IC/96/167 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE DIMENSION OF TENSOR PRODUCTS OF k-ALGEBRAS ARISING FROM PULLBACKS S. Bouchiba Department of Mathematics, Faculty of Sciences I, University of Fes, Fes, Morocco, F. Girolami Dipartimento di Matematica, Universita degli Studi Roma Tre, 00146 Roma, Italy and S. Kabbaj 1 International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT The purpose of this paper is to compute the Krull dimension of tensor products of A;—algebras arising from pullbacks. We also state a formula for the valuative dimension. MIRAMARE - TRIESTE September 1996 1 Permanent address: Department of Mathematics, Faculty of Sciences I, University of Fes, Fes, Morocco. 1 0. Introduction All rings and algebras considered in this paper are commutative with identity elements and, unless otherwise specified, are to be assumed to be non-trivial. All ringhomomorphisms are unital. Let A; be a field. We denote the class of commutative A;—algebras with finite transcendence degree over k by C. Also, we shall use t.d.(A) to denote the transcendence degree of a A;—algebra A over k, A[n] to denote the polynomial ring A[Xi, ...,Xn], and p[n] to denote the prime ideal p[X\, ...,Xn] of A[n], where p is a prime ideal of R. Recall that an integral domain R of finite (Krull) dimension n is a Jaffard domain if its valuative dimension, dimv(R), is also n. Priifer domains and noetherian domains are Jaffard domains. We assume familiarity with this concept, as in [1], [6] and [10]. Suitable background on pullbacks is [4], [11], [12] and [16]. Any unreferenced material is standard, as in [12] and [17]. In [20] Sharp proved that if K and L are two extension fields of k, then dim(K <S>k L) = min(t.d.(J<Q, t.d.(L)). This result provided a natural starting point to investigate dimensions of tensor products of somewhat general A;—algebras. This was concretized by Wadsworth in [21], where the result of Sharp was extended to AF-domains, i.e. integral domains A such that ht(p) + t.d.(A/p) = t.d.(A), for all prime ideals p of A. He showed that if A\ and A2 are AF-domains, then dim(Ai <S>k A2) = min(dim(Ai) + t.d.(A2), dim(A2) + t.d.(Ai)). He also stated a formula for dim(A(E)kR) which holds for an AF-domain A, with no restriction on R. We recall, at this point, that an AF-domain is a (locally) Jaffard domain [13]. In [5] we were concerned with AF-rings (a A;—algebra A is said to be an AF-ring provided ht(p) + t.d.(A/p) = t.d.(Ap), for all prime ideals p of A). A tensor product of AF-domains is perhaps the most natural example of an AF-ring. We then developed quite general results for AF-rings, showing that the results do not extend trivially from integral domains to rings with zero-divisors. Our aim in this paper is to extend Wadsworth's results in a different way, namely to tensor products of A;—algebras arising from pullbacks. In order to do this, we use previous deep invenstigations of the prime ideal structure of various pullbacks, as in [1], [2], [3], [4], [6], [8], [9], [10] and [16]. Moreover, in [21] dimension formulas for the tensor product of two particular pullbacks are established and a conjecture on the dimension formulas of more general pullbacks is raised; in this paper such conjecture is resolved. Before presenting our main result of section 1, Theorem 1.9, it is convenient to recall from [21] some notation. Let A £ C and let d,s be integers with 0 < d < s. Put D(s,d,A) = max{ htp[s] + mm(s,d + t.d.