IDEAL THEORY IN TWO-DIMENSIONAL REGULAR LOCAL DOMAINS AND BIRATIONAL EXTENSIONS

William Heinzer and David Lantz

Department of Mathematics, Purdue University, West Lafayette, IN 47907.

E-mail: heinzer@math.purdue.edu

Department of Mathematics, Colgate University, 13 Oak Drive, Hamilton,

NY 13346-1398. E-mail: dlantz@center.colgate.edu

0. Introduction.

(0.1) Let (R, m) be a two-dimensional regular local domain with infinite residue field R/m. Associated to an m-primary ideal I in R is its Hilbert polynomial

P_{I}(n) = e_{0}(I)\binom{n + 1}{2} − e_{1}(I)n + e_{2}(I) ,

the integer-valued polynomial giving the length of the R-module R/I^{n} for sufficiently large positive integers n. The coefficient e_{0} is well known to be a positive integer, the multiplicity of I, and in our context, the coefficients e_{1} and e_{2} are known to be nonnegative integers.

A well-known result of Rees [Re1, Theorem 3.2] implies that for each mprimary ideal I of R the integral closure of I is the unique largest ideal containing I and having the same multiplicity. A result of Shah in [Sh, Theorem 1] implies the existence of a unique largest ideal I_{{1}} containing I and having the same coefficients e_{0} and e_{1} of its Hilbert polynomial. We call I_{{1}} the e_{1}-ideal associated with I. If I = I_{{1}}, we call I a first coefficient ideal or an e_{1}-ideal.

There is an interplay between the internal structure of the ideals in R and the external structure of certain birational extensions of R. In this connection, for an m-primary ideal I, the blowup of I,

B(I) = Proj(R[It]) = {R[I/a]_{P} : a ∈ I − 0, P ∈ Spec(R[I/a])} ,

is the projective model over R (in the sense of [ZS, page 120]) consisting of the local domains containing R that are minimal with respect to domination among all the local domains containing R in which the extension of I is principal. There is a nonempty finite subset of B(I) consisting of local domains in which I generates an ideal primary for the maximal ideal; each of these local domains is one-dimensional and their intersection D is a one-dimensional semilocal domain called the first coefficient domain of I. As noted in [HJL, (1.3) and (3.2)], we have ID ∩R = I_{{1}}; indeed, since all powers I^{n} of I have the same blowup, we have I^{n}D ∩ R = (I^{n})_{{1}}, for each positive integer n.

Our goal in this paper is a better understanding of e_{1}-ideals and their first coefficient domains over a two-dimensional regular local domain. The situation where the first coefficient domain is a semilocal PID is well understood in view of the Zariski theory concerning complete ideals and prime divisors on R (see, e.g., [Z], [ZS, Appendix 5] or [Hu]). In particular, if V is a DVR birationally dominating R which is a spot over R (i.e., in Zariski’s terminology a prime divisor of the second kind on R; in [A2] a hidden prime divisor of R), then the ideals of R contracted from V form a descending chain m = a_{0} > a_{1} > a_{2} > … of complete m-primary ideals, the valuation ideals of R with respect to V. The Zariski theory associates to the prime divisor of the second kind V a unique simple (i.e., not factorable into a product of proper ideals) complete ideal b. One way of characterizing b is that b is maximal among m-primary ideals c of R with the property that all powers of c are contracted from V. We have b = a_{n} for some n. For example, b = m if and only if V is the ord-valuation domain R[y/x]_{mR[y/x]} where m = (x, y)R. If n > 0, then certain of the ideals a_{0}, … a_{n−1} are also simple complete ideals. If we label the simple complete ideals in this chain as b_{0} = m, b_{1}, …, b_{s} = b = a_{n}, then Zariski proves that each of the valuation ideals a_{i}, i ≥ 0, is a product of powers of b_{0}, …, b_{s} [ZS, page 392]. For example, if m = (x, y)R and V is the integral closure of R[x^{2}/y^{3}]_{mR[x^{2}/y^{3}]}, then b = (x^{2}, xy^{2}, y^{3})R is the simple complete ideal associated to V, and m = a_{0} > a_{1} = b_{1} = (x, y^{2})R > a_{2} = m^{2} > b is the beginning of the chain of ideals {a_{i}} of R contracted from V. The result of Zariski just mentioned implies that each a_{i} is a power product of m, b_{1} and b. More detailed information as to which products of the b_{j} are actually contracted from V is given by Noh in [No, Theorem 3.1].

(0.2) To describe the same situation from a different starting point, let I be a complete m-primary ideal of R. The first coefficient domain D of I is then a semilocal PID which is the intersection of the Rees valuation domains of I, i.e., the DVR’s on B(I) that dominate R. In this case, D is uniquely determined as the largest one-dimensional semilocal subdomain E of the fraction field of R having the property that all the powers of I are contracted from E (see (3.4) below). If V_{1}, …, V_{n} are the Rees valuation domains of I, then the Zariski theory implies that I has the form

(∗)  b_{1}^{r_{1}} … b_{n}^{r_{n}}

where the r_{j} are positive integers and b_{j} is the simple complete ideal of R associated to V_{j}, j = 1, …, n.

