Television Production in Transition: Independence, Scale, Sustainability and the Digital Challenge (Palgrave Global Media Policy and Business) 1st ed. 2021 Edition Gillian
Wu-Yi Hsiang University of California, Berkeley, USA/ Hong Kong University of Science and Technology, Hong Kong
Tzuong-Tsieng Moh Purdue University, USA
Ming-Chang Kang National Taiwan University, Taiwan (ROC)
S S Ding Peking University, China
M Miyanishi University of Osaka, Japan
Published
Vol. 11 Affine Algebraic Geometry: Geometry of Polynomial Rings by M Miyanishi
Vol. 10 Linear Algebra and Its Applications by T-T Moh
Vol. 9 Lectures on Lie Groups (Second Edition) by W-Y Hsiang
Vol. 8 Analytical Geometry by Izu Vaisman
Vol. 7 Number Theory with Applications by W C Winnie Li
Vol. 6 A Concise Introduction to Calculus by W-Y Hsiang
Vol. 5 Algebra by T-T Moh
Vol. 2 Lectures on Lie Groups by W-Y Hsiang
Vol. 1 Lectures on Differential Geometry by S S Chern, W H Chen and K S Lam
Series on University Mathematics – Vol. 11
Affine Algebraic Geometry
Geometry of Polynomial Rings
Masayoshi Miyanishi
Osaka University, Japan & Kwansei Gakuin University, Japan
Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Names: Miyanishi, Masayoshi, 1940– author.
Title: Affine algebraic geometry : geometry of polynomial rings / Masayoshi Miyanishi, Osaka University, Japan & Kwansei Gakuin University, Japan.
Description: New Jersey : World Scientific, [2024] | Series: Series on university mathematics, 1793-1193 ; vol. 11 | Includes bibliographical references and index.
Identifiers: LCCN 2023031983 | ISBN 9789811280085 (hardcover) | ISBN 9789811280092 (ebook for institutions) | ISBN 9789811280108 (ebook for individuals)
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TothelateProfessorMasayoshiNagata
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Preface
OneoftheinnovationsbroughtintoalgebraicgeometrybyA.Grothendieck throughhispublicationsincluding ´ El´ementsdeG´eom´etrieAlg´ebrique is theestablishmentofabijectivecorrespondencebetweenaffineschemesand commutativerings,bywhichonecanintroducealgebro-geometricmethodstocommutativealgebraandcreateviceversaanewfieldinalgebraic geometrywhereonestudiesgeometryofaffinedomainsoverafield.
Anaffinedomain A overafield k isthequotientringofapolynomialring k[x1,...,xn]byanideal I.Ageometricapproachtostudyanaffinevariety X =Spec A withcoordinatering A isconsideredtostudygeometricallya big ringlike A whichisnotafiniteunionoflocalrings.1
AroundthesameperiodtherewasalsoaworldwiderevivalofinteresttowardtheEnriques-Kodairaclassificationofalgebraicsurfaces.In themid-1970s,S.Iitakaintroducedthe logarithmicKodairadimension of noncompletealgebraicvarietiesandproposedaprojectofclassifyingnoncompletesurfaceswithexpectationsthatlogarithmicKodairadimension shouldworkasKodairadimensiondidintheclassificationofsmoothprojectivesurfaces.Iitaka’sstudentsincludingS.TsunodaandY.Kawamata, andthepeopleincludingT.FujitaandtheauthorshowedthatIitaka’sexpectationdidworktoacertaindegreeandbringsomeresultsbeyondthe expectation.3 Theirapproachisnowdevelopedinto logarithmicgeometry, whichisastudyofpairs(V,D)ofacompletevariety V andaneffective divisor D.Inmostcases,byHironaka’sresolutionofsingularities, V is madetobesmoothand D adivisorwithsimplenormalcrossings.The studysofarshowsthatgeometrychangesaccordingtowhatkindofsingularitiesisadmittedon V and D.Foranaffinevariety X,wefindeasilysuch apair(V,D)byembedding X intoaprojectivespaceviatheembedding X → An → Pn andtakingtheclosure X as V and D = V \ X.Bythis approach,oneisabletoobservethegeometricbehaviorof X atinfinity, i.e.,on D ornear D 4
∂(x1,...,xn) then C[x1,...,xn]= C[f1,...,fn].Underthisassumption,themapping
φ : An → An , (x1,...,xn) → (f1,...,fn) inducesalocalanalyticisomorphismbetweeneverypointoftheorigin An anditsimageofthetarget An.So,theconjectureasksiftheselocalanalytic isomorphismsareinducedbyapolynomialisomorphism.Unfortunately,
Aring R is noetherian ifeveryascendingchainofideals
I0 ⊆ I1 ⊆···⊆ In ⊆ In+1 ⊆···
ceasestoincrease.Namely,thereexistsaninteger N suchthat In = In+1 forevery n ≥ N .Thisconditioniscalledthe ascendingchainconditionfor ideals (ACC,forshort).TheACCisequivalenttotheconditionthatevery idealisfinitelygenerated.