(A/p)) / p prime ideal of A}. Our main result is the following : given R\ = ip~1(D\) and R2 = <p~1(D2) two pullbacks issued from T\ and T2 respectively. Assume that Di, T{ are AF-domains and hi (Mi) = di for i = 1,2. Then dim(i?i (g>k R2) = max|htMi[t.d.(i? 2 )] + D(t.d.(Di), dim(Di), R2) , htM 2 [t.d.(i2i)] + D(t.d.(D 2 ), dim(JD2), It turns out ultimately from this theorem and via a result of Girolami [13] that one may compute (Krull) dimensions of tensor products of two A;—algebras for a large class of (not necessarily AF-domains) A;—algebras. The purpose of Section 2 is to prove the following theorem : with the above notation, dimv(R1 (g>k R2) = minjdim^i?! + t.d.(R2), d i m ^ + t.d.(i?i)| In Section 3 Theorem 3.1 asserts that, with mild restrictions, tensor products of pullbacks preserve Jaffard domains. Theorem 3.2 states, under weak assumptions, a formula similar to that of Theorem 1.9. It establishes a satisfactory analogue of [4,Theorem 5.4] (also [1,Proposition 2.7] and [9, Corollary 1]) for tensor products of pullbacks issued from AF-domains. We finally focus on the special case in which R\ = R2. Some examples illustrate the limits of our results and the failure of Wadsworth's results for non-AF-domains. 1. The Krull dimension The discussion which follows, concerning basic facts (and notations) connected with the prime ideal structure of pullbacks and tensor products of A;—algebras, will provide some background to the main theorem of this section and will be of use in its proof. Notice first that we will be concerned with pullbacks (of commutative A;—algebras) of the following type : R --+ D T --+ K I I where T is an integral domain with maximal ideal M, K = T/M, <p is the canonical surjection from T onto K, D is a proper subring of K and R = Lp~1(D). M = (R : T) and D = R/M. Clearly, Let p be a prime ideal of R. If M <£ p, then there is a unique prime ideal q in T such that q f] R = p and Tq = Rp. However, if M C p, there is a unique prime ideal q in D such that p = (p~1(q) and the following diagram of canonical homomorphisms Rp > Dq I I —> is a pullback. K Moreover, hip = hiM + hiq (see [11] for additional evidence). We recall from [8] and [1] two well-known results describing how dimension and valuative dimension behave under pullback : with the above notation, dimi? = max{dimT, dimD + dimT^}, and dim^i? = max{dinii,T, dim^D + dim^T^ + t.d.(K : D)}. However, while dimi?[n] seems not to be effectively computable in general, questions of effective upper and lower bounds for dimi?[n] were partially answered. The following lower bound will be useful in the sequel : dimi?[n] > dimD[n] + dimT^ + min(n, t.d.(K : D)), where the equality holds if T is supposed to be a locally Jaffard domain with hiM = dimT (cf. [9]). At last, it is a key result [13] that R is an AF-domain if and only if so are T and D and t.d.(K : D) = 0. A combination of this result and Theorem 1.9 allows one to compute dimensions of tensor products of two A;—algebras for a large class of (not necessarily AF-domains) A;—algebras. We turn now to tensor products. Let us recall from [21] the following functions : let A,A\ and A2 G C. Let p G Spec(A), p\ G Spec(Ai) and P2 G Spec(A2). Let d,s be integers with 0 < d < s. Set • SP1,P2 = {P e Spec(A! (E)k A2) I • 8(p1,p2) =max{htP • A(s,d,p) = htp[s] + mm(s,d + t.d.