EXAMPLES OF INTEGRAL DOMAINS INSIDE

POWER SERIES RINGS

William Heinzer, Christel Rotthaus and Sylvia Wiegand

Abstract. We present examples of Noetherian and non-Noetherian integral domains which can be built inside power series rings. Given a power series ring R^{∗} over a Noetherian integral domain R and given a subfield L of the total quotient ring of R^{∗} with R ⊆ L, we construct subrings A and B of L such that B is a localization of a nested union of polynomial rings over R and B ⊆ A := L ∩ R^{∗}. We show in certain cases that flatness of a related map on polynomial rings is equivalent to the Noetherian property for B. Moreover if B is Noetherian, then B = A. We use this construction to obtain for each positive integer n an explicit example of a 3-dimensional quasilocal unique factorization domain B such that the maximal ideal of B is 2-generated, B has precisely n prime ideals of height two, and each prime ideal of B of height two is not finitely generated.

1. Introduction. This paper is a continuation of our study of a technique for

constructing integral domains by (1) intersecting a power series ring with a field to

obtain an integral domain A as in the abstract, and (2) approximating the domain

A with a nested union of localized polynomial rings to obtain an integral domain

B as in the abstract. Classical examples such as those of Akizuki [A] and Nagata

[N, pages 209-211] use the second (nested union) description of this construction.

It is possible to also realize these classical examples as the intersection domains of

the first description [HRW6].

In this paper we observe that, in certain applications of this technique, flatness

of a map of associated polynomial rings implies the constructed domains are Noe-

therian and that A = B. We also in the present paper apply this observation to the

construction of examples of both Noetherian and non-Noetherian integral domains.

We begin by describing the technique.

The authors would like to thank the National Science Foundation and the National Security Agency for support for this research. In addition they are grateful for the hospitality and cooperation of Michigan State, Nebraska and Purdue, where several work sessions on this research were conducted.

Typeset by AMS-TEX 2 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND

1.1 General Setting. Let R be a commutative Noetherian integral domain. Let a be a nonzero nonunit of R and let R^{∗} be the (a)-adic completion of R. Then R^{∗} is isomorphic to R[[y]]/(y − a), where y is an indeterminate; thus we consider R^{∗} as R[[a]], the “power series ring” in a over R. The intersection domain (type 1 above) and the approximation domain (type 2) of the construction are inside R^{∗}. Let τ_{1}, …, τ_{n} ∈ aR^{∗} be algebraically independent over the fraction field K of R and let \underline{τ} abbreviate the list τ_{1}, …, τ_{n}. By Theorem 2.2, also known as [HRW1, Theorem 1.1 ], A\underline{τ} := K(τ_{1}, …, τ_{n}) ∩ R^{∗} is simultaneously Noetherian and computable as a nested union B\underline{τ} of certain associated localized polynomial rings over R using \underline{τ} if and only if the extension T := R[\underline{τ}] := R[τ_{1}, …, τ_{n}] ↪^{ψ} R_{a}^{∗} is flat.

In the case where ψ : T ↪ R_{a}^{∗} is flat, so that the intersection domain A\underline{τ} is Noetherian and computable, we construct new “insider” examples inside A\underline{τ}. We choose elements f_{1}, …, f_{m} of T, considered as polynomials in the τ_{i} with coefficients in R and abbreviated by \underline{f}. Assume that f_{1}, …, f_{m} are algebraically independent over K; thus m ≤ n. If S := R[\underline{f}] := R[f_{1}, …, f_{m}] ↪^{φ} T = R[\underline{τ}] is flat, we observe in Section 3 that the “insider ring” A\underline{f} := K(\underline{f}) ∩ R^{∗} is Noetherian and computable; that is, A\underline{f} is equal to an approximating union B\underline{f} of localized polynomial rings constructed using the f_{i}. Moreover, we can often identify conditions on the map φ which imply B\underline{f} and A\underline{f} are not Noetherian. Thus the “insider” examples A\underline{f} and B\underline{f} are inside intersection domains A\underline{τ} known to be Noetherian; the new insider is Noetherian if the associated extension S → T of polynomial rings is flat. The insider examples are examined in more detail in Section 3.

In Section 2 we give background and notation for the construction and for flatness of polynomial extensions in greater generality: Suppose that \underline{x} := (x_{1}, …, x_{n}) is a tuple of indeterminates over R and that \underline{f} := (f_{1}, …, f_{m}) consists of elements of the polynomial ring R[\underline{x}] that are algebraically independent over K. We consider flatness of the following map of polynomial rings.

(1.2)  φ : S := R[\underline{f}] ↪ T := R[\underline{x}].

In Section 4 we continue the analysis of the flatness of (1.2) and the nonflat locus. We discuss results of [P], [W] and others.

DOMAINS INSIDE POWER SERIES RINGS 3

In Section 5 we present for each positive integer n an insider example B such that:

(1) B is a 3-dimensional quasilocal unique factorization domain,

(2) B is not catenary,

(3) the maximal ideal of B is 2-generated,

(4) B has precisely n prime ideals of height two,

(5) Each prime ideal of B of height two is not finitely generated,

(6) For every non-maximal prime P of B the ring B_{P} is Noetherian.

2. Background and Notation.

We begin this section by recalling some details for the approximation to the intersection domain A_{τ} of (1.1).

2.1 Notation for approximations. Assume that R, K, a, τ_{1}, …, τ_{n}, \underline{τ} and A\underline{τ} are as in General Setting 1.1. Then the (a)-adic completion of R is R^{∗} = R[[x]]/(x − a) = R[[a]]. Write each τ_{i} := ∑_{j=1}^{∞} b_{ij}a^{j}, with the b_{ij} ∈ R. There are natural sequences {τ_{ir}}_{r=0}^{∞} of elements in A, called the r^{th} endpieces for the τ_{i}, which “approximate” the τ_{i}, defined by:

(2.1.1)  For each i ∈ {1, …, n} and r ≥ 0, τ_{ir} := ∑_{j=r+1}^{∞}(b_{ij}a^{j})/a^{r}.

Now for each r, U_{r} := R[τ_{1r}, …, τ_{nr}] and B_{r} is U_{r} localized at the multiplicative system 1+aU_{r}. Then define U\underline{τ} := ∪_{r=1}^{∞}U_{r} and B\underline{τ} := ∪_{r=1}^{∞}B_{r}. Thus U\underline{τ} is a nested union of polynomial rings over R and B\underline{τ} is a nested union of localized polynomial rings over R. The definition of the U_{r} (and hence also of B_{r} and U\underline{τ} and B\underline{τ}) are independent of the representation of the τ_{i} as power series with coefficients in R [HRW1, Proposition 2.3].

