On faithfully flat fibrations by a punctured line

Journal of Algebra 415 (2014) 13–34

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Journal of Algebra www.elsevier.com/locate/jalgebra

On faithfully flat fibrations by a punctured line Neena Gupta Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India

a r t i c l e

i n f o

Article history: Received 19 December 2013 Available online xxxx Communicated by Luchezar L. Avramov MSC: primary 14R25, 13F20 secondary 13E05, 13E15, 13B30

a b s t r a c t Let R be a Noetherian normal domain. In this paper, we present the structure of a faithfully flat R-algebra A whose 1 generic and codimension one fibres are of the form K[X, f (X) ]. We also investigate minimal sufficient conditions for such an algebra to be finitely generated over R. © 2014 Elsevier Inc. All rights reserved.

Keywords: Polynomial algebra Localisation Fibre ring Codimension-one Faithful flatness Finite generation Noetherianness

1. Introduction Let R be an integral domain and A an R-algebra such that tr.degR (A) = 1. Recall that for a prime ideal P in R of height r, k(P ) denotes the residue field RP /P RP and A ⊗R k(P ) is called the fibre ring (or fibre) of codimension r at the point P of Spec R. We say that the fibre ring at P ∈ Spec R “misses n points” if A ⊗R k(P ) is 1 of the form k(P )[X, (X−α1 )···(X−α ] for some indeterminate X and n distinct elements n) E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jalgebra.2014.05.025 0021-8693/© 2014 Elsevier Inc. All rights reserved.

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α1 , . . . , αn ∈ k(P ). Thus, if the fibre ring is A∗ (i.e., isomorphic to the Laurent polynomial 1 ring k(P )[X, X ]), then we say that it misses one point. It is known that if R is a Noetherian normal domain and A a faithfully flat R-algebra  n whose fibre rings are all A∗ , then A ∼ for some invertible ideal I of R ([1, = n∈Z I Theorem 3.11], [3, Theorem 4.8]). Thus, if R is a UFD, then A is itself A∗ . R.V. Gurjar asked the author to explore the structure of A∗∗ -fibrations (i.e., algebras whose fibres miss two points). We consider a general version of his question. Q.1. Let R be a Noetherian normal domain. What can we say about the structure and properties of a faithfully flat R-algebra A each of whose fibre rings misses at least one point and some of the fibre rings miss at least two points? We also consider a closely related problem. In [2], an R-algebra B has been defined to be locally A1 in codimension-one (over R) if BP (= B ⊗R RP ) is a polynomial algebra in one variable over RP for every height one prime ideal P of R. The structure and properties of a faithfully flat algebra B over a Noetherian normal domain R which is locally A1 in codimension-one have been extensively studied by Bhatwadekar, Dutta and Onoda in [2]. Bhatwadekar and Gupta have described in [3] the structure and properties of a faithfully 1 flat algebra A over a Noetherian normal domain R satisfying AP ∼ ] with = RP [X, aP X+b P aP , bP ∈ RP , for every height one prime ideal P of R. They showed that the R-algebra A is of the form B[I −1 ], where B is faithfully flat and locally A1 in codimension-one over R and I is an invertible ideal of B. These results lead us to the more general question (first raised to the author by A.K. Dutta): Q.2. Let R be a Noetherian normal domain. What can one say about the structure and properties of a faithfully flat R-algebra A satisfying the condition AP ∼ = RP [X, fP 1(X) ] with fP (X) ∈ RP [X] for every height one prime ideal P of R? In this paper we investigate the above two questions. Q.2 is addressed in Theorem A below (Theorem 5.2): Theorem A. Let R be a Noetherian normal domain with field of fractions K and A be a faithfully flat R-algebra such that: 1 (i) A ⊗R K = K[X, f (X) ] for some f (X) ∈ K[X] \ K and X transcendental over K. (ii) For each prime ideal P in R of height one, AP ∼ = RP [X, fP 1(X) ] for some fP (X) ∈ RP [X].

Let B := A ∩ K[X]. Then B is a faithfully flat R-algebra which is locally A1 in codimension-one over R and there exists an invertible ideal I of B such that A = B[I −1 ]. Moreover, A is Noetherian (respectively, finitely generated over R) if and only if B is Noetherian (respectively, finitely generated over R).

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Theorem A reduces the study of the R-algebra A to the study of an R-algebra B which is faithfully flat and locally A1 in codimension-one over R. As already mentioned such algebras have been studied deeply by Bhatwadekar, Dutta and Onoda in [2]. For instance, it is known that B is finitely generated over R if and only if B ∼ = SymR (Q) for some invertible ideal Q of R [5, Theorem 3.4]. From Theorem A and an analogous result for the ring B proved in [2], we deduce the following (Theorem 5.3): Theorem B. Let R be a Noetherian normal local domain with maximal ideal m and A be as in Theorem A. Then the following are equivalent: 1 A∼ ] for some g(X) ∈ R[X] \ R. = R[X, g(X) A is a finitely generated R-algebra. A[1/a] is a finitely generated R[1/a]-algebra for some a(= 0) ∈ R. The image of fP (X) (as in (ii) of Theorem A) in k(P )[X] is transcendental over k(P ) for all but finitely many prime ideals P in R of height one. (V) R/m  A/m.

(I) (II) (III) (IV)

Moreover, if R is complete and A is Noetherian then the above equivalent conditions hold. The following result (Theorem 5.4) addresses Q.1: Theorem C. Let R be a Noetherian normal domain and A be a faithfully flat R-algebra such that for each prime ideal P in R of ht(P ) ≤ 1, A ⊗R k(P ) ∼ = k(P )[X, fP 1(X) ] for −1 ∼ some fP (X) ∈ k(P )[X] with degX fP (X) ≥ 1. Then A = B[I ], where B ∼ = SymR (Q) for some invertible ideal Q of R and I is an invertible ideal of B. In particular, A is 1 finitely generated over R and, for every maximal ideal m of R, Am ∼ ] for = Rm [X, gm (X) some gm (X) ∈ Rm [X]. As a step towards the proof, we prove a result, which may be of independent interest, relating faithful flatness, finite generation and Noetherianness of an R-algebra B with the corresponding property of a localisation B[f −1 ] for some f ∈ B \ R (Theorem 3.14). We shall denote B[f −1 ] by Bf . Theorem D. Let R be a Noetherian normal domain. Let Δ denote the set of all height one prime ideals of R. Let B be an R-algebra such that (i) (ii) (iii) (iv)

BP ∼ = RP [X] ∀P ∈ Δ,  B = P ∈Δ BP , JB ∩ R = J for every ideal J of R, and Bf is faithfully flat over R for some f ∈ B \ R.

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Then B is faithfully flat over R. Moreover, if Bf is Noetherian (respectively, finitely generated over R) then B is Noetherian (respectively, finitely generated over R). We now give a layout of the paper. In Section 2, we shall give a criterion for a faithfully flat algebra over a Noetherian domain to be Noetherian. In Section 3, we shall prove Theorem D. In Section 4, we shall prove some auxiliary results on algebras with punctured fibres and in Section 5, we shall prove Theorems A, B and C. We recall some standard notation to be used throughout the paper. For a ring R, ∗ R will denote the multiplicative group of units of R. For a prime ideal P of R, and an R-algebra A, AP denotes the ring S −1 A, where S = R \ P and k(P ) denotes the residue field RP /P RP . The notation A = R[1] will mean that A is isomorphic, as an R-algebra, to a polynomial ring in one variable over R. 2. On the Noetherian property of certain faithfully flat algebras For convenience, we recall a well-known result on flatness (cf. [4, Lemma 4.1]). Lemma 2.1. Let R be a Noetherian ring and M be an R-module. Then M is flat over R if and only if TorR 1 (M, R/P ) = 0 for every prime ideal P of R. Consequently, we have the following result. Lemma 2.2. Let R be a Noetherian domain with field of fractions K and S = R \ {0}. Let B be a faithfully flat R-algebra and f ∈ B be such that Bf is also a faithfully flat R-algebra. (Note: this implies that f ∈ B \ R or f ∈ B ∗ .) Suppose that, for each prime ideal P of R, B ⊗R k(P ) is an integral domain. Then B is an integral domain with B = S −1 B ∩ Bf . Proof. Set M := Bf /B. We first show that M is R-flat. We have the short exact sequence: 0 → B → Bf → M → 0.