(A/p)) • D(s,d,A) = max{A(s, d,p) / p G Spec(A)} Pl = P n A1 and p2 = P n A2} / P £ SPljP2} 4 One can easily check that dim(Ai 0k -A2) = max{$(pi,p2)/ Pi G Spec(Ai) and P2 G Spec(A 2 )} (see [21, page 394]). Let P G Spec(Ai 0k A2) with p1 C P C\ A1 and P2 ^ -P H A2. It is known [21] that P is minimal in SPl)P2 if and only if it is a minimal prime divisor of p\ 0 A2 + A\ 0 p2. This result will be used to prove a special chain lemma for tensor products of A;—algebras, which establishes a somewhat analogue of the Jaffard's special chain theorem for polynomial rings (see [7] and [15]). These facts will be used frequently in the sequel without explicit mention. The proof of our main theorem requires some preliminaries. The following two lemmas deal with properties of polynomial rings over pullbacks, which are probably well-known, but we have not located references in the literature. Lemma 1.1. Let T be an integral domain with maximal ideal M, K = T/M, tp the canonical surjection from T onto K, D a proper subring of K and R = ip~x (D). Then htp[n] = ht(p[n]/M[nJ) -/- htM[n], for each positive integer n and each prime ideal p of R such that M C p. Proof. Since M C p, there is a unique q G Spec(Z)) such that p = Lp~1(q) and the following diagram is a pullback / 1M I ' -^ By [1, Lemma 2.1 (c)] MTM = MRp is a divided prime ideal of Rp. By [1, Lemma 2.2] htp[n] = htpRp[n] = ht(pRp[n]/MRp[n]) + htMRp[n] = ht(p[n]/M[n]) + htM[n]. <0> Lemma 1.2. Let T be an integral domain with maximal ideal M, K = T/M, tp the canonical surjection from T onto K, D a proper subring of K and R = Lp~1 (D). Assume TM and D are locally Jaffard domains. Then htp[n] = htp -/- min(n, t.d.(K:D), for each positive integer n and each prime ideal p of R such that M C p. Proof. Since M C p, there is a unique q G Spec(Z)) such that p = Lp~1(q) and the following diagram is a pullback Rp —> I Dq I 1M ' -^ By [3, Corollary 2.10] htp[n] = dim(Rp[n]) - n. Furthermore, dim(Rp[n]) = htM + dim(Dq[n]) + min(n, t.d.(K:D)) = htM + dim.Dq + n + min(n, t.d.(K:D)) = htp + n + min(n, t.d.(A': D), completing the proof. <0> The following corollary is an immediate consequence of (1.2) and will be useful in the proof of the theorem. Corollary 1.3. Let T be an integral domain with maximal ideal M, K = T/M, ip the canonical surjection from T onto K, D a proper subring of K and R = tp~1 (D). Assume TM is a locally Jaffard domain. Then htMfn] = htM + min(n, t.d.(K:D), for each positive integer n. <0> We next analyse the heights of ideals of A\ 0k A2 of the form p\ 0k A2, where p\ £ A\) and A2 is an integral domain. L e m m a 1.4. Let A\, A2 £ C and pi be a prime ideal of A\. Assume A2 is an integral domain. Then ht(p\ 0k A2) = htpi[t.d.(A2)\. Proof. Put £2 = t.d.(A2). Let Q be a minimal prime divisor of pi 0 A2 in A\ 0 A<iThen Q is minimal in Spi^, and hence t.d.((Ai 0 A2)/Q) = t.d.(A\/p\) + t<i by [21, Proposition 2.3]. Furthermore, Q survives in A\ 0 F2, where F2 is the quotient field of A2, whence htQ + t.d.((Ai 0 A2)/Q) = t2 + htpi[t 2 ] + t.d.(Ai/pi) by [21, Remark l.b], completing the proof. <0> With the further assumption that A2 is an AF-domain, we obtain the following. L e m m a 1.5. (Special chain lemma) Let A\, A2 £ C and pi be a prime ideal of A\. Assume A2 is an AF-domain. Let P £ Spec(A\ 0k A2) such that p\ = P f] A\. Then htP = ht(Pl 0k A2) + ht(P/(Pl 0k A2)). Proof. Since A2 is an AF-domain, by [21, Remark l.b] htP + t.d.({A10 A2)/P) = t2 + ^] + t.d.(A\/pi), where t2 = t.d.(A2). A similar argument with (Ai/pi) 0k A2 in 6 place of Ai (g)k A2 shows that ht(P/(pi ®kA2)) + t.d.((Ai ®A2)/P) = t2 + t.d.