Proc. Amer. Math. Soc. 124 (1996) 3595–3604.

INTEGER-VALUED POLYNOMIALS ON KRULL RINGS

Sophie Frisch

Abstract. If R is a subring of a Krull ring S such that R_{Q} is a valuation ring for every finite index Q = P ∩ R, P in Spec^{1}(S), we construct polynomials that map R into the maximal possible (for a monic polynomial of fixed degree) power of P S_{P}, for all P in Spec^{1}(S) simultaneously. This gives a direct sum decomposition of Int(R, S), the S-module of polynomials with coefficients in the quotient field of S that map R into S, and a criterion when Int(R, S) has a regular basis (one consisting of 1 polynomial of each non-negative degree).

Introduction

If A is an infinite subset of a domain S, we write Int(A, S) for the S-module of polynomials with coefficients in the quotient field of S that – when acting as a function by substitution of the variable – map A into S. For Int(S, S), the ring of integer-valued polynomials on S, we write Int(S). Beyond the fact (known of old) that the binomial polynomials \binom{x}{n} = \frac{x(x−1)...(x−n+1)}{n!} form a basis of the free ℤ-module Int(ℤ), the study of Int(S) originated with Pólya [16] and Ostrowski [15], who let S be the ring of integers in a number field (their results have been generalized to Dedekind rings by Cahen [4]). Int(R, S) for R ≠ S has only begun to attract attention more recently [2, 3, 6, 8, 11, 13].

We will treat Pólya’s and Ostrowski’s questions in the case where R ≠ S and S is a Krull ring; in particular the question when Int(R, S) is a free S-module that admits a regular basis, and the related one of determining the highest power of P S_{P}, where P is a height 1 prime ideal of S, that a monic polynomial of fixed degree can map R into. Following Pólya, we call a sequence of polynomials (g_{n})_{n∈ℕ_{0}} regular, if deg g_{n} = n for all n. One basic connection between a module of polynomials and the modules of leading coefficients should be kept in mind:

0.1 Lemma. Let R be a unitary subring of a field K, M an R-submodule of K[x], and I_{n} = { leading coefficients of n-th degree polynomials in M } ∪ {0}.

(i) If (g_{n})_{n∈ℕ_{0}} is a regular sequence of monic polynomials in K[x] such that I_{n}g_{n} ⊆ M for all n, then M = ∑_{n=0}^{∞} I_{n}g_{n} (direct sum).

(ii) A regular set of polynomials in M is an R-basis if and only if the leading coefficient of the n-th degree polynomial generates I_{n} as an R-module.

(iii) M has a regular R-basis if and only if each I_{n} is non-zero and cyclic.

Proof. (i) If (g_{n})_{n∈ℕ_{0}} is as stated, then ∑_{n=0}^{∞} I_{n}g_{n} ⊆ M and the sum is direct, since deg(g_{n}) = n makes the g_{n} linearly independent over K. An induction on N = deg f

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 13B25, 13F05; Secondary 13F20, 11C08.

shows that f ∈ M implies f ∈ ∑_{n=0}^{N} I_{n}g_{n}. Indeed, for N = 0, f ∈ I_{0} = g_{0}I_{0}, and if N > 0 and a_{N} is f’s leading coefficient, then a_{N} ∈ I_{N}, so h = f − a_{N}g_{N} ∈ M and h ∈ ∑_{n=0}^{N−1} I_{n}g_{n} by induction hypothesis. (ii) and (iii) are easy. □

1. Polynomials mapping a Set into a Discrete Valuation Ring Throughout section one, v is a discrete valuation on a field K with value-group Γ_{v} = ℤ and v(0) = ∞, and R_{v} its valuation ring with maximal ideal M_{v}. In a kind of generic local regular basis theorem, we will establish the connection (well-known in special cases) between Int(A, R_{v}) and the maximal power of M_{v} that a monic polynomial of degree n can map A into, for all A⊆K for which this maximum exists for every n. A subset A of the quotient field of a domain R is called R-fractional if there exists a d ∈ R \ {0} such that dA ⊆ R.

1.0 Lemma. If R is an integrally closed domain with quotient field L, A ⊆ L and f non-constant ∈ L[x] then f(A) is R-fractional if and only if A is.

Proof. Let f ∈ L[x], deg f = n > 0. If f(A) is R-fractional there is a non-zero d ∈ R, with df(a) ∈ R for every a ∈ A. Let c ∈ R \ {0}, such that cf ∈ R[x], and set g = cdf = c_{n}x^{n} + … + c_{0}. For every a ∈ A, g(a) ∈ R implies that c_{n}a is integral over R, therefore c_{n}a ∈ R and c_{n}A ⊆ R. The converse is clear. □

Since a set B ⊆ K is R_{v}-fractional if and only if min_{b∈B} v(b) exists, Lemma 1.0 shows that A being R_{v}-fractional is necessary and sufficient for min_{a∈A} v(f(a)) to exist for any non-constant f ∈ K[x]. To exclude polynomials identically zero on A, for which min_{a∈A} v(f(a)) = ∞, we need deg f < |A|, so that the conditions on A in Lemma 1.1 below are necessary.

