Dimension theory of tensor products of AF-rings

0.2 MB·

by  S. Kabbaj

DIMENSION THEORY OF TENSOR PRODUCTS OF AF-RINGS
6 SALAHKABBAJ
1
0
2 Abstract. AF-ringsarealgebrasoverafieldkwhichsatisfytheAltitudeFormula
overk.ThispapersurveysafewworksintheliteratureontheKrullandvaluative
n
dimensionsoftensorproductsofAF-rings.ThefirstsectionextendsWadsworth’s
a
classicalresultsontheKrulldimensionofAF-domainstothelargerclassofAF-
J
rings.Italsoprovidesformulasforcomputingthevaluativedimensionwitheffect
8 onthetransferofthe(locally)Jaffardproperty. Thesecondsectionstudiestensor
2 productsofAF-ringsoverazero-dimensionalring. Mostresultsonalgebrasover
afieldareextendedtothesegeneralconstructions. Thethirdsectionestablishes
] formulasfortheKrullandvaluativedimensionsoftensorproductsofpullbacks
C
issued from AF-domains. Throughout, examples are provided to illustrate the
A scopeandlimitsoftheresults.
.
h
t
a Contents
m
Introduction 1
[
1. TensorproductsofAF-ringsoverafield 3

1.1. Krulldimension 3
1
1.2. Transferofthe(locally)Jaffardproperty 6
v
2. TensorproductsofAF-ringsoverazero-dimensionalring 8
4
2.1. Tensoriallycompatiblealgebras 9
5
2.2. Krulldimension 10
6
2.3. Transferofthe(locally)Jaffardproperty 12
7
3. TensorproductsofpullbacksissuedfromAF-domains 13
0
3.1. Krulldimension 13
.
1 3.2. Valuativedimension 15
0 3.3. Someapplicationsandexamples 15
6 References 17
1
:
v Introduction
i
X
Throughoutthispaper,allringsand,foragivenfieldk,allk-algebrasareassumed
r to be commutative with identity element and have finite transcendence degree over
a
k. ForaringA,weshalluseSpec(A)andMax(A)todenote,respectively,thesets
of all prime ideals and maximal ideals of A. Also, we will denote by A[n] the
polynomialringA[X ,...,X ]andbyp[n]theprimeidealp[X ,...,X ]inA[n],for
1 n 1 n
anyp∈Spec(A)andpositiveintegern.
A finite-dimensional domain R is said to be Jaffard if dim(R[n])=n+dim(R)
for all n≥1; equivalently, if dim(R)=dim (R), where dim(R) denotes the Krull
v
dimensionofRanddim (R)denotesitsvaluativedimension(i.e.,thesupremum
v
of dimensions of the valuation overrings of R). Since this notion does not carry
overtolocalizations,RissaidtobelocallyJaffardifR isJaffardforeachp∈Spec(R)
p
Date:January29,2016.
2010MathematicsSubjectClassification. 13C15,13B24,13F05,13H05,13F20,13B30,13E05,13D05.
Keywordsandphrases. Tensorproductofalgebras,Krulldimension,valuativedimension,Jaffard
ring,locallyJaffardring,altitudeformula,AF-ring,pullback.
1
2 SALAHKABBAJ
(equivalently,ht(p[n])=ht(p),∀p∈Spec(A)). TheclassofJaffarddomainscontains
most of the well-known classes of rings involved in dimension theory such as
Noetheriandomains,Pru¨ferdomains,universallycatenariandomains,andstably
strong S-domains. Analogous definitions are given in Cahen’s paper [16] for a
finite-dimensional arbitrary ring (i.e., possibly with zero-divisors). We assume
familiaritywiththeseconceptsasin[1,2,7,11,12,13,17,19,20,25,26,27,29],and
anyunreferencedmaterialisstandardasin[21,28,30].
Letkbeafieldandlett(A)denotethetranscendencedegreeoverkofak-algebra
A. IfAisnotadomain,thenbydefinitiont(A):=max t(A/p)|p∈Spec(A) .
n o
Definition0.1. Ak-algebraAisanAF-ringifitsatisfiestheAltitudeFormulaover
k;thatis,ht(p)+t(A/p)=t(A ),foreachp∈Spec(A).
p
ExamplesandbasicpropertiesofAF-ringsareprovidedatthebeginningofthe
nextsection. In1977,Sharpprovedin[32]that
dim(K ⊗ K )=min t(K ),t(K )
1 k 2 (cid:16) 1 2 (cid:17)
for any field extensions K and K of k. In 1978, Sharp and Vamos generalized
1 2
thisresulttothetensorproductofafinitenumberoffieldextensionsofk[33]. In
1979, Wadsworth extended their results on field extensions to the larger class of
AF-domains[35];namely,heprovedthatifD andD areAF-domains,then
1 2
dim(D ⊗ D )=min dim(D )+t(D ),t(D )+dim(D ) .