(1)

Fix a prime ideal P of R. Since Bf is flat over R, we have TorR 1 (R/P, Bf ) = 0 and hence, tensoring (1) with R/P , we have the exact sequence: 0 → TorR 1 (R/P, M ) → B/P B → Bf /P Bf → M/P M → 0.

(2)

Since B is R-flat, B/P B → B ⊗R k(P ), and hence B/P B is an integral domain. In particular, taking P = 0, B is an integral domain. Since Bf is faithfully flat over R, we have P Bf = Bf and hence f ∈ / P B. Therefore, the canonical map B/P B → Bf /P Bf is injective. Thus by (2), TorR 1 (R/P, M ) = 0. Hence, by Lemma 2.1, M is a flat R-module.

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Now let x ∈ S −1 B ∩ Bf . Then there exist b1 , b2 ∈ B, s ∈ S and an integer n ≥ 0 such that x = b1 /s = b2 /f n . If n = 0, then x = b2 ∈ B. Hence, suppose that n > 0. Then f n b1 = sb2 . Since M is a flat R-module, we have TorR 1 (R/sR, M ) = 0. Hence the canonical map B/sB → Bf /sBf is injective. Therefore, since f n b1 = sb2 , we have b1 ∈ sB. Hence x ∈ B. This completes the proof. 2 We now give a criterion for a faithfully flat R-algebra to be a Noetherian ring. Proposition 2.3. Let R be a Noetherian ring and B be a faithfully flat R-algebra such that, for each prime ideal P of R, B ⊗R k(P ) is either a PID or a field. Suppose that there exists f ∈ B such that Bf is a Noetherian faithfully flat R-algebra. Then B is a Noetherian ring. Proof. Suppose that B is not Noetherian. Then the set I := {I | I is an ideal of R and B/IB is not Noetherian} contains the ideal (0) and so is non-empty. Since R is Noetherian, I contains a maximal element, say P . Since B/P B is faithfully flat over R/P and Bf /P Bf is faithfully flat over R/P , we can replace B by B/P B and R by R/P and assume that B/IB is Noetherian for any non-zero ideal I of R. We shall now show that every prime ideal of B is finitely generated which will contradict, by Cohen’s criterion (cf. [8, Theorem 3.4]), that B is not Noetherian. Let Q be any prime ideal of B and P = Q ∩ R. Suppose that P = (0). Then B/P B is Noetherian by our reduction. Hence Q/P B is a finitely generated ideal of B/P B. Since P B is a finitely generated ideal of B, it follows that Q is finitely generated. Now suppose that P = (0). In this case, R is an integral domain with Q ∩ R = (0). Then, by Lemma 2.2, it follows that B is an integral domain. Let K be the field of fractions of R and S = R \ {0}. By hypothesis, S −1 B is either a field or a PID. Suppose that S −1 B is a field. Then since Q ∩ R = (0), QS −1 B is prime ideal of S −1 B and hence QS −1 B = (0). Since R is an integral domain and B is flat over R, we have Q = (0). In particular, Q is finitely generated. Now suppose that S −1 B is a PID. Then QS −1 B = gS −1 B for some g ∈ Q. Since B = S −1 B ∩ Bf by Lemma 2.2, we have gB = gS −1 B ∩ gBf , i.e.,   gB = gS −1 B ∩ B ∩ (gBf ∩ B) = Q ∩ (gBf ∩ B).

(3)

Let gBf = N1 ∩ · · · ∩ Nm be a minimal primary decomposition of gBf in the Noetherian √ ring Bf . For 1 ≤ i ≤ m, let pi = Ni . We now show that for each i, 1 ≤ i ≤ m, either Ni ∩ R = (0) or Ni = pi = QBf . Suppose that, for some i, Ni ∩ R = (0). Since R is an integral domain, it follows that pi ∩R = (0), and hence pi S −1 Bf is an associated prime ideal of gS −1 Bf . But as gS −1 B =

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QS −1 B is itself a prime ideal, we have QS −1 Bf = pi S −1 Bf = Ni S −1 Bf . Therefore Ni = pi = QBf . Note that at most one of the Ni ’s can be equal to QBf .  Set J := Ni =QBf Ni and J1 := J ∩ B. Then, from (3), it follows that gB = Q ∩ J1 . Since Ni ∩ R = (0) whenever Ni = QBf , it follows that J ∩ R = (0). Choose s(= 0) ∈ J ∩ R ⊆ J1 . As B/sB is a Noetherian ring by our reduction, its quotient B/J1 is also a Noetherian ring. Hence Q/gB = Q/(Q ∩ J1 ) ∼ = (Q + J1 )/J1 is finitely generated. Thus, Q is finitely generated. 2 Remark 2.4. The following observations show the necessity of the hypotheses in Proposition 2.3. (i) Even if there exists an f ∈ B such that Bf is a Noetherian faithfully flat R-algebra, a faithfully flat R-algebra B need not be Noetherian without appropriate fibre conditions. For instance, taking R to be a field k, B = k[X, XY, XY 2 , . . . , XY n , . . .](⊂ k[X, Y ]) and f = X, we see that Bf = k[X, X −1 , Y ] is a Noetherian ring and B, Bf are both faithfully flat R-algebras, but B is non-Noetherian. Further, an example in [7] shows the necessity of the restriction on the dimension of the fibre rings; it will not suffice to assume that all the fibre rings of B are Noetherian factorial regular domains. (ii) Without the hypothesis that there exists an f ∈ B for which Bf is both Noetherian and faithfully flat over R, a faithfully flat R-algebra B need not be Noetherian even if each fibre ring is either a polynomial ring in one variable or a field. Consider R = Z and B = Z[{X/p | p prime in Z}]. Then B is a faithfully flat R-algebra such that B ⊗R k(P ) ∼ = k(P )[X] ∀P ∈ Spec Z but B is non-Noetherian since X is contained in infinitely many height one prime ideals pB, p prime in Z. Again taking R = C[[t]], B = C[[t]][X, X/t, . . . , X/tn , . . .] and f = X, we see that B is faithfully flat R-algebra whose generic fibre B[1/t] = C((t))[X], a one dimensional polynomial ring, while the closed fibre B/tB = C, a field. However, B[1/f ] is not faithfully flat over R as tB[1/f ] ∩ R = tR. Here, although B[1/f ](= C((t))[X, 1/X]) is Noetherian (in fact a PID), but B is not. 3. On algebras which are locally A 1 in codimension-one Throughout the rest of the paper, R will denote a Noetherian normal domain with field of fractions K and Δ the set of all prime ideals in R of height one. As in [2], we call an integral domain B containing R to be “semi-faithfully flat over R” if  (1) B = P ∈Δ BP . (2) JB ∩ R = J for every ideal J of R. In [2], properties of semi-faithfully flat algebras, which are locally A1 in codimension-one over R, were investigated. A general structure of such an R-algebra B was described [2, Theorem 7.2]. In this section, we shall recall the general structure of B and prove that

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B is faithfully flat over R whenever Bf is faithfully flat over R for some f ∈ B \ R. We further relate the Noetherianness and finite generation of B with that of Bf . Note that 1 the generic fibre of Bf over R is of the form K[X, f (X) ] with f (X) ∈ K[X] \ K. Thus, Bf is a typical example of an R-algebra whose generic fibre is a punctured line. Throughout this section, B will denote a semi-faithfully flat R-algebra which is locally in codimension-one. Let X ∈ B be such that B ⊗R K = K[X], i.e., X is a generic variable of B. We recall from [2], some objects associated with the semi-faithfully flat algebra B and the fixed generic variable X of B. For each P ∈ Δ, fix XP ∈ BP such that BP = RP [XP ]. Then X = aP XP + cP for some aP , cP ∈ RP . Now set e(P ) := vP (aP ),

where vP (aP ) is the valuation of aP in RP .