The following theorem is the basis for our construction of examples.

2.2 Theorem. [HRW1, Theorem 1.1 ] Let R be a Noetherian integral domain with fraction field K. Let a be a nonzero nonunit of R. Let τ_{1}, …, τ_{n} ∈ aR[[a]] = aR^{∗} be algebraically independent over K, abbreviated by \underline{τ}. Let U\underline{τ} and B\underline{τ} be as in (2.1). Then the following statements are equivalent:

(1) A\underline{τ} := K(\underline{τ}) ∩ R^{∗} is Noetherian and A\underline{τ} = B\underline{τ}.

(2) U\underline{τ} is Noetherian.

4 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND

(3) B\underline{τ} is Noetherian.

(4) R[\underline{τ}] → R_{a}^{∗} is flat.

Since flatness is a local property, the following two propositions are immediate corollaries of [HRW5, Theorem 2.1]; see also [P, Théorème 3.15]. 2.3 Proposition. Let T be a Noetherian ring and suppose R ⊆ S are Noetherian subrings of T. Assume that R → T is flat with Cohen-Macaulay fibers and that R → S is flat with regular fibers. Then S → T is flat if and only if, for each prime ideal P of T, we have ht(P) ≥ ht(P ∩ S).

As a special case we have: 2.4 Proposition. Let R be a Noetherian ring and let x_{1}, …, x_{n} be indeterminates over R. Assume that f_{1}, …, f_{m} ∈ R[x_{1}, …, x_{n}] are algebraically independent over R. Then

(1) φ : S := R[f_{1}, …, f_{m}] ↪ T := R[x_{1}, …, x_{n}] is flat if and only if, for each prime ideal P of T, we have ht(P) ≥ ht(P ∩ S).

(2) For Q ∈ Spec T, φ_{Q} : S → T_{Q} is flat if and only if for each prime ideal P ⊆ Q of T, we have ht(P) ≥ ht(P ∩ S).

2.5 Definitions and Remarks. (1) The Jacobian ideal J of the extension (1.2) is the ideal generated by the m × m minors of the m × n matrix J given below:

J := (\frac{∂f_{i}}{∂x_{j}} )_{i,j} .

(2) For the extension (1.2), the nonflat locus of φ is the set F, where

F := {Q ∈ Spec(T) : the map φ_{Q} : S → T_{Q} is not flat }.

For convenience, we also define the set F_{min} and the ideal F of T:

F_{min} := { minimal elements of F} and F := ∩{Q : Q ∈ F}.

By [M2, Theorem 24.3], the set F is closed in the Zariski topology and hence is equal to V(F), the set of primes of T that contain the ideal F. Thus the set F_{min} is a finite set and consists precisely of the minimal primes of the ideal F.

Moreover, Proposition 2.4 implies F_{min} ⊆ {Q ∈ Spec T : ht Q < ht(Q ∩ S)} and for every prime ideal P ⊊ Q ∈ F_{min}, ht P ≥ ht(P ∩ S).

DOMAINS INSIDE POWER SERIES RINGS 5

(3) In general for a commutative ring T and a subring R, we say that elements f_{1}, …, f_{m} ∈ T are algebraically independent over R if, for indeterminates t_{1}, …, t_{m} over R, the only polynomial G(t_{1}, …, t_{m}) ∈ R[t_{1}, …, t_{m}] with G(f_{1}, …, f_{m}) = 0 is the zero polynomial. 2.6 Example and Remarks. (1) Let k be a field, let x and y be indeterminates over k and set f = x, g = (x − 1)y. Then k[f, g] ⟶^{φ} k[x, y] is not flat. Proof. For the prime ideal P := (x − 1) ∈ Spec(k[x, y]), we see that ht(P) = 1, but ht(P ∩ k[f, g]) = 2; thus the extension is not flat by Proposition 2.4.

(2) The Jacobian ideal J of f and g in (1) is given by:

J = (det ( \begin{matrix}\frac{δf}{δx} & \frac{δf}{δy} \\ \frac{δg}{δx} & \frac{δg}{δy} \end{matrix} )) = (det ( \begin{matrix}1 & 0 \\ y & x − 1 \end{matrix} )) = (x − 1).

(3) In this example the nonflat locus is equal to the set of prime ideals Q of k[x, y] which contain the Jacobian ideal (x − 1)k[x, y], thus J = F.

We record in Proposition 2.7 observations about flatness that follow from wellknown properties of the Jacobian. 2.7 Proposition. Let R be a Noetherian ring, let x_{1}, …, x_{n} be indeterminates over R, and let f_{1}, …, f_{m} ∈ R[x_{1}, …, x_{n}] be algebraically independent over R. Consider the embedding φ : S := R[f_{1}, …, f_{m}] ↪ T := R[x_{1}, …, x_{n}]. Let J denote the Jacobian ideal of φ and let Q ∈ Spec T. Then

(1) Q does not contain J if and only if φ_{Q} : S → T_{Q} is essentially smooth.

(2) If Q does not contain J, then φ_{Q} : S → T_{Q} is flat. Thus J ⊆ F.

(3) F_{min} ⊆ {Q′ ∈ Spec T : J ⊆ Q′ and ht(Q′ ∩ S) > ht Q′}.

Proof. For item 1, we observe that our definition of the Jacobian ideal J given in (2.5) agrees with the description of the smooth locus of an extension given in [E], [S, Section 4].

To see this, let u_{1}, …, u_{m} be indeterminates over R[x_{1}, …, x_{n}] and identify

R[x_{1}, …, x_{n}] with \frac{R[u_{1}, …, u_{m}][x_{1}, …, x_{n}]}{(\{u_{i} − f_{i}\}_{i=1,...,m})}.

Since u_{1}, …, u_{m} are algebraically independent, the ideal J generated by the minors of J is the Jacobian ideal of the extension (1.2) by means of this identification. We make this more explicit as follows.