(A1/p1), whence h t P = htpi[t 2 ] + ht(P/(pi <S>k ^2))- The proof is complete via Lemma 1.4. <0> An important case of Lemma 1.5 occurs when A2 = k[X\,..., Xn] and hence if P is a prime ideal of A\ ® A2 = Ai[Xi, ...,Xn] with p = P f] A\, then h t P = htp[n] + htP/p[n\. Our special chain lemma may be then viewed as an analogue of the Jaffard's special chain theorem (see [7] and [15]). Notice for convenience that Jaffard's theorem holds for any (commutative) ring, while here we are concerned with A;—algebras. To avoid unnecessary repetition, let us fix notation for the rest of this section and also for much of section 2 and 3. Data will consist of two pullbacks of A;—algebras Ri —> D1 R2 —> D2 Ti —> Id T2 —^ K2 where, for i = 1,2, Ti is an integral domain with maximal ideal Mi, Ki = TijMi, <fi is the canonical surjection from Ti onto Ki, Di is a proper subring of Ki and Ri = ip~ (Di). Let di = dimTi, d\ = dimDi, ti = t.d.(T^), ri = t.d.(A^) and Si = t.d.(Di). The next result deals with the function S(p\,p2) according to inclusion relations between pi and Mi (i = 1,2). L e m m a 1.6. Assume T\ and T2 are AF-domains. If pi G Spec(R\) and p2 G Spec(R2) are such that M\ (£_ p\ and M2 (£_ p2, then S(p\,p2) Proof. = min(htpi -\-t2, t\ + htp2) < min(d\ -\-t2,t\ -\-d2). By [1, Lemma 2.1 (e)], for i = 1,2, there exists qi G Spec(Ti) such that Pi = qi n Ri and Tiqi = RiPi. So that R\Pl and R2p2 are AF-domains, whence S(p\,p2) = min(htpi + t2, t\ + \A,p2) by [21, Theorem 3.7]. Further, htpi < d\ and htp2 ^ d2, completing the proof. <0> L e m m a 1.7. Assume T\ and T2 are AF-domains. Let P G Spec(R\ (£ikR2), p\ = PC\R\ and p2 = Pf\R2. If Mx C Pl and M2 <£ p2, then htP = htM^t^ Proof. Since M2 <f_ p2, R2p2 ht(P/(pi (g) R2j). 1.1. Hence + M(P/(Mi ® R2)). is an AF-domain. By Lemma 1.5 h t P = Since M1 C p1, htpi[t 2 ] = ht(pi[t 2 ]/Mi[t 2 ]) + htMi[t 2 ] by Lemma htP = ht(pi[t 2 ]/Mi[t 2 ]) + htMi[t 2 ] + ht(P/(pi <8> R2)) = ht((pi <8> R2)/{M1 ® i?2)) + htMi[*2] + ht(P/(pi (8) i?2)) < htMi[<2] + ht(P/(Mi <8>i22)) = ht(Mi (8) i?2) + ht(P/(Mi (8) i22)) < htP. 0 A similar argument with the roles of p\ and p2 reversed shows that if M\ (£_ p\ and M2 C p 2 , then htP = htM2[*i] + ht(P/(i?i <8> M 2 )). Now, we state our last preparatory result, by giving a formula for dim((i?i/Mi) ® (R2/M2)) and useful lower bounds for dim((i?i/Mi) (g) R2) and dim(i?i (g) (R2/M2j). L e m m a 1.8. ^ss^rrie Ti ; T2; I?i anJ D 2 are AF-domains with dimT\ = htM\ and dvmT2 = htM2. Then a) dim((Ri/M\) b ) dim(Ri (g) i ? 2 ) > c/2 -/- min(si ®(R2/M2)) ,r2 — s2) + min(si > d\ + min(s2,ri c) rfim((i2i/Mi) (8) (R2/M2j) + <i2, <i^ + s2). — s\) -/- min(si + d'2ld'x -\-s2). = min(s1 +d'2,dt1 +s2). Proof, a) Since i?i/Mi = £>! is an AF-domain, by [21, Theorem 3.7] dim((i2i/Mi) ®R2) = D(sud'1,R2) = max{A(sudt1,p2)/p2 G Spec(i?2)}. Let p 2 G Spec(i?2) such that M2 C p 2 . Then there is a unique q2 G Spec(D2) such that p 2 = (,c^" (q2) and the following diagram is a pullback —> K2 By Lemma 1.2 htp2[si] = htp 2 + min(si, r2 — s2). Since R2/p2 and D2/q2 are isomorphic A;—algebras, t.d.(i? 2 /p 2 ) = t.d.(D2/q2) A(s1,d'1,p2) = s2 - htp2 + htM 2 , so that = htp2[si] + min(si,<i'1, t.d.(i? 2 /p 2 )) = htp 2 + min(si,r 2 — s2) + min(si,c/'1 + s2— htp 2 + htM 2 ) = min(si,r 2 — s2) + min(si+ htp2,d'1 + s2 + htM 2 ) = htM 2 + min(si,r 2 — s2) + min(si+ htq2,d'1 -\-s2) = d2 + min(si,r 2 — s2) + min(si+ h.tq2,d'1 + s2). b) As in (a) with the roles of R\ and R2 reversed. c) It is immediat from [21, Theorem 3.7]. <0> The facts stated above provide motivation for setting: a1 = d1 + min(t2,r1 - si) + d2 + min(si,r 2 - s2) + min(si +d'2,d'1 + s2); ot2 = d2 -\- min(ti, r2 — s2) + d\ + min(,s2, r\ — s\) + min(si + d'2, d[ -\- s2); 0,3 = di + d2 + min(ri, r2) + min(si + d'2, d[ + s2). We shall use these numbers in the proof of the next theorem and in the section 3. We now are able to state our main result of this section. Theorem 1.9. Assume T\, T2, D\ and D2 are AF-domains with dimT\ = htM\ and dimT2 = htM2. Then dim{R1 ® fc i? 2 ) = maxihtM1[t.d.