Table Of ContentEMBEDDING DIMENSION AND CODIMENSION
OF TENSOR PRODUCTS OF ALGEBRAS OVER A FIELD
7
1
S.BOUCHIBA(⋆)ANDS.KABBAJ(⋆)
0
2
ToDavidDobbsontheoccasionofhis70thbirthday
n
a
J Abstract. Letkbeafield.Thispaperinvestigatestheembeddingdimensionand
codimensionofNoetherianlocalringsarisingaslocalizationsoftensorproducts
9
ofk-algebras. Weuseresultsandtechniquesfromprimespectraanddimension
1
theorytoestablishananalogueofthe“specialchaintheorem”fortheembedding
dimensionoftensorproducts,witheffectiveconsequenceonthetransferordefect
]
C ofregularityasexhibitedbythe(embedding)codimensiongivenbycodim(R):=
embdim(R)−dim(R).
A
.
h
t
a
m 1. Introduction
[ Throughout, all rings are commutative with identity elements, ring homomor-
phisms areunital, and k stands for a field. The embeddingdimension of a Noe-
1
therianlocalring(R,m),denotedbyembdim(R),istheleastnumberofgenerators
v
5 of m or, equivalently, the dimension of m/m2 as an R/m-vector space. The ring
6 RisregularifitsKrulldimensionandembeddingdimensionscoincide. The(em-
3
bedding) codimension of R measures the defect of regularity of R and is given
5 by the formula codim(R):=embdim(R)−dim(R). The concept of regularity was
0
. initiallyintroduced byKrull and becameprominentwhenZariski showed thata
1
local regular ring corresponds to a smooth point on an algebraic variety. Later,
0
Serreprovedthataringisregularifandonlyifithasfiniteglobaldimension. This
7
1 allowed to see thatregularityis stable under localizationand then the definition
: got globalized as follows: a Noetherian ring is regular if its localizations with
v
respecttoallprime idealsareregular. The ringRisa complete intersectionif its
i
X m-completionisthequotientringofalocalregularringmoduloanidealgenerated
r byaregularsequence; RisGorensteinifitsinjectivedimensionisfinite; andRis
a Cohen-Macaulayifthegradeandheightofmcoincide. Allthesealgebro-geometric
notionsareglobalizedbycarryingovertolocalizations.
Theseconceptstransfertotensorproductsofalgebrasoverafieldundersuitable
assumptions. IthasbeenprovedthataNoetheriantensorproductofalgebras(over
afield)inheritsthenotionsof(locally)completeintersectionring,Gorensteinring,
andCohen-Macaulayring[7,19,33,36]. Inparticular,aNoetheriantensorproduct
of any two extension fields is a complete intersection ring. As to regularity and
unlike the above notions, a Noetherian tensor product of two extension fields of
Date:January20,2017.
2010MathematicsSubjectClassification. 13H05,13F20,13B30,13E05,13D05,14M05,16E65.
Key words and phrases. Tensor product of k-algebras, regular ring, embedding dimension, Krull
dimension,embeddingcodimension,separableextension.
(⋆)SupportedbyKFUPMunderDSRResearchGrant#RG1212.
1
2 S.BOUCHIBAANDS.KABBAJ
k is not regular in general. In 1965, Grothendieck proved a positive result in
caseoneofthetwoextensionfieldsisafinitelygeneratedseparableextension[18].
Recently,wehaveinvestigatedthepossibletransferofregularitytotensorproducts
ofalgebrasoverafieldk. IfAandBaretwok-algebrassuchthatAisgeometrically
regular;i.e.,A⊗ FisregularforeveryfiniteextensionFofk(e.g.,Aisaseparable
k
extensionfieldoverk),weprovedthatA⊗ BisregularifandonlyifBisregularand
k
A⊗ BisNoetherian[8,Lemma2.1]. Asaconsequence,weestablishednecessary
k
andsufficient conditions for aNoetherian tensor productof twoextension fields
of k to inherit regularityunder (pure in)separabilityconditions [8, Theorem 2.4].
Also,Majadas’relativelyrecentpapertackledquestionsofregularityandcomplete
intersectionoftensorproductsofcommutativealgebrasviathehomologytheory
ofAndre´ andQuillen[25]. Finally,itisworthwhilerecallingthattensorproducts
of rings subject to the above concepts were recently used to broaden or delimit
the context of validity of some homological conjectures; see for instance [20, 22].