In the present paper we pursue the study of e_{1}-ideals and first coefficient domains begun in [HJL]. In particular, we consider implications of the Zariski theory for these broader classes of ideals and integral domains. Our objective, only partially realized, is to identify the first coefficient domains over a two-dimensional regular local domain and the ideals of which they are first coefficient domains.

In Section 1 we illustrate with several examples properties that onedimensional spots birationally dominating a two-dimensional regular local domain may have or fail to have. We also observe in Proposition 1.1 that the condition of being a spot descends from an integral extension.

In Section 2 we consider implications of residual transcendence. As part of Theorem 2.2, we prove that if R is a two-dimensional RLR of characteristic p > 0 with algebraically closed residue field and D is a one-dimensional local domain birationally dominating R such that the integral closure of D is a prime divisor on R, then D is the first coefficient domain of an ideal of R.

In Section 3 we examine asymptotic behavior of ideals and implications for first coefficient domains. Suppose (R, m) is a local domain that is the intersection of its localizations at height-one primes and D is a one-dimensional semilocal domain birationally dominating R. In Theorem 3.3 we prove that if J is an m-primary ideal of R such that JD is principal and J^{n}D ∩ R = J^{n} for each positive integer n, then the first coefficient domain of J is a localization of D. In particular, if D is local, then D is the first coefficient domain of J.

As usual, we abbreviate “regular local domain” by RLR and “rank-one discrete valuation domain” by DVR. The words “local” and “semilocal” include the hypothesis of Noetherian. The symbol < between sets denotes proper inclusion. For an ideal I in a Noetherian domain R the blowup of I and the first coefficient domain of I are defined as in (0.1) above. The Rees valuation domains of I are the localizations of the integral closure of the first coefficient domain of I at its maximal ideals. It is convenient to extend some familiar terminology to the case of rings that are not necessarily Noetherian or that have more than one maximal ideal: A ring D containing a domain R having a unique maximal ideal m is said to birationally dominate R if D is contained in the fraction field of R and for each maximal ideal N of D, N ∩ R = m. An extension ring D of a ring R is said to be affine over R if D is finitely generated as an algebra over R. We say that a ring D with finitely many maximal ideals is a semispot over a subring R if D is a ring of fractions of a ring containing and affine over R. If such a D has only one maximal ideal, then we call it a spot over R.

1. One-dimensional birational spots.

We are interested in considering one-dimensional semilocal domains D that birationally dominate a two-dimensional RLR R. A DVR V birationally dominating R is a spot over R if and only if the the residue field of V is not algebraic as an extension of R/m [A1, Proposition 3, page 336]. An interesting property of such a DVR V (also proved in [A1]) is that the residue field F of V is ruled as an extension field of R/m, i.e., F is obtained as a simple transcendental extension of a field intermediate between R/m and F. In view of the fact that R is a two-dimensional RLR, it follows that F is a simple transcendental extension of a finite algebraic extension of R/m.

In general, if D is a one-dimensional semilocal domain birationally dominating R, then the integral closure D′ of D is a semilocal PID birationally dominating R. If R is complete, then D′ is necessarily a semispot over R; but for certain R (such as R = k[x, y]_{(x,y)k[x,y]} where x, y are indeterminates over the field k) there exist DVR’s birationally dominating R that are not spots over R (cf., e.g., [HRS]).

We begin by proving a result (Corollary 1.3) that implies that if D is a one-dimensional semilocal domain birationally dominating a two-dimensional RLR R and if the integral closure D′ of D is a semispot over R, then D is a semispot over R and D′ is a finitely generated D-module.

Proposition 1.1. Let R be a Noetherian ring, and let V be a semispot over R. Suppose R ⊆ D ⊆ V with D quasilocal and V integral over D. Then D is a spot over R and V is a finitely generated D-module.

Proof. Since V is a semispot over R, there exist elements a_{1}, …, a_{n} ∈ V such that V is a ring of fractions of R[a_{1}, …, a_{n}]. Let b_{1}, …, b_{m} be the coefficients of monic polynomials over D satisfied by a_{1}, …, a_{n}; set B = R[b_{1}, …, b_{m}] and A = B[a_{1}, …, a_{n}]. Let Q be the center of D on B, and let A_{1} and B_{1} be the rings of fractions of A and B at the multiplicative set B − Q. Then B_{1} is local, with maximal ideal Q_{1} = QB_{1}, and A_{1} is a finite integral extension of B_{1}. Hence A_{1} has only finitely many maximal ideals. Let P_{1}, …, P_{r} denote the centers on A_{1} of the maximal ideals of V, and let S = A_{1} − (⋃_{i=1}^{r} P_{i}). Since V is a ring of fractions of R[a_{1}, …, a_{n}], we have S^{−1}A_{1} = V. Choose a ∈ S such that a is in each maximal ideal of A_{1} distinct from P_{1}, …, P_{r} (if any — otherwise let a = 1). Then 1/a is in V and hence is integral over D. Let c_{1}, …, c_{p} be the coefficients of a monic polynomial over D satisfied by 1/a; let (B_{2}, Q_{2}) be the localization of B_{1}[c_{1}, …, c_{p}] at the center of D on this ring, and set A_{2} = B_{2}[1/a, A_{1}].