P Thus BP = RP [ X−c ], where pRP = P RP . Also set pe(P )

   Δ0 := P ∈ Δ  RP [X]  BP . Let Σ0 be the set of all finite subsets of Δ0 . For Γ = {P1 , . . . , Pn } ∈ Σ0 , set

RΓ :=

RP ,

P ∈Δ\Γ

BΓ := SΓ

−1

B ∩ RΓ [X],

where SΓ := R \



P

and

P ∈Γ

IΓ := P1 (e(P1 )) ∩ · · · ∩ Pn (e(Pn )) . The following technical result is proved in [2, Theorem 7.2]. Theorem 3.1. For each Γ ∈ Σ0 , there exists cΓ ∈ R such that BΓ =



IΓ n

−1

(X − cΓ )n ,

(4)

n≥0

and for any Γ1 , Γ2 ∈ Σ0 , with Γ1 ⊆ Γ2 , we have IΓ1 ⊇ IΓ2 and cΓ2 −cΓ1 ∈ IΓ1 . Moreover, B = lim BΓ . −→ Γ ∈Σ 0

The following result follows from [2, Theorem 7.12].

(5)

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Proposition 3.2. Suppose that B is faithfully flat over R. Then the following are equivalent: (I) (II) (III) (IV)

B is finitely generated over R. B[1/a] is a finitely generated R[1/a]-algebra for some a(= 0) ∈ R. Δ0 is a finite (possibly empty) set. B∼ = SymR (J) for some invertible ideal J of R.

The following result occurs in [3, Corollary 3.11] (see also [2, Theorem 3.10]). Theorem 3.3. Let R be a Noetherian normal local domain with maximal ideal m. Let B be a faithfully flat R-algebra which is locally A1 in codimension-one over R. Suppose that R/m  B/mB. Then B = R[1] . The next theorem is proved in [2, Theorem 3.7]. Theorem 3.4. Let R be a complete Noetherian normal local domain. Suppose that B is a faithfully flat R-algebra. Then B is Noetherian if and only if B ∼ = R[X]. We shall deduce certain properties of B from the study of Bf where f ∈ B \R. For this purpose, we define certain ideals of R associated to an element f ∈ B \ R. Throughout the rest of this section, we fix an element f ∈ B \ R.  BΓ , where BΓ = n≥0 (IΓ n )−1 (X − cΓ )n . Set By Theorem 3.1, B = lim −→ Γ ∈Σ 0

xΓ := X − cΓ . For any Γ ∈ Σ0 such that f ∈ BΓ , we have f = a0Γ + a1Γ xΓ + · · · + adΓ xΓ d ,

 −1 where aiΓ ∈ IΓ i for 0 ≤ i ≤ d.

(6)

Corresponding to Γ (and f ), we now define two ideals of R as follows:   CΓ (f ) := a1Γ IΓ , · · · , adΓ IΓ d R and   CΓ (f ) := a0Γ , a1Γ IΓ , · · · , adΓ IΓ d R = a0Γ R + CΓ (f ). The above notation will be fixed throughout this section. We now prove a few properties of the ideals CΓ (f ) and CΓ (f ). Lemma 3.5. Suppose that CΓ (f ) = R for some Γ (∈ Σ0 ) such that f ∈ BΓ . Then, for all  > 0, CΓ (f  ) = R. Proof. Without loss of generality, we may assume that R is a local domain with maximal ideal m. Since Γ is fixed, we denote aiΓ by ai . Let f  = b0 + b1 xΓ + · · · + bm xΓ m . Since

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CΓ (f ) = R, there exists a least integer r such that ar IΓ r = R. Then br = ar  + u for some u satisfying uIΓ r ⊆ m. Thus br IΓ r = R and hence CΓ (f  ) = R. 2 The next lemma considers the case when Γ is a singleton {P }. Lemma 3.6. Suppose that Bf is faithfully flat over R. Then C{P} (f  )RP = RP for every  > 0 and P ∈ Δ. Proof. Fix P ∈ Δ and set e := e(P ), c := c{P } and ai := ai{P } . We have BP = RP [ X−c pe ], where pRP = P RP . Now f = a0 + a1 (X − c) + · · · + ad (X − c)d , where ai ∈ p−ei RP for 0 ≤ i ≤ d, and C{P} (f )RP = (a0 , a1 pe , · · · , ad ped )RP . Since (Bf )P is faithfully flat over RP , we have P (Bf )P = (Bf )P . Hence f ∈ / P BP = pBP . Thus, C{P} (f )RP  pRP (= P RP ), i.e., C{P} (f )RP = RP . Hence the result follows by Lemma 3.5. 2 We now prove a crucial observation. Lemma 3.7. Suppose that Bf is faithfully flat over R. Let Γ1  Γ2 in Σ0 be such that f ∈ BΓ1 ⊆ BΓ2 and let P ∈ Γ2 \ Γ1 . Then CΓ 2 (f ) ⊆ P and a0Γ2 = 0 (where a0Γ2 is as defined in (6)). Proof. For simplicity of notation, we write aiΓ1 as ai and aiΓ2 as bi for 0 ≤ i ≤ d, xΓj as xj , cΓj as cj and IΓj as Ij for j = 1, 2. Writing f as an element in BΓ1 and BΓ2 , we have f = a0 + a1 x1 + · · · + ad x1 d = b0 + b1 x2 + · · · + bd x2 d

(7)

where ai ∈ (I1 i )−1 and bi ∈ (I2 i )−1 for 0 ≤ i ≤ d. Since x1 = X − c1 = X − c2 + (c2 − c1 ) = x2 + (c2 − c1 ), we have, from (7), for 0 ≤ i ≤ d,

d bi = ai + (i + 1)ai+1 (c2 − c1 ) + · · · + ad (c2 − c1 )d−i . i

(8)

Now, as ai ∈ (I1 i )−1 and I1 RP = RP , we have ai ∈ RP for all i, 0 ≤ i ≤ d and hence by (8), bi ∈ RP , for all i, 0 ≤ i ≤ d. Therefore, CΓ 2 (f )RP = (b1 I2 , · · · , bd I2 d )RP ⊆ P RP and hence CΓ 2 (f ) ⊆ P . As CΓ 2 (f )RP = C{P} (f )RP = RP by Lemma 3.6, it follows that b0 ∈ / P ; in particular, a0Γ2 (= b0 ) = 0. 2 The proof of the following lemma follows from the argument in [6, Lemma 2.8].