An R-module M issaidtobe finitelygenerated over R,orsimplya finite R-moduleif M = Rm1 + ··· + Rmn forafinitesetofgenerators {m1,...,mn}.Thequotientmodule M/N ofafinite R-module M byan R-submodule N isafinite R-module,andasubmodule N isalsofiniteif R isnoetherian(seeLemma 1.1.11).Conversely,ifan R-submodule N of M andthequotientmodule M/N arefinite,thensoisthe R-module M . Asubset S ofaring R isa multiplicativeset (ora multiplicativelyclosed set)if
(i) 0 ̸∈ S, 1 ∈ S,and (ii) s,t ∈ S implies st ∈ S
If p isaprimeidealof R,thenthecomplement S := R \p isamultiplicative set.Infact,thedefinitionofprimeidealimplies,bycontrapositive, ab ̸∈ p if a ̸∈ p and b ̸∈ p.If S isamultiplicativesetof R,the ringofquotients
(or ringoffractions) S 1R isdefinedastheset
S 1R = a s | a ∈ R,s ∈ S , where a/s denotestheequivalenceclassintheproduct R × S underthe equivalencerelation
(a,s) ∼ (b,t)ifandonlyif u(at bs)=0forsome u ∈ S.
Hence a/s isconsideredasausualfractionwhosenumeratorisanarbitrary element a ∈ R andwhosedenominatorisanelement s ∈ S.But a/s = b/t if(a,s) ∼ (b,t).If R isanintegraldomain,theaboveequivalencerelation holdsifandonlyif at = bs.Theset S 1R hasaringstructureforwhich additionandmultiplicationaredefinedrespectivelyby
Thereisanaturalringhomomorphism i : R → S 1R definedby i(a)= a/1.
Thekernelof i isanideal
I0 = {a ∈ R | as =0forsome s ∈ S}.
Withthisringhomomorphism i,thereisabijectivecorrespondencebetween theideals I of R suchthat I ⊇ I0 and I ∩ S = ∅ andthesetofidealsof S 1R.Hereweconsideronly proper ideals,where I isa proper idealof R if I ⫋ R.Thebijectivecorrespondenceisgivenby
I → I(S 1R)= a s | a ∈ I,s ∈ S ,J → i 1(J).
Thiscorrespondencerestrictstoabijectionbetween
{p | aprimeidealof R suchthat p ∩ S = ∅} and
{P | aprimeidealof S 1R}.
Wedenotetheideal i 1(J)by J ∩ R byabuseofnotation.Thisobservation impliesthat S 1R isnoetherianifsois R
Forexample,let S = R \ p foraprimeideal p of R.Wedenote(S 1R) by Rp and p(S 1R)by pRp.Then pRp isthebiggestprimeidealwith respecttoinclusion.Hence pRp isaunique maximal ideal1 of Rp.Aring R iscalleda localring ifitcontainsauniquemaximalideal m inthesense thatanyproperideal I of R iscontainedin m.Bythenotation(R, m)we
1Anideal m of R isa maximal idealif I isaproperidealof R suchthat I ⊇ m then I = m
meanthat R isalocalringwithmaximalideal m.Amaximalideal m is aprimeideal.Infact,supposethat ab ∈ m.Let I = m + aR,whichisan idealsuchthat I ⊇ m.Henceeither I = R or I = m.If I = m then a ∈ m. Supposethat I = R.Then1= ax + z with z ∈ m.Then b = abx + bz ∈ m. So, a ∈ m or b ∈ m.