1 k 2 (cid:16) 1 2 1 2 (cid:17)
Moreover,heestablishedaformulafordim(D⊗ R)whichholdsforanAF-domain
k
D,withnorestrictionontheringR. Healsoprovedthatforanyprimeidealpof
anAF-ringAand, foranyn≥1,ht(p[n])=ht(p)(i.e.,AislocallyJaffard). In[22],
Girolami studied the class of AF-domains with respect to the class of k-algebras
which are stably strong S-domains and examined the behavior of the notion of
AF-domain for certain pullback constructions. An upper bound was then given
forthevaluativedimensionofthetensorproductoftwok-algebras;moreexactly,
ifA andA arek-algebras,then
1 2
dim (A ⊗ A )≤min dim (A )+t(A ),t(A )+dim (A ) .
v 1 k 2 (cid:16) v 1 2 1 v 2 (cid:17)
This paper surveys a few works in the literature on the Krull and valuative
dimensionsoftensorproductsofAF-rings. ThefirstsectionextendsWadsworth’s
classicalresults on the Krull dimension of AF-domains to the larger class of AF-
rings. Italsoprovidesformulasforcomputingthevaluativedimensionwitheffect
onthetransferofthe(locally)Jaffardproperty. Thesecondsectionstudiestensor
productsofAF-ringsoverazero-dimensionalring. Mostresultsonalgebrasover
a field are extended to these general constructions. The third section establishes
formulas for the Krull and valuativedimensions of tensor products of pullbacks
issued from AF-domains. Throughout, examples are provided to illustrate the
scopeandlimitsoftheresults.
The three main papers involved in this survey are [8, 9, 10], which were co-
authoredwithSamirBouchiba(UniversityofMeknes)andFloridaGirolami(Uni-
versityofRome)andpublishedin1997/1999.
DIMENSIONTHEORYOFTENSORPRODUCTSOFAF-RINGS 3
1. TensorproductsofAF-ringsoverafield
Thissectionisdevotedto[8]. Firstitextendssome classicalresults(onthe Krull
dimension) known for the class of AF-domains to the class of AF-rings over a
field. Thenitprovidesformulasforcomputingthevaluativedimensionoftensor
productsemanatingfromAF-ringswitheffectonthepossibletransferofthenotion
of(locally)Jaffardringtotheseconstructions.
Throughoutthissectionkwilldenoteafieldandbyaringwemeanak-algebra.
Also,algebras(resp.,tensorproducts)aretakenover(resp.,relativeto)k. Forthe
reader’sconvenience,wefirstrecallsomebasicpropertiesofAF-rings.
Remark 1.1 ([22, 35]). Let A denote the class of AF-rings over k and let n be a
positiveinteger. Then:
(1) AnyfinitelygeneratedalgebraanditsintegralextensionsbelongtoA.
(2) IfA∈A,thenS−1A∈A,foreverymultiplicativesubsetSofA.
(3) IfA∈A,thenA[n]∈A.
(4) IfA ,...,A ∈A,thenA ⊗···⊗A ∈A.
1 n 1 n
(5) IfA ,...,A ∈A,thenA ×···×A ∈A.
1 n 1 n
(6) IfA∈A,thenAislocallyJaffard.
(7) TheclassAisnotstableunderfactorrings. However,ifAisacatenarian
AF-domain,thenA/p∈A, ∀p∈Spec(A).
1.1. Krulldimension. ThissubsectionaimsatextendingWadsworth’sresultson
AF-domainstotheclassofAF-rings. Thefirsttechnicalresultlinksthetranscen-
dencedegreeofalocalizationofatensorproducttothetranscendencedegreesof
itsrespectivecomponents.
Lemma1.2. LetA ,...,A beAF-ringsandletP∈Spec(A ⊗···⊗A ). Then
1 n 1 n
t(cid:16)(A1⊗···⊗An)P(cid:17)= X t(Aipi)
1≤i≤n
wherep :=P∩A fori=1,...,n.
i i
As an immediate consequence of this lemma, we obtain the following known
resultforAF-domains.
Corollary1.3. LetD ,...,D beAF-domainsandletP∈Spec(D ⊗···⊗D ). Then
1 n 1 n
t(cid:16)(D1⊗···⊗Dn)P(cid:17)=t(D1⊗···⊗Dn)= X t(Di).
1≤i≤n
The following simple statement has important consequences on some of the
followingresults.
