Contemporary Mathematics

Integral dependence and weak subintegrality

Leslie G. Roberts

Abstract. In [RRS] we introduced a generic weakly subintegral extension R ⊆ R[z] = S, based on systems of subintegrality. Here we find a presentation of S as an R-algebra, and show that this universal extension can also be based on a system of equations of integral dependence for z.

1. Introduction

Let A ⊆ B be an extension of commutative rings. An element b ∈ B is defined to be weakly subintegral over A if the extension A ⊆ A[b] is weakly subintegral in the sense of Yanagihara [Y]. That is, A[b] is integral over A, the induced map Spec(A[b]) →Spec(A) is a bijection, and the induced maps on residue field extensions are purely inseparable. The elements of B that are weakly subintegral over A form a ring _{B}^{∗}A which is called the weak normalization of A in B. In [RRS, section 2] we introduced (building on earlier work in [RS], [RRSa] on subintegral extensions of ℚ-algebras) a “universal weakly subintegral extension” based on systems of subintegrality. In this note we explore the possibility of using equations of integral dependence to construct such an extension. This was done in the hope of gaining new insight into weak subintegrality. For example, using systems of subintegrality we construct our universal extension by giving a subring, and using equations of integral dependence we construct the universal extension as a quotient. Our main result (Theorem 3.5) is that the two extensions are isomorphic. This was initially a surprise, but with hindsight perhaps should not be, since (2.6) says that weak subintegrality can be approached equally well via equations of integral dependence as by systems of subintegrality. Section 2 reviews the system of subintegrality approach, in preparation for section 3 where we study subintegrality via equations of integral dependence. The final section illustrates some curious facts about the equations of integral dependence.

Throughout, ℕ denotes the positive integers and ℤ^{+} non-negative integers. The binomial coefficient \binom{k}{i} will be taken to be 0 if 0 ≤ k < i.

2. Review of systems of subintegrality

In [RRS], it was shown that b ∈ B is weakly subintegral over A if and only if:

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

This work was partially supported by an NSERC grant.

(2.1) There exist p ∈ ℤ^{+}, elements c_{1}, …, c_{p} ∈ B, and N ∈ ℕ such that b^{k} + ∑_{i=1}^{p} \binom{k}{i}c_{i}b^{k−i} ∈ A for k ∈ ℕ, k ≥ N.

We called the set-up in (2.1) a system of subintegrality for b over A, SOSI for short [RR]. If b is weakly subintegral over A then there is a SOSI for b over A with N = 1 [RRS, 2.1]. For simplicity we will assume in the rest of this note that N = 1. Then there is the following (arguably) more conceptual way of looking at SOSI’s. Following [RRS, section 3], in the ring B[[T]] define a “multiplication” ∗ by setting T^{m} ∗ T^{n} = \binom{m+n}{n}T^{m+n}, and then extending by bilinearity and completion to f ∗ g for all f, g ∈ B[[T]]. Then B[[T]] with the usual addition and ∗ as multiplication is a commutative ring. For b ∈ B, define Exp(b) ∈ B[[T]] by Exp(b) = ∑_{i≥0} b^{i}T^{i}. Exp has the useful property that Exp(a + b) = Exp(a)∗Exp(b) for a, b ∈ B, thereby serving as an “exponential function” for commutative rings that do not necessarily contain ℚ. If f = ∑_{i≥0} b_{i}T^{i} and g = ∑_{i≥0} c_{i}T^{i}, then the coefficient of T^{k} in h ∗ g is ∑_{i=0}^{k} \binom{k}{i}b_{k−i}c_{i}. Now, let f =Exp(b) and g = ∑_{i=0}^{p} c_{i}T^{i} ∈ B[T] (with c_{0} = 1). Then it follows that the coefficient of T^{k} in g∗Exp(b) is b^{k} + ∑_{i=1}^{p} \binom{k}{i}c_{i}b^{k−i}. Thus (2.1) (with N = 1) is equivalent to:

(2.2) There exists a polynomial g(T) = ∑_{i=0}^{p} c_{i}T^{i} ∈ B[T] such that g(0) = 1 and g∗Exp(b) ∈ A[[T]].

