(Translations of mathematical monographs 185) Kenji Ueno - Algebraic Geometry 1_ From Algebraic Varieties to Schemes-American Mathematical Society (1999)

Prévia do material em texto

Selected Title s i n Thi s Serie s 
185 Kenj i U e n o , Algebrai c Geometr y 1 : Prom algebrai c varietie s t o schemes , 
1999 
184 A . V . Mel'nikov , Financia l markets , 199 9 
183 Haj im e Sato , Algebrai c topology : a n intuitiv e approach , 199 9 
182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d 
conservation law s fo r differentia l equation s o f mathematica l physics , 199 9 
181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups . 
Part 2 , 199 9 
180 A . A . Mi lyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d 
optimal control , 199 8 
179 V . E . Voskresenskii , Algebrai c group s an d thei r birationa l invariants , 
1998 
178 Mi t su o Morimoto , Analyti c functional s o n th e sphere , 199 8 
177 Sator u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8 
176 L . M . Lerma n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e 
Hamiltonian system s wit h simpl e singula r point s (topologica l aspects) , 199 8 
175 S . K . Godunov , Moder n aspect s o f linea r algebra , 199 8 
174 Ya-Zh e Che n an d Lan-Chen g Wu , Secon d orde r ellipti c equation s an d 
elliptic systems , 199 8 
173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l 
properties o f distribution s o f stochasti c functionals , 199 8 
172 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finit e groups . 
Part 1 , 199 8 
171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8 
170 Vikto r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c 
integrals, 199 7 
169 S . K . Godunov , Ordinar y differentia l equation s wit h constan t 
coefficient, 199 7 
168 Junjir o Noguchi , Introductio n t o comple x analysis , 199 8 
167 Masay a Yamaguti , Masayosh i Hata , an d Ju n Kigami , Mathematic s 
of fractals , 199 7 
166 Kenj i U e n o , A n introductio n t o algebrai c geometry , 199 7 
165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g 
problem i n Galoi s theory , 199 7 
164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis , 
1997 
163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I . 
Yadrenko, Probabilit y theory : Collectio n o f problems , 199 7 
162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d 
methods i n linea r statistica l models , 199 7 
161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c 
dynamics, 199 7 
(Continued in the back of this publication) 
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CO Transla t ion s o f 
° MATHEMATICA L 
MONOGRAPHS 
pmm4 
CO 
CO 
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| « w s » ^ 
Volume 18 5 
S Algebrai c Geometr y 1 
From Algebraic Varietie s 
to Scheme s 
*y? Kenj i Uen o 
Translated b y 
Goro Kat o 
| America n Mathematica l Societ y 
J/-? Providence , Rhod e Islan d 
10.1090/mmono/185
Editorial Boar d 
Shoshichi Kobayash i (Chair ) 
Masamichi Takesak i 
mam 1 
D A I S U K I K A ( A L G E B R A I C G E O M E T R Y 1 ) 
by Kenj i Uen o 
with financial suppor t 
from th e J a p a n Associatio n fo r Mathemat ica l Science s 
Copyright © 199 7 b y Kenj i Ueno . 
Originally publishe d i n Japanes e 
by Iwanam i Shoten , Publishers , Tokyo , 199 7 
Transla ted fro m th e Japanes e b y Gor o Ka t o 
2000 Mathematics Subject Classification. Primar y 14-01 . 
ABSTRACT. Thi s i s th e firs t i n a serie s o f thre e book s b y th e author , aime d a t 
introducing th e reade r t o Grothendieck' s schem e theor y a s a method fo r studyin g 
algebraic geometry . Thi s first boo k contain s th e definitio n an d mai n propertie s o f 
schemes, togethe r wit h necessar y materia l fro m th e theor y o f algebrai c varietie s 
and categor y theory . Th e autho r als o include s man y examples . 
The boo k i s aime d a t graduat e an d uppe r leve l undergraduat e student s wh o 
want t o lear n moder n algebrai c geometry . 
L i b r a r y o f Congre s s Ca ta log ing - in -Pub l i ca t io n D a t a 
Ueno, Kenji , 1945 -
[Daisu kika . English ] 
Algebraic geometr y / Kenj i Uen o ; translated b y Gor o Kato . 
p. cm . — (Translation s o f mathematica l monographs , ISS N 0065-928 2 ; 
v. 185 ) (Iwanam i serie s i n moder n mathematics ) 
Includes index . 
Contents: 1 . Pro m algebrai c varietie s t o scheme s 
ISBN 0-8218-0862- 1 (v . 1 : pbk . : acid-free ) 
1. Geometry , Algebraic . I . Title . II . Series . III . Series : Iwanam i serie s i n 
modern mathematics . 
QA564.U3513 199 9 
516.3/5—dc21 99-2230 4 
CIP 
© 199 9 b y th e America n Mathematica l Society . Al l right s reserved . 
The America n Mathematica l Societ y retain s al l right s 
except thos e grante d t o th e Unite d State s Government . 
Printed i n th e Unite d State s o f America . 
@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s 
established t o ensur e permanenc e an d durability . 
Information o n copyin g an d reprintin g ca n b e foun d i n th e bac k o f thi s volume . 
Visit th e AM S hom e pag e a t URL : http:/ /www.ams.org / 
10 9 8 7 6 5 4 3 2 0 6 0 5 0 4 0 3 0 2 0 1 
To the memory of Hisao Miyauchi 
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Contents 
Preface i x 
Preface t o th e Englis h Translatio n xii i 
Summary an d Goal s x v 
Chapter 1 . Algebrai c Varietie s 1 
1.1. Algebrai c Set s 1 
1.2. Hilbert' s Nullstellensat z 6 
1.3. AfEn e Algebrai c Varietie s 1 2 
1.4. Multiplicit y an d Loca l Intersectio n Multiplicit y 2 6 
1.5. Projectiv e Varietie s 2 9 
1.6. Wha t i s Missing? 3 8 
Summary 4 0 
Exercises 4 1 
Chapter 2 . Scheme s 4 3 
2.1. Prim e Spectru m 4 3 
2.2. Affin e Scheme s 5 1 
2.3. Ringe d Spac e an d Schem e 7 5 
2.4. Scheme s an d Morphism s 8 8 
Summary 9 6 
Exercises 9 7 
Chapter 3 . Categorie s an d Scheme s 10 1 
3.1. Categorie s an d Functor s 10 1 
3.2. Representabl e Functor s an d Fibr e Product s 11 5 
3.3. Separate d Morphism s 13 1 
Summary 13 6 
Exercises 13 6 
Solutions t o Problem s 13 9 
Solutions t o Exercise s 14 7 
Index 15 3 
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Preface 
It ha s ofte n bee n sai d tha t algebrai c geometr y i s a difficul t field 
in mathematics . Ther e certainl y wa s a tim e whe n algebrai c geom -
etry wa s a difficul t geometry . I n particular , th e theor y o f algebrai c 
curves o f the Italia n schoo l fro m th e lat e nineteent h centur y throug h 
the first hal f o f th e twentiet h centur y wa s indee d difficult . Intuitiv e 
arguments proceede d withou t rigorou s proofs . Legen d ha s i t tha t 
one o f th e leader s o f th e Italia n school , Enriques , onc e said , "I t i s a 
nobleman's wor k t o find theorems , an d i t i s a slave' s wor k t o prov e 
them. Mathematician s ar e noblemen. " Thei r sharpnes s o f intuitio n 
might wel l convinc e u s o f tha t legend , bu t i t wa s nearl y impossibl e 
for commo n mathematician s t o follo w th e arguments . 
The plan s t o provid e mathematicall y soli d foundation s fo r suc h 
intuitionistic algebrai c geometr y wer e carrie d ou t b y va n de r Waer -
den, Zariski , Weil , Chevalley , an d others , usin g abstrac t algebr a a s i t 
developed i n th e 1930's . Zarisk i an d Wei l provide d a foundatio n fo r 
algebraic geometr y fo r thei r tim e period . Base d o n thei r foundation , 
Weil was able to prove the Riemann hypothesi s fo r a n algebraic curv e 
defined ove r a finite field, establishin g th e closel y relate d theor y o f 
abelian varietie s ove r a field of positive characteristic , an d Zarisk i es-
tablished birationa l geometr y ove r a fiel d o f arbitrar y characteristic . 
The theorem s o f Wei l an d Zarisk i wer e amon g th e mai n result s o f 
their era . 
An importan t aspec t o f Grothendieck' s late r foundatio n wa s t o 
adapt th e categorica l approac h t o algebrai c geometr y b y rewritin g 
the ver yfoundations . Thi s vie w i s a n ultimat e exampl e o f Bour -
baki's "structurism. " Initiall y ther e wa s resistanc e t o acceptin g thi s 
approach. However , mor e algebrai c geometor s bega n t o appreciat e 
solving problems b y reaching the essence of the matte r thoroug h gen -
eralization. Nowadays , Grothendieck' s schem e theor y i s considere d 
as th e mos t natura l an d flexible theor y availabl e i n algebrai c geom -
etry. Grothendieck' s clai m tha t no t onl y a n objec t i n a n absolut e 
ix 
x PREFAC E 
situation, bu t als o a n objec t i n a relativ e situatio n mus t b e studied , 
has bee n considere d t o b e mos t natural , thank s t o th e usefulnes s o f 
the represen t able functo r theory . 
Let m e elaborat e o n my earlie r usag e o f "relative. " Conside r a 
simple exampl e o f a polynomia l wit h coefficient s i n integer s 
(1) f(xu...,x n) = 0 . 
We can conside r th e commo n zero s of this equatio n no t onl y a s ratio-
nal numbers , rea l numbers , o r comple x numbers , bu t also , throug h 
reduction o f equation (1 ) a t a prim e numbe r p (i.e. , with coefficient s 
in th e finit e fiel d Z/pZ) , w e can conside r th e commo n zero s o f equa -
tion (1 ) a s p-adic numbers . Furthermore , fo r a homomorphis m fro m 
the rin g o f integers t o a commutative rin g i? , we can regard equatio n 
(1) a s havin g coefficient s i n R. Throug h suc h relativ e consideratio n 
as above , th e natur e o f th e geometr y o f th e commo n zero s o f equa -
tion (1 ) become s clear . Eve n thoug h suc h a relativ e consideratio n 
had bee n mad e earlie r fo r individua l problems , i t wa s Grothendiec k 
who systematically introduce d suc h a vision to algebraic geometry , t o 
solve Weil' s conjecture s o n congruenc e zet a functions . H e obtaine d 
fruitful results , an d expande d hi s theor y i n "Elements de Geometrie 
Algebrique" (ofte n abbreviate d a s EGA) . However , muc h o f hi s in -
complete theor y ha s bee n publishe d a s semina r notes . On e ha s n o 
difficulties studyin g Grothendieck' s theory . 
This boo k develop s Grothendieck' s schem e theor y a s a metho d 
for studyin g algebrai c geometry . Ou r goa l i s t o develo p an d appl y 
scheme cohomolog y t o th e theorie s o f algebrai c curve s an d algebrai c 
surfaces. I n th e prefac e o f EGA , Grothendiec k eve n claime d tha t a 
knowledge of classical algebraic geometry may hinde r th e reader fro m 
studying schem e theory. I t migh t hav e been a necessary thing for hi m 
to sa y a t th e time , a s he wanted hi s radica l theor y t o b e understood . 
However, thing s ar e reverse d nowadays . On e canno t understan d an d 
apply schem e theor y withou t knowin g classica l algebrai c geometry . 
Therefore w e will not begi n wit h schemes , bu t rathe r w e will first de -
scribe the classical notion o f algebraic varieties, which was introduce d 
in th e mi d twentiet h century . 
The majo r par t o f thi s boo k wil l be devote d t o preparing fo r th e 
definition o f a scheme . W e wil l describ e shea f theor y fro m a n ele -
mental viewpoint , wit h a s fe w prerequisite s a s possible . Pro m thi s 
scheme-theoretic foundation , w e will urge th e reade r onwar d toward s 
a unifie d understandin g i n "Algebraic Geometry 2 " an d "Algebraic 
Geometry 3" , in which we study no t onl y algebrai c geometry bu t als o 
PREFACE xi 
the theor y o f complex analyti c spaces . Unfortunately , muc h prepara -
tion i s require d t o reac h thi s highe r view . I believe , though , tha t a 
careful reade r wil l have littl e difficult y i n understandin g thi s book . 
