aa r X i v : . [ m a t h . A T ] F e b AF-domains and their generalizations
Samir Bouchiba
Department of Mathematics, University Moulay Ismail, Meknes 50000, Morocco ——————————————————————————————
Abstract
In this paper, we are concerned with the study of the dimension theory of tensor products of algebras overa field k . We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) andprove that any k -algebra A such that the polynomial ring in one variable A [ X ] is an AF-domain is in fact aGAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimensionof A ⊗ k B for any k -algebra A such that A [ X ] is an AF-domain and any k -algebra B generalizing the maintheorem of Wadsworth in [16]. MSC (2000): 13C15; 13B24. ——————————————————————————————
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 (for nondomains, t.d.( A ) := sup n t.d.( Ap ) : p ∈ Spec( A ) o ), A [ n ] to denote the polynomialring A [ X , ..., X n ] and p [ n ] to denote the prime ideal p [ X , ..., X n ] of A [ X , ..., X n ] for eachprime ideal p of A . Also, we use Spec( A ) to denote the set of prime ideals of a ring A and ⊂ to denote proper set inclusion. All k -algebras considered throughout this paper are assumedto be of finite transcendence degree over k . Any unreferenced material is standard as in [8],[12], [13] and [14].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. Infact, in [15], 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’s EGA[10, Remarque 4.2.1.4, p. 349]). This formula is rather surprising since, as one may expect,the structure of the tensor product should reflect the way the two components interact and ————— E-mail address : [email protected] keywords:
Krull dimension, tensor product, prime ideal, AF-domain. A ⊗ k B and to widen the scope of algebras A and B for which dim( A ⊗ k B ) depends only on indi-vidual characteristics of A and B . The algebras which proved tractable for Krull dimensioncomputations turned out to be those domains A which satisfy the altitude formula over k (AF-domains for short), that is, ht ( 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 [16, 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 [16, Theorem 3.7].On the other hand, in [11], 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 chain iffor each Q i , the ideal ( Q i ∩ A )[ n ] belongs to C . Subsequently, based on the thorough andbrilliant work of J. Arnold in [1], Brewer et al. gave an equivalent and simple version ofJaffard’s theorem. Actually, they showed that, for each positive integer n and each primeideal P of A [ n ], ht ( P ) = ht ( q [ n ]) + ht ( Pq [ n ] ) [7, Theorem 1], where q := P ∩ A . Taking intoaccount the natural isomorphism B [ n ] ∼ = k [ n ] ⊗ k B for each k -algebra B , we generalized in[5] this special chain theorem to tensor products of k -algebras. Effectively, we proved thatif 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. [5, Lemma 1.5]). It turns out that this very geometrical property totallycharacterizes the AF-domains. In fact, we proved, in [4], that the following statements areequivalent for a domain A which is a k -algebra:2) 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, it is then natural to generalize the AF-domain notion by setting the followingdefinitions: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 satisfies 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. The object of this paper is to handle 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?It is significant, in this regard, to note that if A is an AF-domain then A [ n ] is an AF-domainfor each integer n ≥
0, and using pullback constructions, Proposition 2.1 shows that thesetwo notions do not coincide by providing a family of k -algebras A such that A is not anAF-domain while there exists a positive integer r such that the polynomial ring A [ r ] is anAF-domain. In the present paper, we give partial results settling in the affirmative theabove question (Q). First, we prove that an AF-domain A is in fact a GAF-domain, thus inparticular, any finitely generated algebra over k which is a domain is a GAF-domain. Also,through Proposition 2.5, we prove that (Q) has a positive answer in the case where A isone-dimensional. Whereas, our main result, Theorem 2.8, tackles the case n = 1 of ( Q ).It computes dim( A ⊗ k B ) for a k -algebra A such that A [ X ] is an AF-domain and for anarbitrary k -algebra B generalizing Wadsworth’s main theorem [16, Theorem 3.7] and furtherasserts that A is a GAF-domain. We end this paper by an example of a GAF-domain A such that, for any positive integer n , the polynomial ring A [ n ] is not an AF-domain.Recent developments on height and grade of (prime) ideals as well as on dimension theoryin tensor products of k -algebras are to be found in [2-6].3 . Main results In this section, we handle the question (Q) set above.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 ] \ { } . If A is an integral domain, then ht ( p )+t.d.( Ap ) ≤ t.d.( A ) for each prime ideal p of A (cf. [14,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 [16, Corollary 3.2].Finally, we refer the reader to the useful result of Wadsworth [16, 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 [2], [5] and [16] the following useful results.Our first result allows to construct a bunch of k -algebras A arising from pullbacks whichare not AF-domains while there exists an integer n ≥ A [ n ] is an AF-domain. Proposition 2.1 [5, Proposition 2.2].