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Lemma 3.8. For a torsion-free R-module M , the following conditions are equivalent:  (I) M = P ∈Δ MP , where M and MP = M ⊗R RP are identified with their images in M ⊗R K. (II) For every a, b ∈ R such that (aR : b) = aR, we have (aM : b) = aM . In particular, if M is either R-flat or a direct limit of finitely generated reflexive  R-modules, then M = P ∈Δ MP . We now prove a special case of our result that if Bf is faithfully flat over R then so is B.  r −1 Lemma 3.9. Let D = xr , where I is a divisorial ideal of R and x is an r≥0 (I ) indeterminate over R. Let g = b0 + b1 x + · · · + b x ∈ D, where b0 ∈ R∗ and br ∈ (I r )−1 for 1 ≤ r ≤ . Then the following statements hold:  (i) Suppose that gvi = j cij wj for some cij ∈ R, vi , wj ∈ D, 1 ≤ i ≤ m, 1 ≤ j ≤ q.  Then there exist zj ∈ D, 1 ≤ j ≤ q, such that vi = 1≤j≤q cij zj for each i. (ii) If Dg is faithfully flat over R, then D is faithfully flat over R. Proof. (i) For each 1 ≤ i ≤ m and 1 ≤ j ≤ q, we write vi and wj in D = as 

vi =

vir xr

and wj =

0≤r≤N





r≥0 (I

r −1 r

)

x

wjr xr ,

0≤r≤N +

where vir ∈ (I r )−1 , wjr ∈ (I r )−1 and N is the maximum of the x-degree of the vi ’s  and . Comparing the coefficients of xr in the relation gvi = j cij wj , we see that b0 vi0 =



cij wj0 ,

j

b0 vir + b1 vi(r−1) + · · · + br vi0 =



cij wjr

for 1 ≤ r ≤  and

cij wjr

for  < r ≤ N.

j

b0 vir + b1 vi(r−1) + · · · + b vi(r−) =

 j

For each j, 1 ≤ j ≤ q, set zj0 := wj0 /b0 , zjr := (wjr − b1 zj(r−1) − · · · − br zj0 )/b0 zjr := (wjr − b1 zj(r−1) − · · · − b zj(r−) )/b0

for 1 ≤ r ≤ , for  < r ≤ N

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and zj :=



zjr xr .

0≤r≤N

Since b0 is a unit, we have, for each j, zj0 ∈ R, zjr ∈ (I r )−1 for 1 ≤ r ≤ N and zj ∈ D.   It is now easy to see that, for each i, vir = j cij zjr for 0 ≤ r ≤ N and vi = j cij zj . (ii) Since D → Dg , it is enough to show that D is a flat R-module. Now, by [8,  Theorem 7.6], it is enough to show that if i ai vi = 0, where ai ∈ R and vi ∈ D, then   there exist cij ∈ R and zj ∈ D such that i ai cij = 0 for each j and vi = j cij zj for  each i. Since Dg is flat over R, there exist cij ∈ R and yj ∈ Dg such that i ai cij = 0  for each j and vi = j cij yj for each i. Hence there exists an integer n ≥ 0 such that,  for each i, g n vi = j cij wj for some wj ∈ D. Now g n = e0 + e1 x + · · · + en xn for some er ∈ (I r )−1 , 0 ≤ r ≤ n. Since e0 (= bn0 ) is a unit, the result follows by (i). 2 We shall next show (Proposition 3.12) that the hypothesis b0 ∈ R∗ can be dropped from Lemma 3.9 (ii). For this we require the following lemma, the proof of which is a modification of [3, Lemma 4.2] for our general setup. Lemma 3.10. Let E be a faithfully flat R-algebra such that E ⊗R K = K[X, 1/f (X)]. Let g(X) ∈ K[X] be such that g(X) | f (X) in K[X]. Let C1 := E ∩ Kg(X) and D1 := E ∩Kg(X)−1 be R-submodules of E. Suppose that, for each P ∈ Δ, (C1 )P and (D1 )P are free R-modules of rank one such that (C1 )P (D1 )P = RP . Then C1 and D1 are finitely generated projective R-modules of rank one. Proof. Let ψ1 : C1 ⊗R D1 → E be the canonical map sending g ⊗ h to gh for g ∈ C1 and h ∈ D1 and set J1 := ψ1 (C1 ⊗R D1 ). Then J1 ⊆ E ∩ K = R. Since C1 → Kg(X) 1 ]. Thus, C1 ⊗R E is and E is R-flat, we have C1 ⊗R E → Kg(X) ⊗R E ∼ = K[X, f (X) a torsion-free E-module of rank one. Now if the canonical map C1 ⊗R E → C1 E is not injective then the kernel of this map is a non-zero torsion-free proper E-submodule of C1 ⊗R E, which contradicts that the rank of C1 ⊗R E is one. Thus, the canonical map C1 ⊗R E → C1 E is an isomorphism. Since, for each P ∈ Δ, C1 D1 RP = J1 RP = RP , there exist non-zero elements a, b ∈ J1  such that b is R/aR-regular. Since C1 = E ∩ Kg(X) and E is R-flat, C1 = P ∈Δ (C1 )P by Lemma 3.8. Therefore, again by Lemma 3.8, b is (C1 /aC1 )-regular. Since E is R-flat and C1 ⊗R E ∼ = C1 E, it follows that b is (C1 E/aC1 E)-regular, i.e., (aC1 E : b) = aC1 E. Since D1 E ⊆ E, we have J1 E ⊆ C1 E. Thus a, b ∈ C1 E and hence ab ∈ aC1 E. Therefore a ∈ (aC1 E : b) = aC1 E. Thus C1 E = E. Since C1 ⊗R E(∼ = C1 E = E) is a free E-module of rank one and E is faithfully flat over R, it follows that C1 is a finitely presented flat and hence a projective R-module of rank one. 2 For convenience, we state below an easy lemma:

24

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Lemma 3.11. Let D be a ring and E be a D-algebra. Suppose that Ea and Eb are flat over D for some a, b ∈ E. Then Ea+b is flat over D. / P or b ∈ / P ; say Proof. Fix P ∈ Spec(Ea+b ). Since a + b is a unit in Ea+b , either a ∈ a∈ / P . Then (Ea+b )P is a localisation of the ring Ea and hence (Ea+b )P is flat over D. Since flatness is a local property, it follows that Ea+b is flat over D. 2  Proposition 3.12. Suppose that B = r≥0 (I r )−1 xr , where I is a divisorial ideal of R and x is an indeterminate over R. If Bf is faithfully flat over R then B is faithfully flat over R. Proof. Since faithful flatness is a local property, it is enough to assume that R is a local domain with maximal ideal m. We prove the result by induction on the dimension of the local ring R. If dim R = 1, then I is a principal ideal of R and hence B is faithfully flat over R. Recall that f = a0 + a1 x + · · · + ad xd where ai ∈ (I i )−1 for 0 ≤ i ≤ d. If a0 ∈ R∗ , then B is faithfully flat over R by Lemma 3.9 (ii). So we assume that a0 ∈ m. For convenience, set a := a0 . We first show that Bf −a is faithfully flat over R. Suppose that a = 0. Then, dim Ra < dim R. By applying induction hypothesis on the local rings of Ra , we see that Ba is faithfully flat over Ra and hence flat over R. Since Bf and Ba are both flat over R, by Lemma 3.11, Bf −a is flat over R. Since Bf is faithfully flat over R, f r ∈ / mB for any integer r > 0. Hence, as a ∈ m, it follows that (f − a)r ∈ / mB for any integer r > 0. Therefore, Bf −a is faithfully flat over R. Now, replacing f by f − a, we assume that a = 0. Set E := Bf . Then E is faithfully flat over R, E ⊗R K = K[x, 1/f ] and f ∈ xK[x]. Set C1 := E ∩ Kx and D1 := E ∩ Kx−1 . We now show that C1 = I −1 x. Note that I −1 x = B ∩ Kx ⊆ C1 . Let λx ∈ C1 . Then there exists an integer  ≥ 0 such that f  λx ∈ B. Fix a prime ideal P ∈ Δ. Then  C{P} (f  )RP = RP by Lemma 3.6. Hence λx ∈ BP . As I is divisorial, B = P ∈Δ BP by Lemma 3.8. Therefore, λx ∈ B. Thus, I −1 x = C1 . We now show that C1 is a finitely generated flat R-module of rank one. This will imply that I −1 is a finitely generated flat R-module, which will then imply that B is faithfully flat over R. Since I is a divisorial ideal, IP is principal for each P ∈ Δ. Hence, for each P ∈ Δ, EP = RP [yP , 1/fP (yP )], where yP = (I −1 )P x and fP (yP )(= f ) ∈ yP RP [yP ]. Note that for each P ∈ Δ, (C1 )P = EP ∩ Kx = RP yP and (D1 )P = EP ∩ Kx−1 = RP yP −1 are free RP -modules of rank one. Hence, (J1 )P = RP for each P ∈ Δ. Therefore, by Lemma 3.10, C1 = E ∩ Kx is a finitely generated flat R-module of rank one. Hence we are done. 2 We prove another special case of our result relating faithful flatness of B with that of Bf .