1.1 Lemma. Let n ∈ ℕ_{0}. If A is an R_{v}-fractional subset of K with |A| > n, then max{min_{a∈A} v(f(a)) | f monic ∈ K[x], deg f = n} exists.

Proof. The case n = 0 is trivial; so let n > 0 and m ∈ ℕ such that A is not contained in any union of n cosets of M_{v}^{m} in K. Such an m exists, since n < |A| and by the Krull Intersection Theorem ⋂_{m∈ℕ} M_{v}^{m} = (0). We show that for every monic f ∈ K[x] of degree n there exists an a_{0} ∈ A with v(f(a_{0})) < nm (and consequently max{min_{a∈A} v(f(a)) | f monic ∈ K[x], deg f = n} < nm).

Let v′ be an extension of v to the splitting field of f over K, R_{v′} its valuation-ring with maximal ideal M_{v′}, and e = [Γ_{v′} : Γ_{v}]. A is not contained in any union of n cosets of M_{v′}^{me} in K′. Pick an a_{0} ∈ A that is not in u + M_{v′}^{me} for any root u of f in K′, then v(f(a_{0})) = v′(f(a_{0})) = ∑_{i=1}^{n} v′(a_{0} − u_{i}) < nm. □

1.2 Theorem. Let A be an infinite, R_{v}-fractional subset of K. For n ∈ ℕ_{0} set γ_{v,A}(n) = max{min_{a∈A} v(f(a)) | f monic ∈ K[x], deg f = n}.

(i) M_{v}^{−γ_{v,A}(n)} = {leading coefficients of degree n polynomials in Int(A, R_{v})} ∪ {0}

(ii) A regular basis of Int(A, R_{v}) is given by (c_{n}g_{n})_{n∈ℕ_{0}}, with g_{n} ∈ K[x] monic, deg g_{n} = n, and c_{n} ∈ K, such that min_{a∈A} v(g_{n}(a)) = γ_{v,A}(n) and v(c_{n}) = −γ_{v,A}(n).

Proof. Let I_{n,v} = {leading coefficients of degree n polynomials in Int(A, R_{v})}∪{0}. The leading coefficient c_{n} of any n-th degree polynomial in Int(A, R_{v}) must satisfy v(c_{n}) ≥ −γ_{v,A}(n), so I_{n,v} ⊆ M_{v}^{−γ_{v,A}(n)}. Now, for n ∈ ℕ_{0}, let g_{n} be monic of degree n in K[x] with min_{a∈A} v(g_{n}(a)) = γ_{v,A}(n) (such things exist by dint of Lemma 1.1) then M_{v}^{−γ_{v,A}(n)}g_{n} ⊆ Int(A, R_{v}), so M_{v}^{−γ_{v,A}(n)} ⊆ I_{n,v}. This shows (i) and also that I_{n,v}g_{n} ⊆ Int(A, R_{v}) for all n ∈ ℕ_{0}. (ii) follows by Lemma 0.1 and the fact that M_{v}^{−γ_{v,A}(n)} = c_{n}R_{v} for every c_{n} ∈ K with v(c_{n}) = −γ_{v,A}(n). □

Before deriving a formula for max{min_{a∈A} v(f(a)) | f monic ∈ K[x], deg f = n}, when A is a subring of R_{v}, we check that the other plausible way of normalizing the polynomials would yield the same value. We also see that polynomials mapping A ⊆ R_{v} into the maximal possible power of M_{v} can be chosen to split with their roots in any set that M_{v}-adically approximates A (for instance in A itself, or, if R_{v} is the localization of a ring R at a prime ideal of finite index, in R). We need a lemma from [7] (but include the proof).

1.3 Lemma. Let f ∈ R_{v}[x], not all of whose coefficients lie in M_{v}, split over K, as f(x) = d(x − b_{1}) · . . . · (x − b_{m}) · (x − c_{1}) · . . . · (x − c_{l}) with v(b_{i}) < 0, v(c_{i}) ≥ 0, and put f_{+}(x) = (x − c_{1}) · . . . · (x − c_{l}). Then for all r ∈ R_{v} v(f(r)) = v(f_{+}(r)).

Proof. For r ∈ R_{v} v(r − b_{i}) = v(b_{i}) and so v(f(r)) = v(d) + ∑_{i=1}^{m} v(b_{i}) + v(f_{+}(r)); we show v(d) = − ∑_{j=1}^{m} v(b_{i}). Consider d^{−1}f(x) = x^{n} + a_{n−1}x^{n−1} + … + a_{0}. Since f ∈ R_{v}[x] \ M_{v}[x], v(d) = − min_{0≤k≤n} v(a_{k}). But a_{k} is the elementary symmetric polynomial of degree n − k in the b_{i} and c_{i}, so the minimal valuation is attained by v(a_{n−m}) = ∑_{i=1}^{m} v(b_{i}). □

1.4 Proposition. Let A ⊆ R_{v} and 0 ≤ n < |A|, then α and γ below are equal:

α = max{ min_{a∈A} v(f(a)) | f ∈ R_{v}[x] \ M_{v}[x], deg f = n},

γ = max{ min_{a∈A} v(f(a)) | f monic ∈ K[x], deg f = n}.