We claim that A_{2} = V. To see this, it suffices to show each s in S is a unit in A_{2}: Assume by way of contradiction that s in S is in a maximal ideal M of A_{2}. Since A_{2} is integral over B_{2}, we have M ∩ B_{2} = Q_{2}, and so Q_{1} = M ∩ B_{1} = (M ∩ A_{1}) ∩ B_{1}. Since A_{1} is integral over B_{1}, M ∩ A_{1} is maximal in A_{1}. Moreover, M ∩ A_{1} survives in A_{2} , so our choice of a assures that M ∩ A_{1} is the center on A_{1} of one of the maximal ideals of V. But this yields s ∈ S ⊆ A_{1} − (M ∩ A_{1}), a contradiction.

Therefore, V is an affine extension of B_{2} and hence a finitely generated D-module. Thus, by Artin-Tate [Ku, Lemma 3.3, page 16], D is an affine extension of B_{2} and hence a spot over R. □

To extend this result to the case where D has finitely many maximal ideals, we use:

Proposition 1.2. Let R be an integral domain. Suppose D is an extension domain of R having only finitely many maximal ideals N_{1}, …, N_{r} and having the property that D_{N_{i}} is a spot over R for each i = 1, …, r. Then D is a semispot over R.

Proof. For each maximal ideal N_{i} of D there is a finite subset T_{i} of D_{N_{i}} such that D_{N_{i}} is a localization of R[T_{i}]. And there is an element s_{i} of D − N_{i} for which s_{i}T_{i} ⊆ D. Let A = R[(⋃_{i=1}^{r} s_{i}T_{i}) ∪ {s_{1}, …, s_{r}}]. If P_{i} denotes the center of D_{N_{i}} on A, then A ⊆ D ⊆ D_{N_{i}} = A_{P_{i}}; so D is the ring of fractions of A at the complement of the union of the P_{i}’s. □

As an immediate corollary of Propositions 1.1 and 1.2, we have:

Corollary 1.3. Let D be a semilocal extension domain of a Noetherian domain R, and let V be a domain integral over D. If V is a semispot over R, then D is also a semispot over R. □

(1.4) It follows from Corollary 1.3 that a one-dimensional semilocal domain D that birationally dominates a two-dimensional RLR R is a semispot over R if and only the integral closure of D is an intersection of prime divisors of the second kind on R, or equivalently, if and only if each DVR birationally containing D is a prime divisor of the second kind on R.

We are interested in the question of which one-dimensional semilocal domains birationally dominating R are first coefficient domains of ideals of R. The first coefficient domains of complete ideals of R are well understood. They are precisely the one-dimensional semilocal PID’s birationally dominating R that are semispots over R. Moreover, if I and J are complete m-primary ideals of R with first coefficient domains D_{I} and D_{J}, respectively, then D_{I} ∩ D_{J} is a PID semispot over R and is the first coefficient domain of IJ. More generally, by the Theorem on Independence of Valuations (e.g., [N, (11.11)] or [ZS, Theorem 18, p. 45]) the intersection of two semilocal PID’s birationally dominating a local domain is again a semilocal PID birationally dominating the local domain. But for arbitrary m-primary ideals I and J of R, the relation of D_{I} and D_{J} with the first coefficient domain of IJ is more delicate. It is not necessarily D_{I} ∩ D_{J}; indeed, in Example 1.5 we show that D_{I} ∩D_{J} need not be a first coefficient domain of R. In this example we make use of the description of the first coefficient domain of an ideal generated by a regular sequence given in [HJL, (3.8)].

Example 1.5. Let k be a field of characteristic 0 and x, y be indeterminates over k; set R = k[x, y]_{(x,y)}. Then the first coefficient domains of the ideals (x^{2}, y^{2})R and (x^{2}, xy + y^{2})R are

D_{1} = k((y/x)^{2}) + M  and  D_{2} = k((y/x) + (y/x)^{2}) + M ,

respectively, where M is the maximal ideal of the ord-valuation domain V = R[y/x]_{mR[y/x]} = k(y/x)+M over R. (The maximal ideals M_{1} and M_{2} of D_{1} and D_{2}, respectively, are contained in M, and a module basis for V over either D_{1} or D_{2} is 1, y/x. Since M_{i}(y/x) ⊆ D_{i} and M_{i}V = M, we have M_{i} = M.) Since k is of characteristic zero, we have k((y/x)^{2}) ∩ k((y/x) + (y/x)^{2}) = k. It follows that the residue field of D_{1} ∩ D_{2} at the center of V on D_{1} ∩ D_{2} is not residually transcendental over the residue field k of R, so D_{1} ∩ D_{2} is not a semispot over R by (1.4) and hence is not the first coefficient domain of an ideal of R.