Suitable background on regular, complete intersection, Gorenstein, and Cohen-
Macaulayrings is [14, 18, 24, 26]. For a geometric treatment of these properties,
wereferthereadertotheexcellentbookofEisenbud[15].
Throughout, givenaring R, I anidealof Randp aprime idealof R, when no
confusionislikely,wewilldenotebyI theidealIR ofthelocalringR andbyκ (p)
p p p R
theresiduefieldofR . Oneofthecornerstonesofdimensiontheoryofpolynomial
p
ringsinseveralvariablesisthespecialchaintheorem,whichessentiallyassertsthat
theheightofanyprimeidealofthepolynomialringcanalwaysberealizedviaa
specialchain of prime ideals passing by the extension of its contraction over the
basicring; namely,ifRisaNoetherianringandPisaprimeidealofR[X ,...,X ]
1 n
withp:=P∩R,then
dim(R[X ,...,X ] )=dim(R )+dim κ (p)[X ,...,X ]
1 n P p R 1 n Pp !
pRp[X1,...,Xn]
AnanalogueofthisresultforNoetheriantensorproducts,establishedin[7],states
that,foranyprimeidealPofA⊗ Bwithp:=P∩Aandq:=P∩B,wehave
k
dim(A⊗ B) =dim(A ) + dim κ (p)⊗ B
k P p A k Pp
whichalsocomesinthefollowingextendedform(cid:16) (cid:17)pAp⊗kB
dim(A⊗ B) =dim(A )+dim(B )+dim κ (p)⊗ κ (q) .
k P p q A k B P(Ap⊗kBq)
Thispaperinvestigatestheembeddingdimensi(cid:16)onofNoether(cid:17)iapAnpl⊗okBcqa+lApr⊗inkqgBqsarising
as localizations of tensor products of k-algebras. We use results and techniques
from prime spectra and dimension theory to establish satisfactory analogues of
the “specialchain theorem” for the embedding dimension in variouscontexts of
tensorproducts,witheffectiveconsequencesonthetransferordefectofregularity
asexhibited bythe (embedding)codimension. Thepapertraversesfoursections
alongwithanintroduction.
InSection2,weintroduceandstudyanewinvariantwhichallowstocorrelate
the embedding dimension of a Noetherianlocal ring B with the fibre ring B/mB
ofalocalhomomorphism f :A−→BofNoetherianlocalrings. Thisenablesusto
provideananalogueofthespecialchaintheoremfortheembeddingdimensionas
EMBEDDINGDIMENSIONANDCODIMENSIONOFTENSORPRODUCTSOFk-ALGEBRAS 3
wellastogeneralizetheknownresultthat“if f isflatandAandB/mBareregular
rings,thenBisregular.”
Section3isdevotedtothespecialcaseofpolynomialringswhichwillbeused
intheinvestigationoftensorproducts. Themainresult(Theorem3.1)statesthat,
foraNoetherianringRandX ,...,X indeterminatesoverR,foranyprimeidealP
1 n
ofR[X ,...,X ]withp:=P∩R,wehave:
1 n
P
embdim(R[X ,...,X ] ) = embdim(R )+ht
1 n P p
p[X1,...,Xn]!
= embdim(R )+embdim κ (p)[X ,...,X ]
p R 1 n Pp
!
pRp[X1,...,Xn]
Then,Corollary3.2assertsthat
codim(R[X ,...,X ] )=codim(R )
1 n P p
andrecoversawell-knownresultonthetransferofregularitytopolynomialrings;
i.e., R[X ,...,X ] is regular if and only if so is R (this result was initially proved
1 n
via Serre’s result on finite global dimension and Hilbert Theorem on syzygies).
Then Corollary 3.3 characterizes regularity in general settings of localizations of
polynomialringsand,intheparticularcasesofNagataringsandSerreconjecture
rings, it states thatR(X ,...,X ) is regular if and only if RhX ,...,X i is regular if and
1 n 1 n
onlyifRisregular.