Let T be an integral domain which is a k -algebra, M a maximal ideal of T , K := TM and ϕ : T → K the canonical surjective homomorphism.Let D be a proper subring of K and A := ϕ − ( D ) . Assume that T and D are AF-domains.Let r := t.d. ( K : k ) and s = t.d. ( D : k ) . Then the polynomial ring A [ r − s ] is an AF-domain. Recall that, by [9], under the hypotheses of Proposition 2.1, A is an AF-domain if and onlyif t.d.( K : D ) = r − s = 0. Thus, whenever r > s , the issued pullback A is not an AF-domain. Proposition 2.2 [2, 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 )]) . roposition 2.3 [16, 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 . We establish the following easy result which is probably well known but we have notlocated references in the literature.
Proposition 2.4.
Let A be ring. Let I ⊆ J be ideals in A . Then ht ( I ) + ht ( JI ) ≤ ht ( J ) . Proof.
If both I and J are prime ideals, then the result easily follows. Fix a prime ideal Q of A that contains J . Let P be a minimal prime ideal of I contained in Q . As ht ( I ) ≤ ht ( P )and ht ( P ) + ht ( QP ) ≤ ht ( Q ), we get ht ( I ) + ht ( QP ) ≤ ht ( Q ) for each minimal prime ideal P of I contained in Q . Hence ht ( I )+max { ht ( QP ) : P is a minimal prime ideal of A over I contained in Q } = ht ( I ) + ht ( QI ) ≤ ht ( Q ). Therefore ht ( I ) + ht ( JI ) ≤ ht ( I ) + ht ( QI ) ≤ ht ( Q )for each prime ideal Q of A containing J . It follows that ht ( I ) + ht ( JI ) ≤ min { ht ( Q ) : J ⊆ Q ∈ Spec( A ) } = ht ( J ), as desired. ✷ We begin by proving that an AF-domain is a GAF-domain.
Proposition 2.5.
Let A be an AF-domain. Then A is a GAF-domain. Proof.
Let P be a prime ideal of A ⊗ k B , p = P ∩ A and q = P ∩ B . By the special chaintheorem for tensor products [5, Lemma 1.5], ht ( P ) = ht ( A ⊗ k q ) + ht ( PA ⊗ k q ) . Also, note that PA ⊗ k q is a prime ideal of A ⊗ k Bq such that PA ⊗ k q ∩ Bq = (0). Then5 A ⊗ k q survives in the localization A ⊗ k k B ( q ) of A ⊗ k Bq , where k B ( q ) denotes the quotientfield of Bq , so that, by a second application of [5, Lemma 1.5], we get ht ( PA ⊗ k q ) = ht ( p ⊗ k Bq ) + ht (cid:16) P/ ( A ⊗ k q ) p ⊗ k ( B/q ) (cid:17) = ht ( p ⊗ k B + A ⊗ k qA ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q ) , as p ⊗ k B + A ⊗ k qA ⊗ k q ∼ = p ⊗ k Bq via Proposition 2.3. Applying Proposition 2.4, it follows that ht ( P ) = ht ( A ⊗ k q ) + ht ( p ⊗ k B + A ⊗ k qA ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p ⊗ k B + A ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) . Then the equality holds, completing the proof. ✷ The following result settles in the affirmative the above question (Q) in the case where A is one-dimensional. Proposition 2.6.
Let A be a one-dimensional domain such that A [ n ] is an AF-domain forsome positive integer n . Then A is a GAF-domain. Proof.