(1.6) Suppose I and J are m-primary ideals of R, where (R, m) is a twodimensional RLR, or more generally, a quasi-unmixed analytically unramified local domain. We want to relate the first coefficient domain D of IJ to the first coefficient domains D_{I} and D_{J} of I and J. A first remark is that since the set of Rees valuation domains of IJ is the union of the sets of Rees valuation domains of I and J, the integral closure of D is the intersection of the integral closures of D_{I} and D_{J}. With each DVR V that is a localization of the integral closure of D_{I} (of which there are only finitely many) we associate a one-dimensional semilocal domain D_{V} = (D_{I})_{P} [B], where P is the center of V on D_{I} and B is the unique local domain on the blowup of J that is dominated by V. In an analogous way we construct D_{W} for each DVR W that is a localization of the integral closure of D_{J}. The first coefficient domain D of IJ is the the intersection of the one-dimensional semilocal domains D_{V} and D_{W} as V and W vary over the sets of the Rees valuation domains of I and J respectively.

(1.7) The proofs of several results below rely on Theorem 3.12 of [HJL]; and on rereading the proof of that result, we feel one point deserves a fuller discussion. The relevant hypotheses in that result are as follows: R is a normal, analytically unramified, quasi-unmixed, local domain with infinite residue field, I is an ideal primary for the maximal ideal of R, D is the first coefficient domain of I, E is a domain birational and integral over D, and a is an element of I for which ID = aD. In the proof, we set S = R[1/a] ∩ D and T = R[1/a] ∩ E, and we assert that D, E are rings of fractions of S, T respectively. This is true under the hypothesis of Theorem 3.12 of [HJL], but in Example 1.8 below we show that for a ∈ m with aD ≠ ID it can happen that D is not a ring of fractions of S = R[1/a]∩D. So we felt these assertions should be given a more explicit justification: The hypothesis that D is the first coefficient domain of I means that there exists an element b of I such that D is an intersection of a finite number of one-dimensional localizations of R[I/b] and hence is itself a ring of fractions of R[I/b]. Moreover, bD = ID = aD. Thus, b/a is an element of R[I/a] that is not in any of the prime ideals of D, so the ring of fractions of R[I/a] with respect to the complement in R[I/a] of the union of the primes in D contains R[I/b] and hence is all of D. Since S = R[1/a] ∩ D ⊇ R[I/a], we see that D is also a ring of fractions of S. Now we turn to T = R[1/a] ∩ E, which is almost integral over S since there is a nonzero conductor from E into D (because R is analytically unramified [Re2, Theorem 1.2]). Since

S = ∩{R[I/a]_{P} : P is a height-one prime }

and since R[I/a] is universally catenary, S is contained in the integral closure of R[I/a]. Moreover, the fact that R is analytically unramified implies that the integral closure of R[I/a] is a finitely generated R[I/a]-module. Therefore S is Noetherian and hence T is integral over S. Since D is a ring of fractions of S, the maximal ideals of D are centered on height-one primes of S. It follows that the maximal ideals of E are centered on height-one primes of T. Since the essential valuation domains of R[1/a] are all localizations of S and of T, it follows that E is a ring of fractions of T.

Example 1.8. Let R = k[x, y]_{(x,y)k[x,y]}, where k is a field and x, y are indeterminates over k. Let V = k(y/x)[x]_{(x)} be the ord-valuation domain of R. Then V = k(y/x) + M, where M is the maximal ideal of V. Let D = k((y^{2} + x^{2})/xy) + M. Then D is the first coefficient domain of the ideal (xy, y^{2} + x^{2})R, a one-dimensional local domain that birationally dominates R, and V is the integral closure of D. Let T = R[1/x]∩V and S = R[1/x]∩D. Then T = R[y/x], so S = R[y/x]∩D. Using that k[y/x]∩k((y^{2}+x^{2})/xy) = k and considering the unique expression of each element of a subdomain of V as the sum of an element of k(y/x) and an element of M, we see that S = k + (M ∩ R[y/x]). Hence D is centered on a maximal ideal of S and is not a localization of S. We also have in this example that S is not Noetherian and T is almost integral but not integral over S. The localization of S at each of its height-one primes contains R[1/x].

2. Residually transcendental elements.

Let (R, m) be a two-dimensional RLR with residue field k = R/m. A first coefficient domain of an m-primary ideal of R is a one-dimensional semispot birationally dominating R. As a partial converse, we observe in Proposition 2.1 that a domain satisfying these hypotheses is at least a ring of fractions of a first coefficient domain of R.