LetAandBbetwok-algebrassuchthatA⊗ BisNoetherianandletPbeaprime
k
ideal of A⊗ B with p:=P∩A and q:=P∩B. Due to known behavior of tensor
k
productsofk-algebrassubjecttoregularity(cf. [8,18,19,33,36]),Section4inves-
tigatesthecasewhenA(orB)isaseparable(notnecessarilyalgebraic)extension
fieldofk. Themainresult(Theorem4.2)assertsthat,ifKisaseparableextension
fieldofk,then
embdim(K⊗ A) =embdim(A ) + embdim K⊗ κ (p) .
k P p k A Pp
Inparticular,ifKisseparablealgebraicoverk,then (cid:16) (cid:17)K⊗kpAp
embdim(K⊗ A) =embdim(A ).
k P p
Then,Corollary4.5assertsthat
codim(K⊗ A) =codim(A )
k P p
and hence K⊗ A is regular if and only if so is A. This recovers Grothendieck’s
k
resultonthetransferofregularitytotensorproductsissuedfromfiniteextension
fields[18,Lemma6.7.4.1].
Section5examinesthemoregeneralcaseoftensorproductsofk-algebraswith
separableresiduefields. The maintheorem(Theorem5.1) statesthatif κ (q)is a
B
separableextensionfieldofk,then
embdim(A⊗ B) = embdim(A ) + embdim(B )
k P p q
+ embdim κ (p)⊗ κ (q)
A k B P(Ap⊗kBq)
Then,Corollary5.2contendsthat (cid:16) (cid:17)pAp⊗kBq+Ap⊗kqBq
codim(A⊗ B) =codim(A )+codim(B )
k P p q
4 S.BOUCHIBAANDS.KABBAJ
recovering known results on the transfer of regularity to tensor products over
perfect fields [33, Theorem 6(c)] and, more generally, to tensor products issued
fromresiduallyseparableextensionfields[8,Theorem2.11].
Thefouraforementionedmainresultsareconnectedasfollows:
Proposition4.1
u
Theorem3.1 ⇓ Theorem5.1
u t
Theorem4.2
OfrelevancetothisstudyisBouchiba,Conde-Lago,andMajadas’recentpreprint
[4]wheretheauthorsprovesomeofourresultsviathehomologytheoryofAndre´
and Quillen. In the current paper, we offer direct and self-contained proofs us-
ing techniques and basic results from commutative ring theory. Early and re-
cent developments on prime spectra and dimension theory are to be found in
[3, 5, 6, 7, 29, 30, 31, 34, 35] for the special case of tensor products of k-algebras,
andin[1,11,17,22,24,26,27]forthegeneralcase. Anyunreferencedmaterialis
standard,asin[24,26].
2. EmbeddingdimensionofNoetherianlocalrings
In this section, we discuss the relationship between the embedding dimen-
sionsofNoetherianlocalringsconnectedbyalocalringhomomorphism. Tothis
purpose, we introduce a new invariant µ which allows to relate the embedding
dimensionofalocalringtothatofitsfibrering.
Throughout,let(A,m,K)and(B,n,L)belocalNoetherianrings, f :A−→Balocal
homomorphism(i.e.,mB:= f(m)B⊆n),andIaproperidealofA. Let
I+m2
µ (I):=dim .
A K m2 !
Note that µ (I) equals the maximal number of elements of I which are part of a
A
minimalbasisofm;sothat0≤µ (I)≤embdim(A)andµ (m)=embdim(A). Next,
A A
letµf(I)denotethemaximalnumberofelementsofIB:= f(I)Bwhicharepartofa
B
minimalbasisofn;thatis,
IB+n2
µf(I):=µ (IB)=dim .
B B L n2 !
Itiseasilyseenthatifx ,...,x areelementsofmsuchthat f(x ),...,f(x )arepartof
1 r 1 r
aminimalbasisofn,thenx ,...,x arepartofaminimalbasisofmaswell. Thatis,
1 r
0≤µf(I)≤µ (I). Moreover,ifJisaproperidealofBandπ:B։B/Jisthecanonical
B A
IB+n2 IB+n2+J
surjection,thenthenaturallinearmapofL-vectorspaces ։ yields
n2 n2+J
π◦f f
µ (I)≤µ (I).