6 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND

Let U_{1} := R[u_{1}, …, u_{m}, x_{1}, …, x_{n}] and I = ({f_{i} − u_{i}}_{i=1,...,m})U_{1}. Consider the following commutative diagram

S := R[f_{1}, …, f_{m}] ⟶  T := R[x_{1}, …, x_{n}]
≅↓ ≅↓
S_{1} := R[u_{1}, …, u_{m}] ⟶ T_{1} := R[u_{1}, …, u_{m}, x_{1}, …, x_{n}]/I

Define as in [E], [S, Section 4]

H = H_{T_{1}/S_{1}} := the radical of ΣΔ(g_{1}, …, g_{s})[(g_{1}, …, g_{s}) : I],

where the sum is taken over all s with 0 ≤ s ≤ m, for all choices of s polynomials g_{,} …, g_{s} from I = ({f_{1} − u_{1}, …, f_{m} − u_{m}})U_{1}, where Δ := Δ(g_{1}, …, g_{s}) is the ideal of T ≅ T_{1} generated by the s × s-minors of (\frac{∂g_{i}}{∂x_{j}} ), and Δ = T if s = 0.

To establish (2.7.1), we show that H = rad(J). Since u_{i} is a constant with respect to x_{j}, we have (\frac{∂(f_{i}−u_{i})}{∂x_{j}}) = (\frac{∂f_{i}}{∂x_{j}} ). Thus J ⊆ H.

For g_{1}, …, g_{s} ∈ I, the s × s-minors of (\frac{∂g_{i}}{∂x_{j}} ) are contained in the s × sminors of (\frac{∂f_{i}}{∂x_{j}} ). Thus it suffices to consider s polynomials g_{1}, …, g_{s} from the set {f_{1} − u_{1}, …, f_{m} − u_{m}}. Now f_{1} − u_{1}, …, f_{m} − u_{m} is a regular sequence in R[u_{1} … u_{r}, x_{1}, …, x_{n}]. Thus for s < m, [(g_{1}, …, g_{s}) : I] = (g_{1}, …, g_{s}). Thus the m × m-minors of (\frac{∂f_{i}}{∂x_{j}} ) generate H up to radical, and so H = rad(J).

Hence by [E] or [S, Theorem 4.1], T_{Q} is essentially smooth over S if and only if Q does not contain J.

Item 2 follows from item 1 because essentially smooth maps are flat. In view of Proposition 2.4 and (2.5.2), item 3 follows from item 2. □

2.8 Remarks. (1) For φ as in (1.2), it would be interesting to identify the set F_{min}. In particular we are interested in conditions for J = F and/or conditions for J ⊊ F.

(2) If char R = 0, then the zero ideal is not in F_{min} and so F ≠ {0}.

(3) In view of (2.7.3), we can describe F_{min} exactly as F_{min} = {Q ∈ Spec T : J ⊆ Q, ht(Q ∩ S) > ht Q and ∀P ⊊ Q, ht(P) ≥ ht(P ∩ S)}.

(4) By (2.8.3), every prime ideal Q of F_{min} contains two primes P_{1} ⊊ P_{2} of S such that Q is minimal above both P_{1}T and P_{2}T.

3. Explicit constructions inside simpler extensions.

DOMAINS INSIDE POWER SERIES RINGS 7

Using Theorem 2.2 and intersection domains inside the completion which are known to be Noetherian, we formulate a shortcut method for the construction of “insider” examples.

3.1 General Method. Let R be a Noetherian integral domain. Let a be a nonzero nonunit of R and let R^{∗} = R[[x]]/(x − a) be the (a)-adic completion of R. Let τ_{1}, …, τ_{n} ∈ aR^{∗}, abbreviated by \underline{τ}, be algebraically independent over the fraction field K of R. Assume that the extension T := R[τ_{1}, …, τ_{n}] ↪^{ψ} R_{a}^{∗} is flat. Thus by Theorem 2.2, D := A\underline{τ} = K(τ_{1}, …, τ_{n}) ∩ R^{∗} is Noetherian and computable as a nested union of localized polynomial rings over R using the τ’s.

Let f_{1}, …, f_{m} be elements of T, abbreviated by \underline{f} and considered as polynomials in the τ_{i} with coefficients in R. Assume that f_{1}, …, f_{m} are algebraically independent over K; thus m ≤ n. Let S := R[\underline{f}] ↪^{φ} T = R[\underline{τ}]; put α := ψ ∘ φ : S → R_{a}^{∗}. That is, we have:

R_{a}^{∗} R ⊆ S := R[\underline{f}]  T := R[\underline{τ}]_{ψ}_{α:=ψφ} _{φ}

Using the f’s in place of the τ’s, we define the ring A := A\underline{f} := K(\underline{f}) ∩ R^{∗} and the approximation rings U_{r}, B_{r}, U\underline{f} and B = B\underline{f}, as in (2.1). Let

F := ∩{P ∈ Spec(T) | φ_{P} : S → T_{P} is not flat }.

Thus, as in (2.5.2), the ideal F defines the nonflat locus of the map φ : S → T. For Q^{∗} ∈ Spec(R_{a}^{∗}), we consider whether the localized map φ_{Q^{∗}∩T} is flat:

(3.1.1)  φ_{Q^{∗}∩T} : S → T_{Q^{∗}∩T}

3.2 Theorem. With the notation of (3.1) we have

(1) For Q^{∗} ∈ Spec(R_{a}^{∗}), the map α_{Q^{∗}} : S → (R_{a}^{∗})_{Q^{∗}} is flat if and only if the map φ_{Q^{∗}∩T} in (3.1.1) is flat.

(2) The following are equivalent:

(i) A is Noetherian and A = B.

(ii) B is Noetherian.

(iii) The map φ_{Q^{∗}∩T} in (3.1.1) is flat for every maximal Q^{∗} ∈ Spec(R_{a}^{∗}).

8 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND

(iv) F R_{a}^{∗} = R_{a}^{∗}.

(3) φ_{a} : S → T_{a} is flat if and only if F T_{a} = T_{a}. Moreover, either of these conditions implies B is Noetherian and B = A.

