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Algebraic Geometry

J.S. MilneVersion 5.10

March 19, 2008

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 (v5.10)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={235+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.

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 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder.

Contents

Contents iii

Notationsvi; Prerequisitesvi; Referencesvi; Acknowledgementsvi

Introduction 1

1 Preliminaries 4

Algebras4; Ideals4; Noetherian rings6; Unique factorization8; Polynomial rings11; Integrality11; Direct limits (summary)14; Rings of fractions15; Tensor Products18; Categories and functors21; Algorithms for polynomials23; Exercises29

2 Algebraic Sets 30

Definition of an algebraic set30; The Hilbert basis theorem31; The Zariski topology32; The Hilbert Nullstellensatz33; The correspondence between algebraic sets and ideals34; Finding the radical of an ideal37; The Zariski topology on an algebraic set37; The coor- dinate ring of an algebraic set38; Irreducible algebraic sets39; Dimension41; Exercises 44

3 Affine Algebraic Varieties 45

Ringed spaces45; The ringed space structure on an algebraic set46; Morphisms of ringed spaces50; Affine algebraic varieties50; The category of affine algebraic varieties51; Explicit description of morphisms of affine varieties52; Subvarieties55; Properties of the regular map defined by specm(˛)56; Affine space without coordinates57; Exercises58

4 Algebraic Varieties 59

affine varieties62; Subvarieties63; Prevarieties obtained by patching64; Products of vari- eties64; The separation axiom revisited69; Fibred products72; Dimension73; Birational equivalence74; Dominating maps75; Algebraic varieties as a functors75; Exercises77

5 Local Study 78

Tangent spaces to plane curves78; Tangent cones to plane curves80; The local ring at a point on a curve80; Tangent spaces of subvarieties ofAm81; The differential of a regular map83; Etale maps84; Intrinsic definition of the tangent space86; Nonsingular points89; Nonsingularity and regularity90; Nonsingularity and normality91; Etale neighbourhoods

92; Smooth maps94; Dual numbers and derivations95; Tangent cones98; Exercises100

6 Projective Varieties 101

iii Algebraic subsets ofPn101; The Zariski topology onPn104; Closed subsets ofAnand P n105; The hyperplane at infinity105;Pnis an algebraic variety106; The homogeneous coordinate ring of a subvariety ofPn108; Regular functions on a projective variety109; Morphisms from projective varieties110; Examples of regular maps of projective vari- eties111; Projective space without coordinates116; Grassmann varieties116; Bezout"s theorem120; Hilbert polynomials (sketch)121; Exercises122

7 Complete varieties 123

Definition and basic properties123; Projective varieties are complete125; Elimination theory126; The rigidity theorem128; Theorems of Chow129; Nagata"s Embedding The- orem130; Exercises130

8 Finite Maps 131

Definition and basic properties131; Noether Normalization Theorem135; Zariski"s main theorem136; The base change of a finite map138; Proper maps138; Exercises139

9 Dimension Theory 141

Affine varieties141; Projective varieties148

10 Regular Maps and Their Fibres 150

Constructible sets150; Orbits of group actions153; The fibres of morphisms155; The fibres of finite maps157; Flat maps159; Lines on surfaces159; Stein factorization165;

Exercises165

11 Algebraic spaces; geometry over an arbitray field 167

Preliminaries167; Affine algebraic spaces170; Affine algebraic varieties.171; Algebraic spaces; algebraic varieties.172; Local study177; Projective varieties.179; Complete varieties.179; Normal varieties; Finite maps.179; Dimension theory179; Regular maps and their fibres179; Algebraic groups179; Exercises180

12 Divisors and Intersection Theory 181

Divisors181; Intersection theory.182; Exercises187

13 Coherent Sheaves; Invertible Sheaves 188

images and inverse images of coherent sheaves.193; Principal bundles193

14 Differentials (Outline) 194

15 Algebraic Varieties over the Complex Numbers 196

16 Descent Theory 199

Models199; Fixed fields199; Descending subspaces of vector spaces200; Descending subvarieties and morphisms201; Galois descent of vector spaces202; Descent data204; Galois descent of varieties207; Weil restriction208; Generic fibres208; Rigid descent

209; Weil"s descent theorems211; Restatement in terms of group actions213; Faithfully

flat descent215

17 Lefschetz Pencils (Outline) 218

Definition218

iv

18 Algebraic Schemes and Algebraic Spaces 221

A Solutions to the exercises 222

B Annotated Bibliography 229

Index232

v

Notations

We use the standard (Bourbaki) notations:ND f0;1;2;:::g,ZDring of integers,RD field of real numbers,CDfield of complex numbers,FpDZ=pZDfield ofpelements,p a prime number. Given an equivalence relation,OE??denotes the equivalence class containing ?. A family of elements of a setAindexed by a second setI, denoted.ai/i2I, is a function i7!aiWI!A. A fieldkis said to be separably closed if it has no finite separable extensions of degree > 1. We useksepandkalto denote separable and algebraic closures ofkrespectively. All rings will be commutative with1, and homomorphisms of rings are required to map