If, furthermore, B ⊆ R_{v}, such that B intersects every coset of M_{v}^{l} that A intersects, for all l ∈ ℕ, then δ below is equal to α and γ; and so is β, if B is also a ring:

β = max{ min_{a∈A} v(f(a)) | f ∈ B[x] \ (M_{v} ∩ B)[x], deg f = n},

δ = max{ min_{a∈A} v(f(a)) | f(x) = ∏_{i=1}^{n}(x − d_{i}), d_{i} ∈ B}.

Proof. Let B be a fixed subset of R_{v} that intersects every coset of every power of M_{v} that A intersects (e.g. B = R_{v}, when only interested in α and γ). For n = 0 all four expressions are equal to 0; now consider a fixed n > 0. Clearly δ ≤ γ and, if B is a ring, δ ≤ β ≤ α. Also γ ≤ α, because, given f monic in K[x], there exists a d ∈ R_{v} such that df = g ∈ R_{v}[x]\M_{v}[x] and for all a ∈ A v(g(a)) = v(d)+v(f(a)) ≥ v(f(a)), and so min_{a∈A} v(g(a)) ≥ min_{a∈A} v(f(a)).

To show α ≤ δ, we fix f ∈ R_{v}[x] \ M_{v}[x] of degree n and construct a monic g that splits with roots in B such that v(g(a)) ≥ min_{a∈A} v(f(a)) for all a ∈ A. Let v′ be an extension of v to the splitting field of f over K. For all a ∈ A, v′(f(a)) = v′(f_{+}(a)) with f_{+}(x) = ∏_{i=1}^{l}(x − c_{i}), where the c_{i} are the roots of f in R_{v′}, by Lemma 1.3. Put s = min_{a∈A} v′(f_{+}(a)). We replace each c_{i} by a d_{i} ∈ B chosen such that ∏_{i=1}^{l}(x − d_{i}) = h(x) satisfies: for all a ∈ A v′(h(a)) ≥ s. If (c_{i} + M_{v′}^{k}) ∩ A ≠ ∅ for all k ∈ ℕ, we pick d_{i} out of (c_{i} + M_{v′}^{s}) ∩ B; otherwise out of (c_{i} + M_{v′}^{k}) ∩ B with k maximal such that (c_{i} + M_{v′}^{k}) ∩ A ≠ ∅. Since the intersection of a residue class of M_{v′}^{k} in R_{v′} with R_{v} is either empty or an entire residue class of a power of M_{v} in R_{v}, and B intersects all of these that A intersects, it is possible to find such d_{i} in B. Now for every a ∈ A either v′(a − d_{i}) ≥ v′(a − c_{i}) for all i and so v′(h(a)) ≥ v′(f_{+}(a)) ≥ s, or v′(a − d_{i}) ≥ s for some i and hence v′(h(a)) ≥ s. To get a polynomial of degree n, set g(x) = (x − d_{0})^{n−l}h(x), d_{0} ∈ B. □

2. Polynomials mapping into a maximal Power of M_{v}

If R is an infinite subring of a discrete valuation ring R_{v}, we will construct polynomials g_{n}(x) = (x − a_{1}) … (x − a_{n}) that map R into the maximal possible (for a monic polynomial of degree n) power of M_{v}, by finding sequences (a_{i}) in R that show a nice distribution among the cosets of M_{v}^{n} ∩ R, to serve as roots.

This generalizes a procedure of Pólya [16] (also used by Gunji and McQuillan [12, 14], Cahen [4] and others) for the special case where R_{v} = R_{Q}, Q being a prime ideal of index q in R such that R_{Q} is a discrete valuation ring: Pick π ∈ Q \ Q^{2} and a complete set of residues r_{0}, ..., r_{q−1} of Q in R and define a_{n} = ∑_{i≥0} r_{c_{i}}π^{i}, if n = ∑_{i≥0} c_{i}q^{i} is the q-adic expansion of n. The resulting polynomials map R into the highest possible power of Q and can be used to give a regular basis of Int(R_{v}) (most clearly stated in [14]). Gilmer [10] has remarked that the construction even works for Int(D), D a quasi-local ring with principal maximal ideal.

The I-sequences below are defined for any commutative ring R. All sequences are indexed by an initial segment of ℕ or ℕ_{0}. Quantifiers over indices of such a sequence are assumed to range over precisely the index-set.

2.0 Definition. For a set I of ideals in a commutative ring R we define an I-sequence in R to be a sequence (a_{n}) of elements in R with the property

∀I ∈ I ∀n, m a_{n} ≡ a_{m} mod I ⟺ [R : I] | n − m.

We define a homogeneous I-sequence to be one with the additional property

∀I ∈ I ∀n ≥ 1 a_{n} ∈ I ⟺ [R : I] | n.

(Any infinite [R : I] we regard as dividing 0, but no other integer.) Note that a_{1}, a_{2}, .. is a homogeneous I-sequence if and only if 0=a_{0}, a_{1}, a_{2}, .. is an I-sequence.

2.1 Proposition. Let I = {I_{n} | n ∈ ℕ} be a descending chain of ideals in a commutative ring R, then there exists an infinite homogeneous I-sequence in R.