B/J B
Proposition2.1. Undertheabovenotation,wehave:
embdim(B)=µf(I)+embdim(B/IB).
B
Inparticular,
embdim(A)=µ (I)+embdim(A/I).
A
EMBEDDINGDIMENSIONANDCODIMENSIONOFTENSORPRODUCTSOFk-ALGEBRAS 5
Proof. The first statement follows easily from the following exact sequence of L-
vectorspaces
IB+n2 n n n/IB
0−→ −→ −→ = −→0.
n2 n2 IB+n2 (n/IB)2
Thesecondstatementholdssinceµ (I)=µidA(I). (cid:3)
A A
Recall that, under the above notation, the following inequality always holds:
dim(B)≤dim(A)+dim(B/mB). The first corollary provides an analogue for the
embeddingdimension.
Corollary2.2. Undertheabovenotation,wehave:
embdim(B)≤embdim(A)−embdim(A/I)+embdim(B/IB).
Inparticular,
embdim(B)≤embdim(A)+embdim(B/mB).
It is well known that if f is flat and both A and B/m B are regular, then B is
regular. The second corollary generalizes this result to homomorphisms subject
togoing-down. Recallthataringhomomorphismh:R−→Ssatisfiesgoing-down
(henceforthabbreviatedGD)ifforanypairp⊆qinSpec(R)suchthatthereexists
Q∈Spec(S)lyingoverq,thenthereexistsP∈Spec(S)lyingoverpwithP⊆Q. Any
flatringhomomorphismsatisfiesGD.
Corollary2.3. Undertheabovenotation,assumethat f satisfiesGD.Then:
(a) codim(B)= µf(m)−dim(A) +codim(B/mB).
B
(b) codim(B)+(cid:16)embdim(A)−µf(cid:17)(m) =codim(A)+codim(B/mB).
B
(c) Bisregulara(cid:16)ndµf(m)=embdim(cid:17)(A)⇐⇒AandB/mBareregular.
B
Proof. The proof isstraightforward via a combination of Proposition 2.1and [26,
Theorem15.1]. (cid:3)
Corollary2.4. Undertheabovenotation,assumethat f satisfiesGD.Then:
(a) codim(B)≤codim(A)+codim(B/mB).
(b) IfB/mBisregular,thencodim(B)≤codim(A).
Proof. The proof is direct via a combination of Corollary 2.2 and the known fact
thatdim(B)=dim(A)+dim(B/mB). (cid:3)
3. Embeddingdimensionandcodimensionofpolynomialrings
Thissectionisdevotedtothespecialcaseofpolynomialringswhichwillbeused,
later, for the investigation of tensor products. The main result of this section
(Theorem3.1)settlesaformulafortheembeddingdimensionforthelocalizations
ofpolynomial ringsoverNoetherianrings. Itrecovers(viaCorollary3.2)awell-
knownresultonthetransferofregularitytopolynomialrings;thatis,R[X ,....,X ]
1 n
is regular if and only if so is R. Moreover, Theorem 3.1 leads to investigate the
regularity of two famous localizations of polynomial rings in several variables;
namely,theNagataringR(X ,X ,...,X )andSerreconjectureringRhX ,X ,...,X i.
1 2 n 1 2 n
Weshowthattheregularityofthesetwoconstructionsisentirelycharacterizedby
theregularityofR(Corollary3.3).
6 S.BOUCHIBAANDS.KABBAJ
Recall that one of the cornerstones of dimension theory of polynomial rings
in several variables is the special chain theorem, which essentially asserts that the
height of any prime ideal P of R[X ,...,X ] can always be realized via a special
1 n
chainof prime idealspassing bythe extension (P∩R)[X ,...,X ]. This resultwas
1 n
firstprovedbyJaffardin[22]and,later,Brewer,Heinzer,MontgomeryandRutter
reformulated it in the following simple way ([12, Theorem 1]): Let P be a prime
P
idealofR[X ,...,X ]withp:=P∩R. Thenht(P)=ht(p[X ,...,X ])+ht .
1 n 1 n
p[X1,...,Xn]!