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Lemma 3.13. Suppose that Bf is faithfully flat over R. Assume further that there exists Γ0 ∈ Σ0 for which f ∈ BΓ0 and a0Γ ∈ R∗ for every Γ (∈ Σ0 ) ⊇ Γ0 . Then B is faithfully flat over R. Proof. Since B → Bf , it is enough to show that B is flat over R. By [8, Theorem 7.6],  it is enough to show that if i ai vi = 0, where ai ∈ R and vi ∈ B, then there exist   cij ∈ R and zj ∈ B such that i ai cij = 0 for each j and vi = j cij zj for each i.  Since Bf is flat over R, there exist cij ∈ R and yj ∈ Bf such that i ai cij = 0 for  each j and vi = j cij yj for each i. Hence there exists an integer  ≥ 0, such that for  each i, f  vi = j cij wj for some wj ∈ B. Since B = lim B by Theorem 3.1, we −→ Γ ∈Σ0 Γ can choose Γ ∈ Σ0 such that f, vi , wj ∈ BΓ for each i, j and Γ ⊇ Γ0 . Note that BΓ =  r −1 xΓ r , where IΓ is a divisorial ideal of R. Let f  = b0 + b1 xΓ + · · · + b xΓ  , r≥0 (IΓ ) where bm ∈ (IΓ m )−1 for 0 ≤ m ≤ . Since a0Γ is a unit in R, we have that b0 is a unit in R. Hence the result follows by Lemma 3.9 (i). 2 We now prove Theorem D. Theorem 3.14. Let R be a Noetherian normal domain and B be a semi-faithfully flat R-algebra which is locally A1 in codimension-one. Suppose that Bf is faithfully flat over R for some f ∈ B \ R. Then the following assertions hold: (i) B is faithfully flat over R. (ii) If Bf is finitely generated over R then B is finitely generated over R. (iii) If Bf is Noetherian then B is Noetherian. Proof. (i) If B = BΓ for some Γ ∈ Σ0 , then B is faithfully flat over R by Proposition 3.12. So we assume that B = BΓ for any Γ ∈ Σ0 . Since faithful flatness is a local property, it is enough to assume that R is a local domain with maximal ideal m. We prove this result by induction on the dimension of the local ring R. If dim R = 1, then clearly B is faithfully flat over R. Since B = BΓ for any Γ ∈ Σ0 , there exists Γ1 , Γ2 ∈ Σ0 such that Γ1  Γ2 and f ∈ BΓ1 ⊆ BΓ2 . Let f = a0Γ2 + a1Γ2 xΓ2 + · · · + adΓ2 xΓ2 d . Set ai := aiΓ2 for 0 ≤ i ≤ d. We note that by Lemma 3.7, a0 = 0. We show that a0 ∈ R∗ . Suppose, if possible, that a0 ∈ m. Then, dim Ra0 < dim R. Hence Ba0 is faithfully flat over Ra0 by applying induction hypothesis on the dimension of the local rings of Ra0 . Since Bf and Ba0 are flat over R, we have that Bf −a0 is flat over R by Lemma 3.11. Since Bf is faithfully flat over R, f r ∈ / mB for any integer r > 0, and since a0 ∈ m, we have (f − a0 )r ∈ / mB for any integer r > 0. Hence, Bf −a0 is faithfully flat over R. In particular, (Bf −a0 )P is faithfully flat over RP for P ∈ Γ2 \ Γ1 . Set e := e(P ). Let P RP = pRP for some p ∈ R. Then, BP = RP [XP ],

where XP = (X − cΓ2 )/pe = xΓ2 /pe .

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Now f − a0 = a1 pe XP + · · · + ad pde XP d and hence (a1 pe , . . . , ad pde )RP = CΓ 2 (f − a0 ) × RP = CΓ 2 (f )RP which is contained in P RP by Lemma 3.7. But this contradicts the fact that (Bf −a0 )P is faithfully flat over RP . Hence, for every Γ  Γ1 , a0Γ ∈ R∗ . Therefore, by Lemma 3.13, B is faithfully flat over R. (ii) By Theorem 3.1, B = lim B . Since B[f −1 ] = lim B [f −1 ] is a finitely −→ Γ ∈Σ0 Γ −→ Γ ∈Σ0 Γ −1 generated R-algebra, there exists Γ0 ∈ Σ0 such that B[f ] = BΓ0 [f −1 ]. We show that B = BΓ0 . If not, there exists Γ1 ∈ Σ0 such that Γ0  Γ1 . Fix P ∈ Γ1 \ Γ0 . Set c1 := cΓ1 and e := e(P ). Now  (BΓ0 )P = RP [X],

(BΓ1 )P = RP

X − c1 pe

 where e > 0.

1 −1 1 Hence, BP [f −1 ] = (BΓ0 [f −1 ])P = RP [X, f −1 ] = RP [ X−c ] = pe , f ]. Since BP [f −1 RP [X, f ] is faithfully flat over RP , f ∈ / P B. Hence, tr.degR/P Bf /P Bf > 0. But 1 as X−c ∈ BP [f −1 ], the image of X in (Bf )P /P (Bf )P is algebraic over RP /P RP and pe hence tr.degR/P Bf /P Bf = 0. This is a contradiction. Thus B = BΓ0 . Therefore, B is faithfully flat over R by (i) and hence B is finitely generated over R by Proposition 3.2. (iii) Since Bf is faithfully flat over R, B is faithfully flat over R by (i). Moreover, by Theorem 3.3, for any prime ideal P of R, either BP ∼ = RP [X] or BP /P BP = RP /P RP . The result now follows from (i) and Proposition 2.3. 2

As a consequence, we deduce the following: Proposition 3.15. Let R be a Noetherian normal domain and B be a semi-faithfully flat R-algebra which is locally A1 in codimension-one over R. Suppose that f is an element of B \R for which Bf is faithfully flat over R. Then the following statements are equivalent: (I) (II) (III) (IV)

B∼ = SymR (J) for some invertible ideal J of R. Bf is a finitely generated R-algebra. Bf [1/a] is a finitely generated R[1/a]-algebra for some a(= 0) ∈ R. The image of f in B ⊗R k(P ) is transcendental over k(P )(= RP /P RP ) for all but finitely many prime ideals P of height one.