We think of the set-up in (2.1) as analogous to an equation of integral dependence. But (2.1) differs from an equation of integral dependence in several ways. First of all (2.1) involves an infinite number of conditions on b. Secondly the coefficients c_{i} are in B, not in A. We found both these concerns somewhat troubling. When discussing such things we have found universal examples helpful. Thus let x_{i} (1 ≤ i ≤ p) and z be independent indeterminates over ℤ and let γ_{n} = z^{n} + ∑_{i=1}^{p} \binom{n}{i}x_{i}z^{n−i} (n ≥ 1). Let R = ℤ[\{γ_{i}\}_{i≥1}]. Then R ⊆ S = ℤ[x_{1}, …, x_{p}, z] can be thought of as a universal weakly subintegral extension. More precisely z is weakly subintegral over R by (2.1) and inductively starting with γ_{1} = z + x_{1} we conclude that S = R[z], so the extension R ⊆ S is weakly subintegral. This extension is universal (or more precisely z is a universal weakly subintegral element) in the sense that if A ⊆ B and b ∈ B is weakly subintegral over A (with SOSI (2.1)) then there is a homomorphism S → B which maps z to b, x_{i} to c_{i} and γ_{i} to b^{k} + ∑_{i=1}^{p} \binom{k}{i}c_{i}b^{k−i} ∈ A (and hence R into A). The key to partially alleviating our first concern is the observation in [RRS, 2.2] that R = ℤ[{γ_{i}}_{1≤i≤2p+1}]. From this it follows that if in (2.1) (with N = 1) b^{k} + ∑_{i=1}^{p} \binom{k}{i}c_{i}b^{k−i} ∈ A for k ∈ ℕ, 1 ≤ k ≤ 2p + 1 then b^{k} + ∑_{i=1}^{p} \binom{k}{i}c_{i}b^{k−i} ∈ A for k ≥ 1. Therefore (2.1) (with N = 1) is equivalent to

(2.3) There exist p ∈ ℤ^{+}, and elements c_{1}, …, c_{p} ∈ B, such that b^{k} + ∑_{i=1}^{p} \binom{k}{i}c_{i}b^{k−i} ∈ A for k ∈ ℕ, 1 ≤ k ≤ 2p + 1.

Our second concern can be resolved by the following lemma (which is adapted from [RRS, section 5]):

Lemma 2.4. Let B be any commutative ring and let S be either all of ℕ or the interval {k ∈ ℕ | 1 ≤ k ≤ d} for some d ∈ ℕ. Suppose we have elements b ∈ B and a_{k}, c_{k} ∈ B for all k ∈ S. Then the following are equivalent:

(2.4.1) a_{k} = b^{k} + ∑_{i=1}^{k} \binom{k}{i}c_{i}b^{k−i} for all k ∈ S.
(2.4.2) c_{k} = (−1)^{k}(b^{k} + ∑_{i=1}^{k}(−1)^{i}\binom{k}{i}a_{i}b^{k−i}) for all k ∈ S.

Proof. If S = ℕ let g = 1+∑_{i≥1} c_{i}T^{i} and f = 1+∑_{i≥1} a_{i}T^{i}. Then (2.4.1) is obtained by comparing the coefficients of T^{k} in the equation f = g∗Exp(b), and (2.4.2) is obtained by comparing the coefficients of T^{k} in the equivalent equation g = f∗Exp(−b). The case of finite S follows from this by setting c_{k} = 0 for k > d, a_{k} = 0 for k > d, and comparing coefficients only up to T^{d}.

Let a_{i} ∈ A for i ∈ ℕ and c_{k} = 0 for k ≥ p + 1. By Lemma 2.4 it follows that weak subintegrality of b over A is equivalent to:

(2.5) There exist p ∈ ℤ^{+} and elements a_{i} ∈ A (i ∈ ℕ) such that b^{k} + ∑_{i=1}^{k}(−1)^{i}\binom{k}{i}a_{i}b^{k−i} = 0 for k ∈ ℕ, k ≥ p + 1.

and also, using the finite version of Lemma 2.4 with d = 2p + 1, to:

(2.6) There exist p ∈ ℤ^{+} and elements a_{i} ∈ A (1 ≤ i ≤ 2p + 1) such that b^{k} + ∑_{i=1}^{k}(−1)^{i}\binom{k}{i}a_{i}b^{k−i} = 0 for k ∈ ℕ, p + 1 ≤ k ≤ 2p + 1.

