JOURNAL OF ALGEBRA 181, 803᎐835Ž. 1996 ARTICLE NO. 0147
On the Class Group and the Local Class Group of a Pullback
Marco FontanaU
Dipartimento di Matematica, Uni¨ersita` degli Studi di Roma Tre, Rome, Italy
View metadata, citation and similar papers at core.ac.ukand brought to you by CORE provided by Elsevier - Publisher Connector Stefania GabelliU
Dipartimento di Matematica, Uni¨ersita` di Roma ‘‘La Sapienza’’, Rome, Italy
Communicated by Richard G. Swan
Received October 3, 1994
0. INTRODUCTION AND PRELIMINARY RESULTS
Let M be a nonzero maximal ideal of an integral domain T, k the residue field TrM, : T ª k the natural homomorphism, and D a proper subring of k. In this paper, we are interested in deepening the study of the Ž.fractional ideals and of the class group of the integral domain R arising from the following pullback of canonical homomorphisms:
R [ y1 Ž.DD6
6 6 Ž.I
6 TksTrM
More precisely, we compare the Picard group, the class group, and the local class group of R with those of the integral domains D and T.We recall that as inwx Bo2 the class group of an integral domain A, CŽ.A ,is the group of t-invertibleŽ. fractional t-ideals of A modulo the principal ideals. The class group contains the Picard group, PicŽ.A , and, as inwx Bo1 ,
U Partially supported by the research funds of Ministero dell’Uni¨ersita` e della Ricerca Scientifica e Tecnologica and by Grant 9300856.CT01 of Consiglio Nazionale delle Ricerche.
803
0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 804 FONTANA AND GABELLI we denote by GŽ.A s C Ž.A rPic Ž.A the local class group of A Žmore details are given in Section 2. . The interest in these groups is due to the fact that the group-theoretic properties of CŽ.A and G Ž.A often give useful information about the divisibility properties of the integral domain A. For instance, it is well known that when A is a Krull domainŽ respec- tively, a Prufer¨ domain.Ž.C A is the usual divisor class groupŽ respectively, ideal class group. of A. Furthermore, if A is a Krull domain, then CŽ.As0 Ž respectively, G Ž.A s 0 . if and only if A is factorial Ž respec- tively, locally factorial.Ž. ; if A is a Prufer¨ domain, then G A s 0, since PicŽ.A s C Ž.A , and C Ž.A s 0 if and only if A is Bezout´ wx Bo1, Bo2 . A general reference to CŽ.R and G Ž.R iswx A3 . When considering a diagram of type Ž.I ,ifT is an integral domain of the form T s k q M, then R s D q M. This construction has been exten- sively studied by several authors for its use in constructing examples with prescribed propertiesŽ cf.wx G0 , G, BG, BR, CMZ. ; in particular, the class group of a D q M has been investigated inwx AR . Section 1 contains the basic technical facts about the relations among several distinguished classes of fractional ideals of R, D, and T. The main part of Section 2 is devoted to the computation of PicŽ.R , C Ž.R , and GŽ.R. In particular, we give several equivalent conditions for the existence of a well defined canonical map from CŽ.D to C Ž.ŽR Theorem 2.3 . . The fact that, for a general diagram of type Ž.I , it is not possible to define a map from CŽ.D to C Ž.R makes a substantial difference for the case R s D q M. As a matter of fact, in this last case R is a faithfully flat D-moduleŽ and thus there exists a canonical group homomorphism from CŽ.Dto C ŽD q M ... One of the main results of this section is Theorem 2.5, in which we prove that, under a mild hypothesis on the group of units of T which turns out to be equivalent to the existence of a canonical homomorphism from CŽ.D to C Ž.R ,if kis the quotient field of D, then there exists a diagram of natural group homomorphisms
000
6 6 6
06 PicŽ.D 6 Pic Ž.R 6 Pic Ž.T 6 0
6 6 6
0 6 CŽ.D 6 C Ž.R 6 C Ž.T
6 6 6
0 6 GŽ.D 6 G Ž.R 6 G Ž.T
6 6 6 000 THE CLASS GROUP OF A PULLBACK 805 which has exact columns and rows and in which the first row splits. We obtain again in particularwx AR, Theorem 3.3 . In Section 3, we deepen the study of the group homomorphism CŽ.R ª CŽ.T, which is known to be not surjective in general, even in the case when R s D q M and R is quasi-localwx AR, Example 3.4 . In particular, we give several sufficient conditions for the surjectivity of CŽ.R ª C Ž.T Ž.Propositions 3.1 and 3.7 . Furthermore, we apply the results of this section and of the subsequent Section 4 to give a negative answer to a question posed by Anderson and Ryckaertwx AR, Question 3.10 . As a matter of fact, Example 4.3 shows that it is possible for the map CŽ.R ª C Ž.T to be surjective with PicŽ.T / C Ž.T . The remainder of Section 4 is devoted to the transfer in pullback diagrams of type Ž.I of the PVMD property and of related propertiesŽ. e.g., Bezout,´ GCD . We thank Evan Houston for several stimulating conversations on some topics treated in this paper and the referee for his careful reading.
Notation. In this paper, we will denote by ‘‘: ’’ the set theoretical inclusion and by ‘‘; ’’ the proper inclusion.
We recall that, in a pullback of the previous type Ž.I the ideal M is the conductor of R ¨ T, that is, M s Ž.R : T . Therefore, for each Q g X SpecŽ.R such that Q W M, there exists a unique Q g SpecŽ.T such that X X X Ž. QlRsQ; moreover, RQQs T . In particular, for each N g Max T , X X X N/M, we have RN l RNs T . Furthermore, if we assume that k is the Ž quotient field of D, then it is easy to see that RMMs T for the general properties concerning the pullback constructions cf.wx F. .
XX LEMMA 0.1. Gi¨en a pullback diagram of type Ž.I , the setÄ N l R : N X gMaxŽ.T , N / M4Ä coincides with the set N g MaxŽ.R : N W M4.
Proof. This is an easy consequence of the fact that SpecŽ.R is homeo- Ž. Ž . morphic to the topological amalgamated sum Spec T @ SpecŽk.Spec D wF, Theorem 1.4x .
For each prime ideal P of an integral domain A, we denote by FP the localizing system Ä4I : I ideal of A, I P . It is well known that the ring of fractions A [ ÄŽ.A : I : I F 4of A with respect to the localizing FP D g P system FPPcoincides with A Žfor the general properties concerning localizing systems cf.wx B, Chap. 2, Sect. 2, Ex. 17᎐25; S. . In the following we will consider with particular interest the pullback diagrams of type Ž.I in which k coincides with the field of fractions of D. When this situation occurs, we will say briefly that we are considering a U pullback diagram of type ŽI .. 806 FONTANA AND GABELLI
Ž U . LEMMA 0.2. Gi¨en a pullback diagram of type I , then T s RF , where X Ä X Ž.4 F[FFN l R :NgMax T . Proof. We note that X X X X T s FÄ4TNN: N g MaxŽ.T s FÄR lR: N g MaxŽ.T 4 X Ä4RX:NMaxŽ.T R . sFFNlRgs F U LEMMA 0.3. Gi¨en a pullback of type ŽI ., then T is R-flat. Proof. The statement follows fromwx Fo, Lemma 6.5 , since we have X Ž. X already noted that, in the present situation, for each N g Max T , TN s X Ž . RN l R cf. alsowx R . It can also be viewed as a corollary of Lemma 0.2 and of the fact that T is R-flat if and only if T s RF for some localizing system F of R with the property that when I g F then IT s T wFP, X Ž. Ä X Remarque 1.2 ax . As a matter of fact, when I g F [ F FN R : N g XX l MaxŽ.T 4, then I N l R for each N g MaxŽ.T , and thus IT s T. U XX LEMMA 0.4. Gi¨en a pullback of type ŽI ., let S [ R _ DÄ N l R : N gMaxŽ.T 4. Then the following statements are equi¨alent: 1 Ž.i TsSRy ; Ž.ii if Q g Spec ŽR . and Q > M, then Q l S / л. Proof. Ž.i « Žii . . Let Q g SpecŽ.R .IfQ>M, then QT s T since M is maximal in T, whence Q l S / л. Ž.ii « Ž.i . By Lemma 0.2, it is sufficient to show that if Q is a prime XX X ideal of R and Q : DÄ N l R : N g MaxŽ.T 4, then Q : N l R for X some N g MaxŽ.T wG,Ž. 4.7x . Since Q l S s л, then Q r M; hence, by XX Lemma 0.1, Q : N l T for some N g MaxŽ.T . X COROLLARY 0.5. With the notation of Lemma 0.4, if M DÄ N lR: X X 1 NgMaxŽ.T , N / Me4Ž..g., if T is quasi-semilocal , then T s SRy. Proof. If T / Sy1 R, then by Lemma 0.4 there exists a prime ideal Q of XX X Rsuch that M ; Q : DÄ N l R : N g MaxŽ.T , N / M4and this fact leads to a contradiction. Let A be an integral domain and H a fractional ideal of A. As usual we ŽŽ .. denote by H¨ the fractional ideal A : A : H of A and we say that H is a di¨isorial ideal of A if H s H¨ . A divisorial ideal H of A is called ¨-finite if there exists a finitely generated fractional ideal K of A such that X H s K and it is called ¨-in¨ertible if Ž HH . s A for some divisorial ideal XX¨¨ Hof A; in this case H s Ž.A : H . It is well known that a divisorial ideal Iof A is ¨-invertible if and only if Ž.I : I s A wxG, Proposition 34.2 . Given Ä a fractional ideal H of A, we denote by Ht the fractional ideal D J¨ : J is a finitely generated fractional ideal of A, J : H4 and we say that H is a THE CLASS GROUP OF A PULLBACK 807
t-ideal if H s Ht. It is clear that H is a t-ideal if and only if J¨ : H for each finitely generated fractional ideal J of A such that J : H and that
H : Ht : H¨ ; therefore every divisorial ideal is a t-ideal. Moreover, if Ž H¨¨s J for some finitely generated fractional ideal J : H e.g. if H is . finitely generated then Ht s H¨ .A t-ideal H is called t-in¨ertible if Ž X . XXŽ. HH t s A for some t-ideal H of A; in this case also H s A : H wJ, Chap. 1, Sect. 4x . It is well known that a t-invertible t-ideal is always divisorial and ¨-invertiblewx J, loc. cit. . Therefore, when H is a t-invertible t-ideal, then ŽŽ ..ŽŽ .. HA:H t sHA:H ¨ sA. Conversely, a divisorial ¨-invertible ideal Hof A is t-invertible if and only if H and Ž.A : H are ¨-finitew J, Sect. 4.1, Corollaire 1 and Sect. 4.4, Theoreme´` 8x . The properties contained in the following two propositions are folklore; we recall them for convenience of the reader.