Proof. Put I_{0} = R. For k ≥ 0, if [I_{k}: I_{k+1}] is finite, let {a_{j}^{(k)}| 0 ≤ j < [I_{k}: I_{k+1}]} be a system of representatives of I_{k} : I_{k+1} with a_{0}^{(k)} = 0, otherwise let (a_{j}^{(k)})_{j∈ℕ_{0}} be a sequence in I_{k} of elements pairwise incongruent mod I_{k+1}, with a_{0}^{(k)} = 0. If I_{N} ∈ I with [R: I_{N}] finite, then every n < [R: I_{N}] has a unique representation n = ∑_{k=0}^{N−1} j_{k}[R: I_{k}] with 0 ≤ j_{k} < [I_{k}: I_{k+1}], and we set a_{n} = ∑_{k=0}^{N−1} a_{j_{k}}^{(k)} . If the indices of ideals in I get arbitrarily large while remaining finite, this defines our I-sequence inductively. Otherwise there exists I_{N} ∈ I of maximal finite index such that either [I_{N}: I_{N+1}] is infinite or I_{m} = I_{N} for m ≥ N. Define a_{n} for n < [R: I_{N}] as above. Then, in the first case, set a_{m} = a_{q}^{(N)} + a_{r} for m = q [R: I_{N}] + r with 0 ≤ r < [R: I_{N}], and a_{m} = a_{r} in the second. □

2.2 Facts.

(i) For I ∈ I of finite index in R, any [R : I] consecutive terms of an I-sequence form a complete set of representatives of R mod I.

(ii) If (a_{i})_{i=1}^{n} is an I-sequence in R then (r − a_{i})_{i=1}^{n} is an I-sequence for every r ∈ R and (a_{n} − a_{n−i})_{i=0}^{n−1} is a homogeneous I-sequence.

The following lemma will be needed for globalization.

2.3 Lemma. If a_{1}, ..., a_{l} is an I-sequence for a chain of ideals I, J ∈ I with [R : J] > l, and b_{1}, ..., b_{l} ∈ R such that b_{n} ≡ a_{n} mod J for 1 ≤ n ≤ l, then (b_{n}) is also an I-sequence, and homogeneous if (a_{n}) is.

Proof. Let I ∈ I and 1 ≤ n, m ≤ l. First suppose n ≡ m mod [R : I]. Then n = m or [R: I] < l. In the latter case J ⊆ I, so b_{n} ≡ a_{n} ≡ a_{m} ≡ b_{m} mod I. Now suppose n ≢ m mod [R: I]. Either J ⊆ I or I ⊆ J. If J ⊆ I then b_{n} ≡ a_{n} ≢ a_{m} ≡ b_{m} mod I. If I ⊆ J then b_{n} ≡ a_{n} ≢ a_{m} ≡ b_{m} mod J (because 0 ≠ n − m < [R : J]), hence b_{n} ≢ b_{m} mod I. Homogeneity is shown similarly. □

From now on, R is always an infinite subring of a discrete valuation ring R_{v}. Note that the definitions of α_{v,R}(n) and v-sequence below depend only on M_{v} and R, and thus not distinguish between equivalent valuations.

2.4 Definition. A v-sequence for R is a {M_{v}^{n} ∩ R | n ∈ ℕ}-sequence in R. In other words, (a_{n}) is a v-sequence for R if and only if for all n ∈ ℕ and all i, j,

a_{i} − a_{j} ∈ M_{v}^{n} ⟺ [R : M_{v}^{n} ∩ R] | i − j

and a homogeneous v-sequence if in addition, for all n ∈ ℕ and all j ≥ 1,

a_{j} ∈ M_{v}^{n} ⟺ [R : M_{v}^{n} ∩ R] | j .

If [R : M_{v}^{n} ∩ R] is infinite, distinct elements of a v-sequence must be incongruent mod M_{v}^{n} ∩ R. Proposition 2.1 guarantees the existence of an infinite homogeneous v-sequence for every infinite subring R of every discrete valuation ring R_{v}.

2.5 Definition. For n ∈ ℕ_{0}, R an infinite subring of R_{v} and q ∈ ℕ, let

α_{v,R}(n) = ∑_{j≥1} [\frac{n}{[R : M_{v}^{j} ∩ R]}]  and  α_{q}(n) = ∑_{j≥1} [\frac{n}{q^{j}} ] .

Infinite indices are allowed; \frac{n}{∞} = 0. Since R is infinite, α_{v,R}(n) is always a finite number. We will frequently use the fact that α_{v,R}(n) > 0 if and only if n ≥ [R : M_{v} ∩ R]. If Q is a prime ideal in a domain D, such that D_{Q} is a discrete valuation ring, we write v_{Q} for the corresponding valuation with value group ℤ.

2.6 Facts.

(i) If Q is a prime ideal of finite index q in R such that R_{Q} is a discrete valuation ring, then α_{v_{Q},R}(n) = α_{q}(n) for all n.

(ii) If v is a discrete valuation, R an infinite subring of R_{v} and v′ an extension of v with [Γ_{v′} : Γ_{v}] = e finite, then α_{v′,R}(n) = e · α_{v,R}(n) for all n.

