Table Of ContentAlgebraic Geometry
J.S. Milne
Version6.02
March19,2017
Thesenotesareanintroductiontothetheoryofalgebraicvarietiesemphasizingthesimi-
larities to the theory of manifolds. In contrast to most such accounts they study abstract
algebraicvarieties,andnotjustsubvarietiesofaffineandprojectivespace. Thisapproach
leadsmorenaturallyintoschemetheory.
BibTeX information
@misc{milneAG,
author={Milne, James S.},
title={Algebraic Geometry (v6.02)},
year={2017},
note={Available at www.jmilne.org/math/},
pages={221}
}
v2.01 (August24,1996). Firstversionontheweb.
v3.01 (June13,1998).
v4.00 (October30,2003). Fixederrors;manyminorrevisions;addedexercises;addedtwo
sections/chapters;206pages.
v5.00 (February20,2005). Heavilyrevised;mostnumberingchanged;227pages.
v5.10 (March19, 2008). Minorfixes; TEXstylechanged, sopagenumberschanged; 241
pages.
v5.20 (September14,2009). Minorcorrections;revisedChapters1,11,16;245pages.
v5.22 (January13,2012). Minorfixes;260pages.
v6.00 (August24,2014). Majorrevision;223pages.
v6.01 (August23,2015). Minorfixes;226pages.
v6.02 (March19,2017). Minorfixes;221pages.
Availableatwww.jmilne.org/math/
Pleasesendcommentsandcorrectionstomeattheaddressonmywebpage.
Thecurvesareatacnode,aramphoidcusp,andanordinarytriplepoint.
Copyright(cid:13)c 1996–2017J.S.Milne.
Singlepapercopiesfornoncommercialpersonalusemaybemadewithoutexplicitpermission
fromthecopyrightholder.
Contents
Contents 3
Introduction 7
1 Preliminariesfromcommutativealgebra 11
a. Ringsandideals,11 ;b. Ringsoffractions,15 ;c. Uniquefactorization,21;d. Integral
dependence,24;e. TensorProducts,30 ;f. Transcendencebases,33;Exercises,33.
2 AlgebraicSets 35
a. Definition of an algebraic set, 35 ; b. The Hilbert basis theorem, 36; c. The Zariski
topology,37;d. TheHilbertNullstellensatz,38;e. Thecorrespondencebetweenalgebraic
setsandradicalideals,39 ; f. Findingtheradicalofanideal,43; g. Propertiesofthe
Zariskitopology,43;h. Decompositionofanalgebraicsetintoirreduciblealgebraicsets,
44;i. Regularfunctions;thecoordinateringofanalgebraicset,47;j. Regularmaps,48;k.
Hypersurfaces;finiteandquasi-finitemaps,48;l. Noethernormalizationtheorem,50;m.
Dimension,52 ;Exercises,56.
3 AffineAlgebraicVarieties 57
a. Sheaves,57;b. Ringedspaces,58;c. Theringedspacestructureonanalgebraicset,59
;d. Morphismsofringedspaces,62;e. Affinealgebraicvarieties,63;f. Thecategoryof
affinealgebraicvarieties,64;g. Explicitdescriptionofmorphismsofaffinevarieties,65;
h. Subvarieties,68;i. PropertiesoftheregularmapSpm.˛/,69;j. Affinespacewithout
coordinates,70;k. Birationalequivalence,71;l. NoetherNormalizationTheorem,72;m.
Dimension,73;Exercises,77.
4 LocalStudy 79
a. Tangentspacestoplanecurves,79 ;b. Tangentconestoplanecurves,81;c. Thelocal
ringatapointonacurve,83;d. TangentspacestoalgebraicsubsetsofAm,84 ;e. The
differentialofaregularmap,86;f. Tangentspacestoaffinealgebraicvarieties,87 ;g.
Tangentcones,91;h. Nonsingularpoints;thesingularlocus,92;i. Nonsingularityand
regularity,94;j. Examplesoftangentspaces,95;Exercises,96.