Lemma1.4. LetAbeanAF-ringandp∈Spec(A). Letp beaminimalprimeidealofA
o
containedinpsuchthatht(p)=ht(p/p ). Then,t(A )=t(A ).
o p po
In ordertoproceedwith the mainresults, letus recallfrom[35] the following
functions: GiventworingsAandBwithp∈Spec(A)andq∈Spec(B),considerthe
function
δ(p,q)=max ht(P)|P∈Spec(A⊗B)withP∩A=pandP∩B=q .
n o
Given a ring A, p∈ Spec(A) and d,s integers with 0≤ d ≤s, consider the two
functions
∆(s,d,p)=ht(p[s])+min s,d+t(A/p)
(cid:16) (cid:17)
4 SALAHKABBAJ
D(s,d,A)=max ∆(s,d,p)|p∈Spec(A) .
n o
ThemainresultofthissectionprovidesaformulafortheKrulldimensionofa
tensorproduct.
Theorem1.5. LetAbeanAF-ringandBanarbitraryring. Then:
(1) δ(p,q)=∆(t(A ),ht(p),q),foranyp∈Spec(A)andq∈Spec(B).
p
(2) dim(A⊗B)=max D(t(A ),ht(p),B)|p∈Spec(A) .
n p o
Noticethat(1)isthemostimportantpartoftheabovetheorem. Itsproofrelies
ontheabovetwolemmasafterreduction-vialocalizationtechniques-tothecase
where B is a field. Then, the result upon dim(A⊗B) derives directly from the
definitionsofδ,∆,andD.
IncasebothAandBareAF-rings, wegetthefollowing moreexplicitformula
fortheKrulldimension.
Corollary1.6. LetAandBbetwoAF-rings. Then:
dim(A⊗B)=max min ht(p)+t(A ),t(A )+ht(q) |p∈Spec(A),q∈Spec(B) .
n (cid:16) q p (cid:17) o
The general case of n AF-rings (n≥2) can be proved by induction on n via
Corollary1.6andLemma1.2. Namely,wehavethefollowingresult.
Corollary1.7. LetA ,...,A beAF-rings. Then:
1 n
dim(A1⊗···⊗An)=maxnmin(cid:16)ht(pi)+Xt(Apj)(cid:17)1≤i≤n|pi∈Spec(Ai)o.
j,i
NoticethatD(s,d,A)isanondecreasingfunctionofthefirsttwoargumentsand,
hence,onecanrestricttheformulasintheabovethreeresultstothemaximalideals.
Wadsworth’swell-knownresult[35,Theorem3.8]ontheKrulldimensionofthe
tensor product of n AF-domains reads as follows: Let D ,...,D be AF-domains
1 n
withn≥2. Then
dim(D1⊗···⊗Dn)= X t(Di) − maxnt(Di)−dim(Di)o1≤i≤n.
1≤i≤n
ThisformuladoesnotholdingeneralforAF-rings,asshownbythefollowing
example.
Example 1.8. Let R :=k[X ,X ,X ] , R :=k[X ,X ], A :=R ×R , and A :=
1 1 2 3 (X1) 2 1 2 1 1 2 2
k[X ,X ] . Clearly,A isanAF-ringwithdim(A )=2andt(A )=3;andA isan
1 2 (X1) 1 1 1 2
AF-domainwith dim(A )=1 and t(A )=2. ByCorollary 1.6, one can checkthat
2 2
dim(A ⊗A )=3(cid:8)t(A )+t(A )−1=4.
1 2 1 2
Thesecondmainresultofthissectionestablishesnecessaryandsufficientcon-
ditions for a tensor product of AF-rings to satisfy Wadsworth’s aforementioned
formula.
Theorem1.9. LetA ,...,A beAF-rings. Then,thefollowingassertionsareequivalent:
1 n
(i) dim(A ⊗···⊗A )= t(A) − max t(A)−dim(A) ;
1 n P1≤i≤n i n i i o1≤i≤n
(ii) Foreachi=1,...,nthereexistsM ∈Max(A)suchthatht(M )=dim(A )for
i i io io
somei ∈{1,...,n}and,foralli,i ,t(A )=t(A)&t(A/M)≤t(A /M ).
o o iMi i i i io io
Next,wegivesomeapplicationsofthisresult.
DIMENSIONTHEORYOFTENSORPRODUCTSOFAF-RINGS 5
Corollary 1.10. Let A ,...,A be AF-rings such that, for each i=1,...,n, there exists
1 n
M ∈Max(A)withht(M)=dim(A)andt(A )=t(A). Then
i i i i iMi i
dim(A1⊗···⊗An)= X t(Ai) − maxnt(Ai)−dim(Ai)o1≤i≤n.
1≤i≤n
Corollary 1.11. Let A ,...,A be AF-rings such that, for each i=1,...,n and for each
1 n
M ∈Max(A),t(A )=t(A). Then
i i iMi i
dim(A1⊗···⊗An)= X t(Ai) − maxnt(Ai)−dim(Ai)o1≤i≤n.