Let R beanintegraldomain.Then S := R \{0} isamultiplicative set.Theringofquotients S 1R isnowafield,whichwecallthe fieldof quotients orthe fieldoffractions anddenoteby Q(R).
Let I beanidealof R.Anidealofthequotientring(ortheresidue ring) R = R/I iswritteninaform J/I,where J isanidealof R suchthat J ⊇ I.If R isnoetherian,soisthering R.
1.1.2 SpectrumofaringandZariskitopology
Let R bearing.Thesetofprimeidealsof R iscalledthe spectrum (spec, forshort)of R anddenotedby
Spec R = {p | aprimeidealof R}
Givenaring R,thespectrumSpec R isalsocalledan affinescheme with the coordinatering R.Wecandefineatopology,calledthe Zariskitopology, inSpec R byassigningclosedsetssatisfyingaxiomsoftopology.Aclosed setis V (I)foranideal I of R,where
V (I)= {p ∈ Spec R | p ⊇ I}.
Axiomsoftopologyforclosedsetsrequire
(i) Spec R and ∅ areclosedsets.
(ii) λ∈Λ V (Iλ)isaclosedset,where {Iλ | λ ∈ Λ} ispossiblyaninfinite set.
(iii) Afiniteunion j∈J V (Ij )isaclosedset.
For(i),wehaveSpec R = V ((0))and ∅ = V (R),where(0)isthezero ideal.For(ii),wehave λ∈Λ V (Iλ)= V ( λ∈Λ Iλ).For(iii),itsufficesto showthat
V (I1) ∪ V (I2)= V (I1I2)= V (I1 ∩ I2), where I1I2 istheidealof R generatedby {a1a2 | a1 ∈ I1,a2 ∈ I2}.Since I1 ∩ I2 ⊇ I1I2 andsince p ⊇ I1I2 implieseither p ⊇ I1 or p ⊇ I2,wehave V (I1) ∪ V (I2) ⊇ V (I1I2) ⊇ V (I1 ∩ I2) ⊇ V (I1) ∪ V (I2), whichprovestheassertion.TheZariskitopologyis T0,butnotnecessarily T1.Namely,iftwodistinctprimeideals p, q aregiven,thereisaclosedset
V (I)whichcontainseitheroneof p or q butnottheother.Onecannot choose p or q.Infact, p ⊂ q ifandonlyif p ∈ V (I)alwaysimply q ∈ V (I) foraclosedset V (I).
Foranideal I,definethe radical of I,denotedby √I,by √I = {a ∈ R | an ∈ I forsome n> 0}
Anideal I isa radicalideal if √I = I.Thenwehavethefollowingtheorem.
Theorem1.1.1. Foranideal I of R wehave
√I = p∈V (I) p.
Proof. If p ∈ V (I)then p ⊃ √I,whence p∈V (I) p ⊇ √I.Weshowthe oppositeinclusion.Ifthereexistsanelement s ∈ ( p∈V (I) p) \ √I,then S = {sn | n ≥ 0} isamultiplicativesetof R suchthat S ∩ I = ∅.Hence I(S 1R)isaproperidealof S 1R.ByZorn’slemma(seeCorollary 1.1.4 below)thereexistsamaximalideal M of S 1R suchthat I(S 1R) ⊂ M.
Let m = R ∩ M(= i 1(M)).Then m isaprimeidealof R suchthat m ⊃ I and m ∩ S = ∅.Namely m ∈ V (I)and m ⊃ p∈V (I) p,whence s ∈ m.This contradicts m ∩ S = ∅.
Corollary1.1.2. Forideals I,J of R, V (I)= V (J) ifandonlyif √I = √J.