5 AlgebraicVarieties 97
a. Algebraicprevarieties,97;b. Regularmaps,98;c. Algebraicvarieties,99;d. Mapsfrom
varietiestoaffinevarieties,101;e. Subvarieties,101 ;f. Prevarietiesobtainedbypatching,
102; g. Products of varieties, 103 ; h. The separation axiom revisited, 108; i. Fibred
products,110;j. Dimension,111;k. Dominantmaps,113;l. Rationalmaps;birational
equivalence,113; m. Localstudy,114; n. E´talemaps,115 ; o. E´taleneighbourhoods,
118 ; p. Smoothmaps,120; q. Algebraicvarietiesasafunctors,121; r. Rationaland
unirationalvarieties,124;Exercises,125.
6 ProjectiveVarieties 127
3
a. AlgebraicsubsetsofPn,127;b. TheZariskitopologyonPn,131;c. Closedsubsetsof
An andPn,132;d. Thehyperplaneatinfinity,133;e. Pn isanalgebraicvariety,133;f.
Thehomogeneouscoordinateringofaprojectivevariety,135;g. Regularfunctionsona
projectivevariety,136;h. Mapsfromprojectivevarieties,137;i. Someclassicalmapsof
projectivevarieties,138 ;j. Mapstoprojectivespace,143;k. Projectivespacewithout
coordinates,143;l. Thefunctordefinedbyprojectivespace,144;m. Grassmannvarieties,
144 ;n. Bezout’stheorem,148;o. Hilbertpolynomials(sketch),149;p. Dimensions,150;
q. Products,152 ;Exercises,153.
7 CompleteVarieties 155
a. Definitionandbasicproperties,155 ;b. Propermaps,157;c. Projectivevarietiesare
complete,158;d. Eliminationtheory,159 ;e. Therigiditytheorem;abelianvarieties,
163; f. Chow’s Lemma, 165 ; g. Analytic spaces; Chow’s theorem, 167; h. Nagata’s
EmbeddingTheorem,168;Exercises,169.
8 NormalVarieties;(Quasi-)finitemaps;Zariski’sMainTheorem 171
a. Normalvarieties, 171; b. Regularfunctionsonnormalvarieties, 174; c. Finiteand
quasi-finitemaps, 176 ; d. The fibresof finitemaps, 182; e. Zariski’s maintheorem,
184 ;f. Steinfactorization,189;g. Blow-ups,190 ;h. Resolutionofsingularities,190;
Exercises,191.
9 RegularMapsandTheirFibres 193
a. Theconstructibilitytheorem,193;b. Thefibresofmorphisms,196;c. Flatmapsand
theirfibres, 199 ; d. Linesonsurfaces, 206; e. Bertini’stheorem, 211; f. Birational
classification,211;Exercises,212.
Solutionstotheexercises 213
Index 219
4
Notations
Weusethestandard(Bourbaki)notations: NDf0;1;2;:::g,ZDringofintegers,RDfield
of real numbers, CD field of complex numbers, Fp DZ=pZD field of p elements, p a
primenumber. Givenanequivalencerelation,Œ(cid:3)(cid:141)denotestheequivalenceclasscontaining(cid:3).
AfamilyofelementsofasetAindexedbyasecondsetI,denoted.ai/i2I,isafunction
i 7!a WI !A. WesometimeswritejSjforthenumberofelementsinafinitesetS.
i
Throughout,kisanalgebraicallyclosedfield. Unadornedtensorproductsareoverk. For
ak-algebraRandk-moduleM,weoftenwriteM forR˝M. ThedualHom .E;k/
R k-linear
_
ofafinite-dimensionalk-vectorspaceE isdenotedbyE .
Allringswillbecommutativewith1,andhomomorphismsofringsarerequiredtomap
1to1.
WeuseGothic(fraktur)lettersforideals:
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
Finally
X DdefY X isdefinedtobeY,orequalsY bydefinition;
X (cid:26)Y X isasubsetofY (notnecessarilyproper,i.e.,X mayequalY);
X (cid:25)Y X andY areisomorphic;
X 'Y X andY arecanonicallyisomorphic(orthereisagivenoruniqueisomorphism).
Areference“Section3m”istoSectionminChapter3;areference“(3.45)”istothis
iteminchapter3;areference“(67)”isto(displayed)equation67(usuallygivenwithapage
referenceunlessitisnearby).
Prerequisites
Thereaderisassumedtobefamiliarwiththebasicobjectsofalgebra,namely,rings,modules,
fields,andsoon.
References
AtiyahandMacDonald1969: IntroductiontoCommutativeAlgebra,Addison-Wesley.
CA:Milne,J.S.,CommutativeAlgebra,v4.02,2017.