In the equivalences (2.3) ⇔ (2.6) or (2.1) ⇔ (2.5) the a_{i} and the c_{i} are related by (2.4.∗) (with c_{k} = 0 for k ≥ p + 1). If the equivalence (2.3) ⇔ (2.6) is applied to the extension R ⊆ S then a_{i} of (2.6) equals γ_{i} so z satisfies equations of integral dependence

(2.7) z^{k} + ∑_{i=1}^{k}(−1)^{i}\binom{k}{i}γ_{i}z^{k−i} = 0 over R for k ∈ ℕ, p + 1 ≤ k ≤ 2p + 1.

3. Weak subintegrality through integral dependence

We now set up a generic version of the equations of integral dependence (2.6). These equations, roughly speaking, force the a_{i} to be algebraically dependent. Thus, unlike the c_{i} in the SOSI case (2.1) (which we replaced by indeterminates x_{i}), the a_{i} cannot be replaced by indeterminates. Instead we propose to replace them by the γ_{i} of section 2. Let S′ = R[X]/⟨\{X^{k} + Σ_{j=1}^{k}(−1)^{j}\binom{k}{j}γ_{j}X^{k−j}\}_{p+1≤k≤2p+1}⟩, where X is an indeterminate over R and the notation ⟨⟩ means “ideal generated by”. First of all note that since z satisfies the equations of integral dependence (2.7) and S = R[z] there is a surjective homomorphism of R-algebras ϕ : S′ → S defined by ϕ(X) = z. The surjectivity of ϕ implies that the natural map R → S′ is an inclusion (which is perhaps not obvious since the generators of the ideal ⟨{X^{k} + Σ_{j=1}^{k}(−1)^{j}\binom{k}{j}γ_{j}X^{k−j}\}_{p+1≤k≤2p+1}⟩ contain terms in R). Define a k-algebra homo-

morphism ξ : S → S′ by ξ(z) = X and ξ(x_{k}) = (−1)^{k}(X^{k} + ∑_{i=1}^{k}(−1)^{i}\binom{k}{i}γ_{i}X^{k−i}). By Lemma 2.4 applied in S we see that ϕξ(x_{k}) = x_{k} for all k, and by definition ϕξ(z) = z, so ϕ ∘ ξ = Id_{S}, but is not so clear that ξ ∘ ϕ = Id_{S′}. This will follow from Theorem 3.5 below. (Once we have established Theorem 3.5, ξ and ϕ will be inverse isomorphisms of R-algebras.) Then R = ℤ[{γ_{i}}_{i≥1}] ⊆ S′ can be thought of as a “universal weakly subintegral extension” constructed by equations of integral dependence. This extension is universal for extensions (2.6) in the sense that if A ⊆ B and b ∈ B is weakly subintegral over A (with equations of integral dependence (2.6)) then there is a homomorphism S′ → B which maps X to b, and γ_{i} to a_{i}. This can be seen as follows: b also satisfies the SOSI in (2.3) (for some c_{i} ∈ B). Define a homomorphism ψ : S → B such that ψ(z) = b, ψ(x_{i}) = c_{i}. Then by (2.4.1), ψ(γ_{i}) = a_{i} and ψ ∘ ϕ is the desired homomorphism from S′ → B.

