Homology, Homotopy and Applications, vol.4(2), 2002, pp.295–314

PRESENTATION DEPTH AND THE LIPMAN-SATHAYE JACOBIAN THEOREM

MELVIN HOCHSTER

(communicated by Clas Löfwall)

Abstract

We give a version of the theorem of Lipman and Sathaye on Jacobian ideals but with substantially weaker hypotheses. Both their version and the result here are very useful in providing explicit test elements in tight closure theory. There are two separate ways in which the hypothesis in the theorem is weakened here: one is that the larger ring is not required to be a domain, although it will be reduced. Second, the regularity condition on the smaller ring is weakened to the point where one need not assume that it is Cohen-Macaulay. Instead, a condition on the ring homomorphism is imposed that may be viewed as a relative analogue of the Serre conditon S_{2}: a family of such conditions is introduced and studied here. The definition is made in terms of a presentation of an algebra, but is independent of the presentation.

To Jan–Erik Roos on his sixty–fifth birthday

1. Introduction

In this paper we shall give a version of the theorem of Lipman and Sathaye on Jacobian ideals proved in [LS] (their Theorem 2). The result presented here has substantially weaker hypotheses than in [LS]. Lipman and Sathaye used their theorem to prove the Briançon-Skoda theorem [BrS] (which was first proved using the analytic criterion for membership in an ideal in [Sk]; see also [LT] for background). Their result has also turned out to be very important in providing explicit test elements in tight closure theory (accounts of the latter are given in [HH1–8], [Ho2-3], [Hu], and [Br]; [HH8], in particular, treats the use of the Lipman-Sathaye theorem in constructing test elements).

In the main theorem, Theorem (2.1) of §2 (the reader may want to consult its statement at this point), there are two separate ways in which the hypothesis in [LS] on the map R → S is weakened: first, while the ring S is still required to be torsion-free and generically étale over R, S is not required to be a domain. Second,

The author was supported in part by a grant from the National Science Foundation. Received March 5, 2001; published on July 12, 2002. 2000 Mathematics Subject Classification: 13B02, 13B21, 13C15. Key words and phrases: Jacobian theorem, algebra presentation, depth, integral extension. \begin{matrix}c \\ ◯ \end{matrix} 2002, Melvin Hochster. Permission to copy for private use granted.

the regularity condition on R is relaxed even to the point where we do not need to assume that R is Cohen-Macaulay: a condition on the map R → S is needed, but it is significantly weaker than assuming that R be Cohen-Macaulay. In this connection, we introduce in (1.2) what might be thought of as a relative analogue of the Serre condition S_{i} that is defined in terms of a presentation S over R, but later shown to be independent of the presentation: cf. Theorem (1.8). It turns out that we can replace the Cohen-Macaulay condition on R in the Lipman-Sathaye theorem by the condition that S be relatively S_{2} presentable over R in the sense defined in §1. A key point is that when R → S has this property, it is preserved when we adjoin an element of the normalization of S: see Theorem (1.9). However, we still do need to assume that R is regular when localized at a prime lying under a height one prime of the normalization of S.

The proof of Theorem (2.1) here is modeled on the “down-to-earth” proof of Lipman-Sathaye, and we shall make frequent references to [LS]. One difference is that we need many sources of elements in “very general position,” particularly indeterminate linear combinations of generators of ideals. We are able to do this by replacing R at the outset by a localization R(t) of a polynomial ring in infinitely many variables over R.

As mentioned above, the next section is devoted to developing the required theory of “presentation S_{i} conditions.” The main result is stated in Theorem (2.1) of §2. The proof is outlined in §2, and then the details of the various steps are supplied in §§3–5.

2. DEPTH PROPERTIES IN PRESENTATIONS

Definition-Discussion 2.1. In this section fix an integer i ⩾ 1, let R be a Noetherian domain that satisfies the Serre condition S_{i} (if i = 1, the fact that R is a domain guarantees this), and let S be a reduced algebraic R-torsion-free extension of R essentially of finite type over R. Thus, if K is the fraction field of R, then K ⊗_{R} S is a finite product of finite algebraic field extensions of K. In later sections we shall assume as well that S is generically étale over R, by which we mean that K ⊗_{R} S is a finite product of finite separable algebraic extensions of K. We have that as R-algebras, S ≅ T/I, where T is a localized polynomial ring in finitely many variables over R, and we shall refer to the corresponding surjection T ↠ S as an L-presentation of S over R, where the “L-” is an indication that we are permitting T to be a localization of a polynomial ring. Informally we may also say that S ≅ T/I is an L-presentation of S. An L-presentation is a presentation if T is a polynomial ring in finitely many variables over R.

Remark 2.2. Notice that the minimal primes of I all have the same height in this situation: since the elements of R are nonzerodivisors in S, the situation is unaffected by localizing at the multiplicative system R − {0}, and thus we may assume that R is a field and S is a finite product of finite field extensions. Suppose that T is a localization of the polynomial ring T_{0} ⊆ T. Then the minimal primes of I must correspond to maximal ideals of T_{0} (or else S would transcendental over R) and so all have the same height.