Proof. (i) Since Q is maximal, (QR_{Q})^{n} ∩ R = Q^{n} for all n. Using the fact that Q contains a generator of QR_{Q} one sees that [R: Q^{n}] = [R_{Q} : (QR_{Q})^{n}] = q^{n} for all n. (ii) For k ∈ ℕ, M_{v′}^{k} ∩ R = (M_{v′}^{k} ∩ R_{v}) ∩ R = M_{v}^{⌈ \frac{k}{e} ⌉} ∩ R, where ⌈x⌉ denotes the smallest integer greater or equal x. Each number [\frac{n}{[R:M_{v}^{j}∩R]}] appears e times, as

[\frac{n}{[R:M_{v′}^{k}∩R]}] for k = (j − 1)e + 1, …, je, in the sum for α_{v′,R}(n). □

In the remainder of section two, v is assumed to have value-group ℤ.

2.7 Lemma. Let (a_{i})_{i=1}^{n+1}, (b_{i})_{i=1}^{n} and (c_{i})_{i=1}^{n} be v-sequences for R, and (c_{i})_{i=1}^{n} homogeneous, then

(a) v(c_{1} · . . . · c_{n}) = α_{v,R}(n) ≤ v(b_{1} · . . . · b_{n}) ≤ α_{v,R}(n) + max_{1≤i≤n} v(b_{i}),

(b) v( ∏_{i=1}^{n}(a_{n+1} − a_{i}))= α_{v,R}(n) ≤ v( ∏_{i=1}^{n}(r − b_{i})) for all r ∈ R.

Proof. v(c_{1} · . . . · c_{n}) = ∑_{j≥1} |{i | 1 ≤ i ≤ n, v(c_{i}) ≥ j}| and similarly for the b_{i}. Since for finite index M_{v}^{j} ∩ R every [R: M_{v}^{j} ∩ R] successive terms of a v-sequence form a complete residue system of R mod M_{v}^{j} ∩ R, we have ∀j ∈ ℕ

|\{i | v(c_{i}) ≥ j\}| = [\frac{n}{[R : M_{v}^{j} ∩ R]}] ≤ |\{i | v(b_{i}) ≥ j\}| ≤ [\frac{n}{[R : M_{v}^{j} ∩ R]}] + 1

This implies (a) (and, since the 1 on the right can only occur if [R: M_{v}^{j} ∩ R] |̸ n, v(b_{1} · . . . · b_{n}) ≤ α_{v,R}(n) + max_{1≤i≤n} v(b_{i}) − max{j | [R : M_{v}^{j} ∩ R] divides n}). By Fact 2.2 (ii) about I-sequences, (b) is a special case of (a). □

2.8 Theorem. Let R be an infinite subring of R_{v}. An R_{v}-basis of Int(R, R_{v}) is given by

f_{0} = 1  and  f_{n}(x) = \frac{∏_{i=1}^{n}(x − a_{i})}{∏_{i=1}^{n}(a_{n+1} − a_{i})}  (n ≥ 1),

where (a_{n})_{n=1}^{∞} is a v-sequence for R.

Proof. An infinite v-sequence (a_{n})_{n=1}^{∞} in R exists by Proposition 2.1 applied to {M_{v}^{n} ∩ R | n ∈ ℕ}. The f_{n}, being a K-basis of K[x], are free generators of the R_{v}-module they generate in K[x], call this module F. Since by Lemma 2.7 every f_{n} maps R to R_{v}, F ⊆ Int(R, R_{v}). For the reverse inclusion we show the stronger statement that Int(A, R_{v}) ⊆ F, where A = {a_{n} | n ∈ ℕ}. Let f ∈ Int(A, R_{v}), f =∑_{j=0}^{N} l_{j}f_{j} with l_{j} ∈K. We show inductively that the l_{j} are in R_{v}. l_{0} =f(a_{1})∈R_{v}. The induction hypothesis is l_{j} ∈ R_{v} for 0 ≤ j < n. Using this and the facts that f_{j}(a_{i}) = 0 for j ≥ i and f_{j}(a_{j+1}) = 1, we see that f(a_{n+1}) = l_{n} + ∑_{j=0}^{n−1} l_{j}f_{j}(a_{n+1}). Since f_{j}(a_{i}) ∈ R_{v} for all i, j (by Lemma 2.7) and f ∈ Int(A, R_{v}), the sum on the right as well as f(a_{n+1}) is in R_{v}, therefore l_{n} ∈ R_{v}. □

Remark. For an infinite subring R of R_{v} and A ⊆ R, the proof of Theorem 2.8 shows that if A contains an infinite v-sequence for R, then Int(A, R_{v}) = Int(R, R_{v}). The converse holds, too (the criterion for Int(A, R_{v}) = Int(R, R_{v}) in [7] is easily seen to be equivalent to A containing an infinite v-sequence for R).

Corollary 1. α_{v,R}(n) = max{ min_{r∈R} v(f(r)) | f monic ∈ K[x], deg f = n} and

M_{v}^{−α_{v,R}(n)} = { leading coefficients of n-th degree polynomials in Int(R, R_{v}) } ∪ {0}.

