aa r X i v : . [ m a t h . A C ] F e b Krull dimension of tensor products of pullbacks
Samir Bouchiba
Department of Mathematics, University Moulay Ismail, Meknes 50000, Morocco
Abstract
This paper is concerned with the study of the dimension theory of tensor products of algebrasover a field k . We answer an open problem set in [6] and compute dim( A ⊗ k B ) when A is a k -algebra arising from a specific pullback construction involving AF-domains and B is an arbitrary k -algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, thatsuch a pullback A is in fact a generalized AF-domain.
1. Introduction
All rings considered in this paper are commutative with identity element and all ringhomomorphisms are unital. Throughout, k stands for a field. We shall use t.d.( A : k ), ort.d.( A ) when no confusion is likely, to denote the transcendence degree of a k -algebra A over k , A [ n ] to denote the polynomial ring A [ X , ..., X n ] and p [ n ] to denote the prime ideal p [ X , ..., X n ] of A [ X , ..., X n ] for each prime ideal p of A . Also, we use Spec( A ) to denotethe set of prime ideals of a ring A and ⊂ to denote proper set inclusion. All k -algebrasconsidered throughout this paper are assumed to be of finite transcendence degree over k .Any unreferenced material is standard as in [11], [15], [16] and [17].Several authors have been interested in studying the prime ideal structure and relatedtopics of tensor products of algebras over a field k . The initial impetus for these investigationswas a paper of R. Sharp on Krull dimension of tensor products of two extension fields.In fact, in [19], Sharp proved that, for any two extension fields K and L of k , dim( K ⊗ k L ) = min(t.d.( K ) , t.d.( L )) (actually, this result appeared ten years earlier in Grothendieck’sEGA [13, Remarque 4.2.1.4, p. 349]). This formula is rather surprising since, as one mayexpect, the structure of the tensor product should reflect the way the two componentsinteract and not only the structure of each component. This fact is what most motivatedWadsworth’s work in [20] on this subject. His aim was to seek geometric properties of primesof A ⊗ k B and to widen the scope of algebras A and B for which dim( A ⊗ k B ) dependsonly on individual characteristics of A and B . The algebras which proved tractable forKrull dimension computations turned out to be those domains A which satisfy the altitudeformula over k (AF-domains for short), that is, ————— Mathematics Subject Classification (2000): Primary 13C15; Secondary 13B24.
E-mail address : [email protected] keywords:
Krull dimension, tensor product, prime ideal, AF-domain. t ( p ) + t.d.( Ap ) = t.d.( A )for all prime ideals p of A . It is worth noting that the class of AF-domains contains the mostbasic rings of algebraic geometry, including finitely generated k -algebras that are domains.Wadsworth proved, via [20, Theorem 3.8], that if A and A are AF-domains, thendim( A ⊗ k A ) = min (cid:16) dim( A ) + t.d.( A ) , t.d.( A ) + dim( A ) (cid:17) . His main theorem stated a formula for dim( A ⊗ k B ) which holds for an AF-domain A , withno restriction on B , namely:dim( A ⊗ k B ) = D (cid:16) t.d.( A ) , dim( A ) , B (cid:17) := max n ht ( q [t.d.( A )]) + min (cid:16) t.d.( A ) , dim( A ) + t.d.( Bq ) (cid:17) : q ∈ Spec( B ) o [20, Theorem 3.7].Our aim in [6] was to extend Wadsworth’s results in a different way, namely to tensorproducts of k -algebras arising from pullbacks. In this regard, we use previous deep investi-gations on prime ideal structure of various pullbacks, as in [1]. Our main result in [6] statesthe following: Let T i be a k -algebra which is an integral domain and M i a maximal ideal of T i suchthat ht ( M i ) = dim ( T i ) , K i = T i M i , ϕ i the canonical surjection from T i to K i , D i a subringof K i , and A i = ϕ − i ( D i ) be the issued pullback, for i = 1 , . Assume that T i and D i areAF-domains, for i = 1 , . Thendim ( A ⊗ k A ) = max n ht ( M [ t.d. ( A )]) + D (cid:16) t.d. ( D ) , dim ( D ) , R (cid:17) ,ht ( M [ t.d. ( A )]) + D (cid:16) t.d. ( D ) , dim ( D ) , R (cid:17)o , where D ( s, d, A ) := max n ht ( p [ s ])+min (cid:16) s, d +t.d.