PROPOSITION 0.6. Let B be a flat o¨erring of an integral domain A and H be a finitely generated fractional ideal of A, then:
Ž.a ŽA:HB .s ŽB:HB .; Ž. Ž . Ž . b HB ¨¨¨s HB . Furthermore, assume thatŽ. A : His¨-finite. Then: Ž. Ž . c HB ¨¨s HB. Proof. Ž.a Seewx B, Chap. 2, Sect. 2, N. 11, p. 41 . Ž . Ž Ž .. Ž Ž . . Ž Ž . b HB : HB¨ s A: A:HB:B:A:HBsB:B:HB Ž. Ž. Ž . sHB ¨¨¨, thus, HB s HB¨. Ž. Ž . c Let A : H s K¨ , with K finitely generated. Then
HB¨¨sŽ.A:Ž.ŽA:HBsA:KB .Ž.Ž.sA:KBsB:KB B:KB B : KB B:KB B: A:HB sŽ.Ž.¨s Ž. Ž¨¨ .¨s Ž .sŽ. Ž .
sŽ.B:Ž.Ž.B:HB s HB ¨.
PROPOSITION 0.7. Let B be a flat o¨erring of an integral domain A. Ž.a If L is an integral t-ideal of B, then L l A is an Ž integral . t-ideal of A. Ž.b If H is a t-in¨ertible t-ideal Ž respecti¨ely, an in¨ertible ideal . of A, then HB is a t-in¨ertible t-idealŽ. respecti¨ely, an in¨ertible ideal of B. Proof. Ž.a Let J be a finitely generated ideal of A such that J : L l A. Then JB is a finitely generated ideal of B and JB : Ž.L l AB:L. Ž. Ž. Since L is a t-ideal, JB ¨¨: L and thus JB: JB ¨: L. Therefore, J¨¨:JBlA:LlA. 808 FONTANA AND GABELLI
Ž.bIfHis a t-invertible t-ideal of A, then H and Ž.A : H are Ž Ž .. Ž Ž .. ¨-finite divisorial ideals of A; thus HA:H ¨ sHA:H t sA. Fur- thermore, by Proposition 0.6, HB is a divisorial ideal of B Žhence a t-ideal of B. and also
B s Ž.Ž.Ž.HAŽ.:HB¨¨sŽ.HA Ž.:HB¨sHB Ž B : HB ..
Hence the conclusion follows from the fact recalled above that a ¨-invert- ible divisorial ideal L of B is t-invertible if and only if L and Ž.B : L are ¨-finite. If H is an invertible ideal of A, then mutatis mutandis we conclude that HB is an invertible ideal of B.
We note that the ‘‘invertible part’’ of Proposition 0.7Ž. b does not need the flatness hypothesis.
1. SOME PROPERTIES OF IDEALS IN PULLBACK CONSTRUCTIONS
U Given a pullback diagram of type ŽI ., ring-theoretic properties of R are often determined by those of D and T. When k is a retract of T, i.e., when T s k q M, the integral domain R is of the form D q M. The ideal-theoretic properties of this ring have been extensively studied by
Gilmerwx G0 Ž cf. alsowx BG. . More general constructions have been studied systematically inwx BR, CMZ, F and C . This section is devoted to deepening the study of the relations among some distinguished classes of fractional ideals of R, D, and T Žmainly, divisorial, t-invertible, ¨-invertible, and invertible ideals. . In particular, we investigate the behaviour of the principal ideal-theoretic operations under U the canonical homomorphisms of the diagram ŽI .. Some of the state- ments of this section improve previous works by Anderson and Ryckaert wxAR, Proposition 2.4 and Nour El Abidine wx NeA2, Lemma 2.10 .
PROPOSITION 1.1. Consider a pullback diagram of type Ž.I , and let H be a fractional ideal of R. Then HT s T if and only if M ; H : T.
In order to prove the preceding statement, we need some preliminary results. Let T, M, k, and D be as usual and suppose that k is the quotient field of D. We denote by RMŽ.the CPI Ž complete pre-image . extension of Rwith respect to M wxBS, Definition 2.1 ; i.e., RMŽ.is defined by the THE CLASS GROUP OF A PULLBACK 809 following pullback diagram:
1 6 RMŽ.[y Ž.DRrM,D
U 6 6 Ž.IM
6 RMMsTksRMMrMR
Ž.where is the canonical projection . Ž. LEMMA 1.2. If H is a nonzero ideal of R M , then HRMMs R if and only if H > MRM . Proof. The statement is an easy consequence of the fact that every Ž. Ž. ideal of RM is comparable with MRM s MR M wxBS, Proposition 2.3 . U LEMMA 1.3. Gi¨en a pullback diagram of type ŽI ., let H be a fractional ideal of R. Then H s HRŽ. M l HT. Proof. First suppose that H : R. We have
H s FÄ4HRN: N g MaxŽ.R
sŽ.FÄ4HRN: N g MaxŽ.R , N W M
FFŽ.Ä4HRN: N g MaxŽ.R , N > M Ž.1.3.1 By Lemmas 0.1, 0.2, and 0.3, we know that:
FÄ4HRN: N g MaxŽ.R , N W M X X sFÄ4HTNX : N g MaxŽ.T , N / M ;
RMMsTsRMŽ.MRŽ. M ;
Ä4NgMaxŽ.R : N > M s Ä4N l R : N g MaxŽ.RMŽ.;
RNlRsRMŽ.N, for each N g MaxŽ.RMŽ.. ThenŽ. 1.3.1 implies that:
X X H s Ž.FÄ4HTNX : N g MaxŽ.T , N / M l HRŽ. M X X sŽ.FÄ4HTNMX : N g MaxŽ.T , N / M l HR l HRŽ. M sHT l HRŽ. M .
If H is a fractional ideal of R, then H˜[ dH : R for some d g R _ Ä40; hence, by the above argument, dH s H˜˜s HT l HR ˜Ž. M s dHT Ž l HRŽ.. M . Thus H s HT l HRŽ. M . 810 FONTANA AND GABELLI
Proof of Proposition 1.1. Since M is a maximal ideal of T, it is obvious that if M ; H : T, then HT s T. Conversely, suppose that HT s T. Without loss of generality, we can assume that k is the quotient field of D. ŽAs a matter of fact, if kXXis the quotient field of D and T [ y1ŽkX., XX then it is easy to see that HT s T if and only if HT s T .. Then H : T Ž. and HRMMMMs HT s T s R . Therefore, by Lemma 1.2, HR M > MRM. By Lemma 1.3, H s HT l HRŽ. M , whence H s T l HRŽ. M . In conclu- Ž. sion, we have H s T l HR M > T l MRM s M.
COROLLARY 1.4. Consider a pullback diagram of type Ž.I and let H be an ideal of R. Then H is incomparable with M if and only if H M and H is contained in at least one maximal ideal of R not containing M.
Proof. This is a straightforward consequence of Proposition 1.1 and Lemma 0.1.
LEMMA 1.5. Consider a pullback diagram of type Ž.I and let H and K be two nonzero fractional ideals of R, then:
Ž.a If H properly contains M and K ; T, then Ž K : H .: T. Ž. Ž. b M;H¨ m R:H ;T; and, in this situation, H¨ s ŽŽR:R:TH ... Ž. Ž. c H¨;TmM; R:H. Ž. Ž. Ž. d HMmM; HqM¨ ; and, in this situation, R :T H s Ž Ž .. Ž Ž .. R :T H q M s R : H q M .
Proof. Ž.a Suppose that xH : K. Set x [ arb with a, b g R and b / 0, so that aH : bK. Since HT s T, we have aT s aHT : bKT : bT, thus arb g T. Ž. Ž . Ž . b We note that R : H s T implies that H¨ s R : T s M. Ž.Ž. Ž. Therefore, if M ; H¨ then, by a , R : H : T, whence R : H ; T. Ž. Ž. Conversely, suppose that T > R : H , then M s R : T : H¨ and M / ŽŽ.Ž.. H¨ otherwise, R : H s R : M = T . Ž. Ž. Ž. Ž. c Let K [ R : H . Then K s K¨¨, and, by b , M ; K s R : H Ž. if and only if H¨ s R : K ; T. Ž. Ž. d It is straightforward that H M if and only if M ; H q M ¨ . Ž.ŽŽ.ŽŽ.. Moreover, it is clear that R :TTH s R : H q M : R : H q M .If Ž. Ž. M;HqM¨ , then the opposite inclusion follows from b .
PROPOSITION 1.6. Consider a pullback diagram of type Ž.I . Ž.a Let I and J be two nonzero fractional ideals of D. Then y1Ž.I and y1 Ž.J are two fractional ideals of R contained in TŽ and properly containing THE CLASS GROUP OF A PULLBACK 811
M .; moreo¨er
1 1 1 1 1 1 y ŽI q J .s y Ž.I q y Ž.J and y ŽIJ .s y Ž.I y Ž.J .
Ž.b Let H and K be two fractional ideals of R contained in T. Then
ŽH q K .s Ž.H q Ž.K and ŽHK .s Ž.Ž.H K ; moreo¨er, if M ; H, then Ž.Ž.Ž.H l K s H l K . Suppose, in addition, that k is the quotient field of D. Ž.c If H is a fractional ideal of R with H ; T, then the following statements are equi¨alent: Ž.i ŽH . is a nonzero fractional ideal of D; Ž.Ž . ii H q M ¨ ; T and H M; Ž. Ž . iii M ; R :T H and H M. Proof. Ž.a Let d g D be a nonzero element such that dI : D. Since 1 :Tªkis surjective and KerŽ. s M, then M ; yŽ.dD and 1 1 1 1 1 y Ž.dD yŽ.I : yŽ.dI : yŽ.D s R. Hence yŽ.I is a fractional 1 1 1 ideal of R, and necessarily M ; y Ž.I ; T. Since yŽ.I , yŽ.J , and 1 1 1 1 y Ž.IqJcontain M it is obvious that yŽ.I q yŽ.J s yŽ.I q J . Since M is a maximal ideal of T, we have:
1 1 1 1 y Ž.I y Ž.JTsy Ž.ITy Ž.JTsT. 1 1 1 1 Hence yŽ.I yŽ.J > M ŽProposition 1.1. , and thus yŽ.I yŽ.J s y1 Ž.IJ . Ž.b The first two equalities hold since is a ring homomorphism. Assume that H > M, and let x g H, y g K with Ž.x s Ž.y . Then ygxqM:HqMsH, whence Ž.H l Ž.K : ŽH l K .. The op- posite inclusion holds in general. Ž.ci Ž.« Ž.ii Let x g D, x / 0, such that x ŽH .: D. Then, by Ž. a , 1 1 1 1 1 Rsy Ž.D=yŽŽ..xHsyŽ.xD yŽŽH ..syŽ.ŽxD H q M ., ŽŽ ..ŽŽ ..y1Ž. Ž . hence R : H q M ¨ s R : H q M = xD > M s R : T , thus Ž.Ž Ž.. HqM¨ ;TLemma 1.5 c . Ž. Ž . Ž. Ž Ž .. ii « iii . Since M ; H q M ¨ ; T, then M ; R : H q M s Ž Ž .. Ž . Ž Ž .. R :TTH q M s R : H Lemma 1.5 c and d . Ž.iii « Ž.i . Let t g T _ M be such that tH : R and let Ž.t s arb with a, b g D, a / 0, and b / 0. Then, D = Ž.tH s Ž.Ž.t H s Žarb .Ž.H. Hence a Ž.H : bD : D, and Ž.H is a fractional ideal of D. Since H M, Ž.H / Ž.0. 812 FONTANA AND GABELLI
We note that, in Proposition 1.6Ž. c , the implications Ž. i « Ž.ii « Žiii . hold in general, without supposing that k is the quotient field of D.