I a m thankfu l t o Yuj i Shimiz u fo r readin g th e manuscript , an d 
for hi s correction s an d advice . 
October 199 6 
Kenji Uen o 
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Preface t o th e Englis h Translatio n 
Algebraic geometry plays an important rol e in several branches of 
science and technology . Th e present boo k i s the firs t o f three volume s 
on scheme theory, the most natura l form of algebraic geometry. Thes e 
three volume s ar e writte n fo r non-specialists , t o explai n th e mai n 
ideas and techniques of scheme theory. Th e original Japanese volumes 
have bee n widel y accepte d a s introductor y book s fo r schem e theory . 
The author hope s the present Englis h edition wil l serve the same role. 
My special thanks are due to Professor Gor o Kato, who undertook 
the difficul t jo b o f translatin g th e Japanes e editio n int o English . I 
also expres s my sincer e thank s t o th e lat e Mr . H . Miyauchi , edito r 
of Iwanami Shote n Publisher . Withou t hi s constan t encouragement , 
the Japanes e editio n woul d neve r hav e appeared . 
May 199 9 
Kenji Uen o 
X l l l 
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Summary an d Goal s 
Algebraic geometr y i s a geometry o f figures which ar e defined b y 
equations. Projectiv e geometr y encourage d th e gradua l developmen t 
of algebrai c geometry . T o pu t i t simply , projectiv e geometr y i s th e 
geometry use d t o stud y th e propertie s o f figures tha t ar e invarian t 
under projectio n fro m a point . 
In th e abov e figure, whe n th e lin e fro m poin t O , th e origi n o f 
projection, t o a poin t P o n a plan e i s paralle l t o th e lowe r plane , 
the poin t P i s no t projecte d ont o th e lowe r plane . Thus , a poin t a t 
infinity wa s introduce d i n projectiv e geometry . Fo r example , i n two-
dimensional projectiv e geometr y th e totality o f points a t infinity , i.e. , 
a lin e a t infinity , need s t o b e added . Wit h thi s adjustment , ther e i s 
more geometric harmony tha n i n Euclidean geometry . Fo r instance , a 
parabola intersects with (mor e precisely, i s tangent to ) a unique poin t 
on the line at infinity . A hyperbola intersect s with two distinct point s 
on th e lin e a t infinity . A n ellipse , a parabola , an d a hyperbol a ar e 
essentially the same geometric object o n the projective plane , namely , 
an irreducibl e quadrati c curve . 
In coordinat e geometr y an d projectiv e geometr y i t i s importan t 
to stud y th e intersectio n point s o f tw o distinc t curves . Fo r example , 
let u s conside r a uni t circl e an d a lin e o n a plane . Th e point s o f 
XV 
xvi SUMMAR Y AN D GOAL S 
ellipse hyperbol a parabol a 
According t o th e locatio n o f the lin e a t infinity , 
the quadrati c curv e become s a n ellipse , 
a hyperbol a o r a parabola . 
intersection ca n b e obtaine d b y solvin g 
x2 + y2 = l, 
ax + by + c = 0 . 
They intersec t i f ther e ar e rea l solution s t o th e syste m o f equations , 
and they do not intersec t i f the system has complex solutions. Ther e is 
no real reason for distinguishing complex solutions from rea l solutions. 
Thus, i t i s more natura l t o consider geometr y ove r complex numbers . 
By considerin g projectiv e geometr y ove r th e comple x numbers , 
one obtain s comple x projectiv e geometry . I n comple x projectiv e ge -
ometry, a n irreducibl e quadrati c curve , lik e a n ellipse , parabola , o r 
hyperbola, alway s intersects with a line at tw o points ( a tangent poin t 
is counted a s two points) . 
In plane coordinate geometry, one can parameterize the unit circl e 
as 
fl-t2 2t \ 
The point (—1,0 ) cannot be expressed in this parametric presentation . 
However, i n th e projectiv e plane , th e poin t (—1,0 ) correspond s t o a 
point a t infinit y o n a projectiv e line , i.e. , one-dimensiona l projectiv e 
space. Thi s correspondenc e i s given ove r th e comple x numbers . 
Thus a n irreducibl e quadrati c curv e may b e considere d a s a one-
dimensional projectiv e line . Figure s whic h hav e a one-to-on e corre -
spondence throug h algebrai c equation s ca n b e identifie d i n th e sens e 
SUMMARY AN D GOAL S xvi i 
A uni t circl e x 2 - f y2 = 1 intersects th e lin e x + y = 2 at th e comple x 
points (l + ^ , 1 - ^ ) an d ( l - ^ , l + ^ ) . 
of algebrai c geometr y rathe r tha n projectiv e geometry . I n thi s sit -
uation, flexibility i n studyin g variou s figures i s increased . Thi s i s a 
simplifying effec t i n the stud y o f geometry . 
In comple x algebrai c geometry , a circl e i s a projectiv e lin e ex -
pressed i n a projective plane , and th e degre e of choice of presentatio n 
is within projective transformations . Tha t is , the presentation ca n b e 
an ellipse , a parabola , o r a hyperbola . Furthermore , thi s notio n ca n 
be extended t o a higher-dimensiona l projectiv e space . B y looking fo r 
an ultimat e generalization , on e reache s th e notio n o f Grothendieck' s 
scheme theory . Th e figur e (o r locus ) i n n-dimensiona l comple x affin e 
space C n define d b y 
(1) / a ( * l , . . . , * n ) = 0 , a e A , 
is considere d a s a presentatio n o f a n origina l figure. Wha t i s im -
portant i s neithe r th e equations , no r th e idea l J = {fa, a € A) i n 
the polynomia l rin g C[^ i , . . . , zn], bu t rathe r th e commutativ e rin g 
R = C[z i , . . . , zn]/J. Thi s commutativ e rin g structur e determine s 
the natur e o f the geometry . A figure define d a s i n (1 ) ca n b e consid -
ered a s the figur e determine d b y presenting R a s the quotien t rin g of 
the rin g C[z\, ..., z n] o f polynomials . A quotient rin g o f th e polyno -
mial ring can be presented i n various ways. Hence , various figures can 
be considere d a s differen t presentation s o f a geometri c object . Thus , 
SUMMARY AN D GOAL S 
point a t 
infinity 
y=*(x + l) 
Q-\l + t2,l + t 2) 
Correspondence betwee n a uni t circl e an d a line . Th e tangen t lin e 
at (—1,0 ) intersect s th e lin e x = 1 at a poin t a t infinity . 
it i s not unnatura l tha t on e should begi n t o construc t geometr y fro m 
a commutativ e ring . 
This boo k i s an introductor y boo k t o scheme theory . Th e theor y 
of schemes requires the knowledge of commutative rings , sheaf theor y 
and homologica l algebra . W e have trie d t o begi n fro m a s elementar y 
a leve l a s possible . Ne w concept s ar e introduce d on e afte r anothe r 
in thi s book ; thi s ma y caus e th e reade r t o hav e difficult y i n finding 
the essenc e of the geometry . Thi s volume, Algebraic Geometry 1 , will 
focus o n the notio n o f a scheme a s a local ringed spac e by using shea f 
theory. Thi s metho d enable s u s t o presen t th e theor y o f varietie s 
(manifolds) i n a systemati c way . I n particular , thi s metho d connect s 
to th e theor y o f comple x analyti c spaces . Thi s connectio n wil l b e 
treated i n Algebraic Geometry 3 . 
The mos t importan t ste p i n understandin g thi s boo k i s to mak e 
sheaf theor y b e secon d nature , an d the n convinc e yoursel f tha t a n 
affine schem e can be defined b y introducing th e shea f o f commutativ e 
SUMMARY AN D GOAL S xi x 
rings over th e prim e spectru m o f a commutative ring . The n yo u ma y 
consider tha t yo u understan d schemes . 
A goa l o f algebrai c geometr y i s no t th e introductio n o f schemes , 
but th e us e of schemes freel y t o stud y geometry . I n orde r t o take ful l 
advantage o f scheme theory , on e needs t o stud y variou s propertie s o f 
schemes in detail. Th e major par t o f scheme theory wil l be treated i n 
Algebraic Geometry 2 , and thi s boo k wil l serve a s a foundation . A s a 
preparation fo r Boo k 2 , we describe som e fundamenta l propertie s o f 
schemes i n Chapte r 3 , using th e languag e o f categorie s an d functors . 
This boo k shoul d b e considere d a s preparatio n fo r Algebraic Ge-
ometry 2 and 3 . Th e definition o f a scheme per se does not take up al l 
the pages of this book. Rathe r tha n get mountain sickness by taking a 
lift directl y t o the top , we have decided t o hik e up th e mountai n ste p 
by step. Eve n with our choice , the path ma y appea r t o be steep. Th e 
reader i s recommended t o find hi s o r he r ow n example s whe n a ne w 
concept i s introduced . W e also provid e variou s problem s throughou t 
this book , helpin g th e reade r t o thin k throug h thes e concepts . W e 
recommend tha t yo u rea d thi s boo k withou t undu e haste . 
We list a few notational assumption s that wil l be used throughou t 
our serie s Algebraic Geometry 1 , 2 and 3 . 
(i) A commutativ e rin g i s assume d t o hav e a n identity , denote d 
by eithe r 1 or 1R. 
(ii) A rin g homomorphis m / : R — » S i s assume d t o satisf y 
/ ( ! * ) = Is -
(iii) W e assum e tha t fo r an y elemen t m o f a n jR-modul e M w e 
have IRTU = m. 
(iv) When an arbitrary elemen t o f an .R-module M ca n be written 
as a linea r combinatio n o f finitely man y element s m i , . . . , mn i n M 
with coefficients i n R, the n M i s said to be a finite R-module. Namely , 
if ther e exist s a n epimorphis m fro m th e finite direc t su m R® n ont o 
M, the n M i s a finite i?-module . 
(v) For a commutative rin g R, i f an i?-module S i s a commutativ e 
ring suc h tha t fo r arbitrar y r e R an d a, b G S w e hav e r(ab) = 
(ra)b = a(rb), the n S i s said t o b e a n R-algebra. 
(vi) Whe n a n i^-algebr a S i s finitely generate d ove r i? , i.e. , whe n 
there exist s a n epimorphis m o f i?-algebra s fro m a polynomia l rin g 
R[xi,..., x n] ont o 5 , the n S i s sai d t o b e a finite (o r finitely gener-
ated) R-algebra. 
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CHAPTER 1 
Algebraic Varietie s 
As our preparatio n fo r schem e theory , we will describe th e classi -
cal treatment o f algebraic geometry ove r a n algebraicall y close d field . 
It was only after th e 1930' s that rigorou s foundations wer e establishe d 
for this classical theory. Becaus e of the preparatory nature , not al l the 
details will be presented i n this chapter . I n particular , Serre' s theory , 
which i s a shortcut t o scheme theory , require s sheaf theory , regardin g 
an algebraic variety over an algebraically close d field a s a local ringe d 
space. Sinc e shea f theor y i s explaine d i n Chapte r 2 , Serre' s theor y 
will no t appea r i n thi s chapter . I t make s a more elegan t theor y i f we 
begin wit h shea f theor y t o develo p ou r theory . However , considerin g 
the introductor y natur e o f thi s treatise , w e conside r i t i s bette r t o 
describe classica l algebrai c geometr y wit h th e fewes t possibl e prereq -
uisites. Eve n thoug h i t migh t b e natura l tha t on e shoul d firs t stud y 
projective varieties , becaus e o f ou r emphasi s o n th e connectio n t o a 
scheme, w e wil l focu s o n affin e varieties . W e briefl y wil l touc h upo n 
projective varietie s a s we discuss classica l geometry . 
In Chapte r 2 we will define a scheme a s a local ringed space . Th e 
reader i s aske d t o defin e a n algebrai c variet y a s a loca l ringe d spac e 
after havin g learne d th e definitio n o f a scheme . 