( Ap ) (cid:17) : p ∈ Spec( A ) o for any k -algebra A and any positive integers 0 ≤ d ≤ s . This theorem allows one to compute the Krulldimension of tensor product of two k -algebras for a large family of (not necessarily AF-domains) k -algebras. Further, we set in [6] the open problem of computing dim( A ⊗ k A )when only T and D are assumed to be AF-domains.On the other hand, in [14], Jaffard proved that, for any ring A and any positive integer n ,the Krull dimension of A [ n ] can be realized as the length of a special chain of A [ n ]. Recallthat a chain C = { Q ⊂ Q ⊂ ... ⊂ Q s } of prime ideals of A [ n ] is called a special chainif for each Q i , the ideal ( Q i ∩ A )[ n ] belongs to C . Subsequently, Brewer et al. gave anequivalent and simple version of Jaffard’s theorem. Actually, they showed that, for each2ositive integer n and each prime ideal P of A [ n ], ht ( P ) = ht ( q [ n ]) + ht ( Pq [ n ] ) [10, Theorem1], where q := P ∩ A . Taking into account the natural isomorphism B [ n ] ∼ = k [ n ] ⊗ k B foreach k -algebra B , we generalized in [6] this special chain theorem to tensor products of k -algebras. Effectively, we proved that if A and B are k -algebras such that A is an AF-domain,then for each prime ideal P of A ⊗ k B , ht ( P ) = ht ( A ⊗ k q ) + ht ( PA ⊗ k q ) = ht ( q [t.d.( A )]) + ht ( PA ⊗ k q ) , where q = P ∩ B (cf. [6, Lemma 1.5]). It turned out that this very geometrical property to-tally characterizes the AF-domains. In fact, we proved, in [4], that the following statementsare equivalent for a domain A which is a k -algebra: a) A is an AF-domain;b) A satisfies SCT (for special chain theorem), that is, for each k -algebra B and eachprime ideal P of A ⊗ k B with q := P ∩ B , ht ( P ) = ht ( q [ t.d. ( A )]) + ht ( PA ⊗ k q ) = ht ( A ⊗ k q ) + ht ( PA ⊗ k q ) [4, Theorem 1.1] . In view of this, we generalized in [5] the AF-domain notion by setting the following defini-tions:
We say that a k -algebra A satisfies GSCT (for generalized special chain theorem) withrespect to a k -algebra B if ht ( P ) = ht ( p ⊗ k B + A ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q ) for each prime ideal P of A ⊗ k B , with p = P ∩ A and q = P ∩ B ,and we call a generalized AF-domain (GAF-domain for short) a domain A such that A sat-isfies GSCT with respect to any k -algebra B . There is no known example in the literature of a k -algebra A which is a domain and whichis not a GAF-domain. This may lead one to ask whether any k -algebra which is a domainis a GAF-domain. We were concerned in [5] with the following question:(Q): Is any domain A which is a k -algebra such that the polynomial ring A [ n ] is an AF-domain, for some positive integer n , a GAF-domain?We gave in [5] partial results settling in the affirmative the above question (Q). First, weproved that an AF-domain A is in fact a GAF-domain, thus in particular, any finitelygenerated algebra over k which is a domain is a GAF-domain. Also, we proved that (Q)has a positive answer in the case where A is one-dimensional. Our main result in [5] tacklesthe case n = 1 of ( Q ). It computes dim( A ⊗ k B ) for a k -algebra A such that A [ X ] is anAF-domain and for an arbitrary k -algebra B generalizing Wadsworth’s main theorem [20,3heorem 3.7] and further asserts that A is a GAF-domain. We ended that paper by anexample of a GAF-domain A such that, for any positive integer n , the polynomial ring A [ n ]is not an AF-domain.Our objective in this paper is twofold. On the one hand, we handle the above-mentionedproblem set in [6] and compute dim( A ⊗ k B ) when A is a pullback arising from the aboveconstruction and B is an arbitrary k -algebra. On the other hand, we prove that the answerto the question (Q) set in [5] is affirmative for such a pullback construction A . Besides, ourmain result, Theorem 2.8, is, in particular, an important step towards determining a generalformula for dim( A ⊗ k B ) in the case where A [ n ] is an AF-domain for some positive integer n and B is an arbitrary k -algebra. It states the following: Let T be a k -algebra which is a domain and M a maximal ideal of T . Let K = TM and D be a subring of K . Let ϕ : T −→ K be the canonical surjective homomorphism and A := ϕ − ( D ) . Assume that T and D are AF-domains and T M is catenarian. Then, A is aGAF-domain and for an arbitrary k -algebra B ,dim ( A ⊗ k B ) = max n D (cid:16) t.d. ( A ) , d, B (cid:17) ; ht ( M ) + max n ht ( q [ t.d. ( A )]) + ht (cid:16) qq [ t.d. ( D )] (cid:17) + min (cid:16) t.d. ( Bq ) , t.d. ( K : D ) (cid:17) + min (cid:16) t.d. ( D ) , dim ( D ) + t.d. ( Bq ) (cid:17) : q ⊆ q ∈ Spec ( B ) oo , where d := sup n ht ( Q ) : Q ∈ Spec ( T ) with M Q o .Direct consequences of this theorem are provided as well as a case where we may drop thecatenarity property of T M is exhibited. An example to illustrate our findings closes thispaper.Recent developments on height and grade of (prime) ideals as well as on dimension the-ory in tensor products of k -algebras are to be found in [2-7]. Concerning the study of thetransfer to tensor products of algebras of the S-property, strong S-property, and catenarity,we refer the reader to [8].