Now we begin the proof of our main result. Let x_{1}, …, x_{p}, y_{i}, (i ≥ 1, p ∈ ℤ_{+}) and z be independent indeterminates over ℤ (z being a generic version of b). Let U^{i} := ℤ[x_{1}, …, x_{p}, y_{1}, …, y_{2p+i}, z] (i ≥ 1), and let I^{i} := ⟨\{z^{k} + Σ_{j=1}^{p}\binom{k}{j}x_{j}z^{k−j} − y_{k}}_{1≤k≤2p+i}⟩U^{i}. Clearly we have U^{i}/I^{i} ≅ ℤ[x_{1}, …, x_{p}, z] = S (where ι_{i} : U^{i} → S sends x_{k} to x_{k}, z to z, and y_{k} to γ_{k}). Let R̃^{i} := ℤ[y_{1}, …, y_{2p+i}] and Ĩ^{i} = I^{i} ∩ R̃^{i}. Then R̃^{i}/Ĩ^{i} is isomorphic to the subring of S generated by {γ_{k}}_{1≤k≤2p+i}. But we know from [RRS, 2.2] that this subring is R = ℤ[{γ_{k}}_{1≤k≤2p+1}], so R̃^{i}/Ĩ^{i} ≅ R for i ≥ 1, these isomorphisms sending y_{k} to γ_{k}. Because the last isomorphism holds for all i we will work with i = 1, and will write simply U, I, R̃, and Ĩ instead of U^{1}, I^{1}, R̃^{1} and Ĩ^{1}. In particular

(3.0) R̃/Ĩ ≅ R.

We now have

Lemma 3.1. In the ring U, I = ⟨\{z^{k} +∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j} −(−1)^{k}x_{k}\}_{1≤k≤p}, \{z^{k} + ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j}\}_{p+1≤k≤2p+1}⟩.

Proof. Let T be another indeterminate, independent of z, the x_{j} and the y_{j}. As in section 2 let Exp(a) = 1 + Σ_{i=1}^{∞}a^{i}T^{i}. Consider

(3.2a) Exp(z) ∗ [(1 + x_{1}T + … + x_{p}T^{p}) − Exp(−z) ∗ (1 + Σ_{j=1}^{∞}y_{j}T^{j})] =

(3.2b) Exp(z) ∗ (1 + x_{1}T + … + x_{p}T^{p}) − (1 + Σ_{j=1}^{∞}y_{j}T^{j})

To simplify the notation a bit, let δ_{k} = z^{k} + ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j} − (−1)^{k}x_{k}, 1 ≤ k ≤ p , δ_{k} = z^{k} + ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j}, k ≥ p + 1 and γ′_{k} = γ_{k} − y_{k}, k ≥ 1. The defining generators of I are {γ′_{k}}_{1≤k≤2p+1} and the proposed new generators of I are {δ_{k}}_{1≤k≤2p+1}. The coefficient of T^{k} (1 ≤ k ≤ 2p+1) in (3.2b) is {γ′_{k}}_{1≤k≤2p+1}. We can rewrite (3.2a) as Exp(z) ∗ ∑_{k=1}^{∞}(−1)^{k+1}δ_{k}. Comparing the coefficients of T^{k}, 1 ≤ k ≤ 2p + 1 in (3.2a) and (3.2b) we conclude that I ⊆ ⟨{δ_{k}}_{1≤k≤2p+1}⟩. Now consider

(3.3a) Exp(−z) ∗ [Exp(z) ∗ (1 + x_{1}T + … + x_{p}T^{p}) − (1 + Σ_{j=1}^{∞}y_{j}T^{j})] =

(3.3b) (1 + x_{1}T + … + x_{p}T^{p}) − Exp(−z) ∗ (1 + Σ_{j=1}^{∞}y_{j}T^{j})

Comparing the coefficients of T^{k} (1 ≤ k ≤ 2p + 1) in (3.3a) and (3.3b) we similarly conclude that ⟨{δ_{k}}_{1≤k≤2p+1}⟩ ⊆ I. Putting the two parts together the lemma follows.

Now let J = I ∩ R̃[z]. We claim

Lemma 3.4. J∩R̃ = Ĩ. Also J = ⟨\{z^{k}+∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j}\}_{p+1≤k≤2p+1}⟩R̃[z].