COROLLARY 1.7. Consider a diagram of type Ž.I , with the notation of the preceding proposition,
1 Ž.a if I is an in¨ertible ideal of D, then y Ž.Iisanin¨ertible ideal of R; Ž.b if I is a fractional ideal of D, then I is finitely generated if and only if y1 Ž.I is finitely generated.
Proof. Ž.a Let J be a fractional ideal of D such that IJ s D. Then 1 1 1 1 Rsy Ž.DsyŽ.IJ s yŽ.I yŽ.J . Ž. b Let I [ xD1 qиии qxDniwith x g k,1FiFn. Since xD iis y1 an invertible ideal of D, then, byŽ. a , Ž.xDi is an invertible ideal and therefore a finitely generated ideal of R for 1 F i F n. Therefore,
y1 y1 y1 y1 Ž.I s ŽxD1 qиии qxDn .Ž.Ž.s xD1 qиии q xDn is a finitely generated fractional ideal of R. The converse is clear since 1 is surjective and hence I s ŽŽ..y I . U PROPOSITION 1.8. Consider a diagram of type ŽI .. Ž.a If I and J are two nonzero fractional ideals of D, then 1 1 1 Ž.a1 yŽŽI : J ..s Žy Ž.I : yŽ..J ; Ž.Žy1Ž.. y1Ž. a2 I ¨¨s I ; Ž.Žy1Ž.. y1Ž. a3 I tts I . Ž.b Let H be a nonzero fractional ideal of R and suppose that H M Ž. and H q M ¨ ; T. Then Ž . ŽŽ .. Ž Ž .. b1 R :T H s D : H ; Ž .ŽŽ .. ŽŽ .. Ž . ŽŽ .. b2 H¨¨s H¨ and, when H > M, H¨s H ¨; Ž . Ž Ž .. Ž Ž .. Ž . Ž Ž .. b3 Htts Ht and, when H > M, Hts H t. Proof. First of all we note that y1Ž.I and y1Ž.J are two fractional ideals of R and Ž.H is a fractional ideal of D ŽProposition 1.6 Ž a and c .. .
1 1 1 Ž.a1 Clearly ŽyŽ.I : yŽ..J : yŽŽI : J ... Conversely, if x g 1 y ŽŽI : J .., then ŽxJ . :I; hence by Proposition 1.6Ž. a ,
1 1 1 1 1 xy Ž.J : y Ž. Ž.xJ :y Ž.I, i.e., x g Ž.y Ž.I : y Ž.J .
1 1 Ž.b1 By Ž. a1 and Lemma 1.5Ž. d , yŽŽD : ŽH ...s Ž R : y Ž ŽH ... Ž Ž .. Ž . ŽŽ .. Ž Ž .. s R : H q M s R :TTH . Hence R : H s D : H . THE CLASS GROUP OF A PULLBACK 813
Ž.a2 is an easy consequence of Ž. a1 :
y1 y1 y1 Ž.I¨s Ž.Ž.D : ŽD : I . s Ž.R : Ž.Ž.D : I R:R:y1Iy1I. sŽ.Ž.Ž.Ž.s Ž.¨
Ž. Ž. Žy1Ž Ž ... y1 ŽŽ Ž .. . Ž . b2 By a2 , H¨¨¨¨: H s H, hence H : ŽŽ .. ŽŽ .. ŽŽ .. H¨¨and we have H ¨s H ¨. Moreover, if M ; H then, by Ž. Ž Ž .. Ž. Lemma 1.5 b , H¨ s R :TTR : H . By applying b1 , we deduce that
Ž.H¨sŽ.Ž.Ž.R:TT Ž.R:HsŽ.D: Ž.R:TH
sŽ.D:Ž.Ž.D:Ž.Hs Ž.H¨.
Ž. Žy1Ž.. y1Ž. a3 We begin by showing that I tt: I . Let x g Žy1Ž.. I t _M. Then there exists a finitely generated fractional ideal H of y1 Ž. Ž . Rsuch that x g H¨ and H : I . Since H is a finitely generated Ž. Ž. Ž . ŽŽ.. fractional ideal of D, by b2 we obtain x g H¨¨: H : It, y1Ž. whence x g It . y1 Ž. Ž. Conversely, let x g It . Then x g J¨ , for some finitely gener- 1 ated fractional ideal J of D, J : I. Since y Ž.J is a finitely generated fractional ideal of R ŽŽ..Ž.Corollary 1.7 b , by applying a2 to the ideal J we y1 Ž. Žy1Ž.. Žy1Ž.. obtain x g J¨¨s J : I t. Ž. Ž. Žy1Ž Ž ... y1 ŽŽ Ž .. . b3 By a3 we deduce that Httt: HsH. Ž . ŽŽ .. ŽŽ .. ŽŽ .. Hence Htt: H , and thus H ttts H . ŽŽ .. Assume that M ; H. Let x g H t; hence x g J¨ , for some finitely 1 generated fractional ideal J of D with J : Ž.H . Since yŽ.J is a finitely generated fractional ideal of R ŽŽ..Ž.Corollary 1.7 b , then by a2 applied to J we have
y1 xD y1 J y1 J y1 H H , Ž.: Ž.¨s Ž. Ž.¨: Ž.Ž. Ž.ts t Ž. thus x g Ht . COROLLARY 1.9. Let P be one of the following properties for an ideal of a domain Ž.a fractional; Ž.b di¨isorial; Ž.c t-ideal; Ž.d in¨ertible; Ž.e ¨-in¨ertible di¨isorial; Ž.f t-in¨ertible t-ideal. 814 FONTANA AND GABELLI
U 1 Consider a diagram of type ŽI .. The map I ¬ y Ž.I establishes a one-to- one correspondence between the set of all the nonzero ideals I of D with the property P and the set of all the ideals H of R with the property P such that
M ; H : H¨ ; T. 1 Proof. For every nonzero fractional ideal I of D, we have M ; y Ž.I 1 ;T and Ž y Ž..I sI. Therefore, by Proposition 1.6Ž. c , we have Žy1 Ž.. Ž. I ¨¨;T. Conversely, if M ; H : H ; T, then H is a nonzero 1 fractional ideal of D ŽŽProposition 1.6 c.. and yŽŽH ..sHqMsH. Ž.Ž Ž Moreover, M ; H¨ ; T if and only if M ; R : H ; T Lemma 1.5 b and c.. . The conclusion now follows from Proposition 1.6 and 1.8. The following result is a straightforward consequence of Corollary 1.9 and Proposition 1.1. U COROLLARY 1.10. Consider a diagram of type ŽI ., and let H be a fractional ideal of R such that H¨ ; T. Then HT s T if and only if H s y1 Ž.I for some nonzero fractional ideal I of D. U COROLLARY 1.11. Consider a diagram of type ŽI . and suppose that H is adi¨isorial ideal of R such that H M and H ; R. If H is a ¨-in¨ertible Ž.ŽŽ.. respecti¨ely t-in¨ertible, in¨ertible ideal of R, then Hisa¨ ¨-in¨ertible Ž.respecti¨ely t-in¨ertible, in¨ertible ideal of D. Proof. We start by noting that Ž.H is a nonzero integral ideal of D ŽŽ..Proposition 1.6 c . Let r g H _ M be such that rR Ž.Ž.:H :HR:H :R and rRŽ.:H M Žbecause r g rR Ž.:H and r f M .. We prove the statement in case of ¨-invertibility; mutatis mutandis, the proof is valid for Ž.Ž. ŽŽ.. the other cases. Set L [ rH R : H : R . Then L¨¨s rH R : H s ŽŽ .. rHR:H ¨ srR and, by Propositions 1.8 and 1.6, we have
rD rR L L L rH R : H Ž.s Ž.s Ž.¨¨s Ž. Ž.¨s Ž. Ž.¨s Ž. Ž Ž . .¨ sŽ.Ž.HŽ.rR Ž:H .¨.