Proof. Clearly (I) =⇒ (II) =⇒ (III). (III) =⇒ (I): By Theorem 3.14 (i), B is faithfully flat over R and B[1/a] is finitely generated R[1/a]-algebra. Hence, by Proposition 3.2, B ∼ = SymR (J) for some invertible ideal J of R. (I) =⇒ (IV): We have B ⊗R K = K[X], where K is a field of fractions of R, X ∈ B and f = a0 + a1 X + · · · + ad X d for some ai ∈ K, 0 ≤ i ≤ d, d ≥ 1. By Proposition 3.2, Δ0 = {P ∈ Δ | RP [X]  BP } is a finite set. Let Δ := {P ∈ Δ | ai ∈ P RP or ai −1 ∈ P RP , 0 ≤ i ≤ d}. Since R is a Noetherian domain, Δ is a finite set. Now for P ∈ ∗ Δ \ (Δ0 ∪ Δ ), BP = RP [X], f ∈ RP [X] and ai ∈ RP for 0 ≤ i ≤ d. Hence, for  P ∈ Δ \ (Δ0 ∪ Δ ), the image of f in B ⊗R k(P )(= k(P )[X]) is transcendental over k(P ).

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(IV) =⇒ (I): We show that Σ0 is a finite set. By Lemma 3.7, if Γ1  Γ2 , such that f ∈ BΓ1 ⊆ BΓ2 , then for every prime ideal P ∈ Γ2 \ Γ1 , CΓ 2 (f ) ⊆ P R, and hence, for P ∈ Γ2 \ Γ1 , the image of f in B ⊗R k(P ) is algebraic over k(P ). Now hypothesis (IV) implies that there exists only finitely many such prime ideals P . Hence, Σ0 is a finite set and B = BΓ for some Γ . By Theorem 3.14 (i), B is faithfully flat over R and hence B is finitely generated over R by Proposition 3.2. 2 We now deduce an analogue of Theorem 3.3. Proposition 3.16. Let (R, m) be a Noetherian normal local domain and B be a semifaithfully R-algebra which is locally A1 in codimension-one over R. Suppose that f is an element of B \ R for which Bf is faithfully flat over R. Then the following are equivalent: 1 (I) Bf ∼ ] for some g(X) ∈ R[X] \ R. = R[X, g(X) (II) Bf is finitely generated over R. (III) R/m  Bf /mBf .

Moreover, if R is complete and Bf is Noetherian then Bf is finitely generated over R. Proof. Clearly (I) =⇒ (II) and (I) =⇒ (III). (II) =⇒ (I): By Theorem 3.14 (i), B is a faithfully flat finitely generated R-algebra. Hence, by Proposition 3.2, B ∼ = R[X]. Therefore (I) holds. (III) =⇒ (I): By Theorem 3.14 (i), B is faithfully flat over R. Since R/m is a field, we have R/m  B/mB. Hence, by Theorem 3.3, B ∼ = R[X]. Thus (I) holds. Now suppose that R is complete and Bf is Noetherian. Then B is Noetherian by Theorem 3.14 (iii) and hence B is finitely generated over R by Theorem 3.4. 2 4. On algebras with punctured fibres over a discrete valuation ring Let (R, t) be a discrete valuation ring with K = R[1/t] and A an R-algebra whose 1 generic fibre is of the form K[X, f (X) ]. We shall describe (Proposition 4.2) minimal sufficient condition on the closed fibre A/tA for which A itself is of the form R[X, 1/g(X)]. We shall also record a technical result (Lemma 4.5) over a UFD which will be used in the proof of our main theorems. We first quote a result from [6, Theorem 2.4]. Theorem 4.1. Let R ⊆ B be integral domains. Let t ∈ R be a prime element in R such that (i) B[1/t] = R[1/t][X], where X is transcendental over R. (ii) B/tB is an integral domain and tB ∩ R = tR.

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(iii) R/tR is algebraically closed in B/tB. (iv) tr.degR/tR (B/tB) > 0. Then B = R[1] . We now prove an analogue of Theorem 4.1 for algebras whose generic fibres are punctured lines. Proposition 4.2. Let R ⊆ A be integral domains. Let t ∈ R be a prime element in R such that 1 (i) A[1/t] = R[1/t][X, f (X) ], where X is transcendental over R and f (X) ∈ R[1/t][X] \ R[1/t]. (ii) A/tA is an integral domain and tA ∩ R = tR. (iii) R/tR is algebraically closed in A/tA. (iv) tr.degR/tR (A/tA) > 0. 1 Then A ∼ ] for some g(X) ∈ R[X] \ R. = R[X, g(X)

Proof. Let B = A ∩ R[1/t][X]. We first show that (I) (II) (III) (IV) (V)

B[1/t] = R[1/t][X]. B/tB is an integral domain and tB ∩ R = tR. R/tR is algebraically closed in B/tB. tr.degR/tR (B/tB) > 0. There exists h ∈ B such that A = B[ h1 ].

(I) is obvious; (II) follows from (ii) and the relation tB = tA ∩ R[1/t][X] = tA ∩ B; (III) follows from (iii). Without loss of generality, we may assume that X ∈ A and let p(X) ∈ R[X] be such 1 tn that A[1/t] = R[1/t][X, p(X) ]. Let n ∈ Z≥0 be the least integer such that p(X) ∈ A. Set h :=

p(X) tn .

1 Then h−1 ∈ A. Now if n = 0, then h = p(X) ∈ A and we have R[X, p(X) ] ⊆ A. −1

n−1

t Suppose that n > 0. Note that h t = p(X) ∈ / A by minimality of n. Thus, t  h−1 in A. n −1 Since t is a prime in A and t = h p(X), it follows that tn | p(X) in A. Thus h ∈ A. Therefore,       1 1 p(X) 1 ⊆ A ⊆ R[1/t] X, = R[1/t] X, , where h = n . R X, h, h p(X) h t

To see (V), note that h ∈ B and B[ h1 ] ⊆ A. Moreover, if u ∈ A, then there exists  ∈ Z≥0 such that h u ∈ R[1/t][X] ∩ A(= B), so that u ∈ B[ h1 ]. Thus (V) holds. Now (IV) follows from (iv), (V) and the fact that t  h in A. By Theorem 4.1, B ∼ = R[X] for some X transcendental over R and hence A ∼ = 1 R[X, g(X) ] for some g(X) ∈ R[X] \ R. 2

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Remark 4.3. The following examples show the necessity of the hypotheses (iii) and (iv) respectively in Proposition 4.2. 2 R[X,X1 ,X2 ] (1) Let R = R[[t]] and A = R[X, X t+1 , X 2t+1 ] ∼ . Then A[1/t] = = (tX1 −X 2 −1,X 1 X2 −1) 1 2 ∼ R[1/t][X, X 2 +1 ], A/tA = (R/tR)[X, X1 , X2 ]/(X + 1, X1 X2 − 1) = C[X1 , X11 ], which is an integral domain and tr.degC (A/tA) > 0 but R/tR(= R) is not algebraically closed in A/tA. (2) Let R = C[[t]] and A = R[X, X/t, X/t2 , . . . , X/tn , . . . , 1/(X − 1)]. Then A[1/t] = C((t))[X, 1/(X − 1)], A/tA = C which is an integral domain, R/tR(= C) is algebraically closed in A/tA but tr.degC (A/tA) = 0. As a consequence of Proposition 4.2, we have the following corollary. Corollary 4.4. Let R be a Noetherian normal domain with field of fractions K and A be a faithfully flat R-algebra such that: 1 (i) A ⊗R K = K[X, f (X) ] for some indeterminate X over K and f (X) ∈ K[X] \ K. (ii) For each prime ideal P of height one, A ⊗R k(P ) ∼ = k(P )[X, fP 1(X) ] for some fP (X) ∈ k(P )[X].