Proof. The second statement can be read off the theorem using Lemma 2.7 (b), the first one then follows by Theorem 1.2. □

Pólya’s Satz IV [16] is a special case: if P is a prime ideal in a domain R such that R_{P} is a discrete valuation ring and [R : P] = q, then (by Proposition 1.4 with B = R and Fact 2.6 i) α_{q}(n) = max{ min_{r∈R} v_{P} (f(r)) | f ∈ R[x] \ P [x], deg f = n}.

Corollary 2. Let g_{n}(x) = ∏_{i=1}^{n}(x − a_{i}^{(n)}), where (a_{i}^{(n)})_{i=1}^{n} is a v-sequence for R when n ≥ [R : M_{v} ∩ R], and let g_{n} be any monic polynomial in R_{v}[x] of degree n for 0 ≤ n < [R : M_{v} ∩ R]. Further let, for n ∈ ℕ_{0}, c_{n} ∈ K with v(c_{n}) = −α_{v,R}(n). Then (c_{n}g_{n})_{n∈ℕ_{0}} is an R_{v}-basis of Int(R, R_{v}).

Proof. For all n∈ℕ_{0}, r∈R, v(g_{n}(r))≥α_{v,R}(n) (by Lemma 2.7, when n≥[R:M_{v}∩R], and because g_{n} ∈ R_{v}[x] and α_{v,R}(n) = 0 otherwise). By the maximality of α_{v,R}(n) (Corollary 1), min_{r∈R} v(g_{n}(r)) = α_{v,R}(n). Therefore (c_{n}g_{n})_{n∈ℕ_{0}} is an R_{v}-basis of Int(R, R_{v}) by Corollary 1 and Theorem 1.2 (ii). □

3. Polynomials mapping a subring into a Krull Ring

Notation. Let S be a domain with quotient field K, such that S = ⋂_{v∈V} R_{v}, V a set of discrete valuations (with value-group ℤ) on K; and R an infinite subring of S. We put I_{n} = {leading coefficients of n-th degree polynomials in Int(R, S)} ∪ {0} and introduce names for recurring additional conditions:

(F) ∀ q ∈ ℕ {Q ⊴ R | [R : Q] = q and Q = M_{v} ∩ R for some v ∈ V} is a finite set.

(C) For every prime ideal Q of finite index in R, the set of M_{v}^{n} ∩ R with n ∈ ℕ, v ∈ V, and M_{v} ∩ R = Q, if not empty, forms a descending chain of ideals.

Note that (C) holds naturally in two cases: when there is only one M_{v} such that M_{v} ∩ R = Q, and when every M_{v}^{n} ∩ R with M_{v} ∩ R = Q is a power of Q.

3.0 Lemma. (Cahen [4]) If R is an infinite subring of a Krull ring S and q ∈ ℕ, then S has at most finitely many height 1 prime ideals P with [R : P ∩ R] = q.

Proof. There exists r ∈ R with r^{q} − r ≠ 0. For every P with Q = R ∩ P of index q in R, r^{q} − r ∈ Q ⊆ P, so the statement follows by the definition of Krull ring. □

3.1 Lemma. Let v ∈ V such that M_{v} ∩ R = Q ≠ (0) and L the quotient field of R. If R_{Q} is a valuation ring, then it is a discrete valuation ring and R_{Q} = R_{v} ∩ L. If Q is also a maximal ideal then, for every n ∈ ℕ, M_{v}^{n} ∩ R is a power of Q.

Proof. For any valuation ring V with quotient field L and maximal ideal M we have L\V = {r ∈ L^{∗} | r^{−1} ∈ M}. Put R_{v} ∩L = R_{w} and M_{v} ∩L = M_{w}, then R_{w} and R_{Q} are valuation rings with quotient field L and maximal ideals M_{w} and QR_{Q}, respectively. R ⊆ R_{w} and M_{w} ∩ R = M_{v} ∩ R = Q imply R_{Q} ⊆ R_{w} and also QR_{Q} ⊆ M_{w}. By the latter inclusion L\R_{Q} = {r ∈ L^{∗} | r^{−1} ∈ QR_{Q}} ⊆ {r ∈ L^{∗} | r^{−1} ∈ M_{w}} = L\R_{w}. This shows R_{Q} = R_{w} = R_{v} ∩ L, so R_{Q} is a discrete valuation ring and every M_{v}^{n} ∩ R_{Q} is a power of QR_{Q}. If Q is maximal, then (QR_{Q})^{k} ∩ R = Q^{k} for all k, so M_{v}^{n} ∩ R is a power of Q. □

3.2 Lemma. (C) implies: For every finite set M of prime ideals of finite index in R and every m ∈ ℕ, there exists a sequence (a_{i})_{i=0}^{m} in R that is a homogeneous v-sequence for all v in V with M_{v} ∩ R ∈ M, simultaneously.