InaNoetheriansetting,thisformulabecomes:
P
dim(R[X ,...,X ] ) = dim(R )+ht
1 n P p
p[X1,...,Xn]! (1)
= dim(R )+dim κ (p)[X ,...,X ]
p R 1 n Pp
!
pRp[X1,...,Xn]
wherethesecondequalityholdsonaccountofthebasicfact P ∩R =0. The
p[X1,...,Xn] p
mainresultofthissection(Theorem3.1)featuresa“specialchaintheorem”forthe
embeddingdimensionwitheffectiveconsequenceonthecodimension.
Theorem3.1. LetRbeaNoetherianringandX ,...,X beindeterminatesoverR. LetP
1 n
beaprimeidealofR[X ,...,X ]withp:=P∩R. Then:
1 n
P
embdim(R[X ,...,X ] ) = embdim(R )+ht
1 n P p
p[X1,...,Xn]!
= embdim(R )+embdim κ (p)[X ,...,X ]
p R 1 n Pp
!
pRp[X1,...,Xn]
Proof. We use induction on n. Assume n=1 and let P be a prime ideal of R[X]
with p:=P∩R and r:=embdim(R ). Then p =(a ,...,a )R for some a ,...,a ∈p.
p p 1 r p 1 r
Weenvisagetwocases;namely,eitherPisanextensionofporanuppertop. For
bothcases,wewilluseinductiononr.
Case 1: P is an extension of p (i.e., P=pR[X]). We prove that embdim(R[X] )
P
=r. Indeed,wehaveP =pR [X] =(a ,...,a )R [X] =(a ,...,a )R[X] .So,
P p pRp[X] 1 r p pRp[X] 1 r P
obviously,ifp =(0),thenP =0. Next,wemayassumer≥1. Onecaneasilycheck
p P
thatthecanonicalringhomomorphismϕ:R −→R[X] isinjectivewithϕ(p )⊆P .
p P p P
Thisforcesembdim(R[X] )≥1. Hence,thereexists j∈{1,...,n},say j=1,suchthat
P
a:=a ∈pwith a ∈P \P2 and,afortiori, a ∈p \p2. By[24,Theorem159],weget
1 1 P P 1 p p
R
embdim(R[X] )=1+embdim [X]
P P
(a) aR[X]!
emRbdim(Rp)=1+embdim ((Ra)(cid:17)(pa)! (2)
Thereforeembdim =r−1andthen,byinductiononr,weobtain
(a) p !
(cid:16) (cid:17)(a)
R R
embdim [X] =embdim . (3)
(a) aRP[X]! (cid:16)(a)(cid:17)(pa)!
Acombinationof(2)and(3)leadstoembdim(R[X] )=r,asdesired.
P
EMBEDDINGDIMENSIONANDCODIMENSIONOFTENSORPRODUCTSOFk-ALGEBRAS 7
Case2: Pisanuppertop(i.e.,P,pR[X]). Weprovethatembdim(R[X] )=r+1.
P
NotethatPR [X]isalsoanuppertop andthenthereexistsa(monic)polynomial
p p
f ∈R[X]suchthat f isirreducibleinκ (p)[X]andPR [X]=pR [X]+fR [X]. Notice
1 R p p p
thatpR[X]+fR[X]⊆Pandwehave
P = PR [X] =(pR [X]+fR [X])
P p PRp[X] p p PRp[X]
= (p[X]+fR[X])R [X] =(p[X]+fR[X])
p PRp[X] P
= p[X] +fR[X] =(a ,...,a ,f)R[X] .
P P 1 r P
Assumer=0. ThenPisanuppertozerowithP = fR[X] . Sothatembdim(R[X] )≤
P P P
1. Further,bytheprincipalidealtheorem[24,Theorem152],wehave
embdim(R[X] )≥dim(R[X] )=ht(P)=1.
P P
Itfollowsthatembdim(R[X] )=1,asdesired.