2. Main results
First, for the convenience of the reader, we catalog some basic facts and results connectedwith the tensor product of k -algebras. These will be used frequently in the sequel withoutexplicit mention.Let A and B be two k -algebras. If p is a prime ideal of A , r = t.d.( Ap ) and x , ..., x r areelements of Ap , algebraically independent over k , with the x i ∈ A , then it is easily seen that x , ..., x r are algebraically independent over k and p ∩ S = ∅ , where S = k [ x , ..., x r ] \ { } . If4 is an integral domain, then ht ( p )+t.d.( Ap ) ≤ t.d.( A ) for each prime ideal p of A (cf. [21,p. 37] ). Now, assume that S and S are multiplicative subsets of A and B , respectively,then S − A ⊗ k S − B ∼ = S − ( A ⊗ k B ), where S = { s ⊗ s : s ∈ S and s ∈ S } . We assumefamiliarity with the natural isomorphisms for tensor products. In particular, we identify A and B with their respective images in A ⊗ k B . Also, A ⊗ k B is a free (hence faithfully flat)extension of A and B . Moreover, recall that an AF-domain A is a locally Jaffard domain,that is, ht ( p [ n ]) = ht ( p ) for each prime ideal p and each positive integer n [20, Corollary 3.2].Finally, we refer the reader to the useful result of Wadsworth [20, Proposition 2.3] whichyields a classification of the prime ideals of A ⊗ k B according to their contractions to A and B .We begin by recalling from [3], [5], [6] and [20] the following useful results. Proposition 2.1 [6, Lemma 1.3].
Let A and B be k -algebras such that B is a domain.Let p be a prime ideal of A . Then, for each prime ideal P of A ⊗ k B which is minimal over p ⊗ k B , ht ( P ) = ht ( p ⊗ k B ) = ht ( p [ t.d. ( B )]) . Proposition 2.2 [20, Proposition 2.3].
Let A and B be k -algebras and let p ⊆ p ′ beprime ideals of A and q ⊆ q ′ be prime ideals of B . Then the natural ring homomorphism ϕ : A ⊗ k Bp ⊗ k B + A ⊗ k q −→ Ap ⊗ k Bq such that ϕ ( a ⊗ k b ) = a ⊗ k b for each a ∈ A and each b ∈ B , is an isomorphism and ϕ ( p ′ ⊗ k B + A ⊗ k q ′ p ⊗ k B + A ⊗ k q ) = p ′ p ⊗ k Bq + Ap ⊗ k q ′ q . Proposition 2.3 [6, Lemma 1.2 and Proposition 2.2].
Let T be an integral domainwhich is a k -algebra, M a maximal ideal of T , K := TM and ϕ : T → K the canonicalsurjective homomorphism. Let D be a proper subring of K and A := ϕ − ( D ) . Assume that T and D are AF-domains. Then1) The polynomial ring A [ t.d. ( K : D )] is an AF-domain.2) ht ( p [ n ]) = ht ( p )+ min (cid:16) n, t.d. ( K : D ) (cid:17) for each prime ideal p of A such that M ⊆ p and each positive integer n . The following easy result is probably well known. We refer the reader to [5] for a detailedproof.
Proposition 2.4 [5, Proposition 2.4].
Let A be a ring. Let I ⊆ J be ideals in A . Then ht ( I ) + ht ( JI ) ≤ ht ( J ) . roposition 2.5. Let A and B be k -algebras such that A is an AF-domain. Let P ∈ Spec ( A ⊗ k B ) , p = P ∩ A and q = P ∩ B . Then1) ht ( P ) = ht ( A ⊗ k q ) + ht ( PA ⊗ k q ) [6, Lemma 1.5].2) ht ( P ) = ht ( q [ t.d. ( A )]) + ht ( p ) + ht ( Pp ⊗ k B + A ⊗ k q ) . Proof.
2) As PA ⊗ k q ∩ Bq = (0), PA ⊗ k q survives in A ⊗ k k B ( q ), where k B ( q ) denotes thequotient field of Bq , so that, applying (1), we get ht ( PA ⊗ k q ) = ht (cid:16) p h t.d. (cid:16) Bq ) i(cid:17) + ht (cid:16) P/ ( A ⊗ k q ) p ⊗ k Bq (cid:17) = ht ( p ) + ht ( Pp ⊗ k B + A ⊗ k q ) , as A is a locally Jaffard domainand p ⊗ k Bq ∼ = p ⊗ k B + A ⊗ k qA ⊗ k q , by Proposition 2.2 . Hence ht ( P ) = ht ( q [t.d.( A )]) + ht ( p ) + ht ( Pp ⊗ k B + A ⊗ k q ), as desired. ✷ A domain A is said to be catenarian if for each chain of prime ideals p ⊆ q of A , ht ( p ) + ht ( qp ) = ht ( q ) (cf. [9]) . Proposition 2.6.
Let A be an AF-domain. If A is catenarian, then Ap is an AF-domainfor each prime ideal p of A . Proof.