1.1. Algebrai c Set s 
Algebraic geometr y i s the geometr y o f forms determine d b y alge-
braic equations . I n the mos t naiv e case , i t i s nothing bu t th e geomet -
ric stud y o f al l the solution s o f equation s 
(1.1) fa(xi y...,xn) = 0 , a = l , . . . , i , 
with coefficient s i n the element s o f a field k. However , thi s i s a rathe r 
vague statement , sinc e simultaneous solution s o f (1.1 ) ma y no t exist . 
In fact , i f k i s the field R o f real numbers , th e equatio n 
(1.2) z ? + .- - + a£ + l = 0 
l 
10.1090/mmono/185/01
2 1. ALGEBRAI C VARIETIE S 
has n o solution s i n rea l numbers . Bu t i n comple x numbers , whic h 
are a n extensio n o f rea l numbers , (1.2 ) indee d ha s man y solutions . 
More generally, fo r an y algebraically close d field /c , one sees that (1.2 ) 
possesses man y solutions . Thi s i s because o f the ver y definition o f a n 
algebraically close d field; ever y nonzer o polynomial i n on e variabl e 
with coefficients i n k has a solution in k. I n fact, i n the case where k is 
an algebraically closed field, the totality of the solutions of (1.1) can be 
captured geometricall y (th e Hilbert Nullstellensatz) . Befor e we study 
the Hilber t Nullstellensatz , w e need t o introduc e som e terminology . 
Let k b e a n algebraicall y close d field. Th e totalit y o f n-tuple s 
( o i , . . . , an) o f element s o f fc i s denote d b y k
n, whic h i s calle d th e 
affine n-space ove r k. A s we shall see , affine n-space s k n ar e no t onl y 
n-dimensional vecto r spaces , bu t als o affin e varieties . Whe n k n i s 
regarded a s a n affin e variety , w e write A n o r A £ rathe r tha n k n. 
We denote th e se t o f al l the solution s i n k o f the syste m o f equa -
tions (1.1 ) b y V ( / i , . . . , / / ) , whic h i s calle d th e algebraic set o r th e 
affine algebraic set o f system (1.1) . Namely , 
V{fu---,fi) = { ( a i , . . . , a n ) e / c
n | / a ( a i , . . . , a n ) = 0, a = 1 , . . . , /} . 
On th e othe r hand , fo r a n arbitrar y elemen t o f the idea l I generate d 
by /i » • • • J /z m th e polynomia l rin g fc[#i,..., x n] o f n variables , w e 
have 
/ ( a i , . . . , o n ) = 0 , ( a i , . . . , a n ) G V ( / i , . . ., ft). 
This i s because / ca n b e writte n a s 
i 
j\X\,..., x n) = y ^ ga\Xi)..., x nj ja\x\,... , x n). 
Q ! = l 
Generally, fo r a n idea l J i n a polynomia l rin g k[x±,.. . ,x n ] , we 
define 
V(J) — {(6i , . . . , bn) £ k
n\ fo r a n arbitrar y elemen t g in J , 
0(&l,. . . ,&n)=O}. 
Then V(J) i s said t o b e the algebrai c set , o r th e affin e algebrai c set , 
determined b y th e idea l J . W e have th e followin g lemma . 
LEMMA 1.1 . When I — ( / i , . . . ,ft), we have 
V(I) = V(f 1,...,fl). 
1.1. ALGEBRAI C SET S 3 
PROOF. W e have already show n that V(fi, ..., fi) C V(I). Con -
versely, le t (61 , . . . , bn) G V(7). Then , sinc e f a G 7 , we have 
/ a ( 6 i , . . . , 6 n ) = 0 , a = 1 , . . . , / . 
That is , V(I)cV(f u...,fi). D 
By this lemma , th e algebrai c se t V ( / i , . . . , /z) determine d b y th e 
system o f equations (1.1 ) i s precisely th e algebrai c se t determine d b y 
the idea l 7 = (/1 , • •. , /z) i n th e polynomia l rin g fc[xi,..., x n] gener -
ated b y / 1 , . . . , // . Therefore , w e mostly stud y th e algebrai c se t V(7 ) 
determined b y th e idea l 7 , rathe r tha n th e syste m o f equations . 
For th e zer o idea l (0) , w e hav e V((0) ) = /c n, whic h i s therefor e 
an algebrai c set . W e wil l ofte n writ e thi s n-dimensiona l afrm e spac e 
over k a s AJJ. 
We next stat e Hilbert' s basi s theorem, whic h guarantees tha t th e 
algebraic se t determine d b y a n idea l i s reall y th e algebrai c se t o f a 
system o f equations . 
THEOREM 1. 2 (Hilbert' s basi s theorem) . Any ideal in the poly-
nomial ring k[x\, ...,xn] is finitely generated. That is, any ideal J 
can be written as 
J = (0i, - •• ,9i), 9a G k[xi,...yxn], a = l , . . . ,Z , 
for some I. • 
This theore m ca n b e generalize d t o th e cas e wher e th e fiel d i s 
replaced b y a Noetherian rin g R, i.e. , the polynomia l R[x\, ..., x n] i s 
a Noetheria n ring . 
EXAMPLE 1.3 . Conside r th e followin g algebrai c se t i n A| : 
V(x2 + y2 + l). 
When th e characteristi c o f th e fiel d k doe s no t equa l 2 (w e writ e 
char/c ^ 2) , ther e exist s a n elemen t i i n k suc h tha t i 2 — —1 . Le t 
X — ix an d Y — iy. The n th e equatio n x 2 + y 2 + 1 = 0 become s 
X2 + Y 2 — 1 = 0 . Defin e a ma p (p fro m A 2, to A 2 a s 
(ai ,a2) >- > (iai ,m 2) . 
Then V(x 2 + y 2 + 1 ) i s mapped t o V (x2 + y 2 - 1 ) throug h th e ma p 
¥>• 
On th e othe r hand , whe n char/ c = 2 , we hav e 
X2+2/2 + l - ( x + t / + l ) 2 . 
4 1. ALGEBRAI C VARIETIE S 
Consequently, w e hav e 
[V(x2 + y2 + l) = V(x + y + l). • 
PROBLEM 1 . An y algebraic set in 1-dimensional affin e space A^, except 
Ajt itself , consist s of finite points . 
Note tha t i f th e idea l 7 = ( / i , . . . , fi) associate d wit h (1.1 ) con -
tains 1 , i.e. , 7 = k[xi, ... ,x n], the n V(I) = 0 , namely , syste m (1.1 ) 
has n o solutions . O n th e othe r hand , i f 7 ^ k[x\ )... ,x n ] , the n w e 
have V(7 ) ^ 0 ; tha t is , (1.1 ) ha s a solution . Thi s latte r assertio n i s 
precisely th e Wea k Hilber t NuUstellensatz . W e wil l discus s Hilbert' s 
NuUstellensatz i n detai l i n th e followin g section . I n wha t follows , w e 
discuss fundamental result s on the correspondence between ideals and 
algebraic sets . 
PROPOSITION 1.4 . For ideals I,J,I\, in the polynomial ring 
k[xi,..., x n] over a field k, A G A, where A is allowed to be an infinite 
set, we have 
(i)V(i)uv(J) = v(mj), 
(hi) V{I) D V{J) for y/lCy/J, 
where X^AeA^ * denotes the ideal of k[xi,.. . ,x n] generated by 
{/A}AGA; an d V ? = { / E fc[xi,... , £ n ] | /
m G 7 for a positive inte-
ger m}\ here VI is called the radica l of I. 
PROOF, (i ) Notic e tha t 1/(7 ) D V(J) fo r 7 C J . Thi s i s because , 
for ( a i , . . . , an) G V( J ), a zero of all the polynomial s i n J i s certainly 
a zer o o f al l the polynomial s i n 7 . Pro m this , we hav e 
V(lnJ)DV{I) an d V{ln J)D V(J) . 
Therefore, 
y ( / ) u y ( j ) c ^ ( / n j ) . 
Conversely, choos e ( a i , . . . , an) G V(7 fl J ) . I f ( o i , . . . , an) £ V(-0 > 
there exist s a polynomia l / € 7 such tha t 
/ ( a i , . . . , a n ) ^ 0 . 
Then, fo r a n arbitrar y elemen t g(xi, ..., x n) G J, w e have h — f-g^ 
I f l J . Therefore , 
/ i (a i , . . . ,a n ) = / ( a i , . . . , a n )p ( a i , . . . , a n) = 0 . 
1.1. ALGEBRAI C SET S 5 
Hence g(a\, ..., a n) = 0 , whic h implie s ( a i , . . . , an) G V(J). Conse -
quently, 
v(inj)cv(i)uv(j)9 
completing th e proo f o f (i) . 
(ii) Sinc e J M C SAGA^A , w e have 
\AGA / 
Therefore, 
f]V(I,)Dv(j2h). 
/xGA \ A e A / 
For eac h A , write I\ i n term s o f generators : 
^A = (h\i, ... ,h\ mx). 
For ( a i , . . . , an) G f]XeA V(I\), w e have 
h\j(a1,...,an) = 0 , j = l , . . . , r a A . 
On th e othe r hand , {h\j}\ eA, i<j<m x generate s th e idea l YLxeA^-
Therefore, (a u...,an) e V($2 XeAI\). 
(hi) I t suffice s t o sho w tha t V(y/1) = V(I). Sinc e y/1 D I, 
V{\/l) C V{I). 
Let / G y/l. The n /m G J fo r some positive integer m. Fo r ( a i , . . . , a n) 
G V(7), w e hav e 
/ ( a i , . . . , a n )
m = 0 . 
Hence, / ( a i , . . . ,a n) = 0 . Tha t is , ( a i , . . . , an) G V(v^0- d 
COROLLARY 1.5 . Fo r finitely many ideals 7 i , . . . , / s m 
/c[xi,... ,xn], we have 
PROOF. On e ca n prov e thi s b y inductio n o n s. • 
In Propositio n 1.4(h) , A nee d no t b e finite, bu t i n genera l th e 
above corollar y hold s onl y fo r finitely man y ideals . W e wil l giv e th e 
following counterexample . 
6 1. ALGEBRAI C VARIETIE S 
EXAMPLE 1.6 . Le t c i , . . . , c n b e a countabl y infinit e collectio n 
of distinc t element s i n a fiel d k. Assum e tha t k i s an infinit e field . 
Consider th e ideal s 
Ij = (x-Cj), j = 1,2,... , 
in k[x]. It i s easy t o see that 
s 
ihn---fMjs=l[(x 
Therefore, th e equalit y 
oo 
3 = 1 
must hold . O n the other hand , w e have 
oo 
U ^ ) = KC2,..}, 
3 = 1 
and V((0) ) = A\. On e can choose ci,C2,.. . so that A\ ^ {ci,C2,.. . }. 
Then 
oo / o o \ 
\jv{i0)^v(n^* • ° 
3 = 1 \3 = 1 J 
1.2. Hilbert' s Nullstellensat z 
One must hav e V(I) ^ 0 i n order for an algebraic set V(I) i n A£ 
to have a geometric meaning . Th e next theore m assure s the nonemp-
tyness. Thi s theore m i s called th e Weak Hilber t Nullstellensatz . 
THEOREM 1.7 . If an ideal I in the polynomial ring k[xi,. .., x n] 
over an algebraically closed field k does not contain the identity (that 
is, I y£ k[x\ ,..., x n]), then V(I) ^ 0 .PROOF. Fo r an idea l / ^ fc[#i,... ,x n], ther e exist s a maxima l 
ideal m containing I. The n V(I) D ^(m). Therefore , i t i s sufficien t 
to prov e tha t V(m) ^ 0. S o we may assume tha t I i s maximal, m . 