Proposition 2.1. Let (R, m) be a two-dimensional RLR and E be a onedimensional semispot birationally dominating R. Then there exists a first coefficient domain D of R such that E is a ring of fractions of D.

Proof. Let a_{1}, …, a_{n}, b be elements of R such that E is a ring of fractions of R[a_{1}/b, …, a_{n}/b]. We may assume that a_{1}, …, a_{n}, b have no common factor in R, so that the ideal I = (a_{1}, …, a_{n}, b)R is m-primary. Let D_{0} denote the first coefficient domain of I. Since E is a semispot over R, the dimension formula [M, (14.D)] shows that for each maximal ideal N of E the image of at least one of the quotients a_{i}/b in E/N is transcendental over R/m. Thus, the center of N on R[a_{1}/b, …, a_{n}/b] is one-dimensional, so that D_{0} ⊆ E_{N}. Since this holds for each maximal ideal N of E, D_{0} ⊆ E. But there may be prime divisors dominating R that contain D_{0} but not E. The intersection D of all these prime divisors and E is an integral extension of D_{0} and hence a first coefficient domain (of an ideal integral over a power of I) by [HJL, Theorem 3.12]. We have D ⊆ E are one-dimensional semilocal domains with E birational over D and D integrally closed in E. Forming the ring of fractions of D with respect to the elements of D that are units of E and applying [N, (33.1)], we see that E is a ring of fractions of D. □

A variant of the process used in this proof is as follows: With R, E, etc. as in Proposition 2.1 and its proof, let (c, d)R be a reduction of I = (a_{1}, …, a_{n}, b)R (or of a power of I if the residue field of R is finite and I fails to have a 2-generated reduction). For each maximal ideal N of E, the image of c/d in E/N is transcendental over R/m, so N ∩ R[c/d] = mR[c/d]. It follows that E is a localization of the integral closure of R[c/d]_{mR[c/d]} in E. To realize E itself as a first coefficient domain in this manner amounts to answering in the affirmative the following question: Does there exist a single element a/b of E such that J = (a, b)R is a reduction of a complete ideal of the form (∗) in (0.2) above, where the r_{j} are positive integers and the b_{j} are the simple complete ideals corresponding to the DVR localizations of the integral closure of E? If so, then E and R[a/b]_{mR[a/b]} have the same integral closure. Thus, E is integral over R[a/b]_{mR[a/b]} and hence a first coefficient domain in its own right. The proof of Theorem 2.2 below is essentially the construction of such an element a/b in a special case.

In the proof of Theorem 2.2 is a reference to R(t), where t is an indeterminate over R. In general, for a ring A, the symbol A(t) denotes the ring of fractions of the polynomial ring A[t] with respect to the multiplicative system of polynomials whose coefficients generate the unit ideal in A (cf. [N, page 18]). In the present local case, this means only that not all of the coefficients of the polynomial are in m. There is a natural epimorphism from R(t) onto the simple transcendental field extension k(t) of k, with kernel generated by m; images under this epimorphism (as well as under other extensions of the epimorphism R → k) are denoted by overbars (vincula).

Theorem 2.2. Let D be a one-dimensional local domain birationally dominating a two-dimensional RLR R. Assume that k = R/m is algebraically closed, that the integral closure D′ of D is a prime divisor on R, and that either (1) R has nonzero characteristic or (2) D contains the maximal ideal of D′. Then there is an m-primary ideal of which D is the first coefficient domain.

Proof. By (1.1), D is a spot over R and D′ is a finitely generated D-module. In view of the last sentence of General Example 3.8 and Theorem 3.12 of [HJL], it is enough to find a 2-generated m-primary ideal (a, b)R of R for which a/b ∈ D and the integral closure of R[a/b]_{mR[a/b]} is D′. Also, since D′ is a prime divisor of the second kind of R, there is a simple complete m-primary ideal b with which D′ is associated, in the sense of the Zariski theory. It will suffice to find elements a, b of R so that a/b ∈ D and the ideal (a, b)R is a reduction of a power of b.

Let (c, d)R be a minimal reduction of b (or of a power of b). Then the residue field of D′ is of transcendence degree 1 over k, generated by the image \overline{c/d} of c/d (because k is algebraically closed [HuS, Remark 3.5]), but algebraic over the residue field of D, and for any other prime divisor of the second kind of R, either c/d is not in that prime divisor or its image in the residue field is not transcendental over the image of k (i.e., c/d is not “residually transcendental” for any other prime divisor of the second kind). Thus, for an element z of D of which the image \overline{z} in the residue field of D (or D′) is transcendental over k, there is an element φ(t) of R(t) such that if \overline{φ}(t) ∈ k(t) is the image of φ(t) in R(t)/mR(t), then \overline{z} = \overline{φ}(\overline{c/d}). We may assume that the numerator and denominator of \overline{φ}(t) are relatively prime polynomials over k. Now z − φ(c/d) is in the maximal ideal of D′, so under assumption (2) of the statement, we immediately have that φ(c/d) ∈ D. To reach a similar (though not identical) conclusion under assumption (1), we note that since D′ is local and is a finitely generated D-module, the maximal ideal of D contains a power of the maximal ideal of D′; so we can raise z − φ(c/d) to a sufficiently high power q, a power of the characteristic of R, to conclude that φ(c/d)^{q} ∈ D. Multiplying the numerator and denominator of φ or φ^{q} by the same power of d, we convert them into forms a = a(c, d) and b = b(c, d) in c, d of the same degree n such that their images in the degree-n piece of the fiber ring F ((c, d)) = R[(c, d)t] ⊗_{R} R/m, a polynomial ring in two variables over k, are relatively prime.