Proof. Since R̃ ⊆ R̃[z] ⊆ U it is clear that J ∩R̃ = I ∩R̃[z]∩R̃ = I ∩R̃ = Ĩ. By Lemma 3.1 I is generated by {δ_{i}}_{1≤i≤2p+1}, and the final assertion is that I ∩R̃[z] is generated by {δ_{i}}_{p+1≤i≤2p+1}. But suppose that h ∈ I ∩ R̃[z]. Since h ∈ I we then have h = ∑_{i=1}^{2p+1} A_{i}δ_{i}, A_{i} ∈ U. Set x_{k} = (−1)^{k}(z^{k}+∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j}), 1 ≤ k ≤ p. Then δ_{k} becomes 0 (1 ≤ k ≤ p) but h and δ_{k} (p + 1 ≤ k ≤ 2p + 1) are unchanged since they are in R̃[z]. This yields h = ∑_{i=p+1}^{2p+1} A′_{i}δ_{i}, where A′_{i} ∈ R̃[z] is the result of replacing x_{i} by (−1)^{k}(z^{k} + ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j}) in A_{i}, p + 1 ≤ i ≤ 2p + 1, and the lemma follows.

Theorem 3.5. Let S = ℤ[x_{1}, …, x_{p}, z] where x_{1}, …, x_{p},and z are independent indeterminates over ℤ, and let R = ℤ[{γ_{k}}_{1≤k≤2p+1}] where γ_{k} = z^{k} + Σ_{j=1}^{p}\binom{k}{j}x_{j}z^{k−j}. Let X be an indeterminate over S. Then there is an isomorphism of R-algebras ϕ : S′ = R[X]/⟨\{X^{k} +Σ_{j=1}^{k}(−1)^{j}\binom{k}{j}γ_{j}X^{k−j}\}_{p+1≤k≤2p+1}⟩ → S defined by ϕ(X) = z We can replace p + 1 ≤ k ≤ 2p + 1 by any interval p + 1 ≤ k ≤ 2p + i, i ≥ 1.

Proof. First recall that S is generated as an R-algebra by z. By (3.0)

S′ ≅ ℤ[y_{1}, …, y_{2p+1}, X]/⟨Ĩ, \{X^{k} + ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}X^{k−j}\}_{p+1≤k≤2p+1}⟩.

Now we can rename X as z since X and z are both indeterminates independent of the y_{i}. (This was not possible at the beginning since the elements of R had z in them, but now the z is gone.) By Lemma 3.4

Ĩ ⊆ ⟨\{z^{k} + ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j}\}_{p+1≤k≤2p+1}⟩ℤ[y_{1}, …, y_{2p+1}, z]

so

S′ ≅ ℤ[y_{1}, …, y_{2p+1}, z]/⟨\{z^{k} + ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}y_{j}z^{k−j}\}_{p+1≤k≤2p+1}⟩.

The latter is equal (by Lemma 3.4) to R̃[z]/J which is isomorphic by Lemma 3.1 to U/I ≅ S. It is clear that the k-algebra isomorphism S′ ≅ S constructed here maps γ_{i} to γ_{i} and X to z, hence equals ϕ. The final observation is follows from the second paragraph of this section.

Note that the equations of integral dependence

\{z^{k} + ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}γ_{j}z^{k−j} = 0\}_{p+1≤k≤2p+1}

of z over R, are the equations of integral dependence derived from the SOSI {z^{k} + ∑_{j=1}^{p} \binom{k}{j}x_{j}z^{k−j}\}_{1≤k≤p} via the equivalence (2.3) ⇔ (2.6). Our final conclusion is that R ⊆ S (≅ S′) serves as a universal weakly subintegral extension based on both the SOSI (2.3) and the equations of integral dependence (2.6).

4. Some additional properties of the equations of integral dependence

The equations of integral dependence in (2.6) are rather special, since they characterize weakly subintegral elements. We would like to understand them better. As a step in this direction we discuss some properties of these equations in the case of z over R in the extension R = ℤ[{γ_{k}}_{1≤k≤2p+1}] ⊆ S = ℤ[x_{1}, …, x_{p}, z].