FT:Milne,J.S.,FieldsandGaloisTheory,v4.52,2017.
Hartshorne1977: AlgebraicGeometry,Springer.
Shafarevich1994: BasicAlgebraicGeometry,Springer.
A reference monnnn (resp. sxnnnn) is to question nnnn on mathoverflow.net (resp.
math.stackexchange.com).
Wesometimesrefertothecomputeralgebraprograms
CoCoA (ComputationsinCommutativeAlgebra)http://cocoa.dima.unige.it/.
Macaulay2 (GraysonandStillman)http://www.math.uiuc.edu/Macaulay2/.
5
Acknowledgements
Ithankthefollowingforprovidingcorrectionsandcommentsonearlierversionsofthese
notes: JorgeNicola´sCaroMontoya,SandeepChellapilla,RankeyaDatta,UmeshV.Dubey,
ShalomFeigelstock,TonyFeng,B.J.Franklin,SergeiGelfand,DanielGerig,DarijGrinberg,
LucioGuerberoff,IsacHede´n,GuidoHelmers,FlorianHerzig,ChristianHirsch,Cheuk-Man
Hwang, Jasper Loy Jiabao, Dan Karliner, Lars Kindler, John Miller, Joaquin Rodrigues,
SeanRostami,DavidRufino,HosseinSabzrou,JyotiPrakashSaha,TomSavage,Nguyen
QuocThang,BhupendraNathTiwari,IsraelVainsencher,SoliVishkautsan,DennisBouke
Westra,FelipeZaldivar,LuochenZhao,andothers.
QUESTION: Ifwetrytoexplaintoalaymanwhatalgebraicgeometryis,itseemstomethat
thetitleoftheoldbookofEnriquesisstilladequate: GeometricalTheoryofEquations....
GROTHENDIECK: Yes! butyour“layman”shouldknowwhatasystemofalgebraicequations
is. ThiswouldcostyearsofstudytoPlato.
QUESTION: Itshouldbenicetohavealittlefaiththataftertwothousandyearseverygood
highschoolgraduatecanunderstandwhatanaffineschemeis...
From the notes of a lecture series that Grothendieck gave at SUNY at Buffalo in the
summerof1973(in167pages,Grothendieckmanagestocoververylittle).
6
Introduction
Thereisalmostnothinglefttodiscoverin
geometry.
Descartes,March26,1619
Just as the starting point of linear algebra is the study of the solutions of systems of
linearequations,
n
X
a X Db ; i D1;:::;m; (1)
ij j i
jD1
thestartingpointforalgebraicgeometryisthestudyofthesolutionsofsystemsofpolynomial
equations,
f .X ;:::;X /D0; i D1;:::;m; f 2kŒX ;:::;X (cid:141):
i 1 n i 1 n
Oneimmediatedifferencebetweenlinearequationsandpolynomialequationsisthattheo-
remsforlinearequationsdon’tdependonwhichfieldk youareworkingover,1 butthosefor
polynomialequationsdependonwhetherornotk isalgebraicallyclosedand(toalesser
extent)whetherk hascharacteristiczero.
Abetterdescriptionofalgebraicgeometryisthatitisthestudyofpolynomialfunctions
andthespacesonwhichtheyaredefined(algebraicvarieties),justastopologyisthestudy
of continuous functions and the spaces on which they are defined (topological spaces),
differentialtopologythestudyofinfinitelydifferentiablefunctionsandthespacesonwhich
theyaredefined(differentiablemanifolds),andsoon:
algebraicgeometry regular(polynomial)functions algebraicvarieties
topology continuousfunctions topologicalspaces
differentialtopology differentiablefunctions differentiablemanifolds
complexanalysis analytic(powerseries)functions complexmanifolds.
The approach adopted in this course makes plain the similarities between these different
areasofmathematics. Ofcourse,thepolynomialfunctionsformamuchlessrichclassthan
theothers,butbyrestrictingourstudytopolynomialsweareabletodocalculusoverany
field: wesimplydefine
d Xa Xi DXia Xi(cid:0)1:
i i
dX
Moreover,calculationswithpolynomialsareeasierthanwithmoregeneralfunctions.
1Forexample,supposethatthesystem(1)hascoefficientsa 2kandthatKisafieldcontainingk.Then
ij
(1)hasasolutioninknifandonlyifithasasolutioninKn,andthedimensionofthespaceofsolutionsisthe
sameforbothfields.(Exercise!)