Let B be a k − algebra and P a prime ideal of A ⊗ k B with p = P ∩ A and q = P ∩ B .Then, applying [2, Theorem 1.1], we get ht ( P ) = max n ht ( q [t.d.( A )]) + ht ( qq [t.d.( Ap )]) + ht ( p [t.d.( Bq )]) : q ∈ Spec( B ) with q ⊆ q o + ht ( Pp ⊗ k B + A ⊗ k q ) . Then, for each minimal prime ideal Q of p ⊗ k B + A ⊗ k q , we get ht ( Q ) = max n ht ( q [t.d.( A )]) + ht ( qq [t.d.( Ap )]) + ht ( p [t.d.( Bq )]) : q ∈ Spec( B ) with q ⊆ q o , so that 6 t ( p ⊗ k B + A ⊗ k q ) = max n ht ( q [t.d.( A )]) + ht ( qq [t.d.( Ap )]) + ht ( p [t.d.( Bq )]) : q ∈ Spec( B ) with q ⊆ q o . It follows that ht ( P ) = ht ( p ⊗ k B + A ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q ) . Then A is a GAF-domain, as desired. ✷ Next, we announce the principal result of this paper. It tackles the case n = 1 of theabove-sited question ( Q ) and generalizes the main theorem of Wadsworth in [16], namely:If A and B are k -algebras such that A is an AF-domain, thendim( 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 [16, Theorem 3.7] . This equality might be rewritten in the following way which evokes our next general formula,dim( A ⊗ k B ) = max n ht ( q [t.d.( A )]) + ht ( p [t.d.( Bq )]) + min (cid:16) t.d.( Ap ) , t.d.( Bq ) (cid:17) : p ∈ Spec( A ) and q ∈ Spec( B ) o ( as A is a locally Jaffard domain) . First, it is worthwhile recalling the following definition and results from [3] and [5]. Let A and B be k -algebras and P be a prime ideal of A ⊗ k B . Let q ∈ Spec( B ) such that q ⊂ P ∩ B . We denote by λ (cid:16) ( ., q ) , P (cid:17) the maximum of lengths of chains of prime idealsof A ⊗ k B of the form P ⊂ P ⊂ ... ⊂ P s = P such that P i ∩ B = q , for i = 0 , , ..., s − A and B are integral domains, then λ (cid:16) ( ., (0)) , P (cid:17) ≤ t.d.( A ) − t.d.( Ap ) + ht ( q [t.d.( Ap )]) + ht ( Pp ⊗ k B + A ⊗ k q ) . Further, recall that, if A is a k -algebra and n ≥ A [ n ] is an AF-domain if and only if ht ( p [ n ]) + t.d.( Ap ) = t.d.( A )for each prime ideal p of A [5, Lemma 2.1]. Theorem 2.7.
Let A be a k -algebra such that the polynomial ring A [ X ] is an AF-domain.Let B be an arbitrary k -algebra. Then the following statements hold:a) If P is a prime ideal of A ⊗ k B , p = P ∩ A and q = P ∩ B , then t ( P ) = max n ht ( q [ t.d.(A) ]) + ht ( p [ t.d. ( Bq )]) + ht ( qq [ t.d. ( Ap )]) + ht ( pp [ t.d. ( Bq )]) : p ⊆ p and q ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q ) .b) dim ( A ⊗ k B ) = max n ht ( q [ t.d.(A) ]) + ht ( p [ t.d. ( Bq )]) + ht ( qq [ t.d. ( Ap )])+ ht ( pp [ t.d. ( Bq )]) + min (cid:16) t.d.( Ap ) , t.d. ( Bq ) (cid:17) : p ⊆ p ∈ Spec ( A ) and q ⊆ q ∈ Spec ( B ) o . c) A is a GAF-domain. Proof. a) Step 1. B is an integral domain.If t.d.( B ) = 0, then B is an algebraic extension field of k , and thus, by [5, Lemma 1.5], weare done. Next, assume that t.d.( B ) ≥