For a maximal idea l m, its residue clas s ring fc[#i,..., x n]/m i s a field 
containing k. Sinc e k i s a n algebraicall y close d field , th e followin g 
lemma implie s k[x±, ... ,x n] = k. Therefore , th e residu e clas s Xj 
(modm) o f Xj for m determines a n element aj i n k. Namely , we have 
* c J i ) -
1.2. HILBERT' S NULLSTELLENSAT Z 7 
Xj — aj G m, since x3 — aj = 0 (modm). Henc e (x\ — a i , . . . , xn — an) 
C m. But (#i — a i , . . . , xn — an) i s a maximal ideal . W e obtain 
m = (x i -alj...,xn -a n). 
Therefore, 
V(m) = {(ai , . . . ,On)} . D 
Before w e prove th e lemma mentione d i n the proof, w e state a 
corollary o n maximal ideals . 
COROLLARY 1.8 . A maximal ideal of the polynomial ring 
k[xi,...,xn] over an algebraically closed field k has the following 
form: 
(xi - a i , . . . , x n - o n), a 3 e fc, j = 1 , . . . ,n. D 
Corollary 1. 8 is often calle d th e Weak Hilber t Nullstellensat z a s 
well. I t is crucial for the field k to be algebraically close d in Theorem 
1.7 and Corollary 1.8 . A s was mentioned earlier , i f k is R, Theore m 
1.7 is not true. 
PROBLEM 2 . A maximal ideal of the polynomial ring R[x] o f one vari-
able ove r the field R of real number s ca n be expressed a s either (x — a), 
a € R, or (x2 + ax + 6), a, 6 e R, a2 - 4 6 < 0. 
The followin g lemm a wa s needed i n the proof o f Theorem 1.7 . 
LEMMA 1.9 . If an integral domain R which is finitely generated 
over a field K [which need not be algebraically closed) is a field, then 
every element in R is algebraic over K. 
PROOF. Fro m th e assumption, ther e exis t zi,...,z m i n R suc h 
that 
(1.3) R = K[z 1,...,zrn]. 
We must sho w tha t z 1 ? . . . , zm are algebraic ove r K. Whe n m = 1, i f 
Z\ i s not algebraic ove r K, z\ i s transcendental ove r K. The n K\z\] 
is isomorphi c t o a polynomial ring . Thi s contradict s th e assumption 
that R i s a field. Henc e z\ i s algebraic ove r K. 
Next, let m > 2. For z\ G R, an extension field K(z\) i s a subfield 
of R. Therefore , w e can write 
R = K(zi)[z 2,-..,Zm]. 
Namely, R is generated b y the m — 1 elements z 2,..., z m ove r if(zi) . 
By th e inductiv e assumption , Z2,...,z m ar e algebrai c ove r K{z\). 
8 1 . ALGEBRAI C VARIETIE S 
Therefore, fo r each Zj there exists a polynomial fj(x) G if (zi)[x] wit h 
coefficients i n K{z\) havin g Zj a s a root . Multiplyin g b y a n elemen t 
of K(z\) i f necessary , w e may assum e tha t fj(x) ha s th e for m 
fj{x) = A 3{zx)x
n> + Bf\ Zl)x^-
1 
where Aj(zi),Bj {z\) G if[zi], j = 2 , . . . , m, / = 1 , . . . ,rij. Pu t 
m 
A{zl) = '[[A j(z1)eK[z1], 
and defin e a subrin g S o f R a s S = tf *i> A(Zl) 
(Since R i s a field, w e hav e 1/A(zi) 6 it! , an d S i s a subrin g o f R 
generated b y z\ an d 1/A{z\) ove r if. ) Then , fro m (1.3 ) w e hav e 
(1.5) / 2 = S[*2,...,*m] . 
Multiply (1.4 ) b y A(z 1)/AJ(zi) an d divid e the resul t b y A(zi). The n 
notice tha t 2̂ - is a roo t o f a moni c polynomia l 
g3{x) = x
n* + b^x^- 1 + bfx^' 2 + • • - - f bf'\ j = 2 , . . . , m , 
with coefficient s i n 5 . (I n commutativ e algebra , Z j i s sai d t o b e 
integral ove r S. A n arbitrar y elemen t o f R i s a roo t o f a moni c 
polynomial wit h coefficient s i n 5. ) Sinc e R i s a field, S i s als o a 
field, a s wil l be show n next . Le t a b e a nonzer o elemen t o f S. The n 
a - 1 € R an d a - 1 i s a roo t o f a moni c polynomia l wit h coefficient s i n 
S. Henc e we hav e 
a"1 + 6 1a"-
/+1 + 6 2a~
/+2 + • • • + b t = 0 , bj € 5 , j = 1 , . . . , L 
That is , 
l + b1a + b2a
2 + •• • +W =0 . 
Consequently, 
a"1 - - (6 i + b 2a + • • • + W "
1 ) 6 5 , 
i.e., the invers e a - 1 o f an arbitrar y nonzer o element a € S belong s t o 
S. Hence , 5 i s a field. 
1.2. HILBERT' S NULLSTELLENSAT Z 9 
If z\ i s transcendenta l ove r K, on e ma y regar d K[zi\ a s a poly -
nomial rin g ove r K. The n a n arbitrar y elemen t a i n K\z\, 1/A{z{)\ 
can b e writ te n a s 
If F(z{) an d A{z\) ar e prim e t o eac h other , on e canno t expres s a - 1 = 
Aiz^/Fiz,) a s 
ISP oe-)^W -
Namely, 5 = i f [zi , 1/A(zi) ] canno t b e a field. But , sinc e 5 i s a field, 
z\ mus t b e algebrai c ove r K. • 
PROBLEM 3 . Prov e the following statement use d in the proof of Lemma 
1.9: a n arbitrar y elemen t i n th e integra l domai n R = S[w±, ..., w{\ whic h 
is generated b y wi,. .. , wi i s integra l ove r S. 
Next w e introduc e som e mor e notation . Fo r a subse t V i n th e n -
dimensional affin e spac e A £ ove r a n algebraicall y close d field fc, defin e 
an idea l I(V) determine d b y V a s follows : 
(1.6) I(V) = { / G k[xu...,xn]\f(au . . . , on ) = 0 fo r 
an arbitrar y e l e m e n t . ( a i , . . . , a n) i n V} . 
On th e on e hand , fo r V determine d b y a n idea l J , i.e. , V(J), w e hav e 
(1.7) J C I(V(J)) 
by th e definition . However , V(f 2) = V(f) fo r / G fc[xi,... , x n ] . 
Hence, I(V(J)) = J nee d no t hold . Hilbert ' s Nullstellensat z clarifie s 
the relationshi p betwee n J an d I ( V ( J ) ) . 
T H E O R E M 1.1 0 (Hilbert ' s Nullstellensatz) . For an ideal J in the 
polynomial ring K[x\, ... , x n ] over an algebraically closed field k, we 
have 
I(V(J)) = yfj. 
PROOF. Fro m th e definitio n (1.6 ) w e clearly hav e y/1 C I(V(J)). 
Therefore, i t i s sufficien t t o prov e tha t / G y/J fo r / G I(V(J)), i.e. , 
/ m G J fo r som e positiv e intege r m. Le t XQ be a ne w variabl e an d 
let J b e th e idea l generate d b y 1 — x o / ( x i , . .. ,x n) an d J i n th e 
polynomial rin g k[xo, ... , x n ] i n n + 1 variables . I f V(J) ^ 0 , fo r 
( a o , . . . , a n) G V ( J) C A;
n+1 w e hav e ( a i , . . . , a n ) G V(J) sinc e J C J. 
10 1 . ALGEBRAI C VARIETIE S 
Then / ( a i , . . . ,a n ) = 0 . O n th e othe r hand , sinc e 1 — XQ/ G J, w e 
have th e contradictio n 
0 = 1 - a0 / ( a i , . . . , a n ) = 1 . 
That is , V{J) — 0 mus t hold . Therefore , b y Theore m 1.7 , w e obtai n 
J = k[xo, • • • , xn]. The n J contain s th e identit y 1 . Therefor e w e ca n 
write 
1 = ft(a 0,...,xn)(l-x0/(xi,...,xn)) 
+ ^29j(X0, • • • , Xn)fj(xi,. . . , X n), 
j = l 
h,gj G k[xo> • • • ,x n], fj G J . Substitut e 1 / / fo r XQ in th e abov e 
equation an d multipl y bot h side s o f th e equatio n b y a certai n powe r 
of / , t o ge t 
i 
J = / ^ 9j\X\t • • • •> xn)jj\x\^...,xn), gj G k[x\ y...,xn\. 
Consequently, f p e J. • 
Thanks t o thi s theorem, t o study th e algebrai c set s V{J) w e may 
focus onl y o n ideal s satisfyin g J = \fj. Ideal s wit h th e propert y 
J = \fj ar e calle d reduced ideals. 
EXERCISE 1.11 . I f subset s V an d W i n A£ satisf y V D W, prov e 
we have tha t J(V ) C I(W), and , furthermore , tha t 
for V = l / (J i )i W^^Mz) -
PROOF. B y the definitio n (1.6 ) o f I(V), i t i s clear tha t / G I(V) 
for / G / (W). Therefore , i f V = K(Ji ) D W = F(J 2 ) , the n Theore m 
1.10 implie s 
V ' J I = J(V(Ji) ) c / (F(J 2 ) ) = v ^ -
Suppose \ / J i = A/^2 ; the n V = W. Hence , fo r V ^ W w e obtai n 
EXERCISE 1.12 . Whe n a n algebrai c se t V(I) equal s neithe r 0 
nor th e entir e n-dimensiona l affin e spac e AJ J itself , prov e tha t it s 
1.2. HILBERT' S NULLSTELLENSAT Z 11 
complement 
V(I)C = A] \ V(I) 
— { (a i , . . ., an) G AJJ | there exist s / G / 
such tha t / ( a i , . . . , an) ^ 0 } 
is not a n algebrai c set . 
PROOF. Suppos e there exists an ideal J i n k[x\,..., x n] satisfyin g 
V(I)C = V(J). Then , sinc e V(I)UV(J) = A£, byProposition 1.4(i) , 
we must hav e 
v{i n J) = v{i)uv{J) = A%. 
Prom Hilbert' s Nullstellensatz , w e hav e y/I D J = (0) . The n th e 
definition o f th e radica l implie s / n J = (0) . I f J ^ (0 ) an d J ^ (0) , 
there ar e polynomial s / an d g suc h tha t / G / an d g € J , / , # ^ 0 . 
Then we have / • g ^ 0 and / • # G In J , whic h contradic t I n J = (0) . 
Therefore, eithe r 7 = (0 ) o r J = (0) , whic h implie s V{I) = A £ o r 
V(I) = 0 , contradictin g ou r assumption . Tha t is , ther e doe s no t 
exist a n idea l J satisfyin g V(I) C = V(J). D 
EXERCISE 1.13 . Sho w tha t th e totalit y o f th e complement s o f 
algebraic set s i n a n n-dimensiona l affin e spac e A£ , 
O = {V(I) C\I i s an idea l o f fc[#i,... , £n]}, 
has th e followin g properties : 
(1) 0 G O an d A £ G O. 
(2) O i n 0 2 G 0, provide d O x £ O an d <9 2 € (9 . 
(3) UAG A °A € O , fo r O x G O an d A G A . 
PROOF. (1 ) Sinc e V(0 ) = A £ an d V(k[x ll... ,x n]) = 0 , w e have 
0 = V(0)
c G 0 an d A £ = V(k[x u . . . , xn])
c G O. 
(2) I f 0 1 = F ( J i )
c an d 0 2 G F(J2)
C , the n 
Oi n o2 = (v(JO u y(J2))
c = (F(Ji n J2))
c G 0. 
(3) I f O x = V{J X)
C, w e have 
{JOx=\JV(Jx)c=(f]V(Jx)) =v(^Jx) GO . D 
AGA A(E A \AE A / \AG A / 
Prom Exercis e 1.13 , b y definin g a subse t O t o b e a n ope n se t i f 
O G (9 , a topolog y ca n b e induce d o n AJJ . Thi s topolog y i s calle d 
the Zariski topology of A£. Then , a closed se t i s just a n algebrai c set . 