Assume that A is catenarian and fix p ∈ Spec( A ). Let p ⊆ q ∈ Spec( A ). Then ht ( qp ) + t.d.( Aq ) = ht ( q ) − ht ( p ) + t.d.( Aq )= t.d.( A ) − ht ( p ) as A is an AF-domain= t.d.( Ap ) . Hence Ap is an AF-domain. ✷ Let A and B be k -algebras and P be a prime ideal of A ⊗ k B . Let q ∈ Spec( B ) suchthat q ⊂ P ∩ B . We denote by λ (cid:16) ( ., q ) , P (cid:17) the maximum of lengths of chains of primeideals of A ⊗ k B of the form P ⊂ P ⊂ ... ⊂ P s = P such that P i ∩ B = q , for i = 0 , , ..., s − roposition 2.7 [3, Lemma 2.4]. Let A and B be k -algebras and P be a prime ideal of A ⊗ k B with p = P ∩ A and q = P ∩ B . Assume that A and B are integral domains. Then λ (cid:16) ( ., (0)) , P (cid:17) ≤ t.d.( A ) − t.d.( Ap ) + ht (cid:16) q h t.d.( Ap ) i(cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ) . Finally, recall that, if A is a k -algebra and n ≥ A [ n ] is an AF-domain if and only if, for each prime ideal p of A , ht ( p [ n ]) + t.d.( Ap ) = t.d.( A ) [6, Lemma 2.1] . Next, we announce the main theorem of this paper. It gives an answer to the above-mentioned problem set in [6] as well as to the question (Q) of [5] and represents an importantstep towards determining a formula for dim( A ⊗ k B ) when A [ n ] is an AF-domain for somepositive integer n , and B is an arbitrary k -algebra. Theorem 2.8.
Let T be a k -algebra which is a domain and M a maximal ideal of T . Let K = TM and D be a subring of K . Let ϕ : T −→ K be the canonical surjective homomor-phism and A := ϕ − ( D ) . Assume that T and D are AF-domains and T M is catenarian. Let B be an arbitrary k -algebra and let P ∈ Spec ( A ⊗ k B ) , p = P ∩ A and q = P ∩ B . Then thefollowing statements hold:1) If M p , then ht ( P ) = h t ( p ) + ht ( q [ t.d. ( A )]) + ht ( Pp ⊗ k B + A ⊗ k q ) .
2) If M ⊆ p , then ht ( P ) = ht ( p ) + max n ht ( q [ t.d. ( A )]) + ht (cid:16) qq [ t.d. ( D )] (cid:17) + min (cid:16) t.d. ( Bq ) , t.d. ( K : D ) (cid:17) : q ⊆ q ∈ Spec ( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) .
3) dim ( A ⊗ k B ) = max n D (cid:16) t.d. ( A ) , d, B (cid:17) ; ht ( M )+ max n ht ( q [ t.d. ( A )])+ ht (cid:16) qq [ t.d. ( D )] (cid:17) min (cid:16) t.d. ( Bq ) , t.d. ( K : D ) (cid:17) + min (cid:16) t.d. ( D ) , dim ( D ) + t.d. ( Bq ) (cid:17) : q ⊆ q ∈ Spec ( B ) oo , where d := sup n ht ( Q ) : Q ∈ Spec ( T ) with M Q o .4) A is a GAF-domain. roof.
1) Let M p . Then, by [1, Lemma 2.1], there exists p ′ ∈ Spec( T ) such that p ′ ∩ A = p , and p ′ satisfies A p = T p ′ . Thus A p is an AF-domain, so that by Proposition 2.5, ht ( P ) = ht ( q [t.d.( A )]) + ht ( p ) + ht ( Pp ⊗ k B + A ⊗ k q ) , as desired .