1≤i≤n
TheabovecorollaryrecoversWadsworth’saforementionedresult.
Corollary1.12([35,Theorem3.8]). LetD ,...,D beAF-domains. Then
1 n
dim(D1⊗···⊗Dn)= X t(Di) − maxnt(Di)−dim(Di)o1≤i≤n.
1≤i≤n
Next,asufficientconditioninvolvestheminimalprimeideals.
Corollary 1.13. Let A ,...,A be AF-rings such that, for each i=1,...,n and for each
1 n
minimalprimeidealp ofA,t(A/p)=t(A). Then
i i i i i
dim(A1⊗···⊗An)= X t(Ai) − maxnt(Ai)−dim(Ai)o1≤i≤n.
1≤i≤n
Corollary1.14. LetA ,...,A beequicodimensionalAF-rings. Then
1 n
dim(A1⊗···⊗An)= X t(Ai) − maxnt(Ai)−dim(Ai)o1≤i≤n.
1≤i≤n
Itisknown[22,Corollary3.3]thatifAisanAF-ring,then
dim(A⊗A)=dim (A⊗A)≤dim(A)+t(A)=dim (A)+t(A).
v v
Thisresultfollows also fromCorollary1.6. Now, applying Theorem1.9toA⊗A
weobtain:
Corollary1.15. LetAbeanAF-ring. Then,thefollowingassertionsareequivalent:
(i) dim(A⊗A)=dim(A)+t(A);
(ii) ∃M,N∈Max(A)withht(M)=dim(A),t(A )=t(A),andt(A/N)≤t(A/M).
N
Next,weprovideanexampleofanAF-ringAwithdim(A⊗A)(cid:8)dim(A)+t(A).
Example1.16. LetKbeafieldextensionofkwitht(K)=2andletA:=K×k[X],where
Xisanindeterminateoverk. Then,AisanAF-ringwithdim(A)=1andt(A)=2.
The maximal idealsof A are (0)×k[X]and K×N with N∈Max(k[X]). Moreover,
ht((0)×k[X])=0;t(A(0)×k[X])=2=t(A);ht(K×N)=1=dim(A);t(A/((0)×k[X]))=2;
andt(A/(K×N))=0.So,byCorollary1.15,weobtaindim(A⊗A)(cid:8)dimA+t(A)=3.
Wewillconcludethissubsectionbyanillustrativeexamplewhichrequiresthe
followingtechnicallemma.
Lemma1.17. LetAbeanAF-ringsuchthatthereexistp,q∈Spec(A)witht(A ),t(A ).
p q
Then,foranyAF-ringB,A⊗BisnotthetensorproductofafinitenumberofAF-domains.
Example1.18. Foreachintegern≥1,thereexisttwoAF-ringsA andA suchthat:
1 2
(1) dim(A ⊗A )=n;
1 2
6 SALAHKABBAJ
(2) A ⊗A isnotthetensorproductofafinitenumberofAF-domains;
1 2
(3) If,inaddition,thereexistsanon-finitelygeneratedseparableextensionof
k,thenneitherA norA isafinitedirectproductofAF-domains.
1 2
Indeed, let K be a separable extension of k. Let V :=K(X)[Y] =K(X)+M ,
1 (Y) 1
V:=K(Y)[X] =K(Y)+M, andV :=K[Y] +M=K+M . Then, V andV are,
(X) 2 (Y) 2 1 2
respectively,one-dimensionalandtwo-dimensionalvaluationdomainsofK(X,Y).
SinceV andV areincomparable,T:=V ∩V isatwo-dimensionalPru¨ferdomain
1 2 1 2
withonlytwomaximalideals, M andM , suchthatT =V andT =V [31,
1 2 M1 1 M2 2
Theorem11.11]. Let I:=M M and R:=T/I. Then, R is a zero-dimensional ring
1 2
(and,afortiori,anAF-ring)withonlytwoprimeideals,p :=M /Iandp :=M /I.
1 1 2 2
Further,t(R/p )=1andt(R/p )=0. ByCorollary1.15,wehave
1 2
dim(R⊗R[n])=dim((R⊗R)[n])=dim(R⊗R)+n=t(R)+n=1+n.
Moreover,byLemma1.17,R⊗R[n]isnotthetensorproductofafinitenumberof
AF-domains;soitsufficestotakeA :=RandA :=R[n−1].
1 2
Now assume that K is not finitely generated over k. So K⊗K is a reduced
[36, Theorem 39], zero-dimensional [32, Theorem 3.1], and non-Noetherian [34,
Theorem11]ring. ThenSpec(K⊗K)isinfinite[34,Lemma0]. Next,letA:=K⊗R.