One ca n obtai n th e Zarisk i topolog y o n a n algebrai c se t V(I) a s th e 
12 1. ALGEBRAI C VARIETIE S 
topology induce d b y th e Zarisk i topolog y o n AJJ . Namely , defin e a 
subset U o f V(I) t o b e ope n i f ther e i s a n ope n se t O in A £ suc h tha t 
O f l V(I) = U. Th e Zarisk i topolog y i s not Hausdorff , bu t i t is a n 
important topology . W e wil l discus s th e Zarisk i topolog y i n detai l i n 
the nex t chapter , o n schemes . 
PROBLEM 4 . Prov e tha t a n arbitrar y close d se t in a one-dimensional 
affine spac e Aj . wit h th e Zarisk i topolog y consist s o f finit e points , beside s 
an empt y se t an d A\ itself. Tha t is , an open se t i s the complemen t o f finite 
points. Prov e als o tha t fo r an y tw o point s a and b in k, arbitrar y ope n set s 
0 i an d 0 2 satisfyin g a E Oi an d 6 6 02 mus t intersect , i.e. , 0 i D 02 / 0 . 
(Namely, th e Zarisk i topolog y o n k does no t satisf y Hausdorff' s axio m o f 
separation. Thi s mean s tha t ther e ar e no t enoug h ope n set s i n th e Zarisk i 
topology. Fo r example , whe n k — C, a Zariski ope n se t i s a n ope n se t fo r 
the usua l topolog y induce d b y th e metri c space , bu t a n ope n set , e.g. , th e 
open dis c {z E C| \z\ < 1}, in th e usua l topolog y i s no t a n ope n se t i n th e 
Zariski topology. ) 
1.3. Affin e Algebra i c Variet ie s 
An algebrai c se t V in an n-dimensiona l affin e spac e A £ ove r an 
algebraically close d fiel d k is said t o b e reducible whe n V is a unio n 
of algebrai c set s V\ and V 2 > 
V = Vi U V2, V ^ V 1 an d V ^ V2. 
When a n algebraic se t i s not reducible , i t is said t o be irreducible. 
An irreducibl e algebrai c se t i s said t o be a n affine algebraic variety. 
Let u s fin d a conditio n fo r a n algebraic se t t o be irreducible . I f an 
algebraic se t V(J) i s reducible , i t can b e expresse d as 
(1.8) V(J) - V{J ±) U V ( J2 ) , V(J) ^ VX7i) , V(J) ± V(J 2). 
Hence, w e hav e V{J) ^ V(J j ) , j = 1,2 . The n fro m Exercis e 1.11 , w e 
obtain 
(1.9) yfj = nv(j))$i{v(jj)) = y/j;. 
Therefore, ther e ar e polynomial s / 1 an d f 2 suc h tha t fj G y/7j, bu t 
fj £ \[J•> j = 1,2 . B y (1.9 ) w e mus t hav e fi - f2 € y/J. Namely , 
\fj i s no t a prime ideal . A s a consequence, th e followin g propositio n 
becomes clear . 
P R O P O S I T I O N 1.14 . An algebraic set V is irreducible if and only 
if the ideal I(V) associated with V is a prime ideal. 
1.3. AFFIN E ALGEBRAI C VARIETIE S 13 
PROOF. W e hav e alread y show n tha t I(V) i s no t a prim e idea l 
for a reducibl e algebrai c se t V. Tha t is , V i s irreducibl e i f I(V) i s 
a prim e ideal . Next , le t V b e irreducible . Suppos e I(V) wer e no t 
a prim e ideal . The n ther e woul d exis t polynomial s f\ an d S2 wit h 
/i>/2 £ I(V) a n d / l • Si £ I{V)- Le t J i b e th e idea l generate d b y 
I(V) an d / i , an d le t J 2 b e the idea l generated b y I(V) an d / 2 . Sinc e 
V(Ji) C V an d I/(J 2) C v : 
But / i • /2 £ /(V ) mean s tha t a t eac h poin t ( a i , . . . , an) o n V , / i o r 
/2 become s 0 . The n w e would hav e 
V = V(J 1)UV(J2), 
which contradict s th e irreducibilit y assumption . Tha t is , I(V) mus t 
be a prim e idea l i f V i s irreducible . D 
Incidentally, th e zer o idea l (0 ) i s a prim e idea l o f the polynomia l 
ring fc[xi,..., x n). Therefore , th e affine spac e A£ is an affine algebrai c 
variety. W e will often denot e th e affin e spac e A £ = k n simpl y b y A n . 
One-dimensional affin e spac e A 1 i s said t o b e a n affine line, an d two -
dimensional affin e spac e A 2 i s said t o b e a n affine plane. 
EXAMPLE 1.15 . A principa l idea l / = (F) i n k[xi, ... ,x n] i s a 
prime idea l onl y whe n th e polynomia l F i s irreducible . The n V(F) 
is sai d t o b e a n affine hypersurface i n A n. I f th e degre e o f F i s 
m, the n V(F) i s said t o b e a n m-dimensional hypersurface. Fo r th e 
cases n = 2 an d n — 3 , V(F) i s sai d t o b e a n affine plane curve 
and a n affine surface^ respectively . A n affin e hypersurfac e V(F) i s 
irreducible i f an d onl y i f th e polynomia l F(x\, ... ,x n) i s irreducibl e 
in fc[xi,...,x n]. D 
For a n algebrai c se t V i n a n n-dimensiona l affin e spac e A£ , th e 
set 
k[V]:=k[xu...,xn]/I(V) 
is calle d th e coordinate ring o f V. Propositio n 1. 4 ca n b e rephrase d 
as follows . 
COROLLARY 1.16 . An algebraic set V is irreducible if and only if 
its coordinate ring k[V] is an integral domain. • 
EXAMPLE 1.17 . Th e coordinat e rin g o f a quadrati c curv e 
C = V{x 2 + y 2 - 1 ) 
14 1. ALGEBRAI C VARIETIE S 
in a n affin e plan e A | i s given b y 
k[C\ = k[x,y]/{x 2+y2-l), 
where cha r A: ^ 2 . Le t i b e a n elemen t o f k suc h tha t i 2 = — 1, an d 
put w = x + z?/ and v = x — iy. The n the coordinate ring k[C] becomes 
k[C} = k[u,v)/(uv-l). 
Notice tha t thi s coordinat e rin g i s isomorphic t o k[u, 1/u]. 
The abov e chang e o f variable s ca n b e phrase d i n term s o f com -
mutative rin g theor y a s a n isomorphism : 
ip: k[x,y]/(x 2 + y2 -1) -^ k[u 9l/u], 
x »- » ^(w + 1A0 , 
y »-> h( u-llu)' 
One ca n observ e tha t th e invers e o f this isomorphis m i s given b y 
y r 1 : fc[u, 1/u] ^ k{x,y]/(x 2+y2-l), 
u H- » x -\- iy. 
Incidentally, i f cha r A; = 2 , the n th e coordinat e rin g k[C] of C = 
V(x2 +1/ 2 — 1 ) i s given b y 
k[C] = k[x,y]/(x + y- 1 ) ~ fc[a;]. 
This i s because , i n char/ c = 2 , we have 
x2 + 2 / 2 - l = ( x + 2 / - l ) 2 . • 
PROBLEM 5 . Prov e that th e coordinate ring k[C] o f a quadratic curve 
C = V(y — x 2) i s isomorphic t o th e polynomia l rin g k[x] of one variable 
over k. 
Generalizing th e concep t o f a chang e o f variable s a s i n Exam -
ple 1.17 , on e ca n defin e a morphis m betwee n algebrai c set s V an d 
W. Th e technica l ter m morphism i s used a s terminolog y fo r a ma p 
in algebrai c geometry . I f a set-theoreti c ma p fro m a n algebrai c se t 
V C A ™ to a n algebrai c se t I f c A J ca n b e expresse d i n term s o f 
polynomials, the map is said to be a morphism from V t o W. Namely , 
for th e coordinat e ring s 
k[V]=k[xu...,xm]/I{V), 
k[W]=k[yx,...,yn]/I(W) 
1.3. AFFIN E ALGEBRAI C VARIETIE S 15 
of V an d W , respectively , a ma p cpfro m V t o W i s sai d t o b e a 
morphism fro m th e algebrai c se t V t o th e algebrai c se t W i f ip can 
be expresse d a s 
(1.10) y 3 = / j (x i , . . . , a ; m ) € fc[a;i,...,z m]. 
That is , fo r a poin t P = ( a i , . . . , a m ) o n V , th e coordinate s o f th e 
image o f P unde r <p ar e expresse d a s 
<p((ai,... ,am)) = ( / i ( a i , . . . , a m ) , . . . , / n ( a i , . . . ,a m)) 
in terms of polynomials. However , there are algebraic relations amon g 
a i , . . . , a m. Henc e th e expressio n i n term s o f th e polynomial s i s no t 
uniquely determined . Fo r example , i f there i s a relatio n lik e 
(ax)
2 = a 2, 
for / (# , 2/) = XT / and p(a; , y) = x 3 w e ge t 
f(aua2) = 0 1 -a 2 = a ? = ^ ( a i , a 2 ) . 
Therefore, th e definitio n (1.10 ) i s not precis e enough. W e will give an 
accurate definitio n later . Conside r th e followin g simpl e example . 
EXAMPLE 1.18 . Conside r a curv e o f degre e thre e a s follows : 
C = V(y 2-x3)GAl 
Let u s denot e th e coordinat e rin g o f the affin e lin e A 1 b y 
^[A1] =k[t}. 
The coordinat e rin g o f the affin e plan e A 2 i s given b y 
fc[A2] = fc[*,y]. 
Then, 
(1.11) x = t 2 an d y = t 3 
determine a map from A 1 t o C. Namely , fo r a point a on A1, conside r 
the poin t (a 2, a3) i n A 2, whic h i s on C. Therefore , w e have a ma p 
V? : A1 - > C , 
a ^ (a 2, a3). 
Then < p i s a morphism fro m A 1 t o C. 
The ma p (1.11 ) determine s a ma p (p from A 1 t o A 2 a s 
o \-> (a 2,a3). 
16 1. ALGEBRAI C VARIETIE S 
Then dp is a morphis m fro m A 1 t o A 2 , an d th e imag e ^ (A 1 ) o f dp i s 
contained i n C. 
Note tha t a morphis m cp : A 1 — > C define d b y (1.11 ) induce s a 
ring /c-homomorphis m <p # fro m th e coordinat e rin g k[C] o f C t o th e 
coordinate rin g k [A1] o f A 1 a s follows : 
<p# : k[C] = k[x,y}/(y 2 - x 3) - fc^ 1] = fc[t], 
7 ( ^ J = / ( x , y ) ( m o d ( y 2 - a ;
3 ) ) . - f(t 2,t3). 
Furthermore, th e followin g rin g /c-homomorphis m i s also induced fro m 
(1.11): 
(p* : k[A 2} = k[x,y] - • /c[A x] = jfe[t] , 
/ ( x , y ) ~ / ( t 2 , * 3 ) . 
Observe tha t (p#{f{x,y)) = f{t 2,t3), a n d k e r y ? # = ( y 2 - x 3 ) . Fo r th e 
canonical surjectio n 
6 # : fc[x,y] - > /c[x,y]/(y 2 - x 3 ) , 
we hav e djft = <pft o t # . • 
PROBLEM 6 . Prov e tha t th e abov e (p : A1 — » C i s a set-theoreti c bisec -
tion, an d tha t th e homomorphis m ^ # : /c[C] —> • A;[A1] i s injective , bu t no t 
surjective. 
Here i s anothe r example . 
E X A M P L E 1.19 . Fo r algebrai c set s 
E = V{y 2 - x 3 + 1 ) C A 2 , D = V((x 32 -x
3 + l,x 3- x\)) C A
3 , 
the mappin g give n b y 
(1.12) xi = x, x 2 = y, x 3 = x
2 
defines a morphis m ip from E t o D. Le t I — (x2 — x\ + 1, x% — x\) an d 
J = [y 2 — x3 + 1) . The n (1.12 ) induce s a /c-homomorphis m betwee n 
the coordinat e ring s 
^ # : k[D] = k[x ux2,x3]/I - • *[£ ] - /c[x,y]/J , 
^ ( ^ i , x 2 , x 3 ) H- > g(x,y,x
2). 