2) Assume that M ⊆ p . Then t.d.( Ap ) = t.d. (cid:16) A/Mp/M (cid:17) ≤ t.d.( D ), and applying [1, Lemma2.1], there exists Q ∈ Spec( D ) such that p = ϕ − ( Q ) and the following diagram A p −→ D Q ↓ ↓ T M −→ K is a pullback diagram. Therefore, as ht ( P ) = ht ( P ( A p ⊗ k B )), we may assume without lossof generality that ( T, M ) is a quasilocal catenarian domain, and thus M is a divided primeideal of A , via [1, Lemma 2.1].First, note that, by Proposition 2.1 and Proposition 2.2, we have, ∀ q ⊆ q ∈ Spec( B ), ht ( q [t.d.( A )]) = ht ( A ⊗ k q ) ht (cid:16) p h t.d.( Bq ) i(cid:17) = ht ( p ⊗ k B + A ⊗ k q A ⊗ k q ) ht (cid:16) qq [t.d.( D )] (cid:17) = ht ( M ⊗ k B + A ⊗ k qM ⊗ k B + A ⊗ k q ) . It follows, by Proposition 2.1, Proposition 2.2, Proposition 2.3 and Proposition 2.4, that, ∀ q ⊆ q ∈ Spec( B ), ht ( p ) + ht ( q [t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) = ht ( M ) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) + ht ( pM ) + ht ( A ⊗ k q ) + ht ( M ⊗ k B + A ⊗ k qM ⊗ k B + A ⊗ k q ) = ht (cid:16) M h t.d.( Bq ) i(cid:17) + ht ( A ⊗ k q ) + ht (cid:16) pM h t.d.( Bq ) i(cid:17) + ht ( M ⊗ k B + A ⊗ k qM ⊗ k B + A ⊗ k q ) = ht ( M ⊗ k B + A ⊗ k q A ⊗ k q ) + ht ( A ⊗ k q ) + ht ( p ⊗ k B + A ⊗ k qM ⊗ k B + A ⊗ k q ) + ht ( M ⊗ k B + A ⊗ k qM ⊗ k B + A ⊗ k q ) ≤ ht ( p ⊗ k B + A ⊗ k q ) . q ⊆ q of B , ht ( p ) + ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p ⊗ k B + A ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) , establishing the direct inequality. The proof of the reverse inequality falls into the followingtwo steps. Step 1. B is an integral domain.Our argument uses induction on dim( T ), ht ( p ) and ht ( q ). First, note that( ∗ ) max n ht ( q [t.d.( A )]) + ht (cid:16) p h t.d.( Bq ) i(cid:17) , ht ( p [t.d.( B )]) + ht (cid:16) q h t.d.( Ap ) i(cid:17)o ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [(t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o , it suffices to take q = q and q = (0). If either dim( T ) = 0 or ht ( p ) = 0, then T = K is a field and, thus, A = D is an AF-domain, and applying Proposition 2.5 and ( ∗ ), weobtain the formula. Also, the case ht ( q ) = 0 is fairly easy via Proposition 2.5 and ( ∗ ).Then, assume that dim( T ) > ht ( p ) > ht ( q ) >
0. Consider a chain of prime ideals Q ⊂ Q ⊂ ... ⊂ Q h = P of A ⊗ k B such that h = ht ( P ). Let r := max { m : Q m ∩ A ⊂ p or Q m ∩ B ⊂ q } . Let Q = Q r , p ′ = Q r ∩ A and q ′ = Q r ∩ B . Hence, ht ( P ) = ht ( Q ) + ht ( PQ ).We are led to discuss the following cases. Case 1. p ′ = p . Then, q ′ ⊂ q , and by inductive assumptions, ht ( Q ) = ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) q ′ q [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ′ ∈ Spec( B ) o + ht ( Qp ⊗ k B + A ⊗ k q ′ ) , and thus ht ( P ) ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) q ′ q [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ′ ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ′ ) . Pp ⊗ k B + A ⊗ k q ′ ∩ Ap = (0), we get by Proposition 2.5, ht ( Pp ⊗ k B + A ⊗ k q ′ ) = ht (cid:16) qq ′ h t.d.( Ap ) i(cid:17) + ht (cid:16) P/ ( p ⊗ k B + A ⊗ k q ′ )( A/p ) ⊗ k ( q/q ′ ) (cid:17) = ht (cid:16) qq ′ h t.d.( Ap ) i(cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ) , since, by Proposition 2.2, Ap ⊗ k qq ′ ∼ = p ⊗ k B + A ⊗ k qp ⊗ k B + A ⊗ k q ′ . It follows that ht ( P ) ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) q ′ q [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ′ ∈ Spec( B ) o + ht (cid:16) qq ′ h t.d.( Ap ) i(cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) q ′ q [t.d.( D )] (cid:17) + ht (cid:16) qq ′ [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ′ ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ), as t.d.( Ap ) ≤ t.d.( D ) ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) . Then the equality holds, as we wish to show.