P
Next,assumer≥1. WeclaimthatpR[X] "P2. DenyandsupposethatpR[X] ⊆
P P P
P
P2. This assumption combined with the fact P =p[X] + fR[X] yields P =
P P P P P2
P
fR[X] as R[X] -modules and hence P = fR[X] by [24, Theorem 158]. Next,
P P P P
let a∈p. Then, as a ∈P = fR[X] , there exist g∈R[X] and s,t∈R[X]\P such
1 P P
thatt(sa− fg)=0. Sothattfg∈p[X],whence tg∈p[X]⊂P as f <p[X]. Itfollows
thattsa=tfg∈P2 and thus a ∈P2 = f2R[X] . We iterate the same processto get
1 P P
a ∈Pn = fnR[X] for each integer n≥1. Since R[X] is a Noetherian local ring,
1 P P P
Pn =(0) and thus a =0 in R[X] . By the canonical injective homomorphism
P 1 P
RTp֒→R[X]P, 1a =0inRp. Thuspp=(0),thedesiredcontradiction.
Consequently,pR[X] =(a ,...,a )R[X] "P2.So,thereexistsj∈{1,...,n},sayj=1,
P 1 r P P
suchthata:=a ∈P \P2 and,afortiori,a∈p \p2. SimilarargumentsasinCase1
1 P P p p
R
leadtothesametwoformulasdisplayedin(2). Thereforeembdim =r−1
(a) p !
(cid:16) (cid:17)(a)
andthen,byinductiononr,weobtain
R R
embdim [X] =1+embdim . (4)
(a) aRP[X]! (cid:16)(a)(cid:17)(pa)!
Acombinationof(2)and(4)leadstoembdim(R[X] )=r+1,asdesired.
P
Now, assume that n≥2 and set R[k]:=R[X ,...,X ] and p[k]=p[X ,...,X ] for
1 k 1 k
P
k:=1,...,n. Let P′ :=P∩R[n−1]. We prove that embdim(R[n] )=r+ht .
P
p[n]!
Indeed,byvirtueofthecasen=1,wehave
P
embdim(R[n]P)=embdim(R[n−1]P′)+ht P′[Xn]!. (5)
Moreover,byinductionhypothesis,weget
P′
embdim(R[n−1]P′)=r+ht p[n−1]!. (6)
8 S.BOUCHIBAANDS.KABBAJ
P′[X ] P
n
Notethattheprimeideals and bothsurviveinκ (p)[n],respectively.
R
p[n] p[n]
Hence,asκ (p)[n]iscatenarianand(R/p)[n−1]isNoetherian,weobtain
R
P P′[X ] P P′ P
ht =ht n +ht =ht +ht . (7)
p[n]! p[n] ! P′[Xn]! p[n−1]! P′[Xn]!
Further,thefactthatκ (p)[X ,...,X ]isregularyield
R 1 n
P
ht =embdim κ (p)[X ,...,X ] . (8)
p[X1,...,Xn]! R 1 n pRp[XP1p,...,Xn]!
So(5),(6),(7),and(8)leadtotheconclusion,completingtheproofofthetheorem.
(cid:3)
AsafirstapplicationofTheorem3.1,wegetthenextcorollaryonthe(embed-
ding) codimension. In particular, it recoversa well-known resulton the transfer
ofregularitytopolynomialrings(initiallyprovedviaSerre’sresultonfiniteglobal
dimension and Hilbert Theorem on syzygies [28, Theorem 8.37]. See also [24,
Theorem171]).
Corollary3.2. LetRbeaNoetherianringandX ,...,X beindeterminatesoverR. LetP
1 n
beaprimeidealofR[X ,...,X ]withp:=P∩R. Then:
1 n
codim(R[X ,...,X ] )=codim(R ).
1 n P p
Inparticular,R[X ,...,X ]isregularifandonlyifRisregular.
1 n
Theorem 3.1 allows us to characterize the regularity for two famous localiza-
tions of polynomial rings; namely, Nagata rings and Serre conjecture rings. Let
R be a ring and X,X ,...,X indeterminates over R. Recall that R(X ,...,X ) =
1 n 1 n
S−1R[X ,...,X ]istheNagataring,whereSisthemultiplicativesetofR[X ,...,X ]
1 n 1 n
consistingofthepolynomialswhosecoefficientsgenerateR. LetRhXi:=U−1R[X],
whereUisthemultiplicativesetofmonicpolynomialsinR[X],andRhX ,···,X i:=
1 n
RhX ,...,X ihX i. Then RhX ,...,X i is called the Serre conjecture ring and is a
1 n−1 n 1 n
localizationofR[X ,...,X ].