Proof. ByTheorem 1.1.1, √I = √J followsif V (I)= V (J).Theconverse isclearbecause V (I)= V (√I)and V (J)= V (√J).
Let S beapartiallyorderedset.Itiscalledan inductiveset ifevery totallyorderedsubsethasanupperbound.Thefollowingresultiscalled Zorn’slemma
Lemma1.1.3. Let S beaninductiveset.Then S hasamaximalelement. Thisresultyieldsanimportantresult.
Corollary1.1.4. Let I beaproperidealofaring R.Thenthereexists amaximalideal m suchthat m ⊇ I.Further,amaximalidealisaprime ideal.
Proof. Let S bethesetofproperidealsof R containing I anddefinea partialorderin S bysetting
J1 ≥ J2 ifandonlyif J1 ⊇ J2
Let J1 ≤ J2 ≤···≤ Jn ≤ Jn+1 ≤··· beatotallyorderedsubsetof S.Set J = n≥1 Jn.Then J isaproperidealof R containing I,and J isclearly anupperboundofthetotallyorderedsubset.Hence S isaninductiveset, and S hasamaximalelement m,whichisamaximalidealof R suchthat m ⊇ I.
Let m beamaximalideal.Supposethat ab ∈ m with a,b ∈ R.Then aR + m isanidealcontaining m.Hence aR + m = m or aR + m = R.If aR + m = m then a ∈ m.If aR + m = R then ax + m =1for x ∈ R and m ∈ m.Then b = b(ax + m)= abx + bm ∈ m.So,either a ∈ m or b ∈ m. Thisimpliesthat m isaprimeideal.
1.1.3 Irreducibledecompositionofatopologicalspace
Atopologicalspace X is noetherian ifadescendingchainofclosedsets F1 ⊇ F2 ⊇···⊇ Fn ⊇ Fn+1 ⊇··· stopsalwaystodecrease,i.e.,thereexists N> 0suchthat Fn = Fn+1 for every n ≥ N .If R isanoetherianringthen X =Spec R isanoetherian space.Infact,write Fi = V (Ii)with Ii = √Ii = p∈Fi p.Thenthe descendingchainofthe Fi correspondstoanascendingchainofradical idealsof R
I1 ⊆ I2 ⊆···⊆ In ⊆ In+1 ⊆··· andtheterminationoftheidealchainimpliestheterminationofthechain ofclosedsets.
Atopologicalspace X is quasi-compact ifanyopencovering U = {Uλ}λ∈Λ of X hasafiniteopensub-covering X = U1 ∪ U2 ∪···∪ Un, where Ui = Uλi with λi ∈ Λ.
Proof. Let U = {Uλ}λ∈Λ beanopencoveringof X.Wemayassume thatforany µ ∈ Λ, λ∈Λ\{µ} Uλ = X.Considerawell-orderingonΛand supposethat
λ1 <λ2 < <λn <λn+1 < andidentify λi with i ∈ Z.Let
Fn = X \ (U1 ∪ U2 ∪···∪ Un), whichisaclosedsetof X satisfying
F1 ⊃ F2 ⊃···⊃ Fn ⊃ Fn+1 ⊃···
Since X isnoetherian,thedescendingchainofclosedsetsceases,i.e.,there exists N> 0suchthat Fn = Fn+1 forall n ≥ N .Then FN = ∅.Hence X = U1 ∪ U2 ∪···∪ Un.
Atopologicalspace X is reducible ifthereisadecomposition X = F1∪F2 fortwoclosedsets F1,F2 with Fi ⫋ X.Otherwise, X iscalled irreducible. If F isaclosedsubsetof X,wecansaythat F isreducible(orirreducible) withrespecttotheinducedtopologyon F .
Lemma1.1.6. Let F beaclosedsubsetofanoetheriantopologicalspace X.Thenthereexistsafinitesetofirreducibleclosedsubsets F1,...,Fn suchthat F = F1 ∪ F2
Fn,F
Fj forall i = j. Theseclosedsubsets Fi aredeterminedbythesubset F uniquelyuptopermutations.