Lemma 4.1. Let F_{n}(X) = X^{n} + ∑_{i=1}^{n}(−1)^{i}\binom{n}{i}γ_{i}X^{n−i}, n ≥ p + 1 be the equations of integral dependence of z over R. (X an indeterminate over R). Then (X − z)^{j} is a factor of F_{p+j}(X) (j ≥ 1).

Proof. The factorization of course takes place in S[X], not RX], since z is not an element of R. Note first that \frac{d}{dX} F_{n}(X) = nF_{n−1}(X): \frac{d}{dX} F_{n}(X) = nX^{n−1} +∑_{i=1}^{n}(−1)^{i}\binom{n}{i}(n−i)γ_{i}X^{n−i−1} = n(X^{n−1} +∑_{i=1}^{n−1}(−1)^{i}\binom{n−1}{i}γ_{i}X^{n−i−1}) = nF_{n−1}(X). Now consider the Taylor series F_{p+j}(X) = F_{p+j}(z) + \frac{d}{dX} F_{p+j}(X) |_{X=z} (X − z) + … + \frac{1}{(j−1)!} \frac{d^{j−1}}{dX^{j−1}} F_{p+j}(X) |_{X=z} (X − z)^{j−1} + (∗)(X − z)^{j} = F_{p+j}(z) + \binom{p+j}{1}F_{p+j−1}(z)(X−z)+…+…+\binom{p+j}{j−1}F_{p+1}(z)(X−z)^{j−1}+(∗)(X−z)^{j} = (∗)(X−z)^{j} as required (since F_{p+j}(z) = 0, j ≥ 1). (The ∗’s indicate some element of S that we do not need to determine.)

In the above calculation the \frac{1}{(j−1)!}’s are not interpreted in R. They become part of binomial coefficients in the next line of the argument and these binomial coefficients are evaluated in ℤ before interpreting as an element of R.

By specialization the analogue of Lemma 4.1 will hold for any weakly subintegral extension R ⊆ R[x]. The following example shows that (X − z)^{j} is the highest power of (X − z) that divides F_{p+j}.

Example 4.2. Let A = ℤ[{z^{i}}_{i≥p+1}] ⊆ B = ℤ[z] and let c_{i} = (−z)^{i} (1 ≤ i ≤ p). This gives a SOSI for z with, N = 1 and a_{i} = 0 (1 ≤ i ≤ p), a_{i} = (−1)^{p}\binom{i−1}{p}z^{i} for i ≥ p + 1, as in [RRa, section 5]. Then replacing γ_{i} by a_{i} in (4.1) we claim that F_{p+j} specializes to F̃_{p+j}(X) = (X − z)^{j}(X^{p} + Σ_{i=1}^{p}\binom{i+j−1}{j−1}z^{i}X^{p−i}) for j ≥ 1. This is readily checked if j = 1. Furthermore F_{n}(0) = (−1)^{n}γ_{n} and this becomes (−1)^{n}(−1)^{p}\binom{n−1}{p}z^{n} upon replacing γ_{i} by a_{i}, and this agrees with F̃_{n}(0). Finally one completes the argument by checking that \frac{d}{dX} F̃_{n}(X) = nF̃_{n−1}(X).

References

[RR] L.Reid, and L.G.Roberts, A new criterion for weak subintegrality, Comm. Algebra 24(10) (1996), 3335-3342.

[RRa] L.Reid, and L.G.Roberts, Monomial subrings and systems of subintegrality, to appear.

[RRSa] L.Reid, L.G.Roberts, and B.Singh, The structure of generic subintegrality, Proc. Indian Acad. Sci. Math.Sci. 105 (1995), 1-22.

[RRS] L.Reid, L.G.Roberts, and B.Singh, On Weak Subintegrality. J. Pure Appl. Algebra 114 (1996), 93-109.

[RS] L.G.Roberts, and B.Singh, Subintegrality, invertible modules and the Picard Group, Compositio Math. 85 (1993), 249-279.

[Y] H.Yanagihara, On an intrinsic definition of weakly normal rings, Kobe J. Math. 2 (1985), 89-98.

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6

    E-mail address: robertsl@mast.queensu.ca