Then ^ i s a set-theoreti c bijection , an d ^ i s a ring-theoreti c isomor -
phism. 
1.3. AFFIN E ALGEBRAI C VARIETIE S 17 
Furthermore, (1.12 ) als o determine s a morphism fro m A 2 t o A 3: 
i>: A 2 - > A 3, 
(a,6) H- > (a,6,a 2), 
and the corresponding /c-homomorphis m betwee n the coordinate ring s 
is given b y 
i># : /c[A3] = k[x ux2yxs] - > /c[A
2] = fc[z,y], 
^(a ; i ,X2,X3) i- » g(x,y,x 2). D 
An elemen t o f the coordinat e rin g fc[V] of an algebrai c se t V ca n 
be regarde d a s a regula r functio n o n V. Fo r a ma p ip : V —> W 
between algebrai c set s V an d W an d a regula r functio n / o n W, i f 
we obtai n a regula r functio n / o ip on V induce d b y ^ , the n ^ i s 
said t o b e a morphism . Fo r a ma p ^ t o b e a morphism , whic h i s 
crucial i n algebrai c geometry , conditio n (1.10 ) need s t o b e satisfied . 
Then, t o / € k[W] ther e correspond s / o ip e k[V], whic h i s th e 
/c-homomorphism ip# : k[W] — > k[V] determine d b y (1.10) . 
Notice als o tha t (1.10 ) induce s a morphis m 
$: A m - > A n , 
( a i , . . . , a m ) »- * ( / i ( a i , . . . , am ) , . . . , / n ( a i , . . . , a m ) ) , 
and a coordinat e rin g /c-homomorphis m 
^ # : k[An] = % l 5 . . . , yn] -+ £[A
m] = k[x u . .. , sm ] , 
#(</l, • • • > Vn) ^ g(fl(xi, • • • , ^ m ), • . . , / n ( ^ l, • • , ^ m ) )-
Therefore, th e morphis m V 7 from the algebrai c variet y V t o th e alge -
braic variety W ma y be considered a s the restriction o f the morphis m 
ip : Am — » A n t o V , where a morphism betwee n affin e space s i s given 
as i n (1.10 ) i n term s o f polynomials . 
Even thoug h w e hav e clarifie d th e definitio n o f a morphis m be -
tween algebrai c sets , ther e i s stil l somethin g unnatura l abou t thi s 
definition. O n on e hand , i t wa s necessar y t o conside r th e morphis m 
between affin e space s containin g th e algebrai c sets . I f a morphis m 
ip from a n algebrai c se t V t o a n algebrai c se t W i s a set-theoreti c 
bijection, an d tp# : k[W] — > k[V] i s a /c-algebr a isomorphism , the n 
the morphis m ip : V — > W i s sai d t o b e a n isomorphism. W e sa y 
V an d W ar e isomorphic. Isomorphi c algebrai c set s ca n b e regarde d 
as algebraic-geometricall y th e same . I n thi s view , i t i s desirabl e t o 
obtain a definition o f a morphism i n terms o f an algebrai c se t an d th e 
18 1. ALGEBRAI C VARIETIE S 
coordinate rin g alone . Fo r this purpose , w e shall stud y th e connec-
tion betwee n point s on an algebraic se t V and maximal ideal s of the 
coordinate rin g k[V] of V. 
To a poin t ( a i , . . . , an) on an algebraic se t V C A
n, there cor -
responds a maxima l idea l [x\ — a i , . . . , xn — a n) o f k[x\ y... ,x n ] . 
Let u s denote th e residue clas s o f a^ by Xj in the coordinate rin g 
k[V] = fc[xi,... , xn]/i~(V). The n (x i — a i , . .. , xn — an) is a maximal 
ideal of fc[V]. Conversely, fo r a maximal idea l m in fc[V], the inverse 
image ^ _ 1 (m) o f m under th e canonical epimorphis m 
if; : k[xu...,xn] - > A;[xi, . . . ,x n]/ /(F) 
is a maximal idea l of the polynomia l rin g k[x\, ..., x n]. B y Corollary 
1.8, on e can write 
^~ 1 ( m ) = ( xi - ^ i , - . - , ^ n - b n ) . 
We will show tha t (&i,... , 6n) G V\ It is sufficient t o show tha t 
(xi -bi,...,x n-bn)D / (V) . 
Since 0 € m and ^(O) = I(V), we obtain V ^ O) D ^_ 1 (0) = 7(F) . 
For a commutativ e rin g i? , we denote th e totality o f maximal 
ideals of R by Spm.R, an d call i t the maximal spectrum o f R. Fro m 
the precedin g paragrap h w e obtain the following fact . 
PROPOSITION 1.20 . For an algebraic set V, there exists a one-to-
one correspondence between the points on V and the maximal spec-
trum Spmfc[V] . For the coordinate ring 
k[V] = k[x 1,...,xn]/I(V), 
a point ( a i , . . . , an) on V corresponds to the maximal ideal of k[V], 
determined by [x\ — a\ ,..., x n — an). • 
Let us ask a question: Fo r a given morphism ip : V —> W between 
algebraic sets , how is a ring homomorphis m induce d b y pi Whe n p 
is given by (1.10), i.e., 
<p((au . . ., am)) = ( / i ( a i , . . . , a m ) , . . . , f n{au . . . , am)) , 
/ j ( a i , . . . , a m ) € k[xi,...,xm], j = l , . . . , n , 
a /c-homomorphis m betwee n the coordinate ring s is given as 
(1.13) 
V* : *[W ] = k\y u..., y n]/I(W) - k\V] = k[x lt...,xm)/I{V), 
g(yi,• • •,yn) ^g(fi(xi,...,x m),...,fn(xi,...,xm)) (modI(V)). 
1.3. AFFIN E ALGEBRAI C VARIETIE S 19 
Let m a b e the maximal idea l of k[V] determined b y (xi — a\,..., x m — 
am). The n ip^~
1(ma) i s a maxima l idea l o f k[W]. Thi s i s becaus e 
fc[V]/ma = &; , and th e induce d A:-algebr a homomorphis m 
k[W]/if*-l{ma) - k[V]/m a = k 
satisfies k[W}/(p^ z~1(ma) = k. Thank s t o th e /c-homomorphis m 
(1.13), ^ # - 1 ( m a ) i s a maxima l idea l o f k[W] generated b y {y\ — 6 1? 
• • • ,2/n ~ K), wher e b 3 = f 0(au . . . , a m ) , j = 1 , . . . ,n , sinc e 
PROBLEM 7 . Fo r / (x i , . . . ,x m) G fc[cci,.. . ,£m] an d a j 6 fc, j = 
1,... ,77i, show that 
/(sci,... , zm) — / (a i , . . . ,am) £ (xi — ai,. . . , xm — am), 
and als o that 
# (x i , . . . ,%) £ (z i - a i , . . . , Xm - a m) 
if and only if g(a\,.. . , am) =0 . 
Prom what w e described above , we naturally expec t th e followin g 
proposition. 
PROPOSITION 1.21 . For a morphism (p from an algebraic set V 
to an algebraic set W, i.e., p : V — » W, there is induced a k-
homomorphism between the coordinate rings 
p* :k[W] ->k[V], 
and the inverse image ^" 1(xna) of the maximal ideal ma determined 
by a point ( a i , . . . , am) £ V is the maximal ideal of k[W] correspond-
ing to the point y?((ai,.. . ,am)) on W. 
Conversely, if a set-theoretic map p> : V — > W and a k-homo-
morphism pft : k[W] — • k[V] are given, and if, for an arbitrary point 
(o i , . . . , am) £ V, p^~
1(xna) is a maximal ideal corresponding to the 
point <p((ai,... , am)) , then p : V — » W is a morphism between the 
algebraic sets. 
PROOF. W e hav e alread y show n tha t a morphis m o f algebrai c 
sets ha s th e propert y state d i n thi s proposition . S o le t p b e a ma p 
from V t o W an d le t p# : k[W] — • k[V] b e a /c-homomorphis m 
satisfying th e propert y o f thi s proposition . Expres s th e coordinat e 
rings a s residue ring s o f polynomia l rings : 
k[W] = k[yi,. . .,y n}/I(W), k[V] = k[x u.- .,x m]/I(V). 
20 1 . ALGEBRAI C VARIETIE S 
The A;-homomorphism (p# is uniquely determined by the image ip^iVj) 
oiy3 = y 3 {modI{W)). The n le t 
¥>#(I/j) = fj(xu...,x m) (mod/(F)) , 
where fj £ k[xi,. .., # m ] . W e wil l show tha t th e ma p (p i s given b y 
V-+W, 
( a i , . . . , a m ) *- » ( / i ( a i , . .. , a m ) , . . . , / n ( a i , . . . , a m ) ) . 
The maximal ideal ma o f k[V] corresponding to ( a i , . . . , am) coincide s 
with (x\ — a i , . . . , xm — am ) , x ^ = Xj (mod/(V)) . Therefore , 
<P*(yj - fj{o>i, •••>flm)) = ^ ( x i , . . . , x m ) - / j ( a i , . . . , a m ) € m a 
implies 
^ # _ 1 ( m a ) = (y i - / i ( a i , . . . , a m ) , 
1/2 ~ / 2 ( a i , . . . , a m ) , . . .,y n - / n ( a i , . . . , a m ) ) . 
By th e assumption , th e maxima l idea l (p^~ l(ma) correspond s t o th e 
point <p((ai,... , am)) o n W . Consequently , 
V ? ( ( a i , . . . , O m ) ) = ( / l ( a i , . • • , f l m ) r - - » / n ( f l lr • • , 0 m ) ) - • 
Based o n th e abov e proposition , le t u s redefine th e notion s o f a n 
affine algebrai c variet y an d a morphism . 
DEFINITION 1.22 . A pai r (V , k[V]) consistin g o f a n algebrai c se t 
V an d its coordinate ring k[V] is said to be an affine algebraic variety, 
or simpl y a n affine variety. Whe n V i s irreducible, (V , k[V}) i s called 
an irreducibl e affin e variety . Furthermore , whe n a set-theoreti c ma p 
(p : V — > W o f algebrai c set s an d th e fc-homomorphism <p # : k[W] —> 
k[V] o f th e coordinat e ring s ar e give n an d satisf y <£^ _1(ma) = m* , 
(where b = <p(a) , a G V, an d m a an d m ^ ar e th e maxima l ideal s o f a 
and b in k[V] an d fc[W], respectively), th e pai r (<£>,</?# ) is said t o b e 
a morphism fro m (V , k[V]) t o (W , fc[W]). We write a morphism a s 
(<p,<p*):(V,k[V])^(W,k\W]). 
If </? i s bijective, an d i f (p i s a fc-isomorphism fro m fc[W] to k[V], the n 
the morphis m (y> , (p#) i s said t o b e a n isomorphism . D 
Note tha t i n th e definitio n o f a morphis m ((£>,(/?# ) o f affin e al -
gebraic varietie s a ma p (p : V — > W an d a fc-homomorphism yft : 
k[W] — > k[V] ar e i n th e reverse d direction . Thi s i s becaus e (p# i s 
regarded a s th e pull-bac k o f a functio n o n W t o a functio n o n V 
through th e ma p (p : V — > W . 
1.3. A F F I N E ALGEBRAI C VARIETIE S 21 
One migh t wonde r wha t i s ne w i n Definitio n 1.22 , an d wha t i s 
necessary fo r tha t mor e involve d definition . Firs t recal l tha t w e de -
fined a n algebrai c se t V a s th e subse t i n th e affin e spac e A n o f th e 
common zero s o f element s o f a n ideal . I n thi s sense , th e definitio n 
of V require s th e affin e space , o r equivalentl y a n idea l J definin g V , 
J C k[xi,...,x n]. However , a s a pair , i t suffice s t o conside r V a s 
a se t o f points , an d k[V] a s a commutativ e algebra , regarde d a s a 
/c-algebra. Precisel y speaking , we can regard affin e algebrai c varietie s 
(V, k[V]) an d (W, k[W]) a s the same i f the morphis m (</> , ip#) o f the m 
is an isomorphism . 