1. By Proposition 2.2 and Proposition 2.3, we get, ∀ p ⊆ p ∈ Spec( A ) and ∀ q ⊆ q ∈ Spec( B ), ht ( q [t.d.( A )]) = ht ( A ⊗ k q ) ,ht ( p [t.d.( Bq )]) = ht ( p ⊗ k Bq ) = ht ( p ⊗ k B + A ⊗ k q A ⊗ k q ) ,ht ( qq [t.d.( Ap )]) = ht ( p ⊗ k B + A ⊗ k qp ⊗ k B + A ⊗ k q ), and ht ( pp [t.d.( Bq )]) = ht ( p ⊗ k B + A ⊗ k qp ⊗ k B + A ⊗ k q ) . It follows that, ∀ p ⊆ p ∈ Spec( A ) and ∀ q ⊆ q ∈ Spec( B ), ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) = ht ( A ⊗ k q ) + ht ( p ⊗ k B + A ⊗ k q A ⊗ k q ) + ht ( p ⊗ k B + A ⊗ k qp ⊗ k B + A ⊗ k q ) + ht ( p ⊗ k B + A ⊗ k qp ⊗ k B + A ⊗ k q ) ≤ ht ( p ⊗ k B + A ⊗ k q ), by Proposition 2.4 . Consequently,max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p and8 ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( p ⊗ k B + A ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) . Our proof of the reverse inequality uses induction on ht ( p ) and ht ( q ). First, note that(1) max n ht ( q [t.d.( A )]) + ht ( p [t.d.( Bq )]) , ht ( p [t.d.( B )]) + ht ( q [t.d.( Ap )]) o ≤ max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p and q ⊆ q are prime ideals of A and B, respectively o , it suffices to take p = pq = q and p = pq = (0) . The case where either ht ( p ) = 0 or ht ( q ) = 0 is fairly easy applying [5, Lemma 1.5]. Then,assume that ht ( p ) > ht ( q ) >
0. Suppose that t.d.( Bq ) ≥ x ∈ B such that x is a transcendental element of Bq over k , and put S := k [ x ] \ { } . Then, x is transcendental over A (we identify x with its image 1 ⊗ k x through thecanonical injection B → A ⊗ k B ) S − ( A ⊗ k B ) ∼ = A ⊗ k S − B ∼ = (cid:16) A ⊗ k k ( x ) (cid:17) ⊗ k ( x ) S − B ∼ = S − A [ x ] ⊗ k ( x ) S − Bq ∩ S = ∅ S − P ∈ Spec (cid:16) S − A [ x ] ⊗ k ( x ) S − B (cid:17) S − A [ x ] is an AF-domain, by hypotheses.Hence ht ( P ) = ht ( S − P ) = ht (cid:16) S − A [ x ] ⊗ k ( x ) S − q (cid:17) + ht (cid:16) S − PS − A [ x ] ⊗ k ( x ) S − q (cid:17) , via [5, Lemma 1.5]= ht (cid:16) ( A ⊗ k k ( x )) ⊗ k ( x ) S − q (cid:17) + ht (cid:16) S − P ( A ⊗ k k ( x )) ⊗ k ( x ) S − q (cid:17) = ht ( A ⊗ k S − q ) + ht ( S − PA ⊗ k S − q )9 ht ( A ⊗ k q ) + ht ( PA ⊗ k q )= ht ( q [t.d.( A )]) + ht ( p [ t.d.( Bq )]) + ht ( Pp ⊗ k B + A ⊗ k q ) , by [5, Lemma 1.5] , since PA ⊗ k q ∩ Bq = (0) and p ⊗ k Bq ∼ = p ⊗ k B + A ⊗ k qA ⊗ k q via Proposition 2.3,so that P/ ( A ⊗ k q ) p ⊗ k ( B/q ) ∼ = Pp ⊗ k B + A ⊗ k q ≤ max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p and q ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q ) , by (1) ≤ ht ( P ) , then the equality holds, as contended . Next, suppose that t.d.( Bq ) = 0. Then, by [16, Proposition 2.3], P is a minimal prime idealof p ⊗ k B + A ⊗ k q . Let Q ∈ Spec( A ⊗ k B ) such that Q ⊂ P and ht ( P ) = 1 + ht ( Q ). Let p ′ = Q ∩ A and q ′ = Q ∩ B . Then either p ′ ⊂ p or q ′ ⊂ q . Three cases arise. Case 1. q ′ ⊂ q and q ′ = (0) . Then t.d.( Bq ′ ) ≥ ≤ ht ( qq ′ )+t.d.( B/q ′ q/q ′ ) ≤ t.d.