(R2)] + D^.d^Dx), htM2[t.d.(R1)] dim(Dx), R2) , + D(t.d.(D2), dim(D2), Ri)\ Proof. Since dim(R1(g)R2) > ht(Mi®i? 2 ) + dim((i?i/Mi)(g)i? 2 ), we have dim(i?i ®i?2) > htMi[t 2 ] + dim((i?i/Mi) ® R2) by Lemma 1.4. Similarly, dim(i?i ® R2) > htM2[*i] + dim(i?i <g)(R2/M1)). Therefore it suffices to show that dim(Ri<g>R2) < max{htMi[t2] + dim((i?i/Mi) ®i? 2 ), htM 2 [ti]+ dim(i?i ®(R2/M2))}. It is well-known that dim(i?i <g> R2) = max{$(pi,p 2 )|pi G Spec(i?i),p 2 G Spec(i? 2 )}. Let pi G Spec(i?i) and p2 G Spec(i? 2 ). There are four cases : 1. If M\ (jL p\ and M2 <f_ p2, by Lemma 1.6 S(p\,p2) = mm(htpi -\-t2,t\ + htp2) < 0:3. 2. If Mi CPl and M 2 ^ p 2 , by Lemma 1.7 <^(pi,p2) < htMi[t 2 ] + dim((i?i/Mi) ® i? 2 ). 3. If Mi (/L pi and M 2 C p 2 , by Lemma 1.7 S(p1,p2) < htM 2 [t x ] + dim(i?i (g) (R2/M2j). 4. If Mi C pi and M 2 C p 2 , then S(p1,p2) < max { htM x [t 2 ] + dim((i?i/Mi) (g) i? 2 ), htM 2 [ti] + dim(i?i(g)(i? 2 /M 2 )),a 3 }. Indeed, put h = 8(p1,p2). Pick a chain Po C Pi C .... C Ph of h-\-1 distinct prime ideals in i?i (g)i?2 with Ph G SPl)P2. If Mi C Po Hi?i and M 2 C Po n P 2 , then h = htPh/P0 < dim((i2i/Mi) ® (R2/M2)) < a3. Otherwise, let i be the biggest integer such that Mi ^ Pif] R\ and let j be the biggest integer such that M 2 (JL PJ n R2. If i ^ j , say i < j , by Lemma 1.7 h t P J = htMi[t 2 ]+ ht(P J /(Mi ® i2 2 )), whence /z < htMi[t 2 ] + ht(P f t /(Mi ® i22)) < htMi[t 2 ] + dim((i2i/Mi) ® i2 2 ). If i = j , since Mi C p 1? there is a unique gi G Spec(Di) such that p\ = Lp^1(qi) and the following diagramm is a pullback 9 RiPl —> Dlqi Since Mi £ Pt n P i , it follows that (Pz n Ri)RiPl C M1T1Ml = (RiPl : Ti M l ) by [1, Lemma 2.1 (c)], whence ht(P z n P i ) < htMi - 1 = di - 1. Similarly, ht(P z n P 2 ) < htM2 — 1 = d2 — 1. Finally, we get via Lemma 1.6 h = htPz + 1 + h t ( P n / P i + i ) < 8(Pi n R!,Pi DR2) + l + dim((i2i/Mi) ® (i2 2 /M 2 ) = min(ht(P z n P i ) + t 2 , t i + ht(Pj n R2)) + 1+ dim((P 1 /M 1 ) ® (R2/M2) = «3. The fourth case is done. Now, let us assume s\ < r 2 — s 2 . Then a.\ = d\ + m i n ( t 2 , r \ — s\) + d2 + s\ + m i n ( s i + d'2, d[ + s 2 ) = d\ +min(t 2 + s\, r\) + d2 +min(si + d'2, d[ + s 2 ) > d\ + <i 2 +min(ri,r 2 )+min(si + c ^ , ^ + s 2 ) = a3. If s 2 < ri — s\, in a similar manner we obtain a 2 > 0:3. Finally, assume r\ — s\ < s 2 and r 2 — s 2 < si, so that «i = a2 = t\ — s\ + t2 — s2+ min(si + d2,d[ + s2) = min(ti +t2 - s2 + d'2,t1 + t2 - sx + d[) = min(dinii,Pi -\-t2,t\ Hence by [13, Proposition 2.1] (g) P 2 ) (i ®R2). Finally, one may easily check, via Corollary 1.3 and Lemma 1.8, that a.\ < htMi[t2]-\dim((Pi/Mi) ® R2) and a2 < htM 2 [ti] + dim(Pi (g) (R2/M2)). <} It is still an open problem to compute dim(Pi (g)P2) when only T\ (or T2) is assumed to be an AF-domain. However, if none of the T{ is an AF-domain (i = 1,2), then the formula of Theorem 1.9 may not hold (see [21, Examples 4.3]). 10 2. The valuative dimension It is worth reminding the reader that the valuative dimension behaves well with respect to polynomial rings, that is, dinii,i?