Then, AP ∼ = RP [X, fP 1(X) ] for some fP (X) ∈ RP [X] for each height one prime ideal P of R. Let R be a ring and U be a transcendental element over R. For a polynomial f (U ) ∈ R[U ], the notation C(f (U )) will denote the ideal of R generated by the coefficients of f . Lemma 4.5. Let R be a unique factorisation domain and let A be a faithfully flat R-algebra 1 1 such that A = R[W, g(W ) ]. Suppose that A ⊗R K = K[X, f (X) ]. Let B := A ∩ K[X]. Then there exists U ∈ A such that (I) (II) (III) (IV)

K[U ] = K[X]. 1 A = R[U, h(U ) ] for some h(U ) ∈ R[U ]. B = R[U ]. A ∩ K(f (X))n = R(h(U ))n ∀n ∈ Z.

1 1 Proof. Since A ⊗R K = K[X, f (X) ] = K[W, g(W ) ], we have K(X) = K(W ). Thus aW +b X = cW +d for some a, b, c, d ∈ K such that ad − bc = 0. Replacing X by λX for some λ ∈ K, we may assume that a, b, c, d ∈ R, such that gcd(a, b) = gcd(c, d) = 1 and ad − bc = μ ∈ R \ 0. If c = 0 then X = (aW + b)/d and hence K[X] = K[W ]. Then setting U := W , we are through. Now suppose that c = 0. We first note that since A is faithfully flat over R, +b C(g(W )) = R. Using X = aW cW +d and ad − bc = μ, we have

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cX − a =

−μ . cW + d

1 Thus cW + d is a unit in K[W, g(W ) ], and hence (cW + d) | g(W ) in K[W ]. Since R is a UFD, gcd(c, d) = 1 and C(g(W )) = R, we have (cW + d) | g(W ) in R[W ]. Moreover, since C(g(W )) = R, we have (c, d)R = R. Let c , d ∈ R be such that cc − dd = 1. Set  W +c U := dcW +d . Since cW + d is a unit in A, we have U ∈ A and

cU − d = Moreover, W =

c −dU cU −d .

a − cX 1 = cW + d μ

and hence K[X] = K[U ].

Thus,

 R[W ] ⊆ R U,



  1 1 ⊆ A = R W, . cU − d h(W )

Now let h(U ) ∈ R[U ] be such that gcd of the coefficients of h(U ) is 1 and h(U )K = f (X)K. Hence  A ⊗R K = K U,

1 h(U )



 = K X,

1 f (X)



 = K W,

 1 . g(W )

1 Therefore, there exists s ∈ R such that s/h(U ) ∈ R[W, g(W ) ]. Then, s/h(U ) =  p(W )/(g(W )) for some p(W ) ∈ R[W ] and  ≥ 0. Since the gcd of coefficients of 1 h(U ) is 1, it follows that s | p(W ) in R[W ] and hence 1/h(U ) ∈ R[W, g(W ) ]. Similarly, 1 1 1 1/g(W ) ∈ R[U, h(U ) ]. Hence, A = R[U, h(U ) ] = R[W, g(W ) ]. We now show that B = R[U ]. Note that U ∈ B = A ∩ K[X] = A ∩ K[U ] = 1 R[U, h(U ) ] ∩ K[U ]. Suppose that p(U )/s ∈ A ∩ K[U ] for some p(U ) ∈ R[U ] and s ∈ R. 

Then p(U )h(U ) /s ∈ R[U ] for some  ≥ 0. Since A is faithfully flat over R, C(h(U )) = 1, and hence s | p(U ) in R[U ]. Thus B = R[U ]. Since h(U )K = f (X)K, we have 1 n n A ∩ K(f (X))n = A ∩ K(h(U ))n = R[U, h(U ) ] ∩ K(h(U )) = R(h(U )) for each integer n ∈ Z. 2 5. Main theorems In this section we shall prove Theorems A, B and C. Recall that R denotes a Noetherian normal domain, K its field of fractions and Δ the set of all height one prime ideals of R. Throughout this section A will denote a faithfully flat R-algebra such that  A ⊗R K = K X,

1 f (X)



for some indeterminate X over K and f (X) ∈ K[X]\K. To the R-algebra A, we associate the following objects:

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C1 := A ∩ Kf (X), D1 := A ∩ Kf (X)−1 , B := A ∩ K[X] and   I := A ∩ f (X)K[X] = B ∩ f (X)K[X] . We note that, for each f ∈ C1 and g ∈ D1 , f g ∈ A ∩ K = R. Therefore we get an R-linear map ψ1 : C1 ⊗R D1 → R defined by ψ1 (f ⊗ g) = f g. Set J1 := ψ1 (C1 ⊗R D1 ). For convenience, we prove below a part of Theorem A separately: Lemma 5.1. Let the notation be as above. Assume further that for each P ∈ Δ, AP = 1 RP [XP , fP (X ] for some fP (XP ) ∈ RP [XP ]. Then the following statements hold: P) (i) C1 and D1 are finitely generated projective R-modules of rank one. (ii) J1 = R. (iii) I = C1 B and hence an invertible ideal of B. Proof. (i) By Lemma 4.5, it follows that for each P ∈ Δ, there exists UP ∈ AP such that AP = RP [UP , 1/hP (UP )] for some hP (UP ) ∈ RP [UP ], BP = RP [UP ], (C1 )P = RP hP (UP ), (D1 )P = RP (hP (UP ))−1 and so (J1 )P = RP . Hence, by Lemma 3.10, C1 and D1 are finitely generated projective R-modules of rank one. (ii) Since by (i), C1 and D1 are finitely generated projective R-modules of rank one, we have that C1 ⊗R D1 is also a finitely generated projective R-module of rank one. Hence the surjective map ψ1 : C1 ⊗R D1 → J1 is an isomorphism. Thus J1 is R-projective of rank one, i.e., an invertible ideal of R. Since, for each P ∈ Δ, (J1 )P = RP , we have ht(J1 ) ≥ 2, and hence J1 = R. (iii) It is enough to show that for each maximal ideal m of R, Im = C1 Bm . Let m be a maximal ideal of R. Replacing R by Rm and B by Bm , we may assume that R is a local ring. By (i), C1 is a finitely generated projective R-module of rank one. Hence, there exists f ∈ A such that C1 = Rf . Since, J1 = R by (ii), we have D1 = Rf −1 . Hence f ∈ A∗ . Now clearly, f ∈ I = A ∩ f (X)K[X]. Let g ∈ A ∩ f (X)K[X]. Since Kf (X) = Kf and f ∈ A∗ , we have g/f ∈ A ∩ K[X], i.e., g/f ∈ B. Hence g ∈ f B. 2 We now prove Theorem A.

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Theorem 5.2. Let R be a Noetherian normal domain with field of fractions K and A be a faithfully flat R-algebra such that: 1 (i) A ⊗R K = K[X, f (X) ] for some X transcendental over K and f (X) ∈ K[X] \ K. 1 (ii) For each P ∈ Δ, AP = RP [XP , fP (X ] for some XP transcendental over RP and P) fP (XP ) ∈ RP [XP ].