( Bq ′ ),so that, by the above discussion, ht ( Q ) = ht ( A ⊗ k q ′ ) + ht ( QA ⊗ k q ′ ) , and hence ht ( P ) = ht ( A ⊗ k q ′ ) + ht ( PA ⊗ k q ′ ) . As q ′ = (0), ht ( qq ′ ) < ht ( q ), then by inductive hypotheses with respect to A ⊗ k Bq ′ , we get ht ( PA ⊗ k q ′ ) = max n ht ( q q ′ [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : q ′ ⊆ q ⊆ q and p ⊆ p are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q ), as, by Proposition 2.3, P/ ( A ⊗ k q ′ ) p ⊗ k ( B/q ′ ) + A ⊗ k ( q/q ′ ) ∼ = P/ ( A ⊗ k q ′ )( p ⊗ k B + A ⊗ k q ) / ( A ⊗ k q ′ ) ∼ = Pp ⊗ k B + A ⊗ k q . ht ( P ) = ht ( q ′ [t.d.( A )]) + max n ht ( q q ′ [t.d.(A)]) + ht ( p [t.d.( Bq )])+ ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p and q ′ ⊆ q ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q )= max n ht ( q ′ [t.d.( A )]) + ht ( q q ′ [t.d.(A)]) + ht ( p [t.d.( Bq )])+ ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p and q ′ ⊆ q ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ max n ht ( q [t.d.( A )]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p and q ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) . Then the equality holds.
Case 2. q ′ = q . Then p ′ ⊂ p . By inductive hypotheses, we get ht ( Q ) = max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( p ′ p [t.d.( Bq )]) : p ⊆ p ′ and q ⊆ q are prime ideals of A and B, respectively o + ht ( Qp ′ ⊗ k B + A ⊗ k q ) . Hence ht ( P ) ≤ max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( p ′ p [t.d.( Bq )]) : p ⊆ p ′ and q ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ′ ⊗ k B + A ⊗ k q ) . Pp ′ ⊗ k B + A ⊗ k q ∩ Bq = (0) and Pp ′ ⊗ k B + A ⊗ k q ∩ Ap ′ = pp ′ , we get, by [5, Lemma 1.5], ht ( Pp ′ ⊗ k B + A ⊗ k q ) = ht ( pp ′ [t.d.( Bq )]) + ht (cid:16) P/ ( p ′ ⊗ k B + A ⊗ k q )( p/p ′ ) ⊗ k ( B/q ) (cid:17) = ht ( pp ′ [t.d.( Bq )]) + ht ( Pp ⊗ k B + A ⊗ k q )since pp ′ ⊗ k Bq ∼ = p ⊗ k B + A ⊗ k qp ′ ⊗ k B + A ⊗ k q via Proposition 2.3. It follows that ht ( P ) ≤ max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( p ′ p [t.d.( Bq )]) : p ⊆ p ′ and q ⊆ q are prime ideals of A and B, respectively o + ht ( pp ′ [t.d.( Bq )]) + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( p ′ p [t.d.( Bq )])+ ht ( pp ′ [t.d.( Bq )]) : p ⊆ p ′ and q ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p and q ⊆ q are prime ideals of A and B, respectively o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) , and then the equality holds. Case 3. q ′ = (0) . Then ht ( P ) = 1 + ht ( Q ) = λ (cid:16) ( ., (0)) , P (cid:17) ≤ t.d.( A ) − t.d.( Ap ) + ht ( q [t.d.( Ap )]) + ht ( Pp ⊗ k B + A ⊗ k q ) , by [3, Lemma 2.4]= ht ( p [ X ]) + ht ( q [t.d.( Ap )]) + ht ( Pp ⊗ k B + A ⊗ k q ) , as A [ X ] is an AF-domain ≤ ht ( p [t.d.( B )]) + ht ( q [t.d.( Ap )]) + ht ( Pp ⊗ k B + A ⊗ k q ) , as t.d.( B ) ≥ max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p ∈ Spec( A ) and q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) , by (1) ≤ ht ( P ) . Then the equality holds, as desired.
