field 160 Preliminaries 160; Affine algebraic spaces 163; Affine algebraic varieties. 164; Algebraic spaces; algebraic varieties. 165; Local study 169; Projective varieties. 171; Complete varieties. 171; Normal varieties; Finite maps. 171; Dimension theory 171; Regular maps and their fibres 172; Algebraic groups 172; Exercises 173 12 Divisors and Intersection Theory 174 Divisors 174; Intersection theory. 175; Exercises 179 13 Coherent Sheaves; Invertible Sheaves 180 Coherent sheaves 180; Invertible sheaves. 182; Invertible sheaves and divisors. 183; Direct images and inverse images of coherent sheaves. 184; Principal bundles 185 14 Differentials (Outline) 186 15 Algebraic Varieties over the Complex Numbers (Outline) 188 16 Descent Theory 191 Models 191; Fixed fields 191; Descending subspaces of vector spaces 192; Descending subvarieties and morphisms 193; Galois descent of vector spaces 194; Descent data 196; Galois descent of varieties 198; Weil restriction 199; Generic fibres 200; Rigid descent 200; Weil’s descent theorems 202; Restatement in terms of group actions 204; Faithfully flat descent 206 17 Lefschetz Pencils (Outline) 209 Definition 209 18 Algebraic Schemes 211 A Solutions to the exercises 212 B Annotated Bibliography 219 Index 221 1 Introduction Just as the starting point of linear algebra is the study of the solutions of systems of linear equations, n X aijXj = bi, i = 1, . , m, (1) j=1 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 ∈ k[X1,...,Xn]. 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 X a Xi = ia Xi−1. dX i i Moreover, calculations (on a computer) with polynomials are easier than with more general functions. Consider a nonzero differentiable function f(x, y, z). In calculus, we learn that the equation f(x, y, z) = C (2) 3 defines a surface S in R , and that the tangent plane to S at a point P = (a, b, c) has equation2 ∂f ∂f ∂f (x − a) + (y − b) + (z − c) = 0. (3) ∂x P ∂y P ∂z P 1 For example, suppose that the system (1) has coefficients aij ∈ k and that K is a field containing k. Then (1) has a solution in kn if and only if it has a solution in Kn, and the dimension of the space of solutions is the same for both fields. (Exercise!) 2Think of S as a level surface for the function f, and note that the equation is that of a plane through (a, b, c) perpendicular to the gradient vector (Of)P of f at P . 2 The inverse function theorem says that a differentiable map α: S → S0 of surfaces is a local isomorphism at a point P ∈ S if it maps the tangent plane at P isomorphically onto the tangent plane at P 0 = α(P ). Consider a nonzero polynomial f(x, y, z) with coefficients in a field k. In this course, we shall learn that the equation (2) defines a surface in k3, and we shall use the equation (3) to define the tangent space at a point P on the surface. However, and this is one of the essential differences between algebraic geometry and the other fields, the inverse function theorem doesn’t hold in algebraic geometry. One other essential difference is that 1/X is not the derivative of any rational function of X, and nor is Xnp−1 in characteristic p 6= 0 — these functions can not be integrated in the ring of polynomial functions. The first ten sections of the notes form a basic course on algebraic geometry. In these sections we generally assume that the ground field is algebraically closed in order to be able to concentrate on the geometry. The remaining sections treat more advanced topics, and are largely independent of one another except that Section 11 should be read first. The approach to algebraic geometry taken in these notes In differential geometry it is important to define differentiable manifolds abstractly, i.e., not as submanifolds of some Euclidean space. For example, it is difficult even to make sense of a statement such as “the Gauss curvature of a surface is intrinsic to the surface but the principal curvatures are not” without the abstract notion of a surface. Until the mid 1940s, algebraic geometry was concerned only with algebraic subvarieties of affine or projective space over algebraically closed fields. Then, in order to give substance to his proof of the congruence Riemann hypothesis for curves an abelian varieties, Weil was forced to develop a theory of algebraic geometry for “abstract” algebraic varieties over arbitrary fields,3 but his “foundations” are unsatisfactory in two major respects: — Lacking a topology, his method of patching together affine varieties to form abstract varieties is clumsy.