One ca n pus h th e abov e ide a furthe r t o conside r a k- algebra R 
and th e totalit y Sp m it! o f al l th e maxima l ideal s o f R. Ca n th e 
pair (Sp m R, R) b e calle d a n affin e algebrai c variety ? Thi s i s a fairl y 
abstract definition , becaus e ther e ar e n o equations an d n o geometri c 
forms. Th e reader migh t wonde r i f our original intention was to stud y 
the geometri c form s define d b y equations . Thi s mor e abstrac t defini -
tion stil l contains equation s i f the algebr a R i s finitely generate d ove r 
fc. When R i s finitely generate d ove r k, ther e i s an isomorphis m 
R ~ k[xi,. .. ,x n]/J 
between R an d a residu e rin g o f th e rin g o f polynomials . B y identi -
fying R wit h th e residu e ring , accordin g t o Propositio n 1.2 0 w e have 
(1.14) SpmR = V(J). 
To b e precise , J i s a reduce d ideal . W e prove d Propositio n 1.2 0 
under th e assumptio n y/j = J. Bu t (1.14 ) stil l hold s eve n whe n 
J i s no t a reduce d ideal . Ther e ar e indee d infinitel y man y way s t o 
express a ring R a s a residue ring . Fo r example , (1.14 ) indicate s tha t 
Spm R i s expressed a s an affine algebrai c set in an n-dimensional affin e 
space A J?. Considerin g Sp m R doe s no t mea n tha t w e are considerin g 
an algebrai c se t embedde d i n n-dimensiona l affin e space , bu t doe s 
indicate tha t on e can explicitl y stud y th e propertie s o f the geometri c 
form o f Spm R. A n explici t presentatio n o f a ring R a s a residue rin g 
of a polynomia l rin g correspond s t o a n embeddin g o f Sp m R int o a n 
affine space . 
PROBLEM 8 . Prov e that (1.14 ) holds for an arbitrary ideal J such that 
R = k[xi,... ,x n]/J-
Let u s consider suc h a generalized notio n (Sp m R, R) o f an affin e 
algebraic variety . I n wha t wil l follow we assume that the k-algebra R 
22 1. ALGEBRAI C VARIETIE S 
is finitely generated over k. Th e followin g lemm a wil l tel l u s ho w t o 
define a morphism o f generalized affin e algebrai c varieties . 
LEMMA 1.23 . For a k-homomorphism ip : S — » R between k-
algebras, the inverse image / 0_1(m) of a maximal ideal m of R is a 
maximal ideal of S. 
PROOF. Th e /c-homomorphis m ip induces a n isomorphis m int o 
the A:-algebra : 
1>:S/il>-1(m)-+R/m = k. 
Since k C S/ip' 1 (m) , ip is surjective. Namely , S/ /ip~1(m) i s a field. • 
By thi s lemma , a fc-homomorphism ip : S — > R o f /c-algebra s 
induces a ma p 
i\)a : SpmR - > SpmR, 
m i— » / 0 - 1(m). 
Therefore, w e have obtaine d th e followin g definition . 
DEFINITION 1.24 . Fo r a finitely generate d algebr a ove r a n alge -
braically closed field A;, the pair (Sp m i?, R) i s said to be an affine alge-
braic variety. Fo r affine algebrai c varieties (Spm f?, R) an d (Sp m 5, S) , 
a pai r (i/j a,ip) consistin g o f a fc-homomorphism tp : S — > R an d it s 
induced ma p 
ipa : S p m i ? ^ S p m S 
is said to be a morphism from (Sp m i?, R) t o (Spm S, S) , an d is written 
as 
(^ a ,^) : {SpmR,R) - > (Spm , 5,5) . 
For a n affin e algebrai c variet y (Sp m i?, it!), an elemen t o f R i s calle d 
a regula r functio n o n th e affin e algebrai c variety . 
Even though Definitio n 1.2 4 doe s no t see m t o diffe r fro m Defi -
nition 1.22 , th e followin g exampl e wil l sho w a crucia l difference : th e 
commutative rin g R i s allowed t o hav e nilpoten t elements . 
EXAMPLE 1.25 . Conside r R n = /c[x]/(x
n+1), n = 0,1 , Sinc e 
Rn ha s a uniqu e maxima l ideal , Spm.R n consist s o f onl y on e point . 
An element o f Rn ca n be regarded a s a polynomial consisting of terms 
of a t mos t degre e n , o r a Taylo r expansio n o f degre e n aroun d th e 
origin. Sinc e Spmi2 n i s a point , a functio n o n a poin t i n th e usua l 
sense needs to be a constant. However , th e pai r (Spm/?™ , Rn) shoul d 
1.3. AFFIN E ALGEBRAI C VARIETIE S 23 
be considered a s "functions " define d i n a neighborhood o f the origio n 
of degree n. 
For rt\ < ri2, we have a canonica l A;-homomorphis m 
V>niln2 : Rn2 = k[x]/(x
n*+l) - R ni = fc[x]/(a;
ni+1), 
and a morphis m 
( C x ,n 2 > ^ ni ,n 2 ) • ( S pm # n i I R ril ) -> ( S p m H n 2 , # n2 ) • • 
As we pointed ou t i n the paragraph followin g Definitio n 1.22 , th e 
directions o f th e map s i/j a and ip are i n revers e wit h eac h other . On e 
can "interpret " a functio n o n Sp m R a s a pull-bac k o f a functio n o n 
Spm 5 b y th e ma p ip a. 
Definition 1.2 4 differ s fro m Definitio n 1.2 2 i n the following sense . 
In Definitio n 1.24 , w e defined Spmi ? fro m a give n commutativ e rin g 
R. The n w e defined a n affin e algebrai c variety . Tha t is , the emphasi s 
is on function s rathe r tha n th e space . Th e underlyin g philosoph y i s 
that on e ca n kno w th e spac e i f on e know s th e functions . Thi s ide a 
leads u s t o th e notio n o f a ringe d space , whic h wil l b e discusse d i n 
the followin g chapter . 
The reade r migh t thin k tha t th e stud y o f commutativ e ring s i s 
sufficient fo r algebrai c geometry , becaus e i n Definition 1.2 4 w e bega n 
with a commutativ e rin g an d define d a morphis m betwee n maxima l 
spectra usin g a rin g homomorphism . A s fa r a s affin e algebrai c va -
rieties ar e concerned , i n som e sens e commutativ e ring s describ e ev -
erything. However , geometri c consideration s ofte n clarif y meaning s 
in commutativ e rin g theory . Namely , commutativ e rin g theor y i s a n 
important devic e fo r th e stud y o f algebrai c geometry . 
One define s a n algebrai c variet y b y gluein g affin e algebrai c vari -
eties. I n th e nex t chapter , th e notio n o f a n affin e algebrai c variet y 
will b e generalize d t o obtai n th e notio n o f a n affin e scheme . The n 
an algebrai c variety wil l be generalized t o obtain a scheme by gluein g 
affine schemes . I n what follows , w e will give examples o f algebraic va-
rieties. I n order t o glue affine algebrai c varieties , we need the concep t 
of a n ope n set . Fo r a n affin e algebrai c variet y (Spmi? , R), define , fo r 
feR, 
D(f) = {me Spm R\f£m}. 
The topolog y havin g {D(f)} a s basis element s o f open set s i s said t o 
be a Zariski topology on Spm R. Tha t is , a subset U of Spm R i s open 
24 1. ALGEBRAI C VARIETIE S 
when U ca n b e writte n a s 
U=\J D(f a). 
Denote th e complemen t o f D(f) b y V(f). The n w e hav e 
V(f) = {meSpmR\fem}, 
which i s a close d se t i n Sp m R. I n general , fo r a n idea l I o f R, se t 
D(I) = { m e Spm R\I £ m} . 
Then D(I) i s an ope n subse t o f Spmi?. Le t V(I) b e the complemen t 
of D(I). The n w e hav e 
V(I) = { m G S p m i ? | / C m } , 
which i s a closed subse t o f Spm R. 
PROBLEM 9 . Le t I b e an ideal of R. Prov e tha t 
D(I)=[JD(f) an d V(I)=f)V(f). 
fei fei 
Moreover, sho w that a n ope n se t U in Spmi ? ca n be written a s D(J) fo r 
some ideal J o f R, an d a closed set F ca n be expressed a s V(J). 
EXAMPLE 1.26 . Fo r a nonnilpoten t elemen t f £ R, conside r a n 
ideal ( 1 — ft) o f the rin g o f polynomial s ove r R. Le t 
(1.15) S = R\t}/(1 - ft). 
We also write S = R[l/f]. I f the finitely generate d R ove r k is written 
as 
R = fc[a;i,...,x n]/J, 
there i s a canonica l /c-algebr a homomorphis m 
i> : k[xi,... ,x n,t] ~ > S. 
Then fo r a maximal idea l m of 5, there i s determined ( a i , . . . , an, 6 ) G 
A:n+1 suc h tha t 
(1.16) ^ _ 1( tn) = (xi - a i , . . . , x n - a n,^ - b). 
Then w e hav e m = i/j(ip~ 1(m)). Le t m 7 b e th e maxima l idea l o f R 
determined b y {x\ — a i , . . . , xn — a n) . Pro m (1.15 ) an d (1.16) , w e 
conclude tha t 
(1.17) l = / 6 (modm') . 
1.3. A F F I N E ALGEBRAI C VARIETIE S 25 
That is , / ^ m' . Sinc e R/xa' = k> f £ m' implie s tha t ther e exist s a 
unique b € k so that (1.17 ) holds . The n the image of {x\ — a\,..., x n — 
an) unde r th e k-homomorphism ip is a maxima l idea l o f 5 . 
Consequently, w e obtai n a one-to-on e correspondenc e betwee n 
SpmS an d D(f) = {m ' e SpmR\f e m'} . Therefore , (D{f),S) ca n 
be regarde d a s an affine variety . • 
EXAMPLE 1.27 . Conside r tw o affine line s 
Uo = (A\k[xj) an d U ± = (A\k[y}). 
We can define th e structure o f an affine variet y on the open se t D(x) 
as i n Example 1.26 : 
U0i = (D(x),k[x,l/x}). 
Similarly, o n the open se t D(y) o f U\ we have a n affin e variet y 
U10 = (D(y),k{y,l/y}). 
A /c-isomorphis m o f rings 
ip : k[y,l/y] - * k[x,l/x], 
f(vMy) »f(x,i/x) 
induces a n isomorphis m o f affin e varietie s (ip a,ip) : C7"oi —> UIQ. B y 
glueing Uo and U\ through this isomorphism, we get a one-dimensional 
projective space ¥\ {projective line) ove r a field k. W e have D(x) = 
A1 \ {0 } and D(y) = A1 \ {0} , and, for b e D(x), ip a(b) = \ G D(y). 
Writing U\ = D(y)\J {co} , we have a s set s 
pi = A ^ j o o } . 
In th e case where k = C, let a sequence {b n} satisf y 
bn e D{x) = C \ {0} , li m \b n\ = +oo. 
n—>oo 
Then c n = l/b n i s the correspondin g poin t i n D(y) throug h tp
a sat -
isfying lim n_,oo \c n\ = 0 . Therefore , th e origio n o f U\ i s denote d a s 
CXD and calle d th e poin t a t infinity . Namely , i t look s a s thoug h i t i s 
located a t infinit y fro m th e view of Uo. • 
The reader may wonder what the coordinate ring of the projective 
line woul d be . W e will answe r thi s questio n later . I t doe s no t mak e 
sense to talk abou t the coordinate ring of an algebraic variety which is 
constructed b y glueing affine varieties . I n the next chapte r th e notion 
of a sheaf, instea d o f the coordinate ring , wil l play an important role . 
26 1. ALGEBRAI C VARIETIE S 
RQ f(x),g(x)ek[x], / ( o ) ^ 0 
1.4. Multiplicit y an d Loca l Intersectio n Multiplicit y 
We wil l provid e a brie f descriptio n o f th e loca l intersectio n mul -
tiplicity o f a curve . The n w e wil l describ e propertie s o f projectiv e 
varieties an d plan e curves . 