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 ) = max n ht ( q q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )])+ ht ( pp [t.d.( Bq )]) : p ⊆ p ∈ Spec( A ) and q ⊆ q ⊆ q ∈ Spec( B ) o + ht ( P/ ( A ⊗ k q ) p ⊗ k ( B/q ) + A ⊗ k ( q/q ) ) ≤ max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )])+ ht ( pp [t.d.( Bq )]) : p ⊆ p ∈ Spec( A ) and q ⊆ q ∈ Spec( B ) o + ht ( Pp ⊗ k B + A ⊗ k q ) ≤ ht ( P ) , then the equality holds establishing the desired formula . b) It is a direct consequence of (a) and [16, Proposition 2.3].c) Let B be a k -algebra. Let p ∈ Spec( A ) and q ∈ Spec( B ). Applying (a), we get13 t ( P ) = max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )]) + ht ( pp [t.d.( Bq )]) : p ⊆ p and q ⊆ q are prime ideals of A and B, respectively o for each minimal prime ideal P of p ⊗ k B + A ⊗ k q . It follows that ht ( p ⊗ k B + A ⊗ k q ) = max n ht ( q [t.d.(A)]) + ht ( p [t.d.( Bq )]) + ht ( qq [t.d.( Ap )])+ ht ( pp [t.d.( Bq )]) : p ⊆ p and q ⊆ q are prime ideals of A and B, respectively o , and thus ht ( P ) = ht ( p ⊗ B + A ⊗ k q ) + ht ( Pp ⊗ k B + A ⊗ k q )for each prime ideal P of A ⊗ k B such that p = P ∩ A and q = P ∩ B . Then A is aGAF-domain, as desired. ✷ Next, we present an example of a GAF-domain A such that A [ n ] fails to be an AF-domainfor any positive integer n . Example 2.8.
Let k be an algebraically closed field. Consider the k -algebra homomorphism ϕ : k [ X, Y ] → k [[ t ]] such that ϕ ( X ) = t and ϕ ( Y ) = s := X n ≥ t n ! . Since s is known to betranscendental over k ( t ), ϕ is injective. This induces an embedding ϕ : k ( X, Y ) → k (( t ))of fields. Put A = ϕ − ( k [[ t ]]). It is easy to check that A is a discrete rank-one valuationoverring of k [ X, Y ] of the forme A = k + p , where p = XA . Note that, for each positiveinteger n , and since A is Noetherian, thus a locally Jaffard domain, ht ( p [ n ]) + t.d.( Ap ) = ht ( p ) + t.d.( Ap ) = 1 < A ) . Then, via [5, Lemma 2.1], for each positive integer n , A [ n ] is not an AF-domain. Let B bean arbitrary k -algebra. Let P be a prime ideal of A ⊗ k B and q = P ∩ B . If P ∩ A = (0),then P survives in k ( X, Y ) ⊗ k B , and thus, by [5, Lemma 1.5], we are done. Now, assumethat P ∩ A = p . Then, as t.d.( Ap ) = 0, P is minimal over p ⊗ k B + A ⊗ k q [16, Proposition2.3]. Moreover, since k is algebraically closed, p ⊗ k B + A ⊗ k q is a prime ideal of A ⊗ k B ,as A ⊗ k Bp ⊗ k B + A ⊗ k q ∼ = Ap ⊗ k Bq is an integral domain, by [17, Corollary 1, p. 198]. Hence P = p ⊗ k B + A ⊗ k q , so that ht ( P ) = ht ( p ⊗ k B + A ⊗ k q ). It follows that A is a GAF-domain,as desired. ✷ eferences [1] J.T. Arnold, On the dimension theory of overrings of an integral domain , Trans. Amer. Math.Soc., 138, 1969,313-326.[2] S. Bouchiba,
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