[n] = dim^i? + n, for each positive integer n and for any ring R [15, Theorem 2]. Whereas dimu(i2i <S> R2) seems not to be effectively computable in general. In [13] the following useful result is proved: given A\ and A2 two A;—algebras with dimAi = dim^Ai and dimA2 = dim^,^ (i.e are Jaffard rings), then dimu(Ai ® -A2) < min(dimAi + t.d.(A2), dimA2 + t.d.(Ai)). This section's goal is to compute the valuative dimension for a large class of (not necessarily Jaffard) A;—algebras. We are still concerned with those arising from pullbacks. The proof of our theorem requires a preliminary result, which provides a criterion for a polynomial ring over a pullback to be an AF-domain. We first state the following. L e m m a 2.1. Let A be an integral domain and n a positive integer. Then A[n] is an AF-domain if and only if, for each prime ideal p of A, htp[n] + t.d.(A/p) Proof. Suppose A[n] is an AF-domain. = t.d.(A). So for each prime ideal p of A htp[n] + t.d.(A[n]/p[n]) = t.d.(A) + n , whence htp[n] + t.d.(A/p) = t.d.(A). Conversely, if Q G Spec(A[n]) and p = Q C\ A, then by [21, Remark l.b] htQ + t.d.(A[n]/Q) = n + htp[n] + t.d.(A/p) since A[n] = A ® k[n]. Therefore, htQ + t.d.(A[n]/Q) = n + t.d.(A) = t.d.(A[n]). 0 Proposition 2.2. Let T be an integral domain with maximal ideal M, K = T/M, and (p the canonical surjection. Let D be a proper subring of K and R = (p~1(D). Assume T and D are AF-domains. Let r = t.d.(K) and s = t.d.(D). Then R[r — s\ is an AF-domain. Proof. Let p G Spec(i?). There are two cases: 1. If M <£ p, then Rp is an AF-domain. So htp + t.d.(R/p) Corollary 3.2] htp = htp[r - s], whence htp[r - s] + t.d.(R/p) = t.d.(R). Further, by [21, = t.d.(R). 2. If M C p , by Lemma 1.1 htp[r - s] = htp + r - s. Moreover t.d.(R/p) htp. Then htp[r - s] + t.d.(R/p) > r + d = t.d.(R). 11 = s + htM- On the other hand, htp[r - s] + t.d.(R/p) < t.d.(R). Hence htp[r - s] + t.d.(R/p) = t.d.(R). So R[r - s] is an AF-domain by Lemma 2.1. <0> We may now present the main result of this section. T h e o r e m 2 . 3 . Let Ti,T 2 ,.Di and .D2 be AF-domains, = htM.2, then dimv{R\ ® R2) = min(dimvRi with dimT\ = htM\ and dimT2 + t 2 , dimvR2 + t\). Proof. By Proposition 2.2 R\[r\ — s\] and i?2[?"2 — s2] are AF-domains. Then R\[r\ — s\] (x) -R2[r2 — $2] is a n AF-ring by [21, Proposition 3.1]. Consequently, by [9, Theorem 2] dim r (i2i[ri - si] ® i? 2 [r 2 - s2]) = dim(i?i[ri - si] ® i? 2 [r 2 - s 2 ]) = min(dim i?i [n - si] + t.d.(R2[r2 - s2]), t.d.(i?i[ri - si]) + dim i? 2 [r 2 - s2]) > m.m(di + dimDifri—si] + r i — s i + t 2 + r 2 —s 2 ,<i 2 + dimD 2 [r 2 — s 2 ] + ri — s i + t i + r 2 — s2) = r1 — s1 + r2 — s2+ mm(d1 + d[ + r1 — s1 + t2} d2 + d'2 + r2 — s2 + t^. It turns out that dim u (i2i <E> -R2) > min(c/i + d[ + r i — si + t2, d2 + <i2 + f2 — s 2 + t i ) . So by [1, Theorem 2.11] divnv(Ri ® i? 2 ) > min(diin u i2i + t2, t\ + dimi,i? 2 ). Therefore by [13, Proposition 2.1] we get d\m.v{R\ ® R2) = min(diin u i2i + t2 , dimi,i? 2 + t\) = t\ — s\ + t2 — s2 + min(si + d'2, d[ + s2). <C> 3. Some applications and examples We may now state a stability result. It asserts that, under mild assumptions on transcendence degrees, tensor products of pullbacks issued from AF-domains preserve Jaffard rings. Theorem 3.1. / / T i , T 2 , D i and D 2 are AF-domains, M\ is the unique maximal ideal of T\ with htM\ = dimT\ and M 2 is the unique maximal ideal of T2 with dimT2 = htM2, then R\ ® R2 is a Jaffard ring if and only if either r\ — s\ < t2 and r 2 — s 2 < s\ or r\ — s\ < s 2 and r 2 — s 2 < t\. Proof. Suppose r\ — s\ < t2 and r 2 — s 2 < s\. Then ot\ = t\ — s\ + t2 — s 2 + min(si + d'2ld'x + s2) = min(diin u i2i + t2,t\-\- dimi,i?2). By Theorem 1.9 and Theorem 2.3 a.