Proof. For item (1), we have α_{Q^{∗}} = ψ_{Q^{∗}} ∘ φ_{Q^{∗}∩T} : S → T_{Q^{∗}∩T} → (R_{a}^{∗})_{Q^{∗}}. Since the map ψ_{Q^{∗}} is faithfully flat, the composition α_{Q^{∗}} is flat if and only if φ_{Q^{∗}∩T} is flat [M1, page 27]. For item (2), the equivalence of (i) and (ii) is part of Theorem 2.2. The equivalence of (ii) and (iii) follows from item (1) and Theorem 2.2. For the equivalence of (iii) and (iv), we use F R^{∗} ≠ R^{∗} ⟺ F ⊆ Q^{∗} ∩ T, for some Q^{∗} maximal in Spec(R^{∗})_{a}) ⟺ the map in (3.1.1) fails to be flat. Item (3) follows from the definition of F and the fact that the nonflat locus of φ : S → T is closed. □

To examine the map α : S → R_{a}^{∗} in more detail, we use the following terminology.

3.3 Definition. For an extension of Noetherian rings φ : A′ ↪ B′ and for d ∈ ℕ, we say that φ : A′ ↪ B′, satisfies LF_{d} if for each P ∈ Spec(B′) with ht(P) ≤ d, the composite map A′ → B′ → B′_{P} is flat.

3.4 Corollary. With the notation of (3.1), we have ht(F R_{a}^{∗}) > 1 ⟺ φ : S → R_{a}^{∗} satisfies LF_{1} ⟺ B = A.

Proof. The first equivalence follows from the definition of LF_{1} and the second equivalence from [HRW4,Theorem 5.5].

3.5 A more concrete situation. Let R := k[x, y_{1}, …, y_{s}], where k is a field and x, y_{1}, …, y_{s} are indeterminates over k with the y_{i} abbreviated by \underline{y}. Let R^{∗} = k[\underline{y}][[x]], the (x)-adic completion of R. Let τ_{1}, …, τ_{n}, abbreviated by \underline{τ}, be elements of xk[[x]] which are algebraically independent over k(x). Let D := A\underline{τ} := k(x, \underline{y}, \underline{τ})∩ R^{∗}. Let T = R[\underline{τ}]. Then T → R_{x}^{∗} is flat, D is a nested union of localized polynomial rings obtained using the τ_{i} and D is a Noetherian regular local ring; moreover, if char k = 0, then D is excellent [HRW3, Proposition 4.1].

We now use the procedure of (3.1) to construct examples inside D. Let f_{1}, …, f_{m}, abbreviated by \underline{f}, be elements of T considered as polynomials in τ_{1}, …, τ_{n} with coefficients in R, that are algebraically independent over k(x, \underline{y}). We assume the constant terms in R = k[x, \underline{y}] of the f_{i} are zero. Let S := R[\underline{f}]. The inclusion map S ↪ T is an injective R-algebra homomorphism, and m ≤ n.

DOMAINS INSIDE POWER SERIES RINGS 9

Let A := Q(S) ∩ R^{∗} and let B be the nested union domain associated to the \underline{f}, as in (2.1). By Theorem 2.2, B is Noetherian and B = A if and only if the map α : S → R_{x}^{∗} is flat. Furthermore, by Theorem 3.2, we can recover information about flatness of α by considering the map φ : S → T.

The following remark describes how the f_{i} are chosen in several classical examples:

3.6 Remark. With the notation of (3.5).

(1) Nagata’s famous example [N1], [N2, Example 7, page 209], [HRW6, Example 3.1], may be described by taking n = s = m = 1, y_{1} = y, τ_{1} = τ, and f_{1} = f and localizing. Then R = k[x, y]_{(x,y)}, T = k[x, y, τ]_{(x,y,τ)}, f = (y + τ)^{2}, S = k[x, y, f]_{(x,y,f)} and A = k(x, y, (y + τ)^{2}) ∩ R^{∗}. The Noetherian property of B is implied by the flatness property of the map S → T_{x}. Thus B = A. In this case, T is actually a free S-module with < 1, y + τ > as a free basis.

(2) An example of Rotthaus [R1],[HRW6, Example 3.3], may be described by taking n = s = 2, and m = 1 and localizing. Then R = k[x, y_{1}, y_{2}]_{(x,y_{1},y_{2})}, T = R[τ_{1}, τ_{2}]_{(m,τ_{1},τ_{2})}, f_{1} = (y_{1} + τ_{1})(y_{2} + τ_{2}), S = R[f_{1}]_{(m,f_{1})} and A = k(x, y_{1}, y_{2}, (y_{1} +τ_{1})(y_{2} +τ_{2}))∩R^{∗}. Since the map from R[f_{1}] → R_{x}[τ_{1}, τ_{2}] = T_{x} is flat, the associated nested union domain B is Noetherian.

(3) The following example is given in [HRW5, Section 4]. Let n = s = m = 2, let f_{1} = (y_{1} + τ_{1})^{2} and f_{2} = (y_{1} + τ_{1})(y_{2} + τ_{2}). It is shown in [HRW5] for this example that B ⊊ A and that both A and B are non-Noetherian.

The following lemma follows from [P, Proposition 2.1] in the case of one indeterminate x, so in the case where T = R[x].

3.7 Lemma. Let R be a Noetherian ring, let x_{1}, …, x_{n} be indeterminates over R, and let T = R[x_{1}, … x_{n}]. Suppose f ∈ T − R is such that the constant term of f is zero. Then the following are equivalent:

(1) R[f] → T is flat.

(2) R[f] → T is faithfully flat.

(3) For each maximal ideal q of R, we have qT ∩ R[f] = qR[f].

(4) The coefficients of f generate the unit ideal of R.

10 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND

Proof. (1) ⟹ (2): It suffices to show for P ∈ Spec(R[f]) that P T ≠ T. Let q = P ∩ R and let k(q) denote the fraction field of R/q. Since R[f] → T is flat, tensoring with k(q) gives injective maps

k(q) → k(q) ⊗_{R} R[f] ≅ k(q)[f′] ⟶^{φ} k(q) ⊗_{R} T ≅ k(q)[x_{1}, …, x_{n}],

where f′ is the image of f in k(q)[x_{1}, …, x_{n}]. The injectivity of φ implies f′ has positive total degree as a polynomial in k(q)[x_{1}, …, x_{n}].

The image p′ of P in k(q)[f′] is either zero or a maximal ideal of k(q)[f′]. It suffices to show p′k(q)[x_{1}, …, x_{n}] ≠ k(q)[x_{1}, …, x_{n}]. If p′ = 0, this is clear. Otherwise p′ is generated by a nonconstant polynomial h(f′) and p′k(q)[x_{1}, …, x_{n}] is generated by h(f′(x_{1}, …, x_{n})) which has total degree equal to deg(h) deg(f′) > 0. Thus (1) implies (2).