Proof. For every Q ∈ M, I_{Q} = {M_{v}^{n} ∩ R | v ∈ V, n ∈ ℕ, M_{v} ∩ R = Q} (if not empty) is a descending chain by (C), so there exists a homogeneous I_{Q}-sequence (a_{i}^{(Q)})_{i=0}^{∞} in R by Proposition 2.1. For each Q with I_{Q} ≠ ∅ let I_{Q} be an element of I_{Q} with [R: I_{Q}] > m. I_{Q} = M_{v}^{n} ∩ R for some v and n and therefore contains Q^{n}. Since different Q are co-prime, there exists, by the Chinese Remainder Theorem, a sequence (a_{i})_{i=0}^{m} in R that is congruent to (a_{i}^{(Q)})_{i=0}^{m} modulo I_{Q} for all Q ∈ M. By Lemma 2.3, this a homogeneous I_{Q}-sequence for all Q ∈ M, i.e., a homogeneous v-sequence for all v with M_{v} ∩ R ∈ M. □

From Lemma 3.0, Lemma 3.1 and the fact that the powers of an ideal Q form a descending sequence, we conclude that the hypothesis of Theorem 3.4 below is satisfied in at least one natural setting:

3.3 Fact. If S is a Krull ring, V = {v_{P} | P ∈ Spec^{1}(S)}, and R an infinite subring such that R_{Q} is a valuation ring for every finite index Q = P ∩ R, P ∈ Spec^{1}(S), then (C) and (F) both hold.

In the following theorem, the case where S is a Dedekind ring and R = S is due to Cahen [4] (also pertinent: [5]). 3.4 Theorem. Let R be an infinite subring of S = ⋂_{v∈V} R_{v}. If (C) and (F) hold, then

I_{n} = ⋂_{v∈V} M_{v}^{−α_{v,R}(n)}  (n ∈ ℕ_{0})

and there exists a regular sequence of monic polynomials (g_{n}) in R[x] such that

Int(R, S) = ∑_{n≥0} I_{n}g_{n},

namely, g_{n}(x) = ∏_{i=1}^{n}(x − a_{i}^{(n)}), where (a_{i}^{(n)})_{i=1}^{n} is a simultaneous v-sequence for all v ∈ V with [R : M_{v} ∩ R] ≤ n.

Proof. Int(R, ⋂_{v∈V} R_{v}) = ⋂_{v∈V} Int(R, R_{v}), therefore I_{n} ⊆ ⋂_{v∈V} M_{v}^{−α_{v,R}(n)} (by Theorem 2.8, Corollary 1). For the reverse inclusion, let c ∈ ⋂_{v∈V} M_{v}^{−α_{v,R}(n)}. Set V_{n} = {v ∈ V | α_{v,R}(n) > 0} = {v ∈ V | [R : M_{v} ∩ R] ≤ n} then {M_{v} ∩ R | v ∈ V_{n}} is finite by (F). Let (a_{i}^{(n)})_{i=1}^{n} in R be a homogeneous v-sequence for all v ∈ V_{n} simultaneously (which exists by Lemma 3.2) and g_{n}(x) = ∏_{i=1}^{n}(x − a_{i}^{(n)}). Then min_{r∈R} v(g(r)) ≥ α_{v,R}(n) for all v ∈ V (by Lemma 2.7 when v ∈ V_{n}, and because α_{v,R}(n) = 0 and g_{n} ∈ R[x] otherwise) which means cg(x) ∈ Int(R, ⋂_{v∈V} R_{v}) and hence c ∈ I_{n}. This completes the proof of the first statement and also shows, for all n ≥ 0, that I_{n}g_{n} ⊆ Int(R, ⋂_{v∈V} R_{v}), so the second follows by Lemma 0.1. □

From now on, S is a Krull ring. By convention, the empty intersection or product of ideals of S equals S. We denote the set of height 1 prime ideals of S by Spec^{1}(S) or P. If P ∈ P, we write α_{P,R} for α_{v_{P} ,R} and, if j ∈ ℕ_{0}, P^{(j)} for (P S_{P})^{j} ∩ S. With this notation we have, for n ∈ ℕ_{0} and P ∈ P:

α_{P,R}(n) = ∑_{j≥1} [\frac{n}{[R : P^{(j)} ∩ R]}] .

3.5 Lemma. Let S be a Krull ring and V = {v_{P} | P ∈ P}. If (C) holds, then Int(R, S) has a regular basis ⟺ ⋂_{{P ∈P \atop [R:P ∩R]≤n}} P^{(α_{P,R}(n))} is principal for all n.

Proof. α_{P,R}(n) ≠ 0 if and only if [R : P ∩ R] ≤ n. Since (F) holds by Lemma 3.0, this only happens for finitely many P for each n. If {a_{P} | P ∈ P} is a set of integers, only finitely many of them non-zero, then ⋂_{P ∈P} (P S_{P})^{−a_{P}} is principal if and only if ⋂_{P ∈P} (P S_{P})^{a_{P}} is, namely if there exists c ∈ K with v_{P} (c) = a_{P} for all P ∈ P. If all a_{P} are non-negative then ⋂_{P ∈P} (P S_{P})^{a_{P}} = ⋂_{a_{P} >0} P^{(a_{P})}. Applied to ⋂_{P ∈P} (P S_{P})^{−α_{P,R}(n)}, which is I_{n} by Theorem 3.4, with Lemma 0.1 (iii) in mind, this proves the claim. □

3.6 Theorem. Let R be an infinite subring of a Krull ring S, P = Spec^{1}(S), P^{∗} = {P ∈ P | [R : P ∩ R] finite} and Q = {R ∩ P | P ∈ P^{∗}}. If R_{Q} is a valuation ring for all Q ∈ Q, then R_{Q} is a discrete valuation ring for all Q ∈ Q and

Int(R, S) has a regular basis ⟺ ∀ q ∈ ℕ ⋂_{{P ∈P \atop [R:R∩P]=q}}P^{(e_{P})} is a principal ideal of S,

where e_{P} is the ramification index of P S_{P} over QR_{Q}, for P ∈ P^{∗}, Q = P ∩ R.