\ < dim(i?i ®i? 2 ) < d\m.v{R\ <S> R2) = min(diin u i2i + t 2 , t\-\- dimi,i?2) = a.\. Hence R\ ® R2 is a Jaffard ring. Likewise for r\ — s\ < s 2 and r 2 — s 2 < t i . Conversely, 12 since i ? i / M i ^ Dx is an AF-domain, by [21, Theorem 3.7] dim((i?i/Mi) ® i?2) = D(si, d[, R2) = max{A(si, ^ , ^ 2 ) ^ 2 £ Spec(i?2)}- If M2 C p 2 , by the proof of Lemma 1.8 it follows that A(si, d'1,p2) = G?2 + min(si, r2 — ^2) + min(si+ h t ^ , d[ + S2) where 52 is the unique prime ideal of D2 such that P2 = ¥2 {iz)- If ^ 2 ^ P2 , since i?2p2 i s a n AFdomain, then A(si, d[,p2) = htp2[^i]+ min(si,c/' 1 + t.d.(i?2/p2)) = htp2+ min(.si, (ii-lt . d ^ i ^ / ^ ) = min(si+ htp 2 ,^i+ t.d.(i? 2 /p2) + htp 2 ) = min(si+ htp 2 ,^i +^2)- In conclusion, since htp2 < c?2 — 1 being M2 the unique maximal ideal of T2 with dimT2 = htM2, we get dim((i?i/Mi) <g> R2) = max{c/2+ min(si,r2 — ^2)+ min(si + d'2ld'x +S2), min(si + c?2 — 1, d\ + ^2)}. Similarly, dim(i?i ® (R2/M2)) = max{c/i+ min(s2, r\ — s\)-\min(si+c/ 2 , ^ + ^ 2 ) , min(s2+<ii — 1, d2-\-t\)}. Moreover by Theorem 2.3 dimv(Ri(E)R2) = min(dinii,i?i +^2, dim^,^ +^1) = t\ — s\ +^2 — S2+ min(si +<i25 ^i + 5 2)- Let us assume s\ -\- d'2 < <ii + S2- Necessarily, s\ + c/2 < ^2 + ^ . Applying Corollary 1.3, we obtain htMi[t 2 ] + dim((i?i/Mi)(g)i? 2 ) = <ii+ mm(t2,r1 - s^ + d2+ min(si,r 2 - s2) + s1 +d'2. On the other hand, d\ + min(s2,ri — s\) -\- s\ -\- d'2 = min(s2 + d\,t\ — s\) + s\ + d'2 = min(c?2 + t i , s 2 + d1 + s1 + <i2) > min(s 2 + <ii - 1,4 + ti). Therefore htM 2 [ti] + dim(i?i(x)(R2/M2)) = c?2+ min(ti, r2 — S2) + <ii+ min(s2, ri — si) + si+c/ 2 . Consequently, dim(i?i(x)i?2)= max{<ii+ min(t2, r\ — si)-\-d2-\- min(si, r2 — S2)-\-s\ -\-d2, c?2+ min(ti, r2 — $2) + d\-\- min(s 2 , r\ — s\) + s\ + <i2} and 6imv{R\ ® R2) = t\ + t 2 — s2 + d!2 = d\ + ri + <^2 + r2 — ^2 + d2. Since R\ ® R2 is a Jaffard ring, then either c/i + min(t2 ,r\ — s\) + c/2 + min(si,r 2 — s 2 ) + si + d2 = d\ + ri + d2 + r 2 — s2 + <i2 or <i2+ min(ti,r 2 — s 2 ) + <ii + min(s2, r\ — si) -\- s\ -\- d'2 = d\ + ri + c/2 + r2 — ^2 + d'2. Hence either r\ — s\ < ^2 an r2 — S2 < si or r\—s\ < S2 and r2 — S2 < ti. Similar arguments run for d\ +S2 < ^i +c/ 2 , completing the proof. <) Our next result states, under weak assumptions, a formula similar to that of Theorem 1.9. It establishes a satisfactory analogue of [4, Theorem 5.4] (also [1, Proposition 2.7] and [9, Corollary 1]) for tensor products of pullbacks issued from AF-domains. Theorem 3.2. Assume T\ and T2 are AF-domains, with dimT\ = htM\ and dimT2 = htM2. Suppose that either t.d^Dx) < t.d.(K2 : D2) or t.d.{D2) < t.d^Kx : £>i). Then dim(R1 (g> R2) = max{htM1[t2] + dim(D1 ® R2) , htM2[t1] + dtm(R1 <g)D2)}Here, since none of Di is supposed to be an AF-domain (i = 1,2), the 13 cc dim(Di <g> Rj) d = T)(si, d[, Rj)v assertion is no longer valid in general ((i,j) = (1,2), (2, 1)). Neither is the "dim(Di <g> D2) = min(si + d'2ld'x + S2)" assertion. Put a'3 = mm(di +^2,^1 + ^2) + dim(JDi (g) D2). Proof. The proof runs parallel with the treatment of Theorem 1.9. An appropriate modification of its proof yields dim(i?i ® R2) <max{htMi[t2] + dim((i?i/Mi) ® R2), htM2[ti]-\- dim(i?i ® (R2/M2)), 03}- Now there is no loss of generality in assuming that t.d.(Di) < t.d.