We show that (a, b) is a reduction of (c, d)^{n}, which will complete the proof. It suffices to show that (a, b)(c, d)^{n} = (c, d)^{2n}, and by Nakayama’s Lemma it suffices to show that the k-vector spaces [(a, b)(c, d)^{n} + m(c, d)^{2n}]/m(c, d)^{2n} and (c, d)^{2n}/m(c, d)^{2n} have the same dimension. The latter is the degree-2n piece of the fiber ring F ((c, d)); its dimension is 2n + 1. The images of the products ac^{i}d^{n−i}, i = 0, …, n, span a subspace of the former of dimension n + 1, and similarly with b in place of a; and since the images of a, b are relatively prime, the intersection of these two subspaces is spanned by the image of ab, so it is one-dimensional. Thus, [(a, b)(c, d)^{n} + m(c, d)^{2n}]/m(c, d)^{2n} has dimension 2(n + 1) − 1 = 2n + 1 as required. □

3. Principal extensions and contracted powers.

(3.1) Suppose D is a one-dimensional semispot birationally dominating a quasi-unmixed, analytically unramified, normal local domain (R, m). In this section we seek conditions for D to be the first coefficient domain of an ideal I of R. If D is the first coefficient domain of I, then ID is principal, and replacing I by the associated e_{1}-ideal of a high power of I, we obtain an m-primary ideal J such that JD is principal and J^{n}D ∩ R = J^{n} for each positive integer n [HJLS, Theorem 3.17]. Thus a necessary condition for D to be a first coefficient domain is the existence of an m-primary ideal J of R with the two properties: (1) JD is principal, and (2) J^{n}D ∩ R = J^{n} for each positive integer n. If D is local, we prove in Theorem 3.3 that this necessary condition is also sufficient, and that D is in fact the first coefficient domain of each ideal J with these two properties.

The case in which V is a prime divisor birationally dominating a twodimensional RLR (R, m) is illustrative. Suppose a is a nonzero element of m and consider the descending chain J_{n} = a^{n}V ∩ R, n = 1, 2, …, of ideals of R. As noted in the introduction, each J_{n} is a complete ideal of R, and from the Zariski theory it follows that J_{n} is a product of powers of the simple complete ideals associated with the finitely many prime divisors that “come out” on the sequence of quadratic transformations of R along V. Let b be the simple complete ideal of R associated to V, and suppose the V -values of a and b are p and q respectively. Then J_{q} = b^{p}. Since all powers of b are contracted from V, for each positive integer r we have J_{q}^{r} = J_{qr}, or equivalently the powers of J_{q} are contracted from V. Moreover, J_{q} has V as its first coefficient domain.

(3.2) It was noted in [HJL, (3.7)] that the first coefficient domain of an ideal I of R can be described using the minimal primes of IR[It] of the Rees algebra R[It] or the minimal primes of t^{−1}R[t^{−1}, It] of the extended Rees algebra R[t^{−1}, It] of I (where t is an indeterminate over R). These primes are in one-to-one correspondence with the maximal ideals of the first coefficient domain D of I: If P is one of these minimal primes, then P does not contain the degree-1 piece of the Rees algebra (or extended Rees algebra), say bt ∉ P where b ∈ I. Then the localization of the (extended) Rees algebra at P is also a localization of R[I/b][bt, (bt)^{−1}] and has the form D_{N}(bt) (cf. the paragraph before Theorem 2.2) for the maximal ideal N of D corresponding to P. [Note: The V (t) in the equations on the last line of [HJL, (3.7)] should be V (bt), for b as above.]

Theorem 3.3. Let (R, m) be a local domain that is the intersection of its localizations at height-one primes, and let D be a one-dimensional semilocal domain that birationally dominates R. Suppose J is an m-primary ideal of R such that JD is principal and J^{n}D ∩ R = J^{n} for each positive integer n. Then the first coefficient domain of J is a localization of D. In particular, if D is local, then D is the first coefficient domain of J.