SinceAisanintegralextensionofR,itiszero-dimensional. Moreover,thereexist
twoprimeidealsofA,P andP suchthatP ∩R=p andP ∩R=p witht(A/P )=1
1 2 1 1 2 2 1
and t(A/P )=0. Since K is the quotient field of R/p and Spec(K⊗K) is infinite,
2 2
by [35, Proposition 3.2], Spec(A) is infinite. So A is not a finite direct product
of AF-domains and the same holds for A[n]. By Corollary 1.15, dim(A⊗A[n])=
dim((A⊗A)[n])=dim(A⊗A)+n=t(A)+n=1+n. Moreover, by Lemma 1.17,
A⊗A[n]isnotthetensorproductofafinitenumberofAF-domains. Soitsuffices
totakeA :=AandA :=A[n−1],completingtheproofoftheexample.
1 2
1.2. Transferofthe(locally)Jaffardproperty. Inthissubsection,wefirstestablish
the transfer of the locally Jaffard property in some context of tensor products.
Then,wegivesomeformulasforcomputingthevaluativedimensionofthetensor
product of an AF-ring and an arbitrary ring. We conclude with the fact that the
tensorproductofanAF-ringandaJaffardringisnotnecessarilyaJaffardring.
Next, we announce the main result of this subsection. Notice that the tensor
productoftwoAF-ringsislocallyJaffard(sinceitisanAF-ring).
Theorem1.19. LetAbeanAF-ringandBalocallyJaffardring. Then,A⊗Bisalocally
Jaffardring.
The proof of this result lies on a very important lemma which correlates the
height of a prime ideal of A⊗B to its traces on A and B via the transcendence
degrees;namely,wehave:
Lemma1.20. Let A be an AF-ring and B an arbitrary ring. Let P∈Spec(A⊗B) with
p:=P∩Aandq:=P∩B. Then
ht(P)+t (A⊗B)/P =t(A )+ht(q[t(A )])+t(B/q).
(cid:16) (cid:17) p p
Next, we give some applications of Theorem 1.19. The first one establishes
a formula for the valuative dimension of A⊗B where A is an AF-ring. To this
purpose,one should firstexaminethevariationofthefunctionDbetweenBand
itsassociatedpolynomialrings.
DIMENSIONTHEORYOFTENSORPRODUCTSOFAF-RINGS 7
Lemma1.21. LetAbeanAF-ring,Banarbitraryring,andp∈Spec(A). Then,forany
n≥1,wehave
D t(A ),ht(p),B[n] =D t(A )+n,ht(p)+n,B .
(cid:16) p (cid:17) (cid:16) p (cid:17)
Thenextresultprovidesaformulaforthevaluativedimension.
Corollary1.22. Let A be anAF-ringand B aring withfinitevaluativedimension≥1.
Then,foranyr≥dim (B)−1,wehave
v
dim (A⊗B)=max ht(q[r])+min t(A ),ht(p)+t(B/q)
v n (cid:16) p (cid:17)o
where(p,q)rangesoverSpec(A)×Spec(B).
ThespecialcasewhereAisanAF-domainyieldsamoresimplifiedformula.
Corollary1.23. LetAbeanAF-domainandBaringwithfinitevaluativedimension≥1.
Then,foranyr≥dim (B)−1,wehave
v
dim (A⊗B)=max ht(q[r])+min t(A),dim(A)+t(B/q)
v n (cid:16) (cid:17)o
whereqrangesoverSpec(B).
ThenexttworesultsfeaturespecialcontextswherethetensorproductisJaffard.
Corollary1.24. LetAbeanAF-domainandBaringsuchthatdim (B)≤t(A)+1. Then,
v
A⊗BisaJaffardring.
Recall that, for any ring B of valuative dimension 2, the ring B[X] is locally
Jaffard[16,Proposition1(ii)]. Also,[1,Example3.2]isanexampleofaJaffardring
BthatisnotlocallyJaffardbutB[X]islocallyJaffard.
Corollary 1.25. Let A be an AF-domain and B a Jaffard ring such that B[X] is locally
Jaffard. Then,A⊗BisaJaffardring.
WeclosethissectionwithanexamplewherethetensorproductofanAF-domain
andaJaffardringisnotnecessarilyaJaffardring. Thisshowsthatasimilarresult
toTheorem1.19doesnothold,ingeneral,forthetransferoftheJaffardproperty.
Example1.26. Wededucethisexamplefrom[1]. LetZ ,Z ,Z ,Z befourindeter-
1 2 3 4
minatesoverk. LetL:=k(Z ,Z ,Z ,Z )andV :=k(Z ,Z ,Z )[Z ] . Then,V is
1 2 3 4 1 1 2 3 4 (Z4) 1
aone-dimensional valuationringofLwith maximalidealM :=Z V . LetV′ be
1 4 1
aone-dimensionalvaluationoverringofk(Z )[Z ,Z ]oftheformV′:=k(Z )+M′
4 2 3 4
andV′ =k[Z ] +M′=k+M′,whereM′ =Z k[Z ] +M′. So,V′ isatwo-di-
2 4 (Z4) 2 2 4 4 (Z4) 2
mensionalvaluationring. Now,letV=k(Z ,Z ,Z )[Z ] =k(Z ,Z ,Z )+M,with
2 3 4 1 (Z1) 2 3 4
M=Z V, and M =M′ +M. Then, V :=V′+M=k+M is a three-dimensional
1 2 2 2 2 2
valuationring.