(K2 : D2) (i.e s\ < r2 — S2). By Lemma 1.1 and Corollary 1.5 ht(Mi (g>R2) + ht(Di <g)M2) = htMi[t 2 ]+ htM 2 [si] = htM x + min(t 2 ,ri - s i ) + htM 2 + min(si, T2 — S2) = min(<ii + t2 + d2 + s\, t\ + d2) > min(c/i +^2,^1 + ^2)- Clearly, «g = min(<ii+t2,ti +<i2) + dim(D1(g)D2) < ht(M1®R2) + ht(Di ®M 2 ) + dim(JDi ® D2) < ht(Mi ® i? 2 )+ dim(£>i ® i2 2 ). • We now move to the significant special case in which R\ = R2. Corollary 3.3. Let T be an AF-domain with maximal ideal M with htM = dimT = d, K = T/M, and ip the canonical surjection. Let D be a proper subring of K and R = Lp~1(D). Assume D is a Jaffard domain. Then dim(R <S> R) = htM[t] -/- dim(D <S> R), where t = t.d.(T). If moreover t.d.(K:D) < t.d.(D), then dim(R (g) R) = dimv(R <g) R) = t + dimvR. Proof. If t.d.(D) < t.d.(K : D), the result is immediate by Theorem 3.2. Assume t.d.(K : D) < t.d.(D). Then dim(i? ® R) > ht(M ® R) + ht(D ® M) + dim(.D ® D) > htM[t]+ htM[s]+ dimD + t.d.(D) = d+ min(t,t.d.(A' : D)) + d+ min(s,t.d.(A' : D)) + dimD + t.d.(D) = min(t + d,t-t.d.(D)) + d+ t.d.(K : D)+ dimD + s = t-s + t + d' = t+ > dimv(R ® R). This completes the proof.<) The following example illustrates the fact that in Theorm 1.9 and Corollary 3.3 the = hi Mi (i = 1,2)" hypothesis cannot be deleted. Example 3.4. Let K be an algebraic extension field of k, T = S~1K[X,Y], S = K[X, Y] - ((X) U (X - 1, Y)) and M = S'^X). R I '[X,Y] 14 where Consider the following pullback —• k(Y —• K(Y I Since S 1K[X, Y] is an AF-domain and the extension k(Y) C K(Y) is algebraic, by [13] R is an AF-domain, so that dim(R (g) R) = dimR + t.d.(R) = 2 + 2 = 4 by [21, Corollary 4.2]. However, htM[2] = h t M = 1 and dim(k(Y) ® R) = min(2,1 + 2) = 2. Hence htM + dim(k(Y) ® R) = 3. <0> Theorem 1.9 allows one, via [13], to compute (Krull) dimensions of tensor products of two A;—algebras for a large class of (not necessarily AF-domains) A;—algebras. The next example illustrate this fact. Example 3.5. Consider the following pullbacks R1 I k(X,Y)[Z](z) —> k(X) R2 —-^ 1 I k(X)[Z](z) —> k(X,Y k I --+ k(X) Clearly, dimi?i = dimi?2 = 1 and dim^i?! = dim^i^ = 2. Therefore none of R\ and R2 is an AF-domain. By Theorem 1.9 dim(i?i <g> R2) = 4. Finally, note that Wadsworth's formula fails since min{dimi?i+ t.d.(i?2), dimi?2 + t.d.(i?i)} = 3.<0> The next example shows that a combination of Theorem 1.9 and Theorem 3.2 allows one to compute dim(i?i ® R2) for more general A;—algebras. Example 3.6. Consider the pullback i?i —> I k I —> k(X) Ri is a one-dimensional pseudo-valuation domain with dim^i?! = 2. Clearly, R\ is not an AF-domain. By Theorem 1.9 dim(i?i ® R\) = 3. Consider now the pullback R2 —^ I k(X,Y,Z)[T\iT) Ri I —^ k(X,Y,Z) We have dimi?2 = 2 and dinii,i?2 = 4. The second pullback does not satisfy conditions of Theorem 1.9. Applying Theorem 3.2, we get dim(i?i <g> R2) = max{htMi[4] + dim(A;(8) R2), htM2[2] + dim(i?i ®i?i)} = max{2 + 2,2 + 3} = 5. <} 15 The next example shows that Corollary 3.3 enables us to construct an example of an integral domain R which is not an AF-domain while R <g> R is a Jaffard ring. Example 3.7. Consider the pullback R —> I k(X) I k(XX)[Z](z) —> k(X,Y) dimi? = 1 and dim^i? = 2. Then R is not an AF-domain. By Corollary 3.3 dim(i?(x) R) = dimv(R ® R) = 5 since t.d.(k(X, Y) : k(Xj) < t.d.(R). 0 References [1] D.F. Anderson, A. Bouvier, D.E. Dobbs, M. Fontana and S. Kabbaj, On Jaffard domains, Expo. Math. 6 (1988) 145-175. [2] J. T. Arnold, On the dimension theory of overrings of an integral domain, Trans. Amer. Math. Soc. 138 (1969) 313-326. [3] J. T. Arnold and R. 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