Proof. Replacing J, if necessary, by a power of J, we may assume that JD = aD where a ∈ J. Let A = R[t^{−1}, Jt] be the extended Rees algebra of the ideal J of R; let D(at) denote the localization of the polynomial ring D[at] at the complement of the union of the extension to D[at] of the maximal ideals of D; and let K be the fraction field of R. Since D[at, (at)^{−1}] is Cohen-Macaulay, it is the intersection of its localizations at height-one primes. It follows that D[at, (at)^{−1}] = K[at, (at)^{−1}] ∩ D(at), and hence that

R[t, t^{−1}] ∩ D[at, (at)^{−1}] = R[t, t^{−1}] ∩ K[at, (at)^{−1}] ∩ D(at)
= R[t, t^{−1}] ∩ D(at) = A .

Let P be a minimal prime of t^{−1}A and let S = A − P. Then A_{P} = S^{−1}(R[t, t^{−1}] ∩ D(at)) = S^{−1}(R[t, t^{−1}]) ∩ S^{−1}D(at). Since R[t, t^{−1}] is the locally finite intersection of its localizations at height-one primes, to show S^{−1}(R[t, t^{−1}]) = K(t), it suffices to show S meets each height-one prime Q of R[t, t^{−1}]: If Q ∩ S = ∅, then Q ∩ A ⊆ P. Since Q ∩ A ≠ 0, we must have Q ∩ A = P. But P ∩ R = m and Q ∩ R < m, a contradiction. Thus S meets each height-one prime of R[t, t^{−1}], so A_{P} = S^{−1}D(at).

Let E be the first coefficient domain of J. The maximal ideals N of E are in one-to-one correspondence with the minimal primes P of t^{−1}A, where A_{P} = E_{N}(at). Since each A_{P} is a localization of D(at), the intersection E(at) of the A_{P} ’s is a ring of fractions of D(at). Intersecting with K shows that E is a ring of fractions of D. □

The following corollary implies the uniqueness property of the intersection of the Rees valuation domains of an ideal mentioned in (0.2).

Corollary 3.4. Let (R, m) be a quasi-unmixed, analytically unramified, normal local domain, and let I be an m-primary ideal of R. The first coefficient domain E of I is the unique largest one-dimensional semilocal domain D birationally dominating R and having the properties that ID is principal and I^{n}D ∩ R is contained in the e_{1}-ideal of I^{n} for each positive integer n.

Proof. By [HJLS, Theorem 3.17] for all sufficiently large positive integers r, the ideal J = I^{r}E ∩ R has the property that E is the first coefficient domain of J and for each positive integer n we have J^{n}E ∩R = J^{n} = I^{rn}E ∩R is the e_{1}-ideal associated to I^{rn}. Therefore J^{n}D is principal and J^{n}D ∩ R = J^{n} for each n. By Theorem 3.3, E is a localization of D. □

Corollary 3.5. Let D be a one-dimensional spot birationally dominating a two-dimensional RLR (R, m). If J is an m-primary ideal in R such that JD is principal and all the powers of J are contracted from D, then D is the first coefficient domain of J, and the integral closure of J is a product of powers of the simple complete ideals associated to the localizations of the integral closure of D.

Proposition 3.6. Let (R, m) be a quasi-unmixed analytically unramified local domain of dimension d ≥ 2, and let J be an m-primary ideal of R. Let D be a one-dimensional semilocal domain birationally dominating R, and let V be a finitely generated birational integral extension of D. If all the powers of J are contracted from D, then for each positive integer n, J^{n}V ∩ R is integral over J^{n}. In particular, if I = JD ∩ R is a normal ideal (i.e., the powers of I are integrally closed), then all the powers of I are contracted from V.

Remark. The hypothesis in Proposition 3.6 (and in Proposition 3.7 below) that V is a finitely generated D-module is necessary (cf., e.g., [HRS, (1.27)]). But if D (or V, by Corollary 1.3) is a (birational) semispot over R, the hypothesis on R assures that V is a finitely generated D-module [Re2, Theorem 1.2].

Proof. For the first assertion, it suffices to show that I is integral over J. Since J is contained in each nonzero prime ideal of D, there exists a positive integer c such that J^{c} is contained in the conductor of V into D. Thus, for all positive integers n, we have

I^{n+c} ⊆ I^{n+c}V ∩ R = J^{n+c}V ∩ R ⊆ J^{n}D ∩ R = J^{n} ⊆ I^{n}.

It follows that the length of R/J^{n} is between those of R/I^{n} and R/I^{n+c}. Now, for n sufficiently large, the length of I^{n}/I^{n+c} is a polynomial in n of degree d − 1, while the lengths of R/I^{n} and R/J^{n} are polynomials in n of degree d. Therefore the Hilbert polynomials of I and J have the same highest degree coefficient, i.e., I and J have the same multiplicity. By [Re1, Theorem 3.2], I is integral over J.