1 n
Corollary3.3. LetRbeaNoetherianringandX ,...,X indeterminatesoverR. LetSbe
1 n
amultiplicativesubsetofR[X ,...,X ]. Then:
1 n
(a) S−1R[X ,...,X ]isregularifandonlyifR isregularforeachprimeidealpofR
1 n p
suchthatp[X ,...,X ]∩S=∅.
1 n
(b) Inparticular,R(X ,...,X )isregularifandonlyifRhX ,...,X iisregularifand
1 n 1 n
onlyifR[X ,...,X ]isregularifandonlyifRisregular.
1 n
Proof. (a)LetQ=S−1PbeaprimeidealofS−1R[X ,...,X ],wherePistheinverse
1 n
imageofQbythecanonicalhomomorphismR[X ,...,X ]→S−1R[X ,...,X ]andlet
1 n 1 n
p:=P∩R. NoticethatS−1R[X ,...,X ] (cid:27)R[X ,...,X ] and
1 n Q 1 n P
Q −1 P
(cid:27)S where S denotes the image of S via the natural
S−1p[X1,...,Xn] p[X1,...,Xn]
homomorphismR[X ,...,X ]→ R[X ,...,X ].Therefore,by(1),weobtain
1 n p 1 n
Q
dim(S−1R[X ,...,X ] )=dim(R[X ,...,X ] )=dim(R )+ht (9)
1 n Q 1 n P p S−1p[X ,...,X ]
(cid:16) 1 n (cid:17)
EMBEDDINGDIMENSIONANDCODIMENSIONOFTENSORPRODUCTSOFk-ALGEBRAS 9
and,byTheorem3.1,wehave
embdim(S−1R[X ,...,X ] ) = embdim(R[X ,...,X ] )
1 n Q 1 n P
= embdim(R )+ht Q . (10)
p S−1p[X ,...,X ]
(cid:16) 1 n (cid:17)
Now,observethattheset{Q∩R|QisaprimeidealofS−1R[X ,...,X ]}isequal
1 n
to the set {p|p is a prime ideal of R such that p[X ,...,X ]∩S=∅}. Therefore, (9)
1 n
and(10)leadtotheconclusion.
(b) Combine (a) with the fact that the extension of any prime ideal of R to
R[X ,...,X ]doesnotmeetthe multiplicativesetsrelatedtothe ringsR(X ,...,X )
1 n 1 n
andRhX ,...,X i. (cid:3)
1 n
4. Embeddingdimensionandcodimensionoftensorproductsissuedfrom
separableextensionfields
Thissectionestablishesananalogueofthe“specialchaintheorem”fortheembed-
ding dimension of Noetherian tensor products issued from separable extension
fields,witheffectiveconsequencesonthetransferordefectofregularity. Namely,
due to known behavior of a tensor product A⊗ B of two k-algebras subject to
k
regularity(cf. [8,18,19,26,33,36]),wewillinvestigatethecasewhereAorBisa
separable(notnecessarilyalgebraic)extensionfieldofk.
Throughout,letAandBbetwok-algebrassuchthatA⊗ BisNoetherianandlet
k
PbeaprimeidealofA⊗ Bwithp:=P∩Aandq:=P∩B. RecallthatAandBare
k
Noetheriantoo;andtheconverseisnottrue,ingeneral,evenifA=Bisanextension
fieldofk(cf. [16,Corollary3.6]or[34,Theorem11]). Weassumefamiliaritywiththe
naturalisomorphismsfortensorproductsandtheirlocalizationsasin[9,10,28]. In
particular,weidentifyAandBwiththeirrespectiveimagesinA⊗ Bandwehave
k
A⊗ B A B
k (cid:27) ⊗ andA ⊗ B (cid:27)S−1(A⊗ B)whereS:={s⊗t|s∈A\p,t∈B\q}.
p⊗ B+A⊗ q p k q p k q k
k k
Throughoutthisandnextsections,weadoptthefollowingsimplifiednotationfor
theinvariantµ:
µ (pA ):=µi (pA ) and µ (qA ):=µj (qB )
P p (A⊗kB)P p P q (A⊗kB)P q
wherei:A −→(A⊗ B) and j:B −→(A⊗ B) arethe canonical(localflat)ring
p k P q k P
homomorphisms.