(2) ⟹ (3): This follows from Theorem 7.5 (ii) of [M2].

(3) ⟹ (4) : If the coefficients of f were contained in a maximal ideal q of R, then f ∈ qT ∩ R[f], but f ∉ qR[f].

(4) ⟹ (1): Let v be another indeterminate and consider the commutative diagram

R[v] ⟶ T [v] = R[x_{1}, …, x_{n}, v]
π↓  _{π′}↓
R[f] ⟶^{φ}  \frac{R[x_{1},...,x_{n},v]}{(v−f(x_{1},...,x_{n}))}.

where π maps v → f and π′ is the canonical quotient homomorphism. By [M1, Corollary 2, p. 152], φ is flat if the coefficients of f −v generate the unit ideal of R[v]. Moreover, the coefficients of f − v as a polynomial in x_{1}, …, x_{n} with coefficients in R[v] generate the unit ideal of R[v] if and only if the nonconstant coefficients of f generate the unit ideal of R. □

We observe in Proposition 3.8 that one direction of (3.7) holds for more than one polynomial: see also [P, Theorem 3.8] for a related result concerning flatness. 3.8 Proposition. Assume the notation of (3.7) except that f_{1}, …, f_{m} ∈ T are polynomials in x_{1}, …, x_{n} with coefficients in R and m ≥ 1. If the inclusion map φ : S = R[f_{1}, … f_{m}] → T is flat, then the nonconstant coefficients of each of the f_{i} generate the unit ideal of R. Proof. Since f_{1}, …, f_{m} are algebraically independent over Q(R) = K, for every 1 ≤ i ≤ m, the inclusion R[f_{i}] ↪ R[f_{1}, …, f_{m}] is flat. If S ⟶ T is flat, so is

DOMAINS INSIDE POWER SERIES RINGS 11

the composition R[f_{i}] ⟶ S = R[f_{1}, …, f_{m}] ⟶ T and the statement follows from Proposition 3.7. □ 3.9 Theorem. Assume the notation of (3.1). If m = 1, that is, there is only one polynomial f_{1} = f, then

(1) The map S → T_{a} is flat ⟺ the nonconstant coefficients of f generate the unit ideal in R_{a},

(2) Either of the conditions in (1) implies the constructed ring A is Noetherian and A = B.

(3) B is Noetherian and A = B ⟺ for every prime ideal Q^{∗} in R^{∗} with a ∉ Q^{∗}, the nonconstant coefficients of f generate the unit ideal in R_{q}, where q := Q^{∗} ∩ R.

(4) If the nonconstant coefficients of f_{1} = f generate an ideal L of R_{a} of height d, then the map S → R_{a}^{∗} satisfies LF_{d−1}, but not LF_{d}.

Proof. Item (1) follows from Lemma 3.7 for the ring R_{a} with x_{i} = τ_{i}.

By Theorems 2.2 and 3.2, the first condition in item (1) implies item (2).

For item (3), suppose the nonconstant coefficients of f generate the unit ideal of R_{q}. Then by Lemma 3.7, R_{q}[f] → R_{q}[τ_{1}, …, τ_{n}] is flat. Since R_{q}[τ_{1}, …, τ_{n}] → R_{Q^{∗}}^{∗} is flat, R_{q}[f] → R_{Q^{∗}}^{∗} is also flat. For the other direction, suppose there exists Q^{∗} ∈ Spec R^{∗} with a ∉ Q^{∗} such that the nonconstant coefficients of f are in qR_{q}, where q = Q^{∗} ∩ R. If R[f] → R_{Q^{∗}}^{∗} were flat, then, since qR_{Q^{∗}}^{∗} ≠ R_{Q^{∗}}^{∗}, we would have qR_{Q^{∗}}^{∗} ∩ R[f] = qR[f]. This would imply f ∈ qR_{Q^{∗}}^{∗} ∩ R[f], but f ∉ qR[f], a contradiction.

For item (4), if Q^{∗} ∈ Spec(R_{a}^{∗}) the map S → (R_{a}^{∗})_{Q^{∗}} is not flat if and only if L ⊆ Q^{∗}. By hypothesis there exists such a prime ideal of height d, but no such prime ideal of height less than d. 3.10 Example. With the notation of (3.5), let m = 1 and assume that n and s are each greater than or equal to d. Then f_{1} = f := y_{1}τ_{1} + ⋯ + y_{d}τ_{d} gives an example where S → T_{x} satisfies LF_{d−1}, but fails to satisfy LF_{d}. For d ≥ 2 this gives examples where A = B, i.e., A is “limit-intersecting”, but is not Noetherian.

The following is a related even simpler example: In the notation of (3.5), let m = 1, n = 1, and s = 2; that is, R = k[x, y_{1}, y_{2}]_{(x,y_{1},y_{2})} and τ ∈ xk[[x]]. If f_{1} = f = y_{1}τ + y_{2}τ^{2}, then the constructed intersection domain A := R^{∗} ∩ k(x, y_{1}, y_{2}, f)

12 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND

is not Noetherian. Thus we have a situation where B = A is not Noetherian. This gives a simpler example of such behavior than the example given in Section 4 of [HRW2].

In dimension two (the two variable case), Valabrega proved the following. 3.11 Proposition [V, Prop. 3]. For R = k[x, y]_{(x,y)} with completion R̂ = k[[x, y]], if L is a field between the fraction field of R and the fraction field F of k[y] [[x]], then A = L ∩ R̂ is a two-dimensional regular local domain with completion R̂.

Example 3.10 shows that the dimension three analog to Valabrega’s result fails. With R = k[x, y_{1}, y_{2}]_{(x,y_{1},y_{2})} the field L = k(x, y_{1}, y_{2}, f) is between k(x, y_{1}, y_{2}) and the fraction field of k[y_{1}, y_{2}] [[x]], but L ∩ R̂ = L ∩ R^{∗} is not Noetherian. 3.12 Remark. With the notation of (3.1), it can happen that φ_{a} : S → T_{a} is not flat, but α : S → R_{a}^{∗} is flat. For example, using the notation of (3.5), let R := k[x, y], where k is a field and x, y are indeterminates over k. Let σ, τ ∈ xk[[x]] be such that x, σ, τ are algebraically independent over k, let T := R[σ, τ], and let S := R[σ, στ]. Then φ_{x} : S → T_{x} is not flat since σT_{x} is a height-one prime such that σT_{x} ∩ S = (σ, στ)S has height 2. To see that R_{x}^{∗} is flat over S, observe that dim R_{x}^{∗} = 1 and if Q^{∗} ∈ Spec R_{x}^{∗}, then Q^{∗} ∩ k[x, σ, στ] = (0). Therefore ht(Q^{∗} ∩ S) ≤ 1.