1to1. Ak-algebra is a ringAtogether with a homomorphismk!A. For a ringA,A?is

the group of units inA: A ?D fa2Ajthere exists ab2Asuch thatabD1g:

We use Gothic (fraktur) letters for ideals:

a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q X dfDY Xis defined to beY, or equalsYby definition; X?Y Xis a subset ofY(not necessarily proper, i.e.,Xmay equalY);

X?Y XandYare isomorphic;

X"Y XandYare 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/).

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, Daniel Gerig, Guido Helmers, Jasper Loy Jiabao, Sean Rostami, David Rufino, Tom Savage, Nguyen Quoc Thang, Dennis Bouke Westra, and others. vi

Introduction

Just as the starting point of linear algebra is the study of the solutions of systems of linear equations, nX jD1a ijXjDbi; iD1;:::;m;(1) the starting point for algebraic geometry is the study of the solutions of systems of polyno- mial equations, f i.X1;:::;Xn/D0; iD1;:::;m; fi2kOEX1;:::;Xn?: Note immediately one difference between linear equations and polynomial equations: the- orems for linear equations don"t depend on which fieldkyou are working over,1but those for polynomial equations depend on whether or notkis algebraically closed and (to a lesser extent) whetherkhas 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 geometryregular (polynomial) functionsalgebraic varieties

topologycontinuous functionstopological spaces differential topologydifferentiable functionsdifferentiable manifolds complex analysisanalytic (power series) functionscomplex 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 ddX Xa iXiDXia iXi?1: Moreover, calculations (on a computer) with polynomials are easier than with more general functions.1 For example, suppose that the system (1) has coefficientsaij2kand thatKis a field containingk. Then

(1) has a solution inknif and only if it has a solution inKn, and the dimension of the space of solutions is the

same for both fields. (Exercise!) 1

2INTRODUCTION

Consider a nonzero differentiable functionf.x;y;z/. In calculus, we learn that the equation f.x;y;z/DC(2) defines a surfaceSinR3, and that the tangent plane toSat a pointPD.a;b;c/has equation 2 @f@x P .x?a/C?@f@y P .y?b/C?@f@z P .z?c/D0:(3) The inverse function theorem says that a differentiable map˛WS!S0of surfaces is a local isomorphism at a pointP2Sif it maps the tangent plane atPisomorphically onto the tangent plane atP0D˛.P/. Consider a nonzero polynomialf.x;y;z/with coefficients in a fieldk. In this course, we shall learn that the equation (2) defines a surface ink3, and we shall use the equation (3) to define the tangent space at a pointPon 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 that1=Xis - these functions can not be integrated in the ring of polynomial functions. The first ten chapters of the notes form a basic course on algebraic geometry. In these chapters we generally assume that the ground field is algebraically closed in order to be able to concentrate on the geometry. The remaining chapters treat more advanced topics, and are largely independent of one another except that chapter 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 ofaffineorprojectivespaceoveralgebraicallyclosedfields. Then, inordertogivesubstance 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,

3but his "foundations" are unsatisfactory in two major respects:

Lacking a topology, his method of patching together affine varieties to form abstract varieties is clumsy. His definition of a variety over a base fieldkis not intrinsic; specifically, he fixes some large "universal" algebraically closed fieldand defines an algebraic variety overkto be an algebraic variety overwith ak-structure. In the ensuing years, several attempts were made to resolve these difficulties. In 1955, Serre resolved the first by borrowing ideas from complex analysis and defining an algebraic variety over an algebraically closed field to be a topological space with a sheaf of functions2 Think ofSas a level surface for the functionf, and note that the equation is that of a plane through .a;b;c/perpendicular to the gradient vector.?f /PoffatP.

3Weil, Andr´e. Foundations of algebraic geometry. American Mathematical Society, Providence, R.I. 1946.

3 that is locally affine.

4Then, in the late 1950s Grothendieck resolved all such difficulties by

introducing his theory of schemes. In these notes, we follow Grothendieck except that, by working only over a base field, we are able to simplify his language by considering only the closed points in the underlying topological spaces. In this way, we hope to provide a bridge between the intuition given by differential geometry and the abstractions of scheme theory.4 Serre, Jean-Pierre. Faisceaux alg´ebriques coh´erents. Ann. of Math. (2) 61, (1955). 197-278.