Ž .y1 Ž Ž .. Ž Ž .. Hence r H ¨¨is ¨-invertible, and so H is ¨-invertible. U Given a diagram of type ŽI ., our next goal is to study the transfer of the properties of invertibility, t-invertibility, and ¨-invertibility from D and T to R. U LEMMA 1.12. Consider a diagram of type ŽI .. Let H be a finitely generated fractional ideal of R. Suppose that Ž.H is a nonzero fractional ideal of D. IfŽŽ..Ž D : H and T : HT . are ¨-finite, then Ž R : His .¨-finite. Proof. Since H is finitely generated, Ž.Ž.ŽT : HT s R : HT Lemma 0.3 and Proposition 0.6Ž.. a . Let J Žrespectively, K .be a finitely generated THE CLASS GROUP OF A PULLBACK 815
fractional ideal of D ŽŽ..respectively, of R contained in R : H such that ŽŽ..Ž Ž.Ž .Ž.. J¨¨sD:Hrespectively, KT s T : HT s R : HT. By Corol- lary 1.7Ž. b , y1Ž.J is finitely generated and, by Propositions 1.6Ž. c and 1.8, y1 Ž. y1Ž. y1ŽŽ Ž ... Ž y1 Ž Ž ... Ž . J : J¨ s D : H s R : H:R:H. Thus Žy1 Ž. . Ž . JqK¨:R:H. Žy1Ž. . Ž . y1Ž. We claim that J q K ¨ s R : H . Since J q K > M, then, 1 by Lemma 1.5Ž. b , we have ŽR : ŽyŽ.J q K ..; T. 1 Let x g ŽŽR : yŽ.J q K ... Then
xK : R « xKT : T
« x g Ž.T : KT s Ž.T : Ž.KT ¨s Ž.T : Ž. Ž.R : HT «xRŽ.:H :T; 1 xyŽ.J:R« Ž.xJ:D « Ž.xg ŽD:J .s ŽD:J¨ .sŽ. ŽH .¨ «Ž.xDŽ.: ŽH .:D. Moreover, by Proposition 1.6Ž. b , we have
Ž.xRŽ.Ž.Ž.:H H s ŽxR:HH .s Ž.Ž.Ž.x ŽR:HH .:xD. Hence
Ž.Ž.Ž.xRŽ:H .: Ž.xD: ŽH .s Ž.xD: ŽH .:D, and we have
y1 xRŽ.:H : Ž.DsR, i.e., x g H¨. ŽŽy1Ž. .. The previous argument shows that R : J q K : H¨ . Therefore ŽŽ..Žy1 . ŽŽ..Žy1 . JqK¨¨=R:Hand hence J q K s R : H . U PROPOSITION 1.13. Consider a pullback of type ŽI .and let H be a finitely generated fractional ideal of R with H ; T. Ž. Ž Ž .. Ž . a If H¨¨ and HT are fractional ¨-in¨ertible ideals, then H¨ is a fractional ¨-in¨ertible ideal. Ž.b If ŽH . and HT are fractional in¨ertible ideals, then H is a fractional in¨ertible ideal. Ž. Ž Ž .. Ž . c If H¨¨ and HT are fractional t-in¨ertible ideals, then H¨ is a fractional t-in¨ertible ideal. Ž. Ž Ž .. Proof. a Since H ¨ is a nonzero fractional ideal of D, then Ž.Ž Ž..Ž.Ž. HMand H q M ¨¨; T Proposition 1.6 c . Since HT ¨s HT ¨ 816 FONTANA AND GABELLI
Ž.Proposition 0.6 , then
H : HT HT:HT HT : HT Ž¨¨ .Ž: ¨ ¨ .Ž.Ž.:Ž. ¨¨¨ ¨
sŽ.Ž.Ž.HT ¨¨: HT s T ŽŽ.. the last equality holds because HT ¨ is a ¨-invertible ideal of T . Ž. Hence H¨¨: H : T. Therefore, by Proposition 1.8, we have
H:H H:H H:H Ž.Ž.Ž¨¨ .: Ž.Ž. ¨ ¨:Ž.Ž. Ž.¨ Ž. Ž. ¨¨
sŽ.Ž.Ž.Ž.H¨¨: Ž.HsD,
ŽŽŽ... the last equality holds because H ¨ is a ¨-invertible ideal of D . Ž.y1Ž. Ž . Hence R : H¨¨: H : D s R, thus H¨¨: H s R and this fact implies that H¨ is a ¨-invertible ideal of R. Ž.b Since ŽH .is an invertible ideal of D, by Propositions 1.6Ž. c and 1.8, we deduce that
D s Ž.HDŽ.: Ž.Hs Ž.HŽ.Ž. ŽR:TTH .sHR Ž:H . :Ž.HRŽ.:H :D and ŽŽHR:H ..sD. On the other hand, since H is finitely generated and HT is invertible, then T s HTŽ.Ž. T : HT s HR:HT. Thus M ; HRŽ.Ž:H Proposition 1.1 . , and therefore we can conclude that HRŽ.:H sR. Ž. Ž. c By part a we know that H¨ is ¨-invertible. Since H is finitely generated, in order to conclude that H¨ is a t-invertible t-ideal, it remains Ž.Ž. ŽŽ..Ž. to show that R : H¨¨s R : H is ¨-finite. Since H and HT ¨are t-invertible ideals, then ŽŽ..ŽD : H and T : HT .are ¨-finite, and so the conclusion follows by applying Lemma 1.12.
2. THE CLASS GROUP OF R
For an integral domain A, we denote by UŽ.A by group of units of A and by TŽ.ŽA respectively Inv Ž.A , Prin Ž..A the group of all the t-invertible t-idealsŽ. respectively invertible ideals, principal ideals of A. Let CŽA.[ TŽ.ArPrin Ž.ŽA respectively PicŽ.A [ Inv Ž.A rPrin Ž.A , G Ž.A [ TŽ.ArInv Ž..A be the class groupŽ respectively the Picard group, the local class groupwx Bo. of A. The aim of this section is to study the relationships among the Picard groups, class groups, and local class groups of the domains occurring in a THE CLASS GROUP OF A PULLBACK 817
U pullback diagram of type ŽI .. In particular, we prove that the main results ofwx AR, Section 3 hold in a more general setting. U Given a diagram of type ŽI .Ž, by Proposition 1.6 a. and Corollary 1.9 we may define a group-homomorphism
1 ␣ : TŽ.D ª T Ž.R , I ¬ y Ž.I , which, by restrictionŽ. respectively, by passing to quotient-groups , deter- mines a group-homomorphism
X Y ␣ : InvŽ.D ª Inv Ž.R Ž.respectively ␣ : G Ž.D ª G Ž.R . By Proposition 0.7, we have a group-homomorphism
 : TŽ.R ª T Ž.T , H ¬ HT , which, by restrictionŽ. respectively, by passing to quotient-groups , defines a group-homomorphism
X Y  : InvŽ.R ª Inv Ž.T Ž.respectively,  : G Ž.R ª G Ž.T
Žcf.wx AR, Proposition 2.2. . U PROPOSITION 2.1. For a diagram of type ŽI ., the following sequences of group-homomorphisms
␣ 6  6 0 ª TŽ.D T Ž.R T Ž.T Ž.2.1.1 X X ␣6 6 0ªInvŽ.D Inv Ž.R Inv Ž.T Ž2.1.2 . Y Y ␣6 6 0ªGŽ.D G Ž.R G Ž.T Ž.2.1.3 are exact.
Proof. It is obvious that ␣ is injective. By Proposition 1.1, ImŽ.␣ : KerŽ. . Conversely, assume that H g T Ž.R and HT s T, then by Propo- Ž sition 1.1 M ; H and moreover H s H¨ ; T because T is not a t-invert- 1 ible t-ideal of R.. By Corollary 1.9, H s y Ž.I for some t-invertible t-ideal I of D, hence H g ImŽ.␣ . The proof of the exactness of Ž 2.1.2 . is analogous. The exactness ofŽ. 2.1.3 follows from the exactness ofŽ. 2.1.1 andŽ. 2.1.2 by standard techniques. We note that the extension to T of every principal fractional ideal of R is still principal. Hence the homomorphism  : TŽ.R ª T Ž.T induces a homomorphism  : CŽ.R ª C Ž.T , wxH ¬ wHT x, and, by restriction, a X homomorphism  : PicŽ.R ª Pic Ž.T . Our next goal is to relate CŽ.R Žrespectively Pic ŽR ..with C ŽD . Žrespectively Pic ŽD ... 818 FONTANA AND GABELLI
U LEMMA 2.2. Consider a diagram of type ŽI . and let H be a t-in¨ertible t-ideal of R. Then there exists a nonzero element z in the quotient field of R X X X X and a t-in¨ertible t-ideal H of R such that H M, H : R, and H s zH . Proof. Let x g R, x / 0, be an element such that xH : R and set Y Y Y H [ xH; clearly H is a t-invertible t-ideal of R.If H M, we are Y Y Y done. Suppose that H : M. Since H is t-invertible, then H is a Y divisorial ¨-finite ideal and H RQ is principal, for every maximal t-ideal Q Y Y of R containing H wxGa, Proposition 1.1 . Since H : M and M is a Ž. Y divisorial hence a t- ideal of R, then necessarily HRMMsyR for some Y Y ygHRM . Let K be a finitely generated ideal of R such that K s H . Y ¨ Then K : KRMMM: H R s yR . By the finiteness of K, we can find Y sgR_Msuch that sK : yR. Hence sH s sK s Ž.sK : yR and thus Y ¨¨ Ž.sryH :R. X Let H [ Ž.ŽsxryH and z [ yrsx .. In order to conclude we must show XXŽ. Ž.Y that H M. As a matter of fact, HRMMMs sry xHR s sryHR s Ž. sryyRMMMssR s R . By Lemma 2.2, we can define a group-homomorphism ␥ : CŽ.R ª C Ž.D , X as follows: for each class wxH g CŽ.R , let H g wxH be a t-invertible XX ␥ Ž. ŽŽ X .. t-ideal such that H : R and H M, and set wxH [ w H ¨ x. In order to verify that ␥ is well defined, we begin by noting that Ž.H X is a nonzero integral ideal of D ŽŽ..Proposition 1.6 c . By Corollary 1.11 we X Y know that ŽŽ H .. is a t-invertible ideal of D. Moreover, if H g wxH , YY¨ X Y H : R, and H M, we must show that wŽŽ H .. xws ŽŽ H .. x.We Y X ¨¨ have H s zH for some nonzero element z of the quotient field of R. Y X Since RMMs T s H T Ms zH T Ms zT Ms zR M, then z is a unit in RM; Y Y hence z s arb with a, b g R _ M. Therefore Ž.Žb H .Žs bH .s Ž X .Ž X.Ž.Ž X. ŽŽ Y.. Ž.ŽŽ Y.. bzH s aH s a H , and hence wH ¨¨xws b H x Ž.ŽŽ X.. ŽŽ X.. s wa H ¨¨xws H x. Finally, since is a group homomorphismŽ. Proposition 1.6 , it is straightforward to verify that ␥ : CŽ.R ª C Ž.D is a group homomorphism. By restricting to PicŽ.R it is easy to verify that ␥ defines a group X homomorphism ␥ [ PicŽ.R ª Pic Ž.D . XXX Set ␦ [ Ž.Ž.Ž.Ž.␥,  : C R ª C D [ C T and ␦ [ Ž␥ ,  .Ž.: Pic R ª PicŽ.D [ Pic Ž.T . U THEOREM 2.3. Consider a diagram of type ŽI .. The following statements are equi¨alent: U Ž.i the map ˜ : UŽ.T ª k rUŽ.D , u ¬ Ž.Ž.u U D is a surjecti¨e group homomorphism; U 1 Ž.ii for each x g k , yŽ.xD is a fractional principal ideal of R; Ž.iii the map ␦ : CŽ.R ª C Ž.D [ C Ž.T is an injecti¨e group homo- morphism; THE CLASS GROUP OF A PULLBACK 819
1 Ž.iv the map ␣ : CŽ.Ž.D ª C R , wxI ¬ wy Ž.Ix is a well defined group homomorphism.
U Proof. Ž.i « Žii . . Let x g k . Then there exist u g UŽ.T and ¨ g U ŽD . such that Ž.u s x¨. Since T s uT s uRT, then uR > M ŽProposition 1 1.1.Ž. . Moreover, uR s x¨D s xD, whence yŽ.xD s uR. U Ž.ii « Ž.i . Let x g k . By the hypothesis and by Proposition 1.1, it 1 follows that y Ž.xD s uR for some u g UŽ.T . Therefore, Ž.uR s Ž.uDsxD. Thus Ž.u ¨ s x for some ¨ g U ŽD .and hence ˜ Ž.u s xUŽ.D. Ž.ii « Žiii . . Let H be a t-invertible t-ideal of R such that H : R and Ž. ŽŽ .. HMLemma 2.2 . Suppose that wH ¨ xwxs D and wHT xwxs T . U Then there exist x g k and y in the quotient field of T, y / 0, such that ŽŽ .. y1 H¨ sxD and HT s yT. Since yHis a t-invertible t-ideal of R and 1 1 1 yHTy sT, then M ; yHy ;TŽ.ŽProposition 1.1 , whence yHy.is a t-invertible t-ideal of D Ž.Corollary 1.9 . Since ␥ is well defined, wŽ.yHy1 x ŽŽ . Žy1 . sw H¨ xwxwxsxD s D . Hence yHis a fractional nonzero princi- 1 1 1 pal ideal of D and yHy sy ŽŽyHy .. is a fractional principal ideal of 1 R. Therefore wxH s wyHy xwxsR. U 1 Ž.iii « Ž.ii . Assume that ␦ is injective. For each x g k , ␦ŽwyŽ.xD x. 1 sŽwxwxxD , T .Žs wxwxD , T ., whence yŽ.xD is a fractional principal ideal of R. Ž.ii « Živ . . Assuming Ž. ii , the conclusion follows easily from Corollary 1.9 and Proposition 1.6. U Ž.iv « Ž.ii . Let x g k , so that wxwxxD s D . Hence ␣ŽwxxD .s 1 1 1 wy Ž.xD xws yŽ.D xwxs R ; thus yŽ.xD is a principal fractional ideal of R.