Let F b e a subfield o f an algebraically closed field k. A polynomial 
f(x) wit h coefficient s i n F ca n b e factore d a s 
m 
(1.18) f{x) = av\[{x-a 3)
n>, a 0 ^ 0 . 
3 = 1 
The multiplicit y o f a roo t aj o f f(x) = 0 i s rij. W e ca n captur e th e 
notion o f multiplicity i n term s o f ring theory a s follows . 
For a n elemen t a o f A: , consider a subse t 
of the quotient fiel d h(x) (i.e. , the field o f rational functions o f a single 
variable) o f the polynomial ring R — k[x\. The n R a i s a commutativ e 
ring, containin g R. (Not e that , a s we will show i n §2.2(b ) o f Chapte r 
2, R a i s th e localization o f R a t th e prim e idea l (x — a).) The n fo r 
3 £ Ra-
X — p 
Therefore, fo r a root aj o f f{x) = 0 , the ideal (/(#) ) o f Raj generate d 
by f(x) i s given a s 
(/Or)) = ((*-<*,•)**) , 
and 
(1.19) R aj/{f{x)) = R ai/((x-aj)
n'). 
The right-han d sid e o f (1.19) , a s a k~ vector space , ha s th e residu e 
classes o f 1 , x — aj,(x — aj) 2,..., (x — aj)71^1 a s basi s elements . 
Hence, 
(1.20) dim kROCj/(f(x))=n3. 
Thus, when th e root s o f the polynomia l (1.18 ) ar e known , th e multi -
plicities o f the root s ca n b e obtaine d ring-theoreticall y a s i n (1.20) . 
PROBLEM 10 . Prov e that, fo r the formal powe r series ring 
k[[x _Qiil] = \Yla^x~a^1 \ 
1.4. MULTIPLICIT Y 27 
of the variable x — otj, the ideal (f(x)) generate d by f(x) a s in (1.18) can 
be expressed as 
(/(*)) = ((*-a,-D, 
and 
dimfc k[[x - OLj\]/{f{x)) = n3. 
One may wonde r i f the abov e ide a can be generalized t o the case 
of a system of equations. Fo r simplicity's sake , consider the following 
system of two equations : 
(1.21) {/(*•* ) = 0, 
[g(xyy) = 0. 
Assume that ther e are no common factors betwee n f(x,y) an d g(x,y) 
£ R = k[x,y]. W e interpret (1.21 ) a s the intersecting point s o f the 
curves Cf. f(x, y) = 0 and Cg: g(x, y) = 0. Let P = (a , b) be a point 
of intersectio n o f the plane curve s Cf an d Cg. The n conside r a rin g 
in the quotient field k(x,y) o f k[x, y] as follows: 
o jG(x,y) 
F(x,y)>G(x,y) e R, F(a,b)^0\ 
Note tha t R C Rp an d Rp i s a local ring . Le t (/, g) b e the ideal of 
Rp generate d b y / an d g. The n defin e 
(1.22) Ip(C f,Cg) = dim kRp/(f,g), 
which is said to be the local intersection multiplicity o f Cf an d Cg at 
P = (a , b). Moreover , thi s loca l intersectio n multiplicit y Ip(Cf,C g) 
can be interpreted a s the multiplicity o f the solution (a , b) of (1.21). 
With th e following example s we will sho w tha t thi s definitio n (1.22 ) 
matches up with our intuition. 
For simplicity , replacin g x — a and y — b by x and y, respectively , 
we will consider th e case where O = (0,0) i s the intersection point . 
EXAMPLE 1.28 . When / an d g are both linear , 
f = az + /3y = 0, 
g = 7x + Sy = 0, 
with a6 — /?7 ^ 0 , Cf an d Cg intersec t a t the origin (Figur e 1.1). 
Then w e expect th e local intersectio n multiplicit y a t the origin 
to be 1. On the other hand , a t RQ w e have (f,g) = (x,y). Hence , we 
get 
Io(Cf,Cg) = 1, 
28 1. ALGEBRAI C VARIETIE S 
FIGURE 1. 1 
**\S > 
0\ x 
y | 
fy=x* 
FIGURE 1. 2 
as expected . 
EXAMPLE 1.29 . Le t n > 2 be a n integer . Conside r 
/ = y = o, 
gn = y - x
n = 0 . 
When n = 2 , the x-axi s (7 / an d C^ 2 hav e a double roo t a t th e origin , 
and when n = 3 , they hav e a triple root a t the origin (se e Figure 1.2) . 
Therefore, th e expecte d loca l intersectio n multiplicitie s ar e 2 and 3 , 
respectively. A t Ro w e hav e (/ , #n) — (y,z
n)- Therefore , on e ca n 
take th e residu e classe s o f 1 , x, x , . . . , x a s basi s element s fo r th e 
fc-vector space Ro/{f,g n)' Tha t is , we obtai n 
Jo (C/ ,C 9 J = n , 
as expected . • 
EXAMPLE 1.30 . Conside r 
/ = y - x = 0 , 
g = y 2-x2(x+l)=0. 
If / i s replaced b y / e — y — x — e,e £ k, the n C/ e intersect s C 5 a t thre e 
distinct points . A s e approache s 0 , thos e thre e point s approac h th e 
1.5. P R O J E C T I V E VARIETIE S 2 9 
F I G U R E 1. 3 
origin. Therefore , w e expec t tha t th e loca l intersectio n multiplicit y 
of Cf an d C g a t th e origi n shoul d b e three . A t i£o ,
 w e hav e 
(/,<*) = (y~x,y 2- x 2(x + 1) ) = {y- x , x 3 ) . 
Therefore, w e ca n tak e th e residu e classe s o f 1 , OC) x a s basi s element s 
for th e fc-veetor spac e Ro/{f,g), i.e. , Io(Cf,C g) =3. D 
PROBLEM 11 . Sho w that on e also gets the same local intersection mul -
tiplicities i n th e abov e thre e example s whe n Ro i s replace d b y th e forma l 
power serie s rin g fc[[x,2/]] of tw o variables . (I n general , on e obtain s th e 
same Ip(Cf,C g) whe n Rp i n (1.22 ) i s replaced b y k[[x — a,y — &]].) 
PROBLEM 12 . Comput e Io(Cf,C g) i n th e followin g cases . 
(1) f = y~2x, g = y 2 - x 2 ( x + l ) , 
(2) f^y 2-x
3, g = y 2 - x 2(x + l), 
(3) f = y 2-x
3, g = y~ax, a 6 k. 
1.5. P r o j e c t i v e Var ie t ie s 
We wil l begi n wit h th e definitio n o f a projectiv e spac e P £ o f di -
mension n ove r a n algebraicall y close d fiel d fc. A s analogou s object s o f 
an affln e algebrai c se t an d a n algebrai c variety , a projectiv e se t an d a 
projective variet y wil l be defined . W e wil l als o giv e a brie f descriptio n 
of a curv e i n th e projectiv e plan e P | . Al l th e object s i n thi s sectio n 
are define d ove r a fixe d algebraicall y close d fiel d fc. 
30 1. ALGEBRAI C VARIETIE S 
(a) Projectiv e Space . Le t W b e th e se t 
^ = fc" +1\{(0,...,0)}, 
i.e., ( n - f l)-dimensiona l affin e spac e k n+l minu s th e origin . The n 
define a n equivalenc e relatio n ~ o n W a s follows : 
( a 0 , a i , . ,.,a n)~ (b 0,bi,... , 6 n ) 
(1.23) <= » ( a 0 , a i , . . . , a n ) = (a& 0 ,a*i, . . . ,ab n) 
for som e elemen t a € k x = k\ {0} . 
One ca n chec k tha t ~ i s a n equivalenc e relatio n o n W. The n th e 
quotient spac e o f W b y thi s equivalenc e relation , i.e. , th e se t o f 
equivalence classes , i s th e n-dimensiona l projective space PjJ , i.e. , 
P™ = W/~. Le t (ct o : a\ : •• • : a n) b e th e equivalenc e clas s de -
termined b y (ao , . . . , an) . Th e class (a o : • • • : an) i s said to be a point 
in P£ . A s w e ca n see , a poin t (a o : • • • : an) i s uniquel y determine d 
by th e ratio s o o : • • • : an. Fo r n = 1 , P£ i s called th e projective line, 
and fo r n = 2 , P| i s called th e projective plane. 
Define subset s U 3;, j = 0 , 1 , . . . , n , o f P£ b y 
(1.24) tf, = {(a 0 :a 1:...:an)e P£|a , ^ 0} . 
For (a o : ai : • • • : an) 6 f/j , w e have 
/ a 0 a i Oj_ i aj+ i a n 
( a 0 : a i : • • • : a n ) = — : — : • • • : — — : 1 : - * — : • • • : — 
\ a j a j a j a j a j 
Therefore, a s sets , th e ma p 
(1.25) <p r. A
n - t / , - , j = 0 , l , . . . , n , 
( a i , . . . , a n ) ^ (« i ' • •' •
 : otj : 1 : aj + i : aj + 2 : • • • : a n ) 
is a bijection . Throug h ^ , w e regar d Uj a s a n n-dimensiona l affin e 
space A n . Whe n Uj HUk ^ 0 , conside r a ma p cpjk = (/P" 1 O </? fc fro m 
(f~1(UJ n f/fc ) t o (pJ
1(Uj n E/fc) . Assum e j < A ; fo r simplicity . Notic e 
that 
y?fc H ĵ n t/fc ) = {(ax, . . . , an ) G A
n | a i + i ^ 0} , 
V]\Uj n t/ fc) - { ( f t , . . . ,/3 n) € A n|/3* ^ 0} . 
1.5. PROJECTIV E VARIETIE S 31 
Then th e ma p (pjk can b e writte n a s 
( a i , . . . , a n) H + (ax : • • • : ak : 1 : a k+i OLn) 
OL\ a<i 
a j + i 
1 : 
a ' j+i 
1 . flfc+l ari 
a j + i 
a J + 2 ajfe 1 QJfe+ i 
a :j'+l 
Let u s denot e th e coordinat e rin g o f A n correspondin g t o Uj b y 
k[xi , . . . , X n ] , j = 0 , 1 , . . . , n . The n (/P ^ becomes a ma p fro m a n 
open se t D(x^ 1) o f A
n t o a n ope n se t D(x^) o f A n, an d ipjk i s 
associated wit h th e isomorphis m 
(1.26) 
fjk 
# :k r(J) T(i) . 
.0) 
„(fc) 
^ + 1 
(4J)Y 
r0 ) /to",...,a#>) 
(fc) ^ , o*r+i)'/ 
(fc) 
JfcT 
xk 1 x fc+i 
r(fc) ' T (fc) ' T (fc) ' 6 j + l x j + i x j + i 
£ ( f e ) > 
7fc)~ 
Consequently, y?jf c i s a n isomorphis m fro m th e affin e algebrai c va -
,(*) O'h riety ^(irA+ J t o th e affin e algebrai c variet y D(x]^ J). Thus , P £ i s 
an algebrai c variet y obtaine d b y gluein g th e n + 1 affine space s Uj , 
j = 0 , 1 , . . . , n , wit h isomorphism s y?jfc . 
Since the superscrip t an d th e subscrip t ar e used in the expressio n 
of th e coordinat e ring , w e introduc e a homogeneou s coordinat e rin g 
k[xo,..., x n] fo r PJJ. The n the coordinate ring k[Uj] = fc[xj , . . . , Xn } 
for th e affin e spac e Uj become s 
Xo X i lj-l X j + i 
Then th e isomorphis m (1.26 ) o f ring s mean s simpl y replacin g X%/XJ 
by Xi/xfc . 
PROBLEM 13 . Expres s the isomorphism ip% usin g homogeneous coor -
dinates. 
32 1. ALGEBRAI C VARIETIE S 
(b) P r o j e c t i v e Se t s an d P r o j e c t i v e Varie t ies . Recal l t ha t 
a poin t (a o : &\ : • • • : an) i n a projectiv e spac e P £ ha s th e propert y