Chapter 1

Preliminaries

In this chapter, we review some definitions and basic results in commutative algebra and category theory, and we derive some algorithms for working in polynomial rings.

Algebras

LetAbe a ring. AnA-algebra is a ringBtogether with a homomorphismiBWA!B. A homomorphism ofA-algebrasB!Cis a homomorphism of rings"WB!Csuch that ".i

B.a//DiC.a/for alla2A.

Elementsx1;:::;xnof anA-algebraBare said togenerateit if every element ofBcan be expressed as a polynomial in thexiwith coefficients iniB.A/, i.e., if the homomorphism ofA-algebrasAOEX1;:::;Xn?!BsendingXitoxiis surjective. We then writeBD .i BA/OEx1;:::;xn?. AnA-algebraBis said to befinitely generated(or offinite-typeover A) if it is generated by a finite set of elements. A ring homomorphismA!Bisfinite, andBis afinite1A-algebra, ifBis finitely generated as anA-module. and we can identifykwith its image, i.e., we can regardkas a subring ofA. When1D0 in a ringA, thenAis the zero ring, i.e.,AD f0g. LetAOEX?be the polynomial ring in the symbolXwith coefficients inA. IfAis an integral domain, thendeg.fg/Ddeg.f /Cdeg.g/, and it follows thatAOEX?is also an integral domain; moreover,AOEX??DA?.

Ideals

LetAbe a ring. AsubringofAis a subset containing1that is closed under addition, multiplication, and the formation of negatives. AnidealainAis a subset such that (a)ais a subgroup ofAregarded as a group under addition; (b)a2a,r2A)ra2a: Theideal generated by a subsetSofAis the intersection of all idealsacontainingA

- it is easy to verify that this is in fact an ideal, and that it consists of all finite sums of the1

The term "module-finite" is also used.

4

IDEALS5

form Prisiwithri2A,si2S. WhenSD fs1;s2;:::g, we shall write.s1;s2;:::/for the ideal it generates. Letaandbbe ideals inA. The setfaCbja2a; b2bgis an ideal, denoted byaCb. The ideal generated byfabja2a; b2bgis denoted byab. Clearlyabconsists of all finite sumsPaibiwithai2aandbi2b, and ifaD.a1;:::;am/andbD.b1;:::;bn/, thenabD.a1b1;:::;aibj;:::;ambn/. Note thatab?a\b. Letabe an ideal ofA. The set of cosets ofainAforms a ringA=a, anda7!aCa is a homomorphism"WA!A=a. The mapb7!"?1.b/is a one-to-one correspondence between the ideals ofA=aand the ideals ofAcontaininga. ifA=pis nonzero and has the property that i.e.,A=pis an integral domain. betweenmandA. Thusmis maximal if and only ifA=mis nonzero and has no proper nonzero ideals, and so is a field. Note that mmaximalH)mprime. The ideals ofA?Bare all of the forma?bwithaandbideals inAandB. To see this, note that ifcis an ideal inA?Band.a;b/2c, then.a;0/D.1;0/.a;b/2cand .0;b/D.0;1/.a;b/2c. Therefore,cDa?bwith aD faj.a;0/2cg;bD fbj.0;b/2cg: THEOREM1.1 (CHINESEREMAINDERTHEOREM).Leta1;:::;anbe ideals in a ringA.

A!A=a1? ??? ?A=an(4)

is surjective, with kernel

QaiDTai.

PROOF. Suppose first thatnD2. Asa1Ca2DA, there existai2aisuch thata1Ca2D

1. ThenxDa1x2Ca2x1maps to.x1moda1;x2moda2/, which shows that (4) is

surjective. For eachi, there exist elementsai2a1andbi2aisuch that a iCbiD1, alli?2:

The product

Q i?2.aiCbi/D1, and lies ina1CQ i?2ai, and so a 1CY i?2a iDA: We can now apply the theorem in the casenD2to obtain an elementy1ofAsuch that y

1?1moda1; y1?0modY

i?2a i:

6CHAPTER 1. PRELIMINARIES

These conditions imply

y

1?1moda1; y1?0modaj, allj > 1:

Similarly, there exist elementsy2;:::;ynsuch that

y The elementxDPxiyimaps to.x1moda1;:::;xnmodan/, which shows that (4) is surjective. It remains to prove thatTaiDQai. We have already noted thatTai?Qai. First suppose thatnD2, and leta1Ca2D1, as before. Forc2a1\a2, we have cDa1cCa2c2a1?a2 which proves thata1\a2Da1a2. We complete the proof by induction. This allows us to assume thatQ i?2aiDT i?2ai. We showed above thata1andQquotesdbs_dbs43.pdfusesText_43

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