We claim that V and V are incomparable. Otherwise, V ⊂V and hence
1 2 2 1
V =(V ) . So,MisadividedprimeidealofV . Thatis,Z V =M =MV . So
1 2 M 2 4 1 1 2M
1=Z Z−1 ∈MV=M, the desired contradiction. Now, V and V have the same
4 4 1 2
quotientfieldL. By[31,Theorem11.11],S:=V ∩V isathree-dimensionalPru¨fer
1 2
domainwithonlytwomaximalideals,N andN ,suchthatS =V andS =V .
1 2 N1 1 N2 2
Next,letF:=k(Z ), f :V −→k(Z ,Z ,Z )bethenaturalringhomomorphism,and
1 1 1 2 3
D:= f−1(F)=F+M . Let g:S−→S/N (cid:27)V /N (cid:27)k(Z ,Z ,Z )bethenaturalring
1 1 1 1 1 2 3
homomorphismandB:=g−1(F). WehaveB=D∩S=D∩V anddim(B)=dim(S)=
2
3. Moreover,by[1,Theorem2.11],weobtain
dim (B)=max dim (S),dim (F)+dim (S )+t.d.(S/N :F) =3.
v n v v v N1 1 o
8 SALAHKABBAJ
Therefore, B is Jaffard. Since B=D∩V and V , V are incomparable, it fol-
2 1 2
lows that B =D and B =V , where {n ,n }=Max(B). Moreover, ht(n [s])=
n1 n2 2 1 2 1
ht(n B [s])=ht(M [s]), for anypositive integer s. SinceV isJaffard,by[5, The-
1 n1 1 1
orem1.7],ht (M [s])=ht (M )+min(s,2). Thenht(n )=1,ht(n [X ])=2,and
D[s] 1 V1 1 1 1 1
ht(n [X ,X ])=3;t(B/n )=t(D/M )=1,andt(B/n )=t(V /M )=0.
1 1 2 1 1 2 2 2
LetA:=k(X). ByTheorem1.5,wehave
dim(A⊗B)=D(t(A),0,B)=max ht(q[X ])+min 1,t(B/q) |q∈Spec(B) .
n 1 (cid:16) (cid:17) o
For q:=n , it yields ht(n [X ])+min(1,t(B/n ))=2+1=3; for q:=n , it yields
1 1 1 1 2
ht(n [X ])+min(1,t(B/n ))=ht(n )=3,andht(q[X ])+min(1,t(B/q))≤3forevery
2 1 2 2 1
primeidealqofBcontainedinn . Consequently,dim(A⊗B)=3. ByCorollary1.22,
2
dim (A⊗B) =max ht(q[X ,X ])+min(1,t(B/q))| q∈ Spec(B) . For q:= n , it is
v n 1 2 o 1
ht(n [X ,X ])+min(1,t(B/n ))=3+1=4. Therefore,dim (A⊗B)=4,dim(A⊗B).
1 1 2 1 v
Consequently,A⊗BisnotaJaffardring,completingtheproofoftheexample.
2. TensorproductsofAF-ringsoverazero-dimensionalring
This section is devoted to [9]. Its purpose is to extend all the known results on
the dimension of tensor products of AF-rings over a field to the general case of
AF-ringsoverazero-dimensionalring.
Throughoutthissection,Rdenotesazero-dimensionalring,andalgebras(resp.,
tensorproducts),whennotspecificallymarked,aretakenover(resp.,relativeto)
R. We denote by (A,λ ) an algebra A and its associated ring homomorphism
A
λ :R→A; and, by λ∗ , the associated spectral map Spec(A)→Spec(R). Notice
A A
thatforanyprimeidealPofA, λ−1(P)isamaximalidealofR. So, wedefinethe
A
transcendencedegreeofthealgebraAoverRasfollows
t.d.(A: R)=max t.d. A/P:R/λ−1(P) |P∈Spec(A) .
λA n (cid:16) A (cid:17) o
We write t(A:R) or just t(A) as an abbreviation for t.d.(A: R), when there is
λA
no ambiguity. Allalong this section, we consider only algebras(A,λ ) suchthat
A
t(A)<∞, which also ensures that dim(A)<∞. If A is an integral domain, p
A
denotesKer(λ ).