Proof. Let P_{q} ={P ∈P |[R:P ∩R]=q}, P ∈P_{q}, Q=P ∩R, L the quotient field of R, then by Lemma 3.1 R_{Q} = S_{P} ∩L and R_{Q} is a discrete valuation ring. v′_{P} = (1/e_{P})v_{P} is equivalent to v_{P} and is an extension of v_{Q} to K with [Γ_{v′_{P}} : Γ_{v_{Q}}] = e_{P}. By the Facts 2.6 (ii) and (i), α_{P,R}(n) = α_{v′_{P} ,R}(n) = e_{P} α_{Q,R}(n) = e_{P} α_{q}(n).

If we call left and right side of the claimed equivalence (l) and (r), respectively, then (l) is equivalent to (l’) ‘∀n ⋂_{{P ∈P \atop [R:P ∩R]≤n}} P^{(α_{P,R}(n))} is principal’ by Lemma 3.5

(whose condition (C) holds by Fact 3.3). We know that ⋂_{{P ∈P \atop [R:P ∩R]≤n}} P^{(α_{P,R}(n))} =

⋂_{q≤n} ⋂_{P ∈P_{q}}P^{(e_{P} α_{q}(n))}. The latter is clearly principal provided all ⋂_{P ∈P_{q}} P^{(e_{P})} are; thus (r) ⇒ (l’).

For (l’) ⇒ (r), suppose ⋂_{q≤n} ⋂_{P ∈P_{q}} P^{(e_{P} α_{q}(n))} = s_{n}S for all n. We see that s_{q}S = ⋂_{P ∈P_{q}} P^{(e_{P})} ∩ ⋂_{l<q} ⋂_{P ∈P_{l}} P^{(e_{P} α_{l}(q))}, because α_{q}(q) = 1. This allows an induction on q: from the formula for s_{q}S we conclude that ⋂_{P ∈P_{q}} P^{(e_{P})} is principal if ⋂_{P ∈P_{l}} P^{(e_{P})} is principal for all l < q. □

Corollary 1. If R ⊆ S is an extension of Krull rings such that ht(P ∩ R) ≤ 1 for all height 1 prime ideals P of S then

Int(R, S) has a regular basis ⟺ ∀ q ∈ ℕ ⋂_{{Q∈Spec^{1}(R) \atop [R:Q]=q}} div(QS) is principal,

where div(QS) means the smallest divisorial ideal containing QS.

Proof. If R ⊆ S is an extension of Krull rings with the stated property and Q is in Spec^{1}(R), then div(QS) = ⋂_{{P ∈Spec^{1}(S) \atop P ∩R=Q}} P^{(e_{P})}, where e_{P} = e(P |Q) is the ramification

index of P S_{P} over QR_{Q} [1, p183]. □

In particular, if R ⊆ S is an extension of Dedekind rings, then

Int(R, S) has a regular basis ⟺ ∀ q ∈ ℕ ∏_{{Q∈Spec(R) \atop [R:Q]=q}} QS is principal.

A different specialization gives Ostrowski’s criterion [15]. If S is a Krull ring,

Int(S) has a regular basis ⟺ ∀ q ∈ ℕ ∏_{{P ∈Spec^{1}(S) \atop [S:P]=q}} P is principal.

When a regular basis exists, we can give a fairly explicit description of one. (For Int(S), S a Dedekind ring, there also is a different construction by Gerboud [9].)

Corollary 2. In the situation of Theorem 3.6, if ⋂_{{P ∈P \atop [R:P ∩R]=q}} P^{(e_{P})} = c_{q}S (q ∈ ℕ) then a regular basis of Int(R, S) is given by f_{0} = 1,

f_{n}(x) = ∏_{q≤n} c_{q}^{−α_{q}(n)} ∏_{i=1}^{n}(x − a_{i}^{(n)})  (n ∈ ℕ) ,

where (a_{i}^{(n)})_{i=1}^{n} ⊆ R is a v_{P} -sequence for all P ∈ P with [R : P ∩ R] ≤ n.

Proof. v_{P} (c_{q}^{−α_{q}(n)}) = −e_{P} α_{q}(n) = −α_{P,R}(n) for the P ∈ P with [R : P ∩ R] = q, and zero for all other P ∈ P, so v_{P} (∏_{q≤n} c_{q}^{−α_{q}(n)}) = −α_{P,R}(n) for all P ∈ P (since α_{P,R}(n) = 0 if n < [R: P ∩ R]). Therefore the f_{n} are an S_{P} -basis of Int(R, S_{P}) for all P ∈ P simultaneously, by Theorem 2.8, Corollary 2. □

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Institut für Mathematik C, Technische Universität Graz, Kopernikusgasse 24, A-8010 Graz, Austria

e-mail: frisch@blah.math.tu-graz.ac.at