Case 2. M ⊆ p ′ ⊂ p . By inductive hypotheses, we get ht ( Q ) = ht ( p ′ )+max n ht ( q [t.d.( A )]) + ht (cid:16) q ′ q [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ′ ∈ Spec( B ) o + ht ( Qp ′ ⊗ k B + A ⊗ k q ′ ) . Note that ht ( p ′ ) = ht ( M ) + ht ( p ′ M ) and that QM ⊗ k B + A ⊗ k q ′ survives in D ⊗ k k B ( q ′ ),where k B ( q ′ ) denotes the quotient field of Bq ′ , since QM ⊗ k B + A ⊗ k q ′ ∩ Bq ′ = (0). Then,by Proposition 2.5, we get 10 t ( QM ⊗ k B + A ⊗ k q ′ ) = ht (cid:16) p ′ M h t.d.( Bq ′ ) i(cid:17) + ht ( Qp ′ ⊗ k B + A ⊗ k q ′ )since p ′ M ⊗ k Bq ′ ∼ = p ′ ⊗ k B + A ⊗ k q ′ M ⊗ k B + A ⊗ k q ′ , by Proposition 2.2= ht ( p ′ M ) + ht ( Qp ′ ⊗ k B + A ⊗ k q ′ ) as D is an AF-domain . Hence ht ( Q ) = ht ( M )+max n ht ( q [t.d.( A )]) + ht (cid:16) q ′ q [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ′ ∈ Spec( B ) o + ht ( QM ⊗ k B + A ⊗ k q ′ ) . It follows that ht ( P ) ≤ ht ( M )+max n ht ( q [t.d.( A )]) + ht (cid:16) q ′ q [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ′ ∈ Spec( B ) o + ht ( PM ⊗ k B + A ⊗ k q ′ )= ht ( M )+max n ht ( q [t.d.( A )]) + ht (cid:16) q ′ q [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ′ ∈ Spec( B ) o + ht (cid:16) qq ′ [t.d.( D )] (cid:17) + ht ( pM ) + ht ( Pp ⊗ k B + A ⊗ k q ) , by Proposition 2.5, since D is an AF-domain, and since, by Proposition 2.2, pM ⊗ k Bq ′ + AM ⊗ k qq ′ ∼ = p ⊗ k B + A ⊗ k qM ⊗ k B + A ⊗ k q ′ ≤ ht ( p )+max n ht ( q [t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ), as desired. Case 3. p ′ ⊂ M and q ′ = (0). Then A p ′ is an AF-domain, and thus ht ( Q ) = ht ( A ⊗ k q ′ ) + ht ( QA ⊗ k q ′ ), so that ht ( P ) = ht ( A ⊗ k q ′ ) + ht ( PA ⊗ k q ′ ). As ht ( qq ′ ) < ht ( q ), we get, by inductive assumptions, ht ( PA ⊗ k q ′ ) = ht ( p ) + max n ht (cid:16) q q ′ [t.d.( A )] (cid:17) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ′ ⊆ q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) . ht ( P ) = ht ( p ) + max n ht ( q ′ [t.d.( A )]) + ht (cid:16) q q ′ [ t.d.( A )] (cid:17) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ′ ⊆ q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p ) + max n ht ( q [t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ), and the equality holds. Case 4. (0) = p ′ ⊂ M and q ′ = (0). Then ht ( Q ) = ht ( p ′ ⊗ k B ) + ht ( Qp ′ ⊗ k B ), viaProposition 2.5. It follows that ht ( P ) = ht ( p ′ ) + ht ( Pp ′ ⊗ k B ), as A p ′ is an AF-domain.Since p ′ ⊂ M , there exists a unique prime ideal P ′ of T such that P ′ ∩ A = p ′ . Then P ′ = (0), so that dim( TP ′ ) < dim( T ), and Ap ′ −→ D ↓ ↓ TP ′ −→ K ∼ = T /P ′ M/P ′ is a pullback diagram. Moreover, by Proposition 2.6, TP ′ is an AF-domain which is catenar-ian, therefore, by inductive assumptions, we get ht ( Pp ′ ⊗ k B ) = ht ( pp ′ ) + max n ht (cid:16) q h t.d.( Ap ′ ) i(cid:17) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht (cid:16) P/ ( p ′ ⊗ k B ) p/p ′ ⊗ k B + A/p ′ ⊗ k q (cid:17) = ht ( pp ′ ) + max n ht (cid:16) q h t.d.( Ap ′ ) i(cid:17) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) . ht ( P ) = ht ( p ′ ) + ht ( pp ′ ) + max n ht (cid:16) q h t.d.( Ap ′ ) i(cid:17) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ h ( P ) . Then the equality holds.
Case 5. p ′ = (0) and q ′ = (0). Then, by [19, Theorem 3.1], ht ( Q ) ≤ t.d.( B ), and thus, as Q r +1 ∩ A = p and Q r +1 ∩ B = q , we get ht ( P ) = ht ( Q r +1 ) + ht ( PQ r +1 ) ≤ B ) + ht ( Pp ⊗ k B + A ⊗ k q ) . Suppose that 1+t.d.( B ) ≤ ht ( p [t.d.( B )]). Therefore, ht ( P ) ≤ ht ( p [t.d.( B )]) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p [t.d.( B )]) + ht (cid:16) q h t.d.( Ap ) i(cid:17) + ht ( Pp ⊗ k B + A ⊗ k q )= ht ( p ⊗ k B ) + ht ( p ⊗ k B + A ⊗ k qp ⊗ k B ) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) . Hence ht ( p [t.d.( B )]) = ht ( p [t.d.( B )]) + ht (cid:16) q h t.d.( Ap ) i(cid:17) , so that q = (0) which leads to acontradiction, as ht ( q ) >
0. It follows that, by Proposition 2.3,1+t.d.( B ) > ht ( p [t.d.( B )]) = ht ( p )+min (cid:16) t.d.( B ) , t.d.( K : D ) (cid:17) , so thatt.d.( B ) > t.d.( K : D ), as ht ( p ) ≥
1. On the other hand,13 t ( P ) = ht ( Q r +1 ) + ht ( PQ r +1 )= λ (cid:16)(cid:16) (0) , (0) (cid:17) , Q r +1 (cid:17) + ht ( PQ r +1 ) ≤ t.d.( A ) − t.d.( Ap ) + ht (cid:16) q h t.d.( Ap ) i(cid:17) + ht ( Q r +1 p ⊗ k B + A ⊗ k q ) + ht ( PQ r +1 )(cf. Proposition 2.7) ≤ ht ( p [t.d.( K : D )]) + ht (cid:16) q h t.d.( Ap ) i(cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ) , as A [t.d.( K : D )] is an AF-domain ≤ ht ( p [t.d.( B )]) + ht (cid:16) q [t.d.( Ap )] (cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) , by ( ∗ ) ≤ ht ( P ) . Then the equality holds.