Theset Fi iscalledthe irreduciblecomponent of F andthedecomposition F = F1 ∪ F2 ∪···∪ Fn iscalledthe irreducibledecomposition
Proof. Weprovefirsttheexistenceofadecomposition.Let S betheset ofclosedsubsets F of X suchthat F isnotafiniteunionofirreducible closedsubsets.Then S isaninductivesetwithrespecttoanorderdefined byreverseinclusionofsubsets.Namely, F ≤ F ′ for F,F ′ ∈ S if F ′ ⊆ F . Givenan(ascending)totallyorderedsubsetof S,thereexistsanupper boundbythenoetherianconditionof X whichceasesdescendingchains ofclosedsubsetsof X.Hence S hasamaximalelement,say F0.Then F0 isreducible.Write F0 = F1 ∪ F2 forproperclosedsubsets F1,F2 of F0.Then F1 >F0 and F2 >F0.Since F0 isamaximalelementof S, F1 and F2 arewrittenasfiniteunionsofirreducibleclosedsubsets.Writethe decompositionsas
F1 = F11 ∪ F12 ∪···∪ F1r F2 = F21 ∪ F22 ∪···∪ F2s, where Fij isirreduciblefor i =1, 2.Thenwehave
F0 = F1 ∪ F2 =(F11 ∪···∪ F1r) ∪ (F21 ∪···∪ F2s), whichisafiniteunionofirreducibleclosedsubsets.Thiscontradictsthe assumptionthat F0 ∈ S Weprovenextthatadecompositionisuniqueuptopermutations.Let F = G1 ∪ G2 ∪···∪ Gm
Since G1 isirreducible, G1 = G1 ∩ Fi forsome1 ≤ i ≤ n.Afterapermutationofindices,wemayassumethat i =1.Then G1 ⊆ F1.Similarly,we have F1 =(F1 ∩ G1) ∪···∪ (F1 ∩ Gm).
Hence F1 = F1 ∩ Gj andhence F1 ⊆ Gj .Thisimpliesthat G1 ⊆ Gj , whence j =1.Namely F1 = G1.Wecanargueasabovebyreplacing F1 by Fj ,andshowthat n = m and Fi = Gi afterasuitablepermutationof indices.
1.1.4 Primeidealdecompositionofradicalideals
Lemma1.1.7. Let R beanoetherianringandlet X =Spec R.Let I be aradicalidealof R andlet F = V (I).If F isirreduciblethen I isaprime ideal.Conversely,if I isaprimeidealthen V (I) isirreducible.
Proof. Supposethat ab ∈ I.Then,forany p ∈ F , ab ∈ I ⊆ p.Hence a ∈ p or b ∈ p.Thisimpliesthat F ⊂ V (a) ∪ V (b),where V (a)= V (aR) and V (b)= V (bR),and F =(F ∩ V (a)) ∪ (F ∩ V (b)), where F ∩ V (a)= V (I + aR)and F ∩ V (b)= V (I + bR).Since F is irreducible, F = F ∩ V (a)or F = F ∩ V (b),i.e., F ⊆ V (a)or F ⊆ V (b). Thisimpliesthat aR ⊆ p∈F p = √I = I, or bR ⊆ I.
Hence a ∈ I or b ∈ I.So, I isaprimeideal. Weprovetheconverse.Supposethat V (I)isreducible,andwrite V (I)= V (I1) ∪ V (I2)with V (I1) ⫋ V (I)and V (I2) ⫋ V (I).Since V (I1) ∪ V (I2)= V (I1I2),itfollowsthat I1I2 ⊆ I.Since I isaprime ideal,either I1 ⊆ I or I2 ⊆ I.Theneither V (I) ⊆ V (I1)of V (I) ⊆ V (I2). Thisisacontradiction.
Corollary1.1.8. Let R beanoetherianringandlet F = V (I) fora radicalideal I.Thenthereexistsauniquelydeterminedsetofprimeideals {p1,..., pn} suchthat
(i) I = p1 ∩ p2 ∩···∩ pn,and (ii) pi ̸⊂ pj foranypair (i,j) with i = j.