U When the map ˜ : UŽ.T ª k rU ŽD .is surjective, ␣ : CŽ.Ž.D ª C R is well defined and then, by restriction, we can define a group-homomor- X phism ␣ : PicŽ.D ª Pic Ž.R . In this situation, by Proposition 2.1, it follows easily that the sequences of natural group homomorphisms
␣ 6  6 0 ª C Ž.D C Ž.R C Ž.T X X ␣6 6 0ªPicŽ.D Pic Ž.R Pic Ž.T are exact. It is known that, in general,  is not surjectivew AR, Example X 3.4x . We will show in the following Theorem 2.5 that  is always surjective, as in the D q M constructions. 820 FONTANA AND GABELLI
U COROLLARY 2.4. Consider a diagram of type ŽI . and suppose that the U map ˜ : UŽ.T ª k rUŽ.D is surjecti¨e. Then the following conditions are equi¨alent: Ž.i  is surjecti¨e;
␣ 6  6 Ž.ii 0 ª C ŽD .C ŽR .C ŽT .ª 0 is exact;
␣ 6  6 Ž.iii 0 ª C Ž.D C Ž.R C Ž.T ª 0 splits; Ž.iv ␦ : C ŽR .ª C ŽD .[ C Ž.T is an isomorphism. Proof. The following diagram with exact rows
6 ␣ 6  6
0 C Ž.D C Ž.R C Ž.T 556 ␦ 6 6 6 6 0CŽ.DC Ž.D[C Ž.TC Ž.T0 commutes. It is clear thatŽ. iv « Ž.iii « Ž.ii « Ž.i « Ž.ii . By the Five LemmaŽ. ii « Živ . . U THEOREM 2.5. Consider a diagram of type ŽI .. Then Ž.a C ŽD .[Pic Ž.T : Im Ž.␦ ; hence Pic Ž.T : Im Ž .. XX Ž.b Pic ŽD .[ Pic Ž.T s Im Ž␦ ., hence  : PicŽ.R ª Pic Ž.T is surjec- ti¨e; U Ž.c If ˜ : U ŽT .ª k rUŽ.D is surjecti¨e, then X ␦ : PicŽ.R ª Pic Ž.D [ Pic Ž.T is an isomorphism, or equi¨alently X X ␣ 6  6 0 ª PicŽ.D Pic Ž.R Pic Ž.T ª 0 is a split exact sequence; therefore the following commutati¨e diagram has exact rows and columns
000
6 6 6
X X 6 ␣6 6 6
0 PicŽ.D Pic Ž.R Pic Ž.T 0
6 6 6
6 ␣6 6
0 CŽ.D C Ž.R C Ž.T
6 6 6
Y Y 6 ␣6 6
0 GŽ.D G Ž.R G Ž.T
6 6 6 000 THE CLASS GROUP OF A PULLBACK 821
Proof. Ž.a Let ŽwxwxI , L .Ž.Ž.g C D [ Pic T . We may assume that I : D and, by arguing as in the proof of Lemma 2.2, that L s KT with Ž. Ž. Ksx1,..., xRri,xgR,1FiFn, and K M. Therefore K is a nonzero fractional ideal of D Ž.Proposition 1.8 . Let y g K q M be a unit y1 Ž. of T and set H [ K¨ l y I . We note that H is divisorial as an intersection of divisorial ideals. We claim that HT s L and Ž.H s Ž.yI. 1 1 1 It is clear that yŽ.I > M, so that yyŽ.ITsyT s T and thus yyŽ.I >MŽ.Proposition 1.1 . Since T is flat over R Ž.Lemma 0.3 , we have y1 Ž. HT s KT¨¨ly ITsKT. Moreover, by Proposition 0.6, L s KT : Ž.Ž. y1Ž. KT¨¨¨¨¨: KT s KT s L s L, whence HT s L. Since y I > M, Ž. Ž . Žy1Ž.. Ž . Ž. by Proposition 1.6, H s K¨¨l y I s K q M l yI and, since Ž.y g ŽK q M ., then ŽH .s Ž.yI. In order to conclude the proof, we want to show that H is a t-invertible ideal of R. Since
H : K¨ and HT s KT s L, then
T s LTŽ.:L sHT Ž T : KT .Žs HR:KT .sHR Ž:KT¨ . :HRŽ.:HT:T.
Thus HRŽ.:HTsTand hence M ; HRŽ.Ž:H Proposition 1.1 . . On the other hand, we know that Ž.H s Ž.yI, hence Ž.H is a t-invertible t-ideal of D, and so by Propositions 1.6 and 1.8 we have
D s Ž. Ž.HDŽ.: Ž.H tsŽ. Ž.HŽ.Ž ŽR:TTH .ttsŽ.HR Ž:H . .
:Ž.Ž.Ž.HRŽ.:H tsŽ.HR Ž.:H t: Ž.RsD.
ŽŽ Ž .. . Ž Ž .. y1 Ž. Therefore HR:H ttsDand HR:H s DsR. Ž.b Let I : D and L : T be two invertible ideals. Then byŽ. a there exists a t-invertible t-ideal H of R such that ␦ŽwxH .Žs wxwxI , L .. Let
H s F¨ with F a finitely generated fractional ideal of R. Then, for some nonzero element x Ž.respectively, y in the quotient field of D Žrespec- tively, R.Ž, we have that xI s H .and yL HT FT FT FT Proposition 0.6 . s s ¨¨sŽ.Ž.¨s ¨ Ž . Since Ž.H is invertible, by Propositions 1.6Ž. c and 1.8, we obtain
D s Ž.HDŽ.Ž.: Ž.HsHR Ž:H .. Ž. On the other hand, since HT s FT ¨ is invertible, it results that
T s HTŽ.Ž.Ž.Ž. T : HT s HT T : FT s HR:FTsHR:HT.
Thus M ; HRŽ.Ž:H Proposition 1.1 . and then HR Ž.:H sR. Hence,
F¨ s H is invertible. 822 FONTANA AND GABELLI
Ž.c The map ␦ X is surjective by partŽ. b . Since ˜ is surjective, then X ␦:CŽ.RªC Ž.D[C Ž.Tis injective Ž Theorem 2.3 . and thus ␦ : PicŽ.R ªPicŽ.D [ Pic Ž.T is also injective. Ž .Ž.Ž. Remark 2.6. Let wxwxI , L g C D [ C T .If LsKT¨ , with K a finitely generated ideal of R, as in the proof of Proposition 2.5Ž. a , it is possible to find a divisorial ideal H of R such that Ž.H s dI wth d g D and HT s L.Ž More precisely, without loss of generality, assuming that
I : D, and K an integral ideal of R, and K M, we have H [ K¨ l 1 yy Ž.I, where y g K q M is a unit in T.. Then H is t-invertible ifŽ and X only if. H is ¨-finite. As a matter of fact, let H be a finitely gener- X ŽŽ X.. Ž X . ated integral ideal of R such that H¨¨s H. Then H and HT ¨ are t-invertibleŽ because, as before, by Proposition 0.6 and 1.8 we have Ž. ŽŽ X.. Ž X .. X yIs H ¨¨and L s HT . By Proposition 1.13, H¨s H is t-invertible.
Remark 2.7. Given a diagram of type Ž.I , Milnor proved that the following Mayer᎐Vietoris sequence is exact:
X ab6 6 Uc6 ␦6 0ªUŽ.RU Ž.D[U Ž.TkPicŽ.R Pic Ž.D [ Pic Ž.T ªPicŽ.k s 0Ž. 2.7.1 1 1 where auŽ .[ Ž Žu .,u ., b ŽŽ¨,w .. [¨y Ž.w, cx Ž.[wyŽ.xD xwBa, Theorem 5.3, p. 481x . By using this fact, we may obtain another proof of Theorem 2.5Ž. b , in the more general setting of a diagram of type ŽI .. More precisely, it is easy to prove that the following are equivalent: U Ž.i b:U ŽD .[U ŽT .ªk is surjective; U Ž.ii ˜ : U ŽT .ª k rUŽ.D is surjective; X Ž.iii ␦ : Pic Ž.R ª Pic Ž.D [ Pic Ž.T is an isomorphism. As a matter of fact,Ž. i « Žii . since ˜ is obtained from b by passing to the quotient-groups modulo UŽ.D , and Ž.Ž. i m iii by noting that b is surjective X U if and only if ␦ is injective.Ž. ii « Ž.i . Let x g k and w g UŽ.T be such that Ž.Žw U D .s xU ŽD .. Then Ž.w ¨ s x for some ¨ g U ŽD ., and thus 1 bŽŽ¨y , w.. s x. We note that Milnor’s Theorem was used by Anderson and Ryckaert in provingwx AR, Proposition 3.1 , which is a particular case of the previous Theorem 2.5Ž. c when k is the quotient field of D. This part of the theorem recovers as special cases alsowxw CMZ, Corollary 4.16 and BR, Proposition 6.x We give next some classes of examples of pullbacks for which the map ˜ is surjective. First we note that, when ˜ is surjective, T is not simply R-flat but it is in fact a ring of fractions of R. THE CLASS GROUP OF A PULLBACK 823
U PROPOSITION 2.8. Consider a diagram of type ŽI .Ž.. If ˜ : U T ª U 1 XX krUŽ.D is surjecti¨e, then T s SRy ,where S s R _ DÄN l R : N g MaxŽ.T 4.
Proof. Let y g T _ R. By the surjectivity of ˜, we can find a, b g R 1 1 with b g UŽ.T such that Ž.y s Žaby.. Thus y s aby q for some 1 1 1 gM, whence y s Ž.a q bbygSRy. The inclusion SRy:Tfol- lows from Lemma 0.2.