Step 2. B is an arbitrary k -algebra.Let P ⊂ P ⊂ ... ⊂ P h = P be a chain of prime ideals of A ⊗ k B such that h = ht ( P ). Let q := P ∩ B . Then P A ⊗ k q ⊂ P A ⊗ k q ⊂ P A ⊗ k q ⊂ ... ⊂ P h A ⊗ k q = PA ⊗ k q is a chain of prime ideals of A ⊗ k Bq and h = ht ( P ) = ht ( PA ⊗ k q ). By Step 1, ht ( P ) = ht ( PA ⊗ k q ) = ht ( p ) + max n ht (cid:16) q q [ t.d.( A )] (cid:17) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ⊆ q ∈ Spec( B ) o + ht (cid:16) P/ ( A ⊗ k q ) p ⊗ k ( B/q ) + A ⊗ k ( q/q ) (cid:17) ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) , then the equality holds establishing the desired formula .
3) First, observe that, by [1, Lemma 2.1], for each p ∈ Spec( A ) such that M p , thereexists a unique Q ∈ Spec( T ) such that Q ∩ A = p , and Q satisfies A p = T Q . Then d = max n ht ( p ) : p ∈ Spec( A ) with M p o . Now, by (1), we havemax n ht ( P ) : P ∈ Spec( A ⊗ k B ) and M p := P ∩ A o =max n ht ( p ) + ht ( q [t.d.( A )]) + ht ( Pp ⊗ k B + A ⊗ k q ) : P ∈ Spec( A ⊗ k B ) with p = P ∩ A , q = P ∩ B such that M p o =max n ht ( p ) + ht ( q [t.d.( A )]) + min (cid:16) t.d.( Ap ) , t.d.( Bq ) (cid:17) : p ∈ Spec( A ) and q ∈ Spec( B )such that M p o (cf. [20, Proposition 2.3]) =max n ht ( q [t.d.( A )]) + min (cid:16) t.d.( A ) , ht ( p ) + t.d.( Bq ) (cid:17) : p ∈ Spec( A ) with M p and q ∈ Spec( B ) o (as A p is an AF-domain) =max n ht ( q [t.d.( A )]) + min (cid:16) t.d.( A ) , d + t.d.( Bq ) (cid:17) : q ∈ Spec( B ) o = D (cid:16) t.d.( A ) , d, B (cid:17) ( ∗∗ ).On the other hand, let M ⊆ p . As done in (2), we may assume that ( T, M ) is a quasilocaldomain, and thus M is a divided prime ideal of A . First, note that ht ( p )+min (cid:16) t.d.( Ap ) , t.d.( Bq ) (cid:17) = ht ( M ) + ht ( pM ) + min (cid:16) t.d.( Ap ) , t.d.( Bq ) (cid:17) = ht ( M ) + min (cid:16) t.d.( D ) , ht ( pM ) + t.d.( Bq ) (cid:17) , as D isan AF-domain . Hence, by (2),max n ht ( P ) : P ∈ Spec( A ⊗ k B ) and M ⊆ p := P ∩ A o =max n ht ( p ) + max n ht ( q [t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) : P ∈ Spec( A ⊗ k B ) with p = P ∩ A and q = P ∩ B such that M ⊆ p o =15ax n ht ( p ) + ht ( q [t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) +min (cid:16) t.d.( Ap ) , t.d.( Bq ) (cid:17) : p ∈ Spec( A ) and q ⊆ q ∈ Spec( B ) such that M ⊆ p o =max n ht ( M ) + ht ( q [t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) +min (cid:16) t.d.( D ) , ht ( pM ) + t.d.( Bq ) (cid:17) : p ∈ Spec( A ) and q ⊆ q ∈ Spec( B ) such that M ⊆ p o =max n ht ( M ) + ht ( q [t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) +min (cid:16) t.d.( D ) , dim( D ) + t.d.( Bq ) (cid:17) : q ⊆ q ∈ Spec( B ) o ( ∗ ∗ ∗ ) . Consequently, combining ( ∗∗ ) and ( ∗ ∗ ∗ ), we get easily the desired formula for dim( A ⊗ k B ).4) Let p ∈ Spec( A ) and q ∈ Spec( B ). Let Q be a minimal prime ideal of p ⊗ k B + A ⊗ k q .Assume that M p . Then, by (1), ht ( Q ) = ht ( p ) + ht ( q [t.d.( A )]). Therefore, ht ( p ⊗ k B + A ⊗ k q ) = ht ( p ) + ht ( q [t.d.( A )])and for any P ∈ Spec( A ⊗ k B ) with p = P ∩ A and q = P ∩ B , via (1), ht ( P ) = ht ( p ⊗ k B + A ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q ) . Now, let M ⊆ p . Then, by (2), ht ( Q ) = ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o , so that ht ( p ⊗ k B + A ⊗ k q ) = ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) +min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o . Thus, via (2), ht ( P ) = ht ( p ⊗ k B + A ⊗ k q )+ ht ( Pp ⊗ k B + A ⊗ k q ) for each P ∈ Spec( A ⊗ k B )such that p = P ∩ A and q = P ∩ B . Therefore A is a GAF-domain, completing the proof. ✷
16e get the following interesting consequence of Theorem 2.8.