4. Flatness of maps of polynomial rings. 4.1 Proposition. Let k be a field, let x_{1}, …, x_{n} be indeterminates over k, and let f_{1}, …, f_{m} ∈ k[x_{1}, …, x_{n}] be algebraically independent over k. Consider the embedding φ : S := k[f_{1}, …, f_{m}] ↪ T := k[x_{1}, …, x_{n}] and let J denote the Jacobian ideal of φ. Then

(1) F_{min} ⊆ {Q ∈ Spec T : J ⊆ Q, ht Q ≤ m − 1 and ht Q < ht(Q ∩ S)}.

(2) φ is flat ⟺ for every Q ∈ Spec(T) such that ht(Q) ≤ m − 1 and J ⊆ Q we have ht(Q ∩ S) ≤ ht(Q).

(3) If ht J ≥ m, then φ is flat.

Proof. For item 1, if ht(Q) ≥ m, then ht(Q ∩ S) ≤ dim(S) = m ≤ ht(Q), so by (2.3) S → T_{Q} is flat. Therefore Q ∉ F_{min}. Item 1 now follows from (2.7.3).

The ( ⟹ ) direction of item 2 is clear [M2, Theorem 9.5]. For ( ⟸ ) of item 2 and for item 3, it suffices to show F_{min} is empty and this holds by item 1. □

DOMAINS INSIDE POWER SERIES RINGS 13

The following is an immediate corollary to (4.1).

4.2 Corollary. Let k be a field, let x_{1}, …, x_{n} be indeterminates over k and let f, g ∈ k[x_{1}, …, x_{n}] be algebraically independent over k. Consider the embedding φ : S := k[f, g] ↪ T := k[x_{1}, …, x_{n}] and let J be the associated Jacobian ideal. Then

(1) F_{min} ⊆ {minimal primes Q of J with ht(Q ∩ S) > ht Q = 1 }.

(2) φ is flat ⟺ for every height-one prime ideal Q ∈ Spec T such that J ⊆ Q we have ht(Q ∩ S) ≤ 1.

(3) If ht(J) ≥ 2, then φ is flat.

In the case where k is algebraically closed, another argument can be used for (4.2.2): Each height-one prime ideal Q ∈ Spec T has the form Q = hT for some element h ∈ T. If ht(P ∩ S) = 2, then Q ∩ S has the form (f − a, g − b)S, where a, b ∈ k. Thus f − a = f_{1}h and g − b = g_{1}h for some f_{1}, g_{1} ∈ T. Now the Jacobian ideal of f, g is the same as the Jacobian ideal of f − a, g − b and an easy computation shows this has h as a factor. Thus Q contains the Jacobian ideal, and so by assumption, ht(Q ∩ S) ≤ 1, a contradiction.

4.3 Examples. Let k be a field of characteristic different from 2 and let x, y, z be indeterminates over k.

(1) With f = x and g = xy^{2} − y, consider S := k[f, g] ⟶^{φ} T := k[x, y]. Then J = (2xy − 1)T. Since ht((2xy − 1)T ∩ S) = 1, φ is flat. Hence J ⊊ F = T.

(2) With f = x and g = yz, consider S := k[f, g] ⟶^{φ} T := k[x, y, z]. Then J = (y, z)T. Since ht J ≥ 2, φ is flat. Again J ⊊ F = T.

We are interested in extending Prop. 4.1 to the case of polynomial rings over a Noetherian domain. In this connection we first consider behavior with respect to prime ideals of R in a situation where the extension (1.2) is flat.

4.4 Proposition. Let R be a commutative ring, let x_{1}, …, x_{n} be indeterminates over R, and let f_{1}, …, f_{m} ∈ R[x_{1}, …, x_{n}] be algebraically independent over R. Consider the embedding φ : S := R[f_{1}, …, f_{m}] ↪ T := R[x_{1}, …, x_{n}].

(1) If p ∈ Spec R and φ_{pT} : S → T_{pT} is flat, then pS = pT ∩ S and the images \overline{f_{i}} of the f_{i} in T/pT ≅ (R/p)[x_{1}, …, x_{n}] are algebraically independent over R/p.

14 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND

(2) If φ is flat, then for each p ∈ Spec(R) we have pS = pT ∩S and the images \overline{f_{i}} of the f_{i} in T/pT ≅ (R/p)[x_{1}, …, x_{n}] are algebraically independent over R/p.

Proof. Item 2 follows from item 1, so it suffices to prove item 1. Assume that T_{pT} is flat over S. Then pT ≠ T and it follows from [M2, Theorem 9.5] that pT ∩ S = pS. If the \overline{f_{i}} were algebraically dependent over R/p, then there exist indeterminates t_{1}, …, t_{m} and a polynomial G ∈ R[t_{1}, …, t_{m}] − pR[t_{1}, …, t_{m}] such that G(\overline{f_{1}}, …, \overline{f_{m}}) ∈ pT. This implies G(f_{1}, …, f_{m}) ∈ pT ∩ S. But f_{1}, …, f_{m} are algebraically independent over R and G(t_{1}, …, t_{m}) ∉ pR[t_{1}, …, t_{m}] implies G(f_{1}, …, f_{m}) ∉ pS = pT ∩ S, a contradiction. □

4.5 Proposition. Let R be a Noetherian integral domain, let x_{1}, …, x_{n} be indeterminates over R and let f_{1}, …, f_{m} ∈ R[x_{1}, …, x_{n}] be algebraically independent over R. Consider the embedding φ : S := R[f_{1}, …, f_{m}] ↪ T := R[x_{1}, …, x_{n}] and let J denote the Jacobian ideal of φ. Then

(1) F_{min} ⊆ {Q ∈ Spec T : J ⊆ Q, dim(T/Q) ≥ 1 and ht(Q ∩ S) > ht Q}.

(2) φ is flat ⟺ ht(Q ∩ S) ≤ ht(Q) for every nonmaximal Q ∈ Spec(T) with J ⊆ Q.

(3) If dim R = d and ht J ≥ d + m, then φ is flat.