Let B = A ∩ K[X], C1 = A ∩ K(f (X)) and I = B ∩ f (X)K[X]. Then (1) C1 is a finitely generated projective R-module of rank one and I = C1 B and hence an invertible ideal of B. (2) A = B[I −1 ]. (3) B(= A ∩ K[X]) is a faithfully flat locally polynomial algebra in codimension-one. (4) There exist c1 , · · · , cn ∈ R such that (c1 , · · · , cn )R = R, IAci = fi Aci for some fi ∈ Aci and Aci = Bci [fi −1 ] for 1 ≤ i ≤ n. (5) A is finitely generated over R if and only if B is finitely generated over R. (6) A is Noetherian if and only if B is Noetherian. Proof. (1) follows from Lemma 5.1 (i) and (iii). (2) Since I = C1 B is an invertible ideal of B, it is enough to prove the case when C1 is a free R-module, say C1 = Rf for some f ∈ A. Since J1 = R by Lemma 5.1 (ii), it follows that D1 = Rf −1 and hence f −1 ∈ A. Therefore, B[f −1 ] ⊆ A. Since Kf = K(f (X)), for any g ∈ A, there exists an integer n ≥ 0 such that f n g ∈ A ∩ K[X] = B, and hence A = B[f −1 ]. (3) By Lemma 4.5, it follows that for each P ∈ Δ, BP is a polynomial algebra in one  variable over RP . Since A is faithfully flat over R, we have A = P ∈Δ AP by Lemma 3.8 and N A ∩ R = N for any ideal N of R. Therefore, since B = A ∩ K[X], we have   B = P ∈Δ (AP ∩ K[X]) = P ∈Δ BP and N B ∩ R = N R for any ideal N of R. Thus B is a semi-faithfully flat R-algebra which is a locally polynomial algebra in codimension-one. We now show that B is faithfully flat over R. Since faithful flatness is a local property, it is enough to assume that R is a local domain with maximal ideal m. Since C1 is a finitely generated projective R-module of rank one, there exists f ∈ A such that C1 = Rf and hence A = Bf by (ii). Since A is faithfully flat over R and f ∈ B \ R, B is faithfully flat over R by Theorem 3.14 (i). (4) follows from (1) and (2). (5) and (6) Since A = B[I −1 ], where I is an invertible ideal of B, if B is finitely generated over R (respectively Noetherian), then clearly A is finitely generated over R (respectively Noetherian). Now suppose that A is finitely generated over R (respectively Noetherian). Let c1 , · · · , cn ∈ R be as in (4). Then Aci = Bci [fi −1 ] is a finitely generated Rci -algebra (respectively Noetherian). Hence, Bci is a finitely generated Rci -algebra (respectively Noetherian) by Theorem 3.14. Since (c1 , · · · , cn )R = R, B is a finitely generated R-algebra (respectively Noetherian). 2

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We now prove Theorem B. Theorem 5.3. Let (R, m) be a Noetherian normal local domain and A be as in Theorem 5.2. Then the following are equivalent: 1 A∼ ] for some g(X) ∈ R[X] \ R. = R[X, g(X) A is a finitely generated R-algebra. A[1/a] is a finitely generated R[1/a]-algebra for some a(= 0) ∈ R. The image of fP (XP ) in A ⊗R k(P ) is transcendental over k(P ) for all but finitely many prime ideals P of height one. (V) R/m  A/m.

(I) (II) (III) (IV)

1 Moreover, if R is complete and A is Noetherian then A ∼ ] for some g(X) ∈ = R[X, g(X) R[X] \ R.

Proof. Since C1 is a finitely generated projective R-module of rank one and R is local, there exists f ∈ A such that C1 = Rf . Hence, A = Bf , where B is a faithfully flat locally polynomial algebra in codimension-one. Thus the proof follows from Propositions 3.15 and 3.16. 2 We now prove Theorem C. Theorem 5.4. Let R be a Noetherian normal domain with field of fractions K and A be a faithfully flat R-algebra such that: 1 (i) A ⊗R K = K[X, f (X) ] for some X transcendental over K and f (X) ∈ K[X] \ K. (ii) For each P ∈ Δ, A ⊗R k(P ) ∼ = k(P )[X, fP 1(X) ] for some fP (X) ∈ k(P )[X] \ k(P ).

Then A ∼ = SymR (Q) for some invertible ideal Q of R and I is an = B[I −1 ], where B ∼ invertible ideal of B. In particular, A is finitely generated over R and, for every maximal 1 ideal m of R, Am ∼ ] for some gm (X) ∈ Rm [X]. = Rm [X, gm (X) Proof. By Corollary 4.4 and Theorem 5.2, we have A = B[I −1 ], where B(= A ∩ K[X]) is a faithfully flat locally polynomial algebra in codimension-one, I = C1 B, where C1 (= A ∩Kf (X)) is a finitely generated projective R-module of rank one. By Theorem 5.2 (4), we may assume that I is a principal ideal generated by f and hence A = Bf . By hypothesis (ii), the image of f in A ⊗R k(P ) is transcendental over k(P ) for each height one prime ideal P of R. Thus, by (IV) =⇒ (I) of Proposition 3.15, B ∼ = SymR (Q) for 1 some invertible ideal Q of R. Hence, for every maximal ideal m of R, Am ∼ ] = Rm [X, gm (X) for some gm (X) ∈ Rm [X]. 2

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Remark 5.5. (i) In view of the statement (IV) of Theorem 5.3, the condition “fP (X) ∈ k(P )[X] \ k(P ) for all P ∈ Δ” in (ii) of Theorem 5.4 can be relaxed as follows: “fP (X) ∈ / k(P ) for all but finitely many P ∈ Δ”. k(P )[X] for all P ∈ Δ and fP (X) ∈ (ii) [3, Example 6.4] provides an example of a faithfully flat algebra A over a Noetherian factorial domain R whose generic fibre is of the form K[X, 1/f (X)] for some f (X) ∈ K[X] \K but for which there are infinitely many P ∈ Δ satisfying A ⊗R k(P ) ∼ = k(P )[X]. Acknowledgments The author thanks Professor S.M. Bhatwadekar for the proof of Lemma 2.2. She also thanks Professor A.K. Dutta for carefully going through the draft. References [1] S.M. Bhatwadekar, A.K. Dutta, On A∗ -fibrations, J. Pure Appl. Algebra 149 (2000) 1–14. [2] S.M. Bhatwadekar, A.K. Dutta, N. Onoda, On algebras which are locally A1 in codimension-one, Trans. Amer. Math. Soc. 365 (9) (2013) 4497–4537. [3] S.M. Bhatwadekar, Neena Gupta, On locally quasi A∗ algebras in codimension-one over a Noetherian normal domain, J. Pure Appl. Algebra 215 (2011) 2242–2256. [4] S.M. Bhatwadekar, Neena Gupta, The structure of a Laurent polynomial fibration in n variables, J. Algebra 353 (1) (2012) 142–157. [5] A.K. Dutta, On A1 -bundles of affine morphisms, J. Math. Kyoto Univ. 35 (3) (1995) 377–385. [6] A.K. Dutta, N. Onoda, Some results on codimension-one A1 -fibrations, J. Algebra 313 (2007) 905–921. [7] Neena Gupta, N. Onoda, On finite generation of Noetherian algebras over two-dimensional regular local rings, preprint. [8] H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1990, reprint of the first paperback edition with corrections.