EMBEDDING 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  whichalsocomesinthefollowingextendedform(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)  Thispaperinvestigatestheembeddingdimensi(cid:16)onofNoether(cid:17)iapAnpl⊗okBcqa+lApr⊗inkqgBqsarising 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(pAp(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⊗kqfBoqllowing 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