Corollary 2.9.
Let T be a k -algebra which is a domain and M a maximal ideal of T . Let K = TM and D be a subring of K . Let ϕ : T −→ K be the canonical surjective homomor-phism and A := ϕ − ( D ) . Assume that T and D are AF-domains and ht ( M ) ≤ . Then,the assertions (1), (2), (3) and (4) of Theorem 2.8 hold for A and any k -algebra B . Proof.
It is direct from Theorem 2.8 since any two-dimensional domain is catenarian. ✷ The following result discusses a case where it is possible to drop the catenarity assumptionof T M in Theorem 2.8. Proposition 2.10.
Let T be a k -algebra which is a domain and M a maximal ideal of T .Let K = TM and D be a subring of K . Let ϕ : T −→ K be the canonical surjective homo-morphism and A := ϕ − ( D ) . Assume that T and D are AF-domains and t.d. ( K : D ) ≤ .Then, the assertions (1), (2), (3) and (4) of Theorem 2.8 hold for A and any k -algebra B . Proof.
The proof runs similar to that of Theorem 2.8. We need only to discuss the casewhere (0) = p ′ ⊂ M and q ′ = (0) (Case 4 of Step 1 of the above proof) where the catenarityproperty of T is required. So, let B be an integral domain and let P ∈ Spec( A ⊗ k B ), p = P ∩ A and q = P ∩ B . Let r := max { m : Q m ∩ A ⊂ p or Q m ∩ B ⊂ q } . Let Q = Q r , p ′ = Q r ∩ A and q ′ = Q r ∩ B , and suppose that (0) = p ′ ⊂ M and q ′ = (0). Hence ht ( P ) = ht ( Q r +1 ) + ht ( PQ r +1 )= λ (cid:16) ( ., (0)) , Q r +1 (cid:17) + ht ( PQ r +1 ) ≤ t.d.( A ) − t.d.( Ap ) + ht (cid:16) q [t.d.( Ap )] (cid:17) + ht ( Q r +1 p ⊗ k B + A ⊗ k q ) + ht ( PQ r +1 ) , by Proposition 2.7 ≤ ht ( p [t.d.( K : D )]) + ht (cid:16) q [t.d.( Ap )] (cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ), as A [t.d.( K : D )] isan AF-domain, by Proposition 2.3 . Now, if t.d.( B ) ≥
2, then we get ht ( P ) ≤ ht ( p [t.d.( B )]) + ht (cid:16) q [t.d.( Ap )] (cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) ,
17o that the desired equality holds. Assume that t.d.( B ) ≤
1. Then B is an AF-domain.Hence ht ( P ) = ht ( p [t.d.( B )]) + ht (cid:16) q [t.d.( Ap )] (cid:17) + ht ( Pp ⊗ k B + A ⊗ k q ), by Proposition 2.5 ≤ ht ( p ) + max n ht ( q [ t.d.( A )]) + ht (cid:16) qq [t.d.( D )] (cid:17) + min (cid:16) t.d.( Bq ) , t.d.( K : D ) (cid:17) : q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) . Then the equality holds, as contended. ✷ Next, for each positive integer n , we provide an example of a finite dimensional valuationdomain ( V, M ), thus universally catenarian (cf. [18]), of Krull dimension n such that V is an AF-domain. Then, given any subring D of K := VM such that D is an AF-domain(in particular any subfield of K ), Theorem 2.8 allows us to compute dim( A ⊗ k B ) for any k -algebra B , where A is the pullback ϕ − ( D ). Example 2.11.
Let V = k ( X , X , ..., X n )[ Y ] ( Y ) = k ( X , X , ...X n ) + M be a rank-onediscrete valuation domain with M := Y V . Let V = k ( X , X , ..., X n − )[ X n ] ( X n ) + M = k ( X , X , ..., X n − ) + M with M := M + X n k ( X , ..., X n − )[ X n ] ( X n ) . It is well known that V is a valuation domainof (Krull) dimension 2 [11, Exercise 13 (2), page 203]. Also, by [12], V is an AF-domain.Let V = k ( X , X , ..., X n − )[ X n − ] ( X n − ) + M = k ( X , X , ..., X n − ) + M with M := M + X n − k ( X , ..., X n − )[ X n − ] ( X n − ) . Then V is a valuation domain ofdimension 3 which is an AF-domain. We may iterate this process to construct a valuationdomain of the form V = K + M which is an AF-domain of dimension r for each positiveinteger r . ✷ Acknowledgement.
The author would like to thank the referee for his/her helpful sugges-tions.
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