RecallthatA⊗ BisCohen-Macaulay(resp.,Gorenstein,locallycompleteinter-
k
section)ifandonlyifsoareAandthefibreringsκ (p)⊗ B(foreachprimeideal
A k
pofA)[7,33]. AlsoifAandthefibreringsκ (p)⊗ BareregularthensoisA⊗ B
A k k
[26,Theorem23.7(ii)]. However,theconversedoesnotholdingeneral;precisely,
if A⊗ B is regular then so is A [26, Theorem 23.7(i)] but the fibre rings are not
k
necessarilyregular(see[8,Example2.12(iii)]).
From [7, Proposition 2.3]and its proof, recallananalogue of the specialchain
theorem(recordedin (1))for thetensor productswhich correlatesthe dimension
of(A⊗ B) tothedimensionofitsfibrerings;namely,
k P
P
dim(A⊗ B) = dim(A ) + ht
k P p p⊗ B!
k
(11)
= dim(A ) + dim κ (p)⊗ B
p A k Pp
(cid:16) (cid:17)pAp⊗kB
10 S.BOUCHIBAANDS.KABBAJ
Our first result reformulates Proposition 2.1 and thus gives an analogue of
the special chain theorem for the embedding dimension in the context of tensor
productsofalgebrasoverafield.
Proposition4.1. LetAandBbetwok-algebrassuchthatA⊗ BisNoetherianandletP
k
beaprimeidealofA⊗ Bwithp:=P∩Aandq:=P∩B. Then:
k
(a) embdim(A⊗ B) =µ (pA )+embdim κ (p)⊗ B .
k P P p A k Pp
(b) codim(A⊗kB)P+ embdim(Ap)−µP(pAp(cid:16)) = (cid:17)pAp⊗kB
(cid:16) (cid:17)
codim(A )+codim κ (p)⊗ B .
p A k Pp
!
(c) (A⊗ B) is regular and µ (pA )=embdim(A ) if and o(cid:16)nly if both(cid:17)ApAp⊗kaBnd
k P P p p p
κ (p)⊗ B areregular.
A k Pp
(cid:16) (cid:17)pAp⊗kB
Recallthatanextendedformofthespecialchaintheorem[7]statesthat
dim(A⊗ B) =dim(A )+dim(B )+dim κ (p)⊗ κ (q) .
k P p q A k B P(Ap⊗kBq)
In this vein, notice that, via Proposition 4.1((cid:16)a), we always(cid:17)phApa⊗vkBeq+tAhpe⊗kqfBoqllowing
inequalities:
embdim(A⊗ B) ≤ embdim(A ) + embdim κ (p)⊗ B
k P p A k Pp
≤ embdim(Ap) + embdim(B(cid:16)q) (cid:17)pAp⊗kB
+ embdim κ (p)⊗ κ (q) .
A k B P(Ap⊗kBq)
Letusstatethemaintheoremofthissection.(cid:16) (cid:17)pAp⊗kBq+Ap⊗kqBq
Theorem4.2. LetKbeaseparableextensionfieldofkandAak-algebrasuchthatK⊗ A
k
isNoetherian. LetPbeaprimeidealofK⊗ Awithp:=P∩A. Then:
k
P
embdim(K⊗ A) = embdim(A ) + ht
k P p K⊗ p!
k
= embdim(A ) + embdim K⊗ κ (p)
p k A Pp
If,inaddition,Kisalgebraicoverk,thenembdim(K⊗ A) =(cid:16)embdim(A(cid:17)K⊗)k.pAp
k P p
Theproofofthistheoremrequiresthe followingtwopreparatorylemmas; the
first of which determines a formula for the embedding dimension of the tensor
product of two k-algebras A and B localized at a special prime ideal P with no
restrictiveconditionsonAorB.
Lemma4.3. LetAandBbetwok-algebrassuchthatA⊗ BisNoetherianandletPbea
k
primeidealofA⊗ Bwithp:=P∩Aandq:=P∩B. AssumethatP =(p⊗ B+A⊗ q) .
k P k k P
Then:
(a) µ (pA )=embdim(A )andµ (qB )=embdim(B ).
P p p P q q
(b) embdim(A⊗ B) =embdim(A )+embdim(B ).
k P p q
