Algebraic Geometry - James Milne

1 Algebraic MilneVersion 19, 2017 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 abstractalgebraic varieties, and not just subvarieties of affine and projective space. This approachleads more naturally into scheme information@misc{milneAG,author={ Milne , James S.},title={ Algebraic Geometry ( )},year={2017},note={Available at },pages={221}} (August 24, 1996). First version on the (June 13, 1998). (October 30, 2003). Fixed errors; many minor revisions; added exercises; added twosections/chapters; 206 (February 20, 2005). Heavily revised; most numbering changed; 227 (March 19, 2008). Minor fixes; TEXstyle changed, so page numbers changed; (September 14, 2009).

2 Minor corrections; revised Chapters 1, 11, 16; 245 (January 13, 2012). Minor fixes; 260 (August 24, 2014). Major revision; 223 (August 23, 2015). Minor fixes; 226 (March 19, 2017). Minor fixes; 221 at send comments and corrections to me at the address on my web curves are a tacnode, a ramphoid cusp, and an ordinary triple 1996 2017 paper copies for noncommercial personal use may be made without explicit permissionfrom the copyright Preliminaries from commutative algebra11a. Rings and ideals, 11 ; b. Rings of fractions, 15 ; c. Unique factorization, 21 ; d. Integraldependence, 24; e. Tensor Products, 30 ; f. Transcendence bases, 33; Exercises, Algebraic Sets35a. Definition of an Algebraic set, 35 ; b.

3 The Hilbert basis theorem, 36; c. The Zariskitopology, 37; d. The Hilbert Nullstellensatz, 38; e. The correspondence between algebraicsets and radical ideals, 39; f. Finding the radical of an ideal, 43; g. Properties of theZariski 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; ; finite and quasi-finite maps, 48; l. Noether normalization theorem, 50 ; , 52 ; Exercises, Affine Algebraic Varieties57a. Sheaves, 57 ; b. Ringed spaces, 58; c. The ringed space structure on an Algebraic set, 59; d. Morphisms of ringed spaces, 62 ; e. Affine Algebraic varieties, 63; f. The category ofaffine Algebraic varieties, 64; g.

4 Explicit description of morphisms of affine varieties, 65 ;h. Subvarieties, 68; i. Properties of the regular mapSpm. /, 69; j. Affine space withoutcoordinates, 70; k. Birational equivalence, 71; l. Noether Normalization Theorem, 72; , 73 ; Exercises, Local Study79a. Tangent spaces to plane curves, 79 ; b. Tangent cones to plane curves, 81 ; c. The localring at a point on a curve, 83; d. Tangent spaces to Algebraic subsets ofAm, 84 ; e. Thedifferential of a regular map, 86; f. Tangent spaces to affine Algebraic varieties, 87 ; cones, 91; h. Nonsingular points; the singular locus, 92 ; i. Nonsingularity andregularity, 94; j. Examples of tangent spaces, 95; Exercises, Algebraic Varieties97a. Algebraic prevarieties, 97; b.

5 Regular maps, 98; c. Algebraic varieties, 99; d. Maps fromvarieties to affine varieties, 101; e. Subvarieties, 101 ; f. Prevarieties obtained by patching,102; g. Products of varieties, 103 ; h. The separation axiom revisited, 108; i. Fibredproducts, 110 ; j. Dimension, 111; k. Dominant maps, 113; l. Rational maps; birationalequivalence, 113; m. Local study, 114; n. Etale maps, 115 ; o. Etale neighbourhoods,118 ; p. Smooth maps, 120 ; q. Algebraic varieties as a functors, 121 ; r. Rational andunirational varieties, 124 ; Exercises, Projective Varieties1273a. Algebraic subsets ofPn, 127; b. The Zariski topology onPn, 131; c. Closed subsets ofAnandPn, 132 ; d. The hyperplane at infinity, 133; an Algebraic variety, 133; homogeneous coordinate ring of a projective variety, 135; g.