Proof. For item 1, suppose Q ∈ F_{min} is a maximal ideal of T. Then ht Q < ht(Q∩S) by (2.4.2). By localizing at R − (R ∩ Q), we may assume that R is local with maximal ideal Q ∩ R := m. Since Q is maximal, T/Q is a field finitely generated over R/m. By the Hilbert Nullstellensatz [M2, Theorem 5.3], T/Q is algebraic over R/m and ht(Q) = ht(m) + n. It follows that Q ∩ S = P is maximal in S and ht(P) = ht(m) + m. But the algebraic independence hypothesis for the f_{i} implies m ≤ n. This is a contradiction. Therefore item 1 follows from (2.7.3).

The ( ⟹ ) direction of item 2 is clear. For ( ⟸ ) of item 2 and for item 3, it suffices to show the set F_{min} is empty, and this follows from item 1. □

As an immediate corollary to (2.7) and (4.5), we have:

4.6 Corollary. Let R be a Noetherian integral domain, let x_{1}, …, x_{n} be indeterminates over R and let f_{1}, …, f_{m} ∈ R[x_{1}, …, x_{n}] be algebraically independent over R. Consider the embedding φ : S := R[f_{1}, …, f_{m}] ↪ T := R[x_{1}, …, x_{n}] and let J be

DOMAINS INSIDE POWER SERIES RINGS 15

the associated Jacobian ideal. Then φ is flat if for every nonmaximal Q ∈ Spec(T) such that J ⊆ Q we have ht(Q ∩ S) ≤ ht(Q).

Also as a corollary of (2.7) and (4.5) we have:

4.7 Corollary. Let R be a Noetherian ring, let x_{1}, …, x_{n} be indeterminates over R and let f_{1}, …, f_{m} ∈ R[x_{1}, …, x_{n}] be algebraically independent over R. Consider the embedding φ : S := R[f_{1}, …, f_{m}] ↪ T := R[x_{1}, …, x_{n}], let J be the Jacobian ideal of φ and let F be the (reduced) ideal which describes the nonflat locus of φ as in (2.4.2). Then J ⊆ F and either F = T, that is, φ is flat, or dim(T/Q) ≥ 1, for all Q ∈ Spec(T) which are minimal over F.

4.8 Proposition. Let R be a Noetherian integral domain containing a field of characteristic zero. Let x_{1}, …, x_{n} be indeterminates over R and let f_{1}, …, f_{m} ∈ R[x_{1}, …, x_{n}] be algebraically independent over R. Consider the embedding φ : S := R[f_{1}, …, f_{m}] ↪ T := R[x_{1}, …, x_{n}] and let J be the associated Jacobian ideal. Then

(1) If p ∈ Spec R and J ⊆ pT, then pT ∈ F, i.e., φ_{pT} S → T_{pT} is not flat.

(2) If the embedding φ : S ↪ T is flat, then for every p ∈ Spec(R) we have J ⊈ pT.

Proof. Item 2 follows from item 1, so it suffices to prove item 1. Let p ∈ Spec R with J ⊆ pT, and suppose φ_{pT} is flat. Let \overline{f_{i}} denote the image of f_{i} in T/pT. Consider

\overline{φ} : \overline{S} := (R/p)[\overline{f_{1}}, …, \overline{f_{m}}] → \overline{T} := (R/p)[x_{1}, …, x_{n}].

By Proposition 4.4, \overline{f_{1}}, …, \overline{f_{m}} are algebraically independent over \overline{R} := R/p. Since the Jacobian ideal commutes with homomorphic images, the Jacobian ideal of \overline{φ} is zero. Thus for each Q ∈ Spec \overline{T} the map \overline{φ}_{Q} : \overline{S} → \overline{T_{Q}} is not smooth. But taking Q = (0) gives \overline{T_{Q}} which is a field separable over the fraction field of \overline{S} and hence \overline{φ_{Q}} is a smooth map. This contradiction completes the proof. □

5. Examples.

5.1 Examples. For each positive integer n, we present an example of a 3-dimensional quasilocal unique factorization domain B such that

(1) B is not catenary,

16 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND

(2) the maximal ideal of B is 2-generated,

(3) B has precisely n prime ideals of height two,

(4) Each prime ideal of B of height two is not finitely generated,

(5) For every non-maximal prime P of B the ring B_{P} is Noetherian.

The notation for this construction is a localized version of the notation of Section 3.5, with s = 1. Thus k is a field, R = k[x, y]_{(x,y)} is a 2-dimensional regular local ring and R^{∗} = k[y]_{(y)}[[x]] is the (x)-adic completion of R. Let τ = ∑_{j=1}^{∞} c_{j}x^{j} ∈ xk[[x]] be algebraically independent over k(x). Let p_{i} ∈ R − xR be such that p_{i}R^{∗} are n distinct prime ideals. For example, we could take p_{i} = y − x^{i}. Let q = p_{1} ⋯ p_{n}. We set f := qτ and consider the injective R-algebra homomorphism S = R[f] ↪ R[τ] = T.

Let B be the nested union domain associated to f as in (2.1). If τ_{r} = ∑_{j=r+1}^{∞} \frac{c_{j}x^{j}}{x^{r}} is the r^{th} endpiece of τ, then ρ_{r} := qτ_{r} is the r^{th} endpiece of f. For each r ∈ ℕ, let B_{r} = R[ρ_{r}]_{(x,y,ρ_{r})}. Then each B_{r} is a 3-dimensional regular local ring and B = ⋃_{r=1}^{∞} B_{r}.

The map α : S → R_{x}^{∗} is not flat since p_{i}R_{x}^{∗} is a height-one prime and p_{i}R_{x}^{∗} ∩ S = (p_{i}, f)S is of height two. By Theorem 2.2, B is not Noetherian. By [HRW4, Theorem 4.5], B is a quasilocal unique factorization domain. Moreover, by [HRW4, Theorem 4.4], for each t ∈ ℕ, x^{t}B = x^{t}R^{∗} ∩ B and R/x^{t}R = B/x^{t}B = R^{∗}/x^{t}R^{∗}. It follows that the maximal ideal of B is (x, y)B. If P ∈ Spec B is such that P ∩ R = (0), then because the field of fractions K(f) of B has transcendence degree one over the field of fractions K of R, ht(P) ≤ 1 and hence because B is a UFD, P is principal. Claim 1. Let I be an ideal of B and let t ∈ ℕ. If x^{t} ∈ IR^{∗}, then x^{t} ∈ I. Proof. There exist elements b_{1}, …, b_{s} ∈ I such that IR^{∗} = (b_{1}, …, b_{s})R^{∗}. If x^{t} ∈ IR^{∗}, there exist α_{i} ∈ R^{∗} such that

x^{t} = α_{1}b_{1} + ⋯ + α_{s}b_{s}.