U PROPOSITION 2.9. Consider a diagram of type ŽI .. Then the map U ˜ :UŽ.TªkrUŽ.D is surjecti¨e in each of the following cases: 1. T is quasi-semilocal; 2. k is a retract of TŽ. i.e., T s k q M . < 6 UUU ŽT . Proof. In Case 2, k ; UŽ.Tkis the identity map and hence U U < U ŽT . :UŽ.Tªk is surjective. In Case 1,
Remark 2.10.Ž. a If T is a quasi-semilocal domain, then PicŽ.T s 0 Žcf. for instancewx K, Theorem 60.Ž.Ž.Ž , then Pic D ( Pic R see also the subsequent Corollary 3.4. . Ž.b When k is a retract of T, Theorem 2.5 recoversw AR, Theorem 3.3x . Our next example demonstrates the existence of other classes of pull- backs having ˜ surjective, different from the classes considered in the previous proposition.
EXAMPLE 2.11. Let Ž.A, m, k [ Arm be a quasilocal domain with chŽ.A s 0 and ch Ž.k s p, with p prime. Let T [ AXwx, where X is an indeterminate over A. For each ␣ g A, the ideal M [ Ž.m, X y ␣ is a maximal ideal of T and TrM is naturally isomorphic to k. In this situation T/kqMfor characteristic reasons, andŽ. obviously T is not quasi-semi- U local, but < U ŽT . : UŽ.T s U ŽA .ª k is surjective. For an explicit example one can consider A [ ޚŽ.Y , the Nagata ring of ޚwith respect to the indeterminate Y, m [ pYŽ.for some prime p and M[ŽŽ.pY,X .. In this case k s Arm and TrM are canonically isomor- Ž. Ž. Ž . phic to the field of rational functions ކp Y ; hence
3. WHEN IS CŽ.R ª C Ž.T SURJECTIVE?
Given a subdomain A of an integral domain B,if Bis a flat A-module then by Proposition 0.7, the map : CŽ.A ª C Ž.B , wxI ¬ wIB x, is a group homomorphismŽ. see Section 2 for more details . A relevant case is when B is a ring of fractions of A. In this situation, if A is a Krull domain, by Nagata’s theorem, is a surjective mapŽ cf.w Fo, Corollary 7.2x. . This statement has been recently generalized by several authors, relaxing the Krull assumptionŽ cf.wx GR, NeA1, AAZ, and AHZ. . U However, even in the case of a pullback diagram of type ŽI . in which T is a ring of fractions of R Ž.Lemma 0.4, Corollary 0.5, and Proposition 2.8 , the canonical map s  : CŽ.R ª C Ž.T is known to be not surjective wxAR, Example 3.4 . In this section, we will investigate the problem of surjectivity of . We recall that a Prufer¨ ¨-multiplication domain Ž.for short, PVMD is an integral domain A such that the set of ¨-finite ideals form a groupŽ which necessarily coincides with CŽ..A wxGr, MZ . A GCD domain Žrespectively, a generalized GCD domain. is an integral domain in which the intersection of two principal ideals is a principal idealŽ. respectively, an invertible ideal Žcf.wxwx AA, Theorem 1; B, Chap. 7, Sect. 1, Exercise 21 and also A1, A2, Z. . Since a generalized GCD domainŽ. for short, a G-GCD domain is charac- terized by the property that each ¨-finite ideal is invertiblew AA, Theorem 1x , then it is clear that a generalized GCD domain is a PVMD. Conversely, a PVMD is a generalized GCD domain if and only if it is a locally GCD domainwx Z, Corollary 3.4Ž cf. alsowx AA. . In terms of class groups, it is known that a GCD domainŽ. respectively, a generalized GCD domain is a PVMD A with CŽ.A s 0 Ž respectively, G Ž.A s 0 .Ž cf.w J, Theorem 4, p. 81; BZ1; AA, Theorem 1; and BZ2, Corollary 1.5 and Corollary 2.3x. . Finally a Bezout´ domain is an integral domain such that every finitely generated ideal is principal and it is characterized by being a Prufer¨ domain with zero class groupwx G . We start with a simple consequence of the fact that PicŽ.T is always contained into the image of  : CŽ.R ª C Ž.ŽT Proposition 2.5Ž.. a : U PROPOSITION 2.1. Consider a pullback diagram of type ŽI .Ž., if G T s 0, then  : CŽ.R ª C Ž.T s Pic Ž.T is surjecti¨e. In particular,  is surjecti¨eif one of the following conditions is satisfied: Ž.a T is a Prufer¨ domain; THE CLASS GROUP OF A PULLBACK 825
Ž.b T is a reflexi¨e domain Ž i.e., e¨ery nonzero ideal of T is di¨isorial, cf.M,H,Q1;wx. Ž.c e¨ery maximal ideal of T is a t-ideal Ž e.g., dim ŽT .s 1J,w Chap.I, Sect.4,Corollaire 3, p.31;x.
Ž.d TN is a GCD domain for each maximal ideal N of TŽ e. g. Tisa generalized GCD domain wxAA, Corollary 1;. Ž.e C ŽT .s0 Že.g., T has finitely many maximal t-ideals, cf. for instance wxBGR, p. 309. .
Proof. Ž.a Since the invertible ideals of a Prufer¨ domain are the nonzero finitely generated fractional ideals, then it is trivial that the local class group of a Prufer¨ domain is zero. Ž.b The local class group of an integral domain A is zero if and only if I, J g TŽ.A implies that IJ g TŽ.A wxBZ1, Theorem 2.1 . If T is reflexive and I, J g TŽ.T then IJ is divisorial and hence a t-invertible t-ideal. Ž. Ž. Ž. c and d . Given an integral domain A,if G AM s0 for each maximal ideal M of A, then GŽ.A s 0wx BZ2, Proposition 2.4 . We recall Ž. Ž. Ž . that, in both cases c and d , G TN s 0 for each maximal ideal N of T ŽŽ.for c seewx Gr, Theorem 5. . Ž.e is trivial.
In the case T s k q M, a result similar to Proposition 3.1 was given in wxAR, Theorem 3.7 . In each of the cases considered in Proposition 3.1, we have an interest- ing ‘‘splitting’’ property for the class group of R. More precisely:
U PROPOSITION 3.2. In a pullback diagram of type Ž.I ,Im Ž. sPic Ž.T if and only if ImŽ.␦ s C ŽD .[ Pic Ž.T . U Moreo¨er, if ˜ : UŽ.T ª k rU ŽD . is surjecti¨e and ImŽ .s Pic Ž.T then the following exact sequence:
␣ 6  6 0 ª C Ž.D C Ž.R Pic Ž.T ª 0
splits, and GŽ.D is canonically isomorphic to GŽ.R .
Proof. This is an easy consequence of Theorem 2.5, Theorem 2.3, and the commutativity of the following diagram: 826 FONTANA AND GABELLI
 C(R) C(T)
␦ C(D) ⊕ C(T)
C(D) ⊕ Pic(T)
Pic(D) ⊕ Pic(T) ␦' ' Pic(R) Pic(T).
The previous result generalizeswx AR, Theorem 3.6 .
U COROLLARY 3.3. Consider a pullback diagram of type ŽI .. Suppose that U GŽ.Ts0and that ˜ : U Ž.T ª k rUŽ.D is surjecti¨e. Then CŽ.R ( C Ž.D [PicŽ.T and G ŽD .( G Ž.R .
U COROLLARY 3.4. Consider a pullback diagram of type ŽI .. Suppose that CŽ.Ts0and that ˜ is surjecti¨e. Then, Pic Ž.D ( Pic Ž.R , C Ž.D ( C Ž.R , and GŽ.D ( G Ž.R .
In addition to the case ImŽ. s Pic Ž.T examined above, there is another relevant case for which  : CŽ.R ª C Ž.T is surjective. As a matter of fact,  is surjective if and only if  : TŽ.R ª T Ž.T is surjective; the following result shows that, when R is a PVMD,  is always surjective and gives a description of KerŽ. .
U PROPOSITION 3.5. Consider a pullback diagram of type ŽI .. If R is a  Ž. Ž. Ž . ÄŽŽ .. PVMD, then : T R ª T T is surjecti¨e and Ker s HR:K ¨:H 4 and K are finitely generated ideals of R and M ; H¨¨: R, M ; K : R . Ž. Proof. Let L be a t-invertible t-ideal of T. Then L s KT ¨ for some finitely generated fractional ideal K of R. Since R is a PVMD, then K¨ is Ž. Ž .Ž . at-invertible t-ideal of R and K¨¨s KTs KT ¨Proposition 0.6 ; thus  is surjective. Ž. Ä Since T is flat over R Lemma 0.3 , then T s RF , where F [ F : F is an integral ideal of R such that FT s T4 is a multiplicative system of THE CLASS GROUP OF A PULLBACK 827
Ž. finite type of R wxFP, Remarque 1.2 . Let H g T R . Then H s G¨ for some finitely generated fractional ideal G of R. Suppose that T s HT s Ž. Ž . Ž . Ž . GT¨¨¨s GT , then G > M Proposition 1.1 . Since R :RTGTs T: GT ŽŽ.. Ž . sT:T GT ¨ s T, then R :R G g F, and hence there exists a finitely Ž. XX generated ideal F g F such that F : R :R G . Set F [ GF; then F is a finitely generated ideal of R and moreover, by Proposition 0.6Ž. c , we have
X X F¨T s Ž.ŽF T ¨¨¨¨s GFT .Žs HFT .Ž.s FT s T ,
XŽ. so that F¨¨> M Proposition 1.1 . Since R is a PMVD, F is a ¨-invertible ŽXŽ..ŽŽ.. ideal of R, and so FR:F¨¨sGF R : F s G¨s H. Conversely, if H and K are finitely generated ideals of R and M ; Ž. Ž. H¨¨:R,M;K:R, we want to show that HR:K¨is in Ker . Since ŽR : ŽHR Ž:K ...is ¨-finite Ž because R is a PVMD. , then by Proposi- tion 0.6Ž. c , Proposition 1.1, and Lemma 1.5 Ž. c
Ž.Ž.Ž.HRŽ.:KT¨¨sŽ.HR Ž.:KT¨sHT Ž T : KT . HT T:KT T. sŽ.¨¨Ž.¨s
It is useful at this point to introduce a weak version of coherence: we say that an integral domain A is ¨-coherent if, for each pair of finitely generated fractional ideals I and J of A, the ideal I¨¨l J is ¨-finite. PROPOSITION 3.6. Let A be an integral domain. Then A is ¨-coherent if and only if, for each nonzero finitely generated fractional ideal I of A, Ž.A : I is ¨-finite. Ž. Proof. Suppose that A is ¨-coherent. Let I s x1,..., xAniwith x a Ž.Ž . nonzero element in the quotient field of A. Then A : I s A : xA1 Ž.y1 y1 lиии l A : xAn sxA1 lиии l xAn is ¨-finite. Conversely, let I and J be two nonzero finitely generated fractional ideals of A, and write Ž.A : I X Ž.XX sI¨¨and A : J s J for some finitely generated fractional ideals I and X Ž ŽŽ . Ž ... Ž Ž XX.. J of A. Then I¨¨l J s A : A : I q A : J s A : I¨¨q J s ŽŽXX.. Ž Ž XX.. XX A:IqJ¨ sA:IqJis ¨-finite, since I q J is finitely gener- ated.