A
Firstof all, observethatthe transcendencedegreeof analgebraA dependson
itsR-modulestructure,asshownbythenextexample.
Example2.1. LetR:=k(X)×kandA:=k(X),wherekisafield. Considerthetworing
homomorphismsλ :R→Aandλ :R→Adefinedbyλ (x,y)=xandλ (x,y)=y.
1 2 1 2
Thent.d.(A: R)=t.d.(k(X):k(X))=0whereast.d.(A: R)=t.d.(k(X):k)=1.
λ1 λ2
The following lemma providessimple generalizationsof well-known factsfor
algebrasoverafield.
Lemma2.2. Let(A,λ )beanalgebra,P∈Spec(A),andp:=λ−1(P). Then:
A A
(1) ht(P)+t(A/P:R)≤t(A :R)=t((A/pA) :R).
P P/pA
(2) ht(P)=ht(P/pA).
(3) ht(P[n])=ht((P/pA)[n]),foreachn≥1.
(4) IfAislocallyJaffard,thensoisA/qA,foreachq∈Spec(R)withqA,A.
DIMENSIONTHEORYOFTENSORPRODUCTSOFAF-RINGS 9
2.1. Tensorially compatible algebras. Let (A ,λ ) and (A ,λ ) be algebras. For
1 1 2 2
i=1,2, we denote by µ :A →A ⊗A the canonical A-algebra homomorphism.
i i 1 2 i
ThealgebraA1⊗A2,whennotspecificallyindicated,hasλA1⊗A2 =µ1◦λ1=µ2◦λ2
asitsassociatedringhomomorphism. Finally,let
Γ(A ,A ):= (P ,P )∈Spec(A )×Spec(A )|λ−1(P )=λ−1(P ) .
1 2 n 1 2 1 2 1 1 2 2 o
Weareinterestedinalgebras(A ,λ )and(A ,λ )suchthatA ⊗A ,0,andcall
1 1 2 2 1 2
suchalgebrastensoriallycompatible. Thenextresultprovidessomeelementaryand
useful characterizations of tensorially compatible algebras. For a more general
result,wereferthereaderto[24,Corollary3.2.7.1].
Proposition2.3. Let(A ,λ )and(A ,λ )bealgebras. Then,thefollowingassertionsare
1 1 2 2
equivalent:
(1) (A ,λ )and(A ,λ )aretensoriallycompatible;
1 1 2 2
(2) λ∗(Spec(A ))∩λ∗(Spec(A )),∅;
1 1 2 2
(3) ∃P Spec(A )suchthatλ−1(P )A ,A ;
1 1 1 1 2 2
(4) ∃P Spec(A )suchthatλ−1(P )A ,A ;
2 2 2 2 1 1
(5) ∃pSpec(R)suchthatpA ,A andpA ,A ;
1 1 2 2
(6) Ker(λ )+Ker(λ ),R.
1 2
Asimilarresultholdsforanyfinitenumberofalgebras,asshownbelow.
Proposition2.4. Let(A ,λ ),...,(A ,λ )bealgebras. Then,thefollowingassertionsare
1 1 n n
equivalent:
(1) A ⊗···⊗A ,0;
1 n
(2) λ∗(Spec(A ))∩λ∗(Spec(A ))∩···∩λ∗(Spec(A )),∅;
1 1 2 2 n n
(3) ∃pSpec(R)suchthatpA ,A,foreachi=1,2,...,n.
i i
Thenextresultestablishesananalogueto[35,Proposition2.3].
Proposition2.5. Let(A ,λ )and(A ,λ )bealgebrasand(P ,P )∈Spec(A )×Spec(A )
1 1 2 2 1 2 1 2
withλ−1(P )=λ−1(P )=p. LetΩ:= Q∈Spec(A ⊗A )|µ−1(Q)=P, i=1,2 . Then
1 1 2 2 n 1 2 i i o
(1) ΩislatticeisomorphictoSpec AP1 ⊗ AP2 .
(cid:16)P1AP1 Rp P2AP2(cid:17)
(2) Q∈ΩisminimalinΩifandonlyift((A ⊗A )/Q)=t(A /P )+t(A /P ).
1 2 1 1 2 2
(3) IfQ ∈Spec(A ⊗A )andµ−1(Q )⊇P (i=1,2),then∃Q∈ΩsuchthatQ⊆Q .
o 1 2 i o i o
Follow two applications of the above result, which extend two known results
onalgebrasoverafield[35]toR-algebras.
Corollary 2.6. Let (A ,λ ) and (A ,λ ) be tensorially compatible algebras and let Q∈
1 1 2 2
Spec(A ⊗A ). Then
1 2
ht(Q)≥ht(µ−1(Q))+ht(µ−1(Q)).