Theclosedsubsets V (p1),...,V (pn) correspondbijectivelywithirreducible components F1,...,Fn of F = V (I).
Proof. Thereisanirreducibledecompositionof F whichisuniquelydetermineduptopermutationsofcomponents F = F1 ∪···∪ Fn,
where Fi = V (pi)foraprimeideal pi.Thenwehave V (I)= V (p1) ∪···∪ V (pn)= V (p1 ∩···∩ pn),
where p1 ∩···∩ pn isaradicalideal.ThenitfollowsbyCorollary 1.1.2 that I = p1 ∩···∩ pn.
Thedecomposition
I = p1 ∩ p2 ∩···∩ pn
inCorollary 1.1.8 iscalledthe primedecomposition2 oftheradicalideal I. Forafixed1 ≤ i ≤ n,write j=i
and j=i
where ∨ pi showsthattheideal pi isomitted.Then j=i pj ⊆ j=i pj ,and hence j=i pj ̸⊂ pi because pi ̸⊂ pj foranypair(i,j).Let ai beanelement of( j=i pj ) \ pi.Foranelement a ∈ R,thesubset
(I : a)= {x ∈ R | ax ∈ I}
iscalledan idealquotient oftheideal I.Itisanidealof R containing I Fortheidealquotient(I : ai)itholdsthat(I : ai)= pi.Infact,if x ∈ pi then aix ∈ ( j=i pj )bythechoiceof ai and aix ∈ pi because x ∈ pi.Hence aix ∈ I and pi ⊆ (I : ai).Conversely,if x ∈ (I : ai)then aix ∈ I and
2Laterweneedafinerdecompositionofideals,calledthe primarydecomposition of ideals.Wedevelopthetheoryintheappendix.
ai ̸∈ pi.Hence x ∈ pi.Thisshowsthat pi =(I : ai).Furthermore,ifan idealquotient(I : a)isaprimeideal p,then p ⊇ I ⊇ n i=1 pi.
Hence p containssome pi.Wesaythattheidealquotient(I : a)isa prime divisor of I if(I : a)isaprimeideal.Thesetofallprimedivisorsof I is denotedbyAss(R/I).Theneach pi isa minimal amongprimedivisorsof I withrespecttotheinclusionorder.Anon-minimalprimedivisorof I is called embedded.
Theradical n = √0ofthezeroideal(0)of R iscalledthe nilradical. Let n = √0= p1 ∩ p2 ∩···∩ pn betheprimedecomposition.Sinceanyprimeideal p of R contains n,the aboveargumentshowsthat p ⊇ pi forsome1 ≤ i ≤ n.Thisimpliesthat X =Spec R = V (p1) ∪ V (p2) ∪···∪ V (pn) andeach V (pi)isanirreduciblecomponentof X
1.1.5 Genericpoint,closedpointandKrulldimension
Let R beannoetherianringandlet X =Spec R.Wehaveaprimeideal px identifiedwitheachpoint x ∈ X.Forasubset S of X,wedenoteby S theclosureof S withrespecttotheZariskitopology.Apoint x of X isa closedpoint if {x} = {x}.Apoint x isa genericpoint if X = {x}. Lemma1.1.9. Thefollowingassertionshold.
(1) S = V (I(S)),where I(S)= x∈S px.
(2) If S consistsofasinglepoint x thentheclosure {x} isirreducible.We have {x} = {y} ifandonlyif x = y
(3) x isthegenericpointofanirreduciblecomponentof V (I) ifandonly if px isaminimalprimedivisorof √I (4) X isirreducibleifandonlyifthenilradical √0 of R isaprimeideal. (5) x isaclosedpointifandonlyif px isamaximalideal.
Proof. (1)Ifaclosedset V (I)contains S then px ⊇ I forevery x ∈ S.This impliesthat I ⊆ I(S),where I(S)isa radicalideal,i.e., I(S)= I(S). Thisimpliesthat I(S)definesthesmallestclosedsubsetof X whichcontains S,i.e.,theclosureof S