For the second assertion, note that J^{n} ⊆ I^{n} ⊆ I^{n}V ∩ R = J^{n}V ∩ R; the last ideal is integral over J^{n}, so if I^{n} is integrally closed, it is equal to I^{n}V ∩ R. □

Proposition 3.7. Let (R, m) be a normal, quasi-unmixed, analytically unramified local domain of dimension d ≥ 2, and J be an m-primary ideal of R. Let D be a one-dimensional semilocal domain birationally dominating R such that the integral closure V of D is a finitely generated D-module. Suppose that J^{n}D ∩ R = J^{n} for each positive integer n, and let I_{n} = J^{n}V ∩ R for each n.

(1) For sufficiently large r, all the powers of I_{r} are contracted from V.

(2) Therefore V is contained in each of the Rees valuation domains of J, and so D is contained in the integral closure of the first coefficient domain of J.

Proof. (1) By [Re2, Theorem 1.4], for sufficiently large r, I_{r} is a normal ideal, so by Proposition 3.6 all the powers of I_{r} are contracted from V.

(2) Since the intersection of the Rees valuation domains is the unique largest one-dimensional semilocal subdomain E of the fraction field of R with the property that the integral closure of J^{n} is J^{n}E ∩R for each n, V is contained in each of the Rees valuation domains of J. □

(3.8) Let (R, m) be a two-dimensional RLR, let D be a one-dimensional semispot birationally dominating R, and let V be the integral closure of D. Let b_{1}, …, b_{n} be the simple complete ideals of R associated to the DVR’s which are localizations of V. Then the associated e_{1}-ideal of an m-primary ideal I of R has the form (∗) as in (0.2) above, where the r_{j} are positive integers, if and only if V is the first coefficient domain of I. By [HJL, Theorem 3.12], D is a first coefficient domain if and only if there exists an ideal J of R such that JD is principal and such that the integral closure of J is of the form (∗). Thus, for example, if V is the ord-valuation domain of R, then D is a first coefficient domain if and only if there exists an ideal J such that JD is principal and such that the integral closure of J is a power of m.

(3.9) With R, D as in Corollary 3.5, there always exist m-primary ideals J with the property that all their powers are contracted from D (for, if J is the product of the simple complete ideals associated to the DVR localizations of the integral closure of D, then all the powers of J are contracted from the integral closure of D and hence also from D). Thus, in this case the issue is whether there exists such a J with JD principal. However, if one passes to a more general situation where R is a two-dimensional excellent normal local domain, then there may exist birationally dominating DVR spots V over R for which there does not exist an ideal J of R such that all the powers of J are contracted from V. By definition, an excellent two-dimensional normal local domain (R, m) with the property that each prime divisor of the second kind on R is the first coefficient domain of an m-primary ideal is said to satisfy Muhly’s condition (N) (cf. [HL, page 291]). If R is a two-dimensional complete normal local domain, Cutkosky proves in [C, Theorem 4] that R satisfies condition (N) if and only if R has torsion divisor class group. Thus, for example, R = ℂ[[x, y, z]], where x^{3} + y^{3} + z^{3} = 0, has prime divisors of the second kind which are not first coefficient domains of an ideal of R.

(3.10) Let (R, m) be a two-dimensional RLR and let D be the first coefficient domain of an ideal I of R. If D is a prime divisor of R and a ∈ m is a nonzero element, then there exists a positive integer n such that D is the first coefficient domain of a^{n}D ∩ R (cf. (3.1)). The case of a general first coefficient domain, however, is different: In Example 1.8, there is no positive integer m for which D is the first coefficient domain of x^{m}D ∩ R. This phenomenon is the reef on which founders the following naive approach to realizing a one-dimensional semispot E birationally dominating R as a first coefficient domain. Let b_{1}, …, b_{s} be the distinct simple complete ideals of R associated with the prime divisors obtained as localizations of the integral closure E′ of E, and let a ∈ R be such that aE′ ∩ R = b_{1} … b_{s}. Let A = R[t^{−1}, t] ∩ E(at). Then A = R[t^{−1}, I_{1}t, I_{2}t^{2}, …], where I_{n} = a^{n}E ∩ R. The integral closure of A is A′ = R[t^{−1}, (I_{1})′t, (I_{2})′t^{2}, …], while the domain A′′ = R[t^{−1}, t] ∩ E′(at) is almost integral over A since there is a nonzero conductor from E′ to E. The following conditions are equivalent: (1) A is Noetherian. (2) A is affine over R. (3) A′ = A′′. When these conditions hold, (I_{1})′ = b_{1} … b_{s} and E is the first coefficient domain of an ideal integral over a power of I_{1}. In Example 1.8, however, for E = D and a = x, we have A′ < A′′. When we have A′ < A′′, there is no positive integer m for which the powers of I_{m} are contracted from E, nor for which E is the first coefficient domain of I_{m}.

ACKNOWLEDGEMENT

This work is the outgrowth of a project with Bernard Johnston. We would like to acknowledge his penetrating questions which initiated the present paper. We also thank the referee for a careful reading of the paper.

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