7
8 INTRODUCTION
Consider a nonzero differentiable function f.x;y;z/. In calculus, we learn that the
equation
f.x;y;z/DC (2)
defines a surface S in R3, and that the tangent plane to S at a point P D .a;b;c/ has
equation2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
@f @f @f
.x(cid:0)a/C .y(cid:0)b/C .z(cid:0)c/D0: (3)
@x @y @z
P P P
Theinversefunctiontheoremsaysthatadifferentiablemap˛WS !S0 ofsurfacesisalocal
isomorphism at a point P 2S if it maps the tangent plane at P isomorphically onto the
tangentplaneatP0D˛.P/.
Consideranonzeropolynomialf.x;y;z/withcoefficientsinafieldk. Inthesenotes,
weshalllearnthattheequation(2)definesasurfaceink3, andweshallusetheequation
(3)todefinethetangentspaceatapointP onthesurface. However,andthisisoneofthe
essentialdifferencesbetweenalgebraicgeometryandtheotherfields,theinversefunction
theoremdoesn’tholdinalgebraicgeometry. Oneotheressentialdifferenceisthat1=X is
notthederivativeofanyrationalfunctionofX,andnorisXnp(cid:0)1 incharacteristicp¤0—
thesefunctionscannotbeintegratedinthefieldofrationalfunctionsk.X/.
These notes form a basic course on algebraic geometry. Throughout, we require the
groundfieldtobealgebraicallyclosedinordertobeabletoconcentrateonthegeometry.
Additionalchapters,treatingmoreadvancedtopics,canbefoundonmywebsite.
The approach to algebraic geometry taken in these notes
Indifferentialgeometryitisimportanttodefinedifferentiablemanifoldsabstractly,i.e.,not
assubmanifoldsofsomeEuclideanspace. Forexample,itisdifficulteventomakesense
ofastatementsuchas“theGausscurvatureofasurfaceisintrinsictothesurfacebutthe
principalcurvaturesarenot”withouttheabstractnotionofasurface.
Untilthemid1940s,algebraicgeometrywasconcernedonlywithalgebraicsubvarieties
ofaffineorprojectivespaceoveralgebraicallyclosedfields. Then,inordertogivesubstance
tohisproofofthecongruenceRiemannhypothesisforcurvesandabelianvarieties,Weil
wasforcedtodevelopatheoryofalgebraicgeometryfor“abstract”algebraicvarietiesover
arbitraryfields,3 buthis“foundations”areunsatisfactoryintwomajorrespects:
˘ Lackingasheaftheory,hismethodofpatchingtogetheraffinevarietiestoformabstract
varietiesisclumsy.4
˘ Hisdefinitionofavarietyoverabasefieldk isnotintrinsic;specifically,hefixessome
large“universal”algebraicallyclosedfield˝ anddefinesanalgebraicvarietyoverk
tobeanalgebraicvarietyover˝ togetherwithak-structure.
Intheensuingyears,severalattemptsweremadetoresolvethesedifficulties. In1955,
Serreresolvedthefirstbyborrowingideasfromcomplexanalysisanddefininganalgebraic
varietyoveranalgebraicallyclosedfieldtobeatopologicalspacewithasheafoffunctions
thatislocallyaffine.5 Then,inthelate1950sGrothendieckresolvedallsuchdifficultiesby
developingthetheoryofschemes.
2ThinkofS asalevelsurfaceforthefunctionf,andnotethattheequationisthatofaplanethrough
.a;b;c/perpendiculartothegradientvector.Of/ off atP.
P
3Weil,Andre´.Foundationsofalgebraicgeometry.AmericanMathematicalSociety,Providence,R.I.1946.
4NordidWeilusetheZariskitopologyin1946.
5Serre,Jean-Pierre.Faisceauxalge´briquescohe´rents.Ann.ofMath.(2)61,(1955).197–278,commonly
referredtoasFAC.
9
Inthesenotes,wefollowGrothendieckexceptthat,byworkingonlyoverabasefield,
weareabletosimplifyhislanguagebyconsideringonlytheclosedpointsintheunderlying
topologicalspaces. Inthisway,wehopetoprovideabridgebetweentheintuitiongivenby
differentialgeometryandtheabstractionsofschemetheory.