6 Regular functions on aprojective variety, 136; h. Maps from projective varieties, 137; i. Some classical maps ofprojective varieties, 138; j. Maps to projective space, 143; k. Projective space withoutcoordinates, 143; l. The functor defined by projective space, 144; m. Grassmann varieties,144 ; n. Bezout s theorem, 148; o. Hilbert polynomials (sketch), 149; p. Dimensions, 150;q. Products, 152 ; Exercises, Complete Varieties155a. Definition and basic properties, 155 ; b. Proper maps, 157; c. Projective varieties arecomplete, 158 ; d. Elimination theory, 159 ; e. The rigidity theorem; abelian varieties,163; f. Chow s Lemma, 165 ; g. Analytic spaces; Chow s theorem, 167; h. Nagata sEmbedding Theorem, 168 ; Exercises, Normal Varieties; (Quasi-)finite maps; Zariski s Main Theorem171a.

7 Normal varieties, 171 ; b. Regular functions on normal varieties, 174 ; c. Finite andquasi-finite maps, 176; d. The fibres of finite maps, 182; e. Zariski s main theorem,184; f. Stein factorization, 189; g. Blow-ups, 190 ; h. Resolution of singularities, 190 ;Exercises, Regular Maps and Their Fibres193a. The constructibility theorem, 193; b. The fibres of morphisms, 196; c. Flat maps andtheir fibres, 199; d. Lines on surfaces, 206; e. Bertini s theorem, 211; f. Birationalclassification, 211; Exercises, to the exercises213 Index2194 NotationsWe use the standard (Bourbaki) notations:NDf0;1;2;:::g,ZDring of integers,RDfieldof real numbers,CDfield of complex numbers,FpDZ=pZDfield ofpelements,paprime number.

8 Given an equivalence relation, denotes the equivalence class containing .A family of elements of a setAindexed by a second setI, , is a functioni7!aiWI!A. We sometimes writejSjfor the number of elements in a finite ,kis an algebraically closed field. Unadorned tensor products are overk. Forak-algebraRandk-moduleM, we often writeMRforR M. The ;k/of a finite-dimensionalk-vector spaceEis denoted rings will be commutative with1, and homomorphisms of rings are required to use Gothic (fraktur) letters for ideals:a b c m n p q A B C M N P Qa b c m n p q A B C M N P QFinallyXdefDY Xis defined to beY, or equalsYby definition;X Y Xis a subset ofY(not necessarily proper, ,Xmay equalY);X Y XandYare isomorphic;X'Y XandYare canonically isomorphic (or there is a given or unique isomorphism).

9 A reference Section 3m is to Section m in Chapter 3; a reference ( ) is to thisitem in chapter 3; a reference (67) is to (displayed) equation 67 (usually given with a pagereference unless it is nearby).PrerequisitesThe reader is assumed to be familiar with the basic objects of algebra, namely, rings, modules,fields, and so and MacDonald 1969: introduction to Commutative Algebra, : Milne , , Commutative Algebra, , : Milne , , Fields and Galois Theory, , 1977: Algebraic Geometry , 1994:Basic Algebraic Geometry , reference monnnn (resp. sxnnnn) is to question nnnn on ( ).We sometimes refer to the computer algebra programsCoCoA(Computations inCommutativeAlgebra) 2(Grayson and Stillman) thank the following for providing corrections and comments on earlier versions of thesenotes: Jorge Nicol as Caro Montoya, Sandeep Chellapilla, Rankeya Datta, Umesh V.

10 Dubey,Shalom Feigelstock, Tony Feng, Franklin, Sergei Gelfand, Daniel Gerig, Darij Grinberg,Lucio Guerberoff, Isac Hed en, Guido Helmers, Florian Herzig, Christian Hirsch, Cheuk-ManHwang, Jasper Loy Jiabao, Dan Karliner, Lars Kindler, John Miller, Joaquin Rodrigues,Sean Rostami, David Rufino, Hossein Sabzrou, Jyoti Prakash Saha, Tom Savage, NguyenQuoc Thang, Bhupendra Nath Tiwari, Israel Vainsencher, Soli Vishkautsan, Dennis BoukeWestra, Felipe Zaldivar, Luochen Zhao, and : If we try to explain to a layman what Algebraic Geometry is, it seems to me thatthe 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 equationsis.