By Proposition 3.6 the ¨-coherent integral domains coincide with the integral domains with the property PU introduced inwx NeA2 . Therefore Mori domainswx Q2, p. 195 , quasi-coherent domains wx BAD , and PVMDs are all examples of ¨-coherent domains; it is known that the class of all ¨-coherent integral domains properly contains the union of these classes wxNeA2, Example 2.8 . 828 FONTANA AND GABELLI
We summarize in a diagram the principal implications among the classes of domains that we are considering:
¨-coherent
PVMD Mori quasi-coherent
G-GCD Krull coherent
CGD Prüfer Noetherian
Bézout
U PROPOSITION 3.7. Consider a pullback diagram of type ŽI .. If R is ¨-coherent, then ␦ : CŽ.R ª C Ž.D [ C Ž.T is surjecti¨e; in particular, the map  : CŽ.R ª C Ž.T is also surjecti¨e. Proof. Let ŽwxwxI , L .Ž.Ž.g C D [ C T . By using the notation and the ŽŽ.. conclusion of Remark 2.6 since R is ¨-coherent, L s KT ¨¨s KT,itis y1Ž. y1Ž. sufficient to show that H [ K¨ l y I is ¨-finite. Since I is ¨-finite, because t-invertible, and K¨ is trivially ¨-finite, then by the property of ¨-coherence we can conclude that H is ¨-finite. U COROLLARY 3.8. Consider a pullback diagram of type ŽI .. Suppose that U ˜ :UŽ.TªkrUŽ.D is surjecti¨e and that R is ¨-coherent, then the diagram
000
6 6 6
X X 6 ␣6 6 6
0 PicŽ.D Pic Ž.R Pic Ž.T 0
6 6 6
6 ␣6 6 6
0 CŽ.D C Ž.R C Ž.T 0
6 6 6
Y Y 6 ␣6 6 6
0 GŽ.D G Ž.R G Ž.T 0
6 6 6 000 has exact columns and splitting exact rows. THE CLASS GROUP OF A PULLBACK 829
Proof. This is an easy consequence of Theorem 2.3, Corollary 2.4, Proposition 3.7, and Proposition 2.5.
We note that Corollary 3.8 generalizesw AR, Proposition 3.9; BR, Propo- sition 6x .
4. PULLBACK AND PRUFER¨ ¨-MULTIPLICATION DOMAINS
In this section we study the transfer in pullback diagrams of type Ž.I of the PVMD property and of related properties involving the annihilation of theŽ. local class group. In the present context, this study is motivated by Propositions 3.5 and 3.7, in which we proved that if R is a ¨-coherent domainŽ.Ž in particular, a PVMD then  : C R.ª CŽT .is surjective. We recall that inwx NeA2, Corollaire 2.11 it was proved that, in a Ž U . pullback of type I , if one supposes that T is quasi-semilocal and TM is a valuation domain, then R is ¨-coherent if and only if T and D are ¨-coherent. Using the techniques from Section 1, it is not difficult to extend this theorem to the non-quasi-semilocal case. However, examples show that, when R is ¨-coherent, k does not need to be the quotient field of D and TM does not need to be a valuation domain. The transfer of the ¨-coherence property will be studied in a forthcoming paper. The results of this section generalize several properties proved for D q M constructions. It is well known that, in a pullback diagram of type Ž.I , R is a Prufer¨ domain if and only if T and D are Prufer¨ domains and k is the field of quotients of D Žcf. for instancewx F, Proposition 2.2 and Theorem 2.4. . The following result generalizeswx AR, Theorem 4.1 , where R is a domain of D q M type.
THEOREM 4.1. Consider a pullback diagram of type Ž.I . Then R is a PVMD if and only if k is the quotient field of D, D and T are PVMDs, and TM is a ¨aluation domain.
Proof. Assume that R is a PVMD. Let x g k, x / 0, thus x s y q M 1 for some y g T _ M. Set H [ yR l R s ŽŽR : R q yRy ... Since R is a PVMD, H is a t-invertible integral divisorial ideal of R, whence H M Žotherwise H s H l M s yR l M s yM and M is not a t-invertible t-ideal of R.Ž. In particular, 0.Ž/ H .: xD l D, and we can find two elements X X X dgŽ.Hand d g D with d / 0Ž and d / 0. such that d s xd . Thus x belongs to the quotient field of D. Since k is the quotient field of D, then Ž. we know that RMMs T ; since M s R : T is a divisorial ideal of a 830 FONTANA AND GABELLI
PVMD, then necessarily RMMs T is a valuation domainwx Gr, Theorem 5 . In order to show that D Ž.respectively, T is a PVMD, we prove that, for each nonzero finitely generated fractional ideal I of D Žrespectively, L of .Ž . y1Ž. Žy1Ž.. T, the ideal I¨¨respectively, L is t-invertible. Since I¨¨s I is ¨-finiteŽ. Corollary 1.7 and Proposition 1.8 and R is a PVMD, then y1 Ž. I¨¨is a t-invertible t-ideal of R, whence I is a t-invertible t-ideal of DŽ.Corollary 1.9Ž respectively, since L s KT for some nonzero finitely generated fractional ideal K of R and R is a PVMD, then K¨ is a Ž. Ž Ž.. t-invertible t-ideal of R and L¨¨¨s KT s KT Proposition 0.6 c ; hence, Ž. . by Proposition 0.7 b , L¨ is a t-invertible t-ideal of T . Conversely, in order to show that R is a PVMD we consider a nonzero finitely generated fractional ideal K of R and we must prove that K¨ is a t-invertible t-ideal of R. Since RMMs T is a valuation domain, then
KRŽ.:KRMMMMMsKR Ž R : KR .s R .
Hence KRŽ.:K M, and we can find x g Ž.R : K such that xK M. Without loss of generality, we can assume that K is a nonzero finitely generated integral ideal of R and K M, because K¨ is t-invertible if and only if yK¨ is t-invertible for every nonzero element y in the quotient field of R. In the present situation, Ž.K is a nonzero finitely generated Ž Ž .. Ž integral ideal of D, hence K ¨ is t-invertible in D since D is a .Ž.Ž PVMD . On the other hand, KT ¨ is also t-invertible in T since T is a .Ž. PVMD ; therefore, by Proposition 1.13 c K¨ is a t-invertible t-ideal of R.
We can apply immediately the previous theorem to the study of the transfer of Bezout,´ GCD, and G-GCD properties in a pullback diagram of type Ž.I . THEOREM 4.2. Consider a pullback diagram of type Ž.I . Then Ž.a RisaG-GCD domain if and only if k is the quotient field of D, D and T are G-GCD domains, and TM is a ¨aluation domain; Ž.b R is a GCD domain if and only if k is the quotient field of D, D and Ž. U Ž. T are GCD domains, TisaM ¨aluation domain, and ˜ : U T ª k rU D is surjecti¨e; Ž.c R is a Bezout´ domain if and only if k is the quotient field of D, D U and T are Bezout´˜ domains, and : UŽ.T ª k rUŽ.D is surjecti¨e. Proof. Suppose that R is a G-GCD domainŽ respectively, a GCD domain, a Bezout´ domain.Ž . Then, in particular, R is a PVMD, and by Theorem 4.1. k is the quotient field of D, TM is a valuation domain, and D and T are PVMDs. Since a PVMD is a ¨-coherent domain, then the group homomorphism ␦ : CŽ.R ª C Ž.D [ C Ž.T is surjective Ž Proposition THE CLASS GROUP OF A PULLBACK 831
3.7. ; therefore, by passing to the quotient-groups, we see that the canoni- cal map GŽ.R ª G Ž.D [ G Ž.T is also surjective. The previous remark implies that if GŽ.R s 0 Ž respectively, C Ž.R s 0 . then G Ž.D s 0 s G Ž.T ŽŽ.Ž..respectively, C D s 0 s C T ; this fact leads easily to the conclusion that if R is a G-GCD domainŽ respectively, a GCD-domain, a Bezout´ domain. , then D and T are the same. Finally, if R is a GCD-domain then ␦:CŽ.RªC Ž.D[C Ž.Tis trivially an isomorphism; hence, by Theorem 2.3, ˜ is surjective. Conversely, by Theorem 4.1, in each case R is a PVMDŽ and it is a Prufer¨´ domain when D and T are Bezout domains. . By Proposition 2.1, if GŽ.Ds0sG Ž.T, then G Ž.R s 0. Hence if D and T are G-GCD do- mains, then R is also a G-GCD domain. If ˜ is surjective, then by Theorem 2.3 ␦ : CŽ.R ª C Ž.D [ C Ž.T is injective. Therefore if CŽ.D s 0sCŽ.T, then C Ž.R s 0; hence if D and T are GCD domains Ž respec- tively, Bezout´ domains. , then R is the same. We note that the previous theorem generalizesw BR, Theorem 7 and Theorem 11xw or AR, Corollary 4.2 x and w AA, Theorem 4 x ; very particular cases are when T is a valuation domainwx Ma, Theorem 4.2.16 or T is a polynomial ring with coefficients in k wCMZ, Theorem 4.13 and Theorem 4.43x . Anderson and Ryckaert inwx AR ask the following questions:
Let T s k q M and R s D q M be integral domains with D a proper subring of k with quotient field k. Does ImŽŽ.Ž..Ž. : C R ª C T s Pic T ? In particular, if T is aPVMD, does CŽ.T s Pic Ž.T ? Equi¨alently, is T a G-GCD domain? As an application of Theorems 4.1 and 4.2 we give next a negative answer to these questions:
EXAMPLE 4.3. Let T1 be a quasi-semi-local Bezout´ domain with at least two maximal ideals M11and N , such that k [ T11rM is a retract of T1. Set k1111111[ T rN , : T ª T rN and let D 1be a quasi-semi-local nonlo- cally factorial Krull k-algebra with quotient field k1. We note that Ž. U Ž. Ž ˜111:UTªkrUD 1is surjective, since T1is quasi-semi-local Proposi- tion 2.9. . y1 Set T [ 11Ž.D . Then, by Theorems 4.1 and 4.2, T is a PVMD non-G-GCD domain, since D1 is a PVMD non-G-GCD domainŽ cf.w Z, Corollary 3.4; Bo1, Section 2; and A1, Theorem 1x. . Ž. Ž. Ž. Ž . More precisely, since C T11s Pic T s 0, then Pic T s Pic D1s 0. Ž. Ž . Ž. Ž . Ž . Whence C T ( C D1111( G T ( G D , with G D / 0. Set M [ M lTand : T ª TrM. Since k11is the quotient field of D , then TMs Ž. T1 M is a valuation domain and TrM s T11rM s k. Moreover, since D 1 1 Ž and T1 are k-algebras, T is also a k-algebra, whence T s k q M after identifying k with its canonical image in T .. 832 FONTANA AND GABELLI
We note also that T is quasi-semi-local, because T11and D are quasi-semi-localŽ cf. for instancewx F, Theorem 1.4. . Let D be any PVMD proper subring of k, with quotient field k; set R [ D q M. In the present situation, R is a PVMDŽ. Theorem 4.1 , and T is a ring of fractions of R U Ž.ŽProposition 2.8 since ˜ : U T .ª k rUŽ.D is obviously surjectiveŽ Pro- position 2.9. . Therefore, by Theorem 2.5Ž. c , the diagram
00
6 6
X X 6 ␣6 6
0 PicŽ.D Pic Ž.R 0
6 6 6
6 ␣6 6 6
0 CŽ.D C Ž.R C Ž.T(C ŽD1 . 0
6 6 6
Y Y 6 ␣6 6 6
0 GŽ.D G Ž.R G Ž.T(G ŽD1 . 0
6 6 6 00 0 has exact columns and splitting exact rows. In conclusion, since GŽ.T ( Ž. Ž. Ž. GD1 /0, the domain R is a PVMD such that  : C R ª C T is surjective, but PicŽ.T / C Ž.T . For an explicit example, we fix an algebraically closed field k and we 2 consider the domain A [ kXwx,Y,ZrŽXY y Z .. Let x, y, and z denote the canonical images of the indeterminates X, Y, and Z in A, set m[Ž.x,y,zAand D1 [ Am. It is known that D1 is a two-dimensional Noetherian integrally closed local domain which is not factorialŽ cf.w ZS, Vol. I, p. 154xw and Hu, Example 7 x. . Let k11be the quotient field of D . X Since tr.deg.k k1 s 2wx ZS, Vol. II, p. 22 , then we can find t, t g k1 algebraically independent over k such that k1 is a finiteŽ. algebraic X Y Y extension of ktŽ ,t.. Let t be a indeterminate over k11, set E [ ktŽ . and F[ktŽ,tXY,t.. We can consider the following valuation domains:
Y X YXY XY Y X Y VE[kt1w xŽ.ttttand WF[ ktw xŽ.qtkŽ. tw txŽ.qtk Ž t , tt .wxŽ..