We have α_{i} = a_{i} + x^{t+1}λ_{i}, where a_{i} ∈ B and λ_{i} ∈ R^{∗}. Thus

x^{t}[1 − x(b_{1}λ_{1} + ⋯ + b_{s}λ_{s})] = a_{1}b_{1} + ⋯ + a_{s}b_{s} ∈ B.

Since x^{t}R^{∗} ∩ B = x^{t}B, γ := 1 − x(b_{1}λ_{1} + ⋯ + b_{s}λ_{s}) ∈ B. Moreover, γ is invertible in R^{∗} and hence also in B. It follows that x^{t} ∈ I. □

DOMAINS INSIDE POWER SERIES RINGS 17

To examine more closely the prime ideal structure of B, it is useful to consider the inclusion map B ↪ A := R^{∗} ∩ K(f) and the map Spec A → Spec B.

5.2 Proposition. With the notation of Example 5.1 and A = R^{∗} ∩ K(f), we have

(1) A is a two-dimensional regular local domain with maximal ideal m_{A} = (x, y)A.

(2) m_{A} is the unique prime of A lying over m_{B} = (x, y)B, the maximal ideal of B.

(3) If P ∈ Spec B is nonmaximal, then ht(P R^{∗}) ≤ 1 and ht(P A) ≤ 1. Thus every nonmaximal prime of B is contained in a nonmaximal prime of A.

(4) If P ∈ Spec B and xq ∉ P, then ht P ≤ 1.

(5) If P ∈ Spec B, ht P = 1 and P ∩ R ≠ 0, then P = (P ∩ R)B.

Proof. By Proposition 3.11 (the result of Valabrega) A := R^{∗} ∩ K(f) is a twodimensional regular local domain having the same completion as R and R^{∗}. This proves item 1. Since B/xB = A/xA = R^{∗}/xR^{∗}, m_{A} = (x, y)A is the unique prime of A lying over m_{B} = (x, y)B. Thus item 2 holds and also item 3 if x ∈ P. To see (3), it remains to consider P ∈ Spec B with x ∉ P. By Claim 1, for all t ∈ ℕ, x^{t} ∉ P R^{∗}. Thus ht(P R^{∗}) ≤ 1. Since A ↪ R^{∗} is faithfully flat, ht(P A) ≤ 1.

For (4), we see by (3) that ht(P A) ≤ 1. Let Q ∈ Spec A be a height-one prime ideal such that P ⊆ Q. Since xq ∉ P, we have B_{P} = S_{P ∩S} = T_{Q∩T} = A_{Q}, where S = R[f] and T = R[τ]. Thus ht(P) ≤ 1. For (5), if x ∈ P, then P = xB and the statement is clear. Assume x ∉ P. Since B_{x} is a localization of (B_{r})_{x}, we have (P ∩ R)B_{r} = P ∩ B_{r} for all r ∈ ℕ. Thus P = (P ∩ R)B. □

We observe that the DVRs B_{xB} and A_{xA} are equal. Moreover, A is the nested union ⋃_{r=1}^{∞} R[τ_{n}]_{(x,y,τ_{n})} of 3-dimensional regular local domains. Since A is a twodimension regular local domain each nonmaximal prime of A is principal. If pA is a height-one prime of A with pA ∉ {p_{1}A, … p_{n}A}, then A_{pA} = B_{pA∩B} and ht(pA ∩ B) = 1. We observe in Claim 2 that p_{i}A ∩ B has height two and is not finitely generated.

Claim 2. Let p_{i} be one of the prime factors of q. Then p_{i}B is prime in B. Moreover

(1) p_{i}B and Q_{i} := (p_{i}, ρ_{1}, ρ_{2}, …)B = p_{i}A ∩ B are the only primes of B lying

over p_{i}R in R,

(2) Q_{i} is of height two and is not finitely generated.

Proof. We use that B = ⋃_{r=1}^{∞} B_{r}, where B_{r} = R[ρ_{r}]_{(x,y,ρ_{r})} is a 3-dimensional regular local ring. For each r ∈ ℕ, p_{i}B_{r} is prime in B_{r}. Hence p_{i}B is a height-one prime ideal of B, for i = 1, …, n. Since ρ_{r} = qτ_{r}, p_{i}A ∩ B_{r} = (p_{i}, ρ_{r})B_{r} is a height-two prime ideal of the 3-dimensional regular local domain B_{r}. Therefore Q_{i} := (p_{i}, ρ_{1}, ρ_{2}, …)B = p_{i}A ∩ B is a nested union of prime ideals of height two, so ht(Q_{i}) ≤ 2. Since p_{i}B is a nonzero prime ideal properly contained in Q_{i}, ht(Q_{i}) = 2. Moreover x ∉ (p_{i}, ρ_{r})B_{r} for each r, so x ∉ Q_{i}. Hence for each r ∈ ℕ, ρ_{r+1} ∉ (p_{i}, ρ_{r})B and Q_{i} is not finitely generated. □

Since x ∉ Q_{i} and B[1/x] is a localization of the Noetherian domain B_{n}[1/x], we see that B_{Q_{i}} is Noetherian. Since the Q_{i} are the only prime ideals of B of height two and B is a UFD, B_{P} is Noetherian for every non-maximal prime P.

This completes the presentation of Examples 5.1. With regard to the birational inclusion B ↪ A and the map Spec A → Spec B, we remark that the following holds: Each Q_{i} contains infinitely many height-one primes of B that are the contraction of primes of A and infinitely many that are not. Among the primes that are not contracted from A are the p_{i}B. In the terminology of [ZS, page 325], P is not lost in A if P A ∩ B = P. Since p_{i}A ∩ B = Q_{i} properly contains p_{i}B, p_{i}B is lost in A. Since (x, y)B is the maximal ideal of B and (x, y)A is the maximal ideal of A and B is integrally closed, a version of Zariski’s Main Theorem [Pe], [Ev], implies that A is not essentially finitely generated as a B-algebra.

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Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395

Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027

Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323