1 2
Corollary2.7. Let(A ,λ )and(A ,λ )betensoriallycompatiblealgebras. Then
1 1 2 2
t(A ⊗A )=max t(A /P )+t(A /P )|(P ,P )∈Γ ≤t(A )+t(A ).
1 2 n 1 1 2 2 1 2 o 1 2
Let(A ,λ )and(A ,λ )betensoriallycompatiblealgebras. Clearly,t(A ⊗A )=
1 1 2 2 1 2
t(A )+t(A )ifandonlyifthereexists(P ,P )∈Γwitht(A )=t(A /P )andt(A )=
1 2 1 2 1 1 1 2
t(A /P ). The second condition holds, for instance, if A and A are integral
2 2 1 2
domainsorifSpec(R)isreducedtoonlyoneprimeideal. Ingeneral,theequality
fails as it is shown by the next example. Moreover, when R is a field, we have
10 SALAHKABBAJ
dim(A ⊗A )≥dim(A )+dim(A )[35, Corollary2.5]. Thisisnottrue,ingeneral,
1 2 1 2
inthezero-dimensionalcase,asshownbelow.
Example2.8. LetR:=R×R, A :=Rand A :=R×R[X]. Consider the two ring
1 2
homomorphismsλ :R→A andλ :R→A definedbyλ (x,y)=xandλ (x,y)=
1 1 2 2 1 2
(x,y). Clearly,A andA aretensoriallycompatible. Weclaimthat
1 2
t(A ⊗A )(cid:8)t(A )+t(A ) and dim(A ⊗A )(cid:8)dim(A )+dim(A ).
1 2 1 2 1 2 1 2
Indeed, one can easily see that t(A )=t(R: R)=t.d.(R:R)=0, and t(A )=
1 λ1 2
t(R×R[X]: R)=max t.d.(R:R),t.d.(R[X]:R) =1. Moreover,byCorollary2.7,
λ2 n o
t(A ⊗A )=t(A )+t A /(0×R[X]) =t.d.(R:R)+t.d.(R:R)=0. Itfollowsthat
1 2 1 (cid:16) 2 (cid:17)
dim(A ⊗A )≤t(A ⊗A )=0(cid:8)t(A )+t(A )=dim(A )+dim(A )=1
1 2 1 2 1 2 1 2
completingtheproofoftheexample.
2.2. Krulldimension. ThissubsectioninvestigatestheKrulldimensionoftensor
products of AF-rings over zero-dimensional rings. We first extend Wadsworth’s
definitionofAF-ringsoverfieldstoAF-ringsoverzero-dimensionalrings. Recall
thatRdenotesazero-dimensionalringandalgebrasaretakenoverR.
Definition2.9. Undertheabovenotation,analgebra(A,λ )isanAF-ringif
A
ht(P)+t(A/P)=t(A ), ∀P∈Spec(A).
P
It is worthwhile observing that this notion of AF-ring is independent of the
structure of algebra defined by the ring homomorphism λ . Indeed, let A be
A
an algebra and let λ and λ′ be two ring homomorphisms defining two different
structuresofalgebraoverRonA. LetP∈Spec(A)andπ:A→A/Pbethecanonical
ring homomorphism. Let p:=Ker(π◦λ)=λ−1(P) and q:=Ker(π◦λ′)=λ′−1(P).
OnecanregardR/pandR/qassubfieldsofA/P. Letk:=R/p∩R/q. Ononehand,
wehave
t(A/P: R)=t.d.(A/P:R/p)=t.d.(A/P:k)−t.d.(R/p:k)
λ
and
t(A/P:λ′ R)=t.d.(A/P:R/q)=t.d.(A/P:k)−t.d.(R/q:k).
Ontheotherhand,wehave
t(A : R) = max t.d.(A/Q:R/p)|Q∈Spec(A)andQ⊆P
P λ n o
= max t.d.(A/Q:k)|Q∈Spec(A)andQ⊆P −t.d.(R/p:k)
n o
and
t(AP:λ′ R) = maxnt.d.(A/Q:R/q)|Q∈Spec(A)andQ⊆Po
= max t.d.(A/Q:k)|Q∈Spec(A)andQ⊆P −t.d.(R/q:k).
n o
Itfollowsthatt(AP:λR)−t(A/P:λR)=t(AP:λ′ R)−t(A/P:λ′ R). Thatis,(A,λ)isan
AF-ringifandonlyif(A,λ′)isanAF-ring.
Next,weprovidesomeexamplesandbasicpropertiesofAF-rings.
Lemma2.10. LetRbetheclassofAF-rings(overR)andletnbeapositiveinteger. Then:
(1) A∈R⇔A/pAisanAF-ringoverthefieldR/p,∀p∈Spec(R)withpA,A.
(2) AnyfinitelygeneratedR-algebrasanditsintegralextensionsbelongtoR.