It is easy to see that VEFŽ.respectively, W is a one-dimensional Ž respec- tively, three-dimensional.Ž valuation domain with quotient field E respec- Y X X . Ž X tively, F , with maximal ideal n EE[ tV respectively, m F[ tktw xŽt.q YXY XY Ž.YŽ. .Ž tktw txŽt .Žqtk t , ttwxt.and with residue field VEErn s k1respec- .Ž . tively, WFFrm s k . Let WEE, m be a valuation domain with quotient field E such that WEFFEl F s W . Set V [ V l F, so that VFs THE CLASS GROUP OF A PULLBACK 833
X Y Ž . Y kt,ttw xŽt .. It is clear that WFF V and V F W F, whence W EE V and VEEW. Moreover, since E is a finite extension of F and k s WFFrm is an algebraically closed field, then necessarily WEErm s k wB, Chap. 6, Sect. 8, Lemma 2x . Set T1 [ VEEl W , M1[ m El T11, N [ n El T1, and 11: T ª T 11rN . Then it is easy to see that T1is a quasi-semi-local Bezout´ domain with quotient field E, with maximal ideals M11and N , and Ž.T V, Ž.T W B, Chap. 6, Sect. 7, Propositions 1 and 2; K, 1 NE11s 1MEs w Theorem 107x . y1 Ž. Set T [ 11D and M [ M1l T. We note that T contains the field k, since T and D contain k, and that T Ž.T . Thus T M T M 11 Ms 1M1 r s 11r sWEErmskand we have T s k q M. Furthermore, T is a PVMD but not a G-GCD domainŽ. Theorems 4.1 and 4.2 . For each proper subring D of k, with quotient field k, we can consider R [ D q M. If we take D to be a PVMD, then R is a PVMD, not a G-GCD domain, with the property CŽ.RªC Ž.T is surjective, but Pic Ž.T / C Ž.ŽT . We observe that it is possible to find explicitly a proper subring D of k which is a PVMD: for instance, if k s ޑ is the algebraic closure of ޑ, we can take D equal to the integral closure in ޑ of a discrete valuation domain of ޑ..
Remark 4.4. After this paper was accepted for publication, we were informed, at the Fes` Conference in June 1995, that D. Nour El Abidine and M. Khalis have recently and independently obtained some of the results presented here, under the assumption that the integral domain T is local.
REFERENCES wxA1 D. D. Anderson, -Domains, overrings, and divisorial ideals, Glasgow Math. J. 19 Ž.1978 , 199᎐203. wxA2 D. D. Anderson, Globalization of some local properties in Krull domains, Proc. Amer. Math. Soc. 85 Ž.1982 , 141᎐145. wxA3 D. F. Anderson, A general theory of class groups, Comm. Algebra 16 Ž.1988 , 805᎐847. wxAA D. D. Anderson and D. F. Anderson, Generalized GCD domains, Comm. Math. Uni¨. St. Pauli 28 Ž.1979 , 215᎐221. wxAAZ D. D. Anderson, D. F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 Ž.1991 , 17᎐37. wxAHZ D. D. Anderson, E. G. Houston, and M. Zafrullah, t-Linked extensions, the t-class group, and Nagata’s theorem, J. Pure Appl. Algebra 86 Ž.1993 , 109᎐124. wxAR D. F. Anderson and A. Ryckaert, The class group of D q M, J. Pure Appl. Algebra 52 Ž.1988 , 199᎐212. wxBAD V. Barucci, D. F. Anderson, and D. E. Dobbs, Coherent Mori domains and the principal ideal theorem, Comm. Algebra 15 Ž.1987 , 1119᎐1156. 834 FONTANA AND GABELLI wxBGR V. Barucci, S. Gabelli, and M. Roitman, On semi-Krull domains, J. Algebra 145 Ž.1992 , 306᎐328. wxBa H. Bass, ‘‘Algebraic K-Theory,’’ Benjamin, New York, 1968. wxBG E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form DqM,Michigan Math. J. 20 Ž.1973 , 79᎐95. wxBS M. B. Boisen and P. B. Sheldon, CPI-extension: overrings of integral domains with special prime spectrum, Canad. J. Math. 29 Ž.1977 , 722᎐737. wxB N. Bourbaki, ‘‘Algebre` Commutative,’’ Hermann, Paris, 1961. wxBo1 A. Bouvier, The local class group of a Krull domain, Canad. Math. Bull. 26 Ž.1983 , 13᎐19. wxBo2 A. Bouvier, Le groupe des classes d’un anneau integre,``in ‘‘107`eme Congres des Societes´´ Savantes, Brest, 1982,’’ Vol. 4, pp. 85᎐92. wxBZ1 A. Bouvier and M. Zafrullah, On the class group, unpublished manuscript. wxBZ2 A. Bouvier and M. Zafrullah, On some class groups of an integral domain, Bull. Soc. Math. Grece` 29 Ž.1988 , 45᎐59. wxBR J. Brewer and E. Rutter, D q M constructions with general overrings, Michigan Math. J. 23 Ž.1976 , 33᎐42. wxC P.-J. Cahen, Couple d’anneaux partageant un ideal,´ Arch. Math. 51 Ž.1988 , 505᎐514. wxCMZ D. Costa, J. Mott, and M. Zafrullah, The D q XDSwx X construction, J. Algebra 53 Ž.1978 , 423᎐439. wxF M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl. 123 Ž.1980 , 331᎐355. wxFP M. Fontana and N. Popescu, Sur une classe d’anneaux qui generalisent´´ les anneaux de Dedekind, J. Algebra, 173 Ž.1995 , 44᎐66. wxFo R. Fossum, ‘‘The Divisor Class Group of a Krull Domain,’’ Springer-Verlag, New YorkrBerlin, 1973. wxGa S. Gabelli, Completely integrally closed domains and t-ideals, Boll. Un. Mat. Ital. B 3Ž.1989 , 327᎐342. wxGR S. Gabelli and M. Roitman, On Nagata’s theorem for the class group, J. Pure Appl. Algebra 66 Ž.1990 , 31᎐42. wxG0 R. Gilmer, ‘‘Multiplicative Ideal Theory,’’ Queen’s Series in Pure and Applied Mathematics, Kingston, 1968. wxG R. Gilmer, ‘‘Multiplicative Ideal theory,’’ Dekker, New York, 1972. wxGr M. Griffin, Some results on ¨-multiplication rings, Canad. J. Math. 19 Ž.1967 , 710᎐722. wxH W. Heinzer, Integral domains in which each nonzero ideal is divisional, Mathematika 15 Ž.1968 , 164᎐169. wxHu H. Hutchins, ‘‘Examples of Commutative Rings,’’ Polygonal, 1981. wxJ P. Jaffard, ‘‘Les systemes`´ d’ideaux,’’ Dunod, Paris, 1960. wxK I. Kaplansky, ‘‘Commutative Rings,’’ The University of Chicago Press, Chicago, 1974. wxMa S. Malik, ‘‘A Study of Strong S-Rings and Prufer¨ ¨-Multiplication Domains,’’ Ph.D. thesis, Florida State University, 1979. wxM E. Matlis, Reflexive domains, J. Algebra 8 Ž.1968 , 1᎐33. wxMZ J. L. Mott and M. Zafrullah, On Prufer¨ ¨-multiplication domains, Manuscripta Math. 35 Ž.1981 , 1᎐26. wxNeA1 D. Nour El Abidine, Sur le groupe des classes d’un anneau integre,` Ann. Uni¨. Ferrara 36 Ž.1990 , 175᎐183. wxNeA2 D. Nour El Abidine, ‘‘Groupe des classes de certain anneaux integres`´ et ideaux transformes,’’´ Doctoral thesis, Univ. of Lyon, 1992. THE CLASS GROUP OF A PULLBACK 835 wxQ1 J. Querre,´ Sur les anneaux reflexifs, Canad. J. Math. 27 Ž.1975 , 1222᎐1228. wxQ2 J. Querre,´` ‘‘Cours d’Algebre,’’ Masson, Paris, 1976. wxR F. Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16 Ž.1965 , 794᎐799. wxRy1 A. Ryckaert, Sur le groupe des classes des anneaux D q M, C.R. Math. Rep. Acad. Sci. Canada 8 Ž.1986 , 363᎐367. wxRy1 A. Ryckaert, ‘‘Sur le groupe des classes d’un anneau integre,’’`´ thesis, Universite Claude Bernard de Lyon I, June 1986. wxS B. Stenstrom,¨ ‘‘Ring of Quotients,’’ Springer-Verlag, New YorkrBerlin, 1976. wxZ M. Zafrullah, On a property of pre-Schreier domains,’’ Comm. Algebra 15 Ž.1987 , 1895᎐1920. wxZS O. Zariski and P. Samuel, ‘‘Commutative Algebra,’’ Van Nostrand, 1958.