Definition 2.3. Let R → S be as in (1.0). We shall say that an L-presentation η : T ↠ S with Ker η = I is relatively S_{i} if, with h = ht I, for every prime ideal P of T containing I, depth of T_{P} ⩾ min {ht P, h + i}. (Note that this implies depth_{T} I = ht I.)

It will be convenient to have a name for the sort of ideal described above.

Definition 2.4. Let T be a Noetherian ring. An ideal I of T all of whose minimal primes have height h will be said to be of CM_{+i} type if for every prime ideal P containing I, depthT_{P} ⩾ min {ht P, h+i}. (Again, note that this implies depth_{T} I = ht I.)

With the terminology of (1.3) and R → S as in (1.0), an L-presentation S ≅ T/I over R is relatively S_{i} if and only if I is of CM_{+i} type in T. Since R is a domain, T has no embedded primes, and (1.1) shows that the ideal I associated with the presentation has the property that all of its minimal primes have the same height. We shall say that an ideal all of whose minimal primes have height h has pure height h. Evidently, a Noetherian ring T is Cohen-Macaulay if and only if every ideal of pure height is of CM_{+i} type for all i.

Although we have given Definition (1.3) without restriction on T, in the sequel we shall almost alway be assuming that T satisfies the Serre condition S_{i} when we discuss ideals of CM_{+i} type. Likewise, in the sequel, when discussing the relatively S_{i} condition in (1.2), we shall almost always be assuming, as indicated in (1.0), that R satisfies the Serre condition S_{i}.

Discussion 2.5 (behavior of depth and dimension under flat base change). For use in the proof of the very important Lemma (1.5) that follows, we note two well known facts: if (T, P) → (W, Q) is a flat local homomorphism of local rings, and B = W/P B is the closed fiber, then dim W = dim T +dim B ([Mat], (13.B) Theorem 19 (2), p. 79), and depth W = depthT +depthB ([Mat], (21.C) Corollary 1, p. 154). This implies at once, for example, that if T satisfies S_{i} and W is flat over T with Cohen-Macaulay fibers (or S_{i} fibers), then W satisfies S_{i}.

We shall make extensive use of the following alternative characterization of when an ideal I is of CM_{+i} type.

Lemma 2.6. Let T be a Noetherian ring that is S_{i}. Let I ⊆ T be an ideal of pure height h. Let f_{1}, …, f_{r} be generators of I. Let t_{μj} be indeterminates over T, 1 ⩽ μ ⩽ h, 1 ⩽ j ⩽ r, and let g_{μ} = ∑_{j=1}^{r} t_{μj}f_{j}. Then the following are equivalent:

(1) I has CM_{+i} type.

(2) In some faithfully flat extension W of T, IW has height equal to its depth h, and contains a regular sequence of length h generating an ideal J such that W/J is S_{i}.

(3) In the polynomial ring W = T [t_{μj} : μ, j], the elements g_{1}, …, g_{h} form a regular sequence generating an ideal J such that W/J is S_{i}.

(4) In the localization W_{U} of the ring W from part (3) at some (resp., every) multiplicative system U consisting of polynomials whose coefficients generate the unit ideal, the elements g_{1}, …, g_{h} form a regular sequence generating an ideal J such that W/J is S_{i}.

Proof. We shall prove (3) ⇒(2) ⇒(1) ⇒(3). Note that each W_{U} as in (4) above is faithfully flat over T, and so we also have (3) ⇒( (4) for all U ) ⇒( (4) for some U ) ⇒(2) as well.

(3) ⇒(2) is obvious. To see that (2) ⇒(1), note that, after a faithfully flat extension T → W, the ideal IW has the same height as I and the same depth as I. Thus, I has height and depth h. Now let P be a prime ideal of R containing I of height greater than h. Then there is a minimal prime Q of P W lying over P, and T_{P} → W_{Q} will be faithfully flat with a zero-dimensional fiber. The depth and dimension of W_{Q} are the same as the depth and dimension of T_{P}: cf. the discussion preceding (1.5). When we kill J in W_{Q}, the depth and dimension drop by h, and we obtain an S_{i} ring. This shows that W_{Q} has the required depth, and, hence, so does R_{P}.

It remains to prove that (1) ⇒(3). The fact that the elements g_{i} form a regular sequence is proved in [Ho1]. Let Q be a prime containing J = (g_{1}, …, g_{h}) in W = T [t_{μj} : μ, j]. First suppose that Q contains I. We must show that the depth of W_{Q} is at least the smaller of ht Q, h + i, since this will give the correct depth once we kill J. Then Q lies over a prime P of T that contains I. Then the depth of R_{P} is at least min {ht P, h+i}, and by [Mat], (13.B) Theorem 19 (2), p. 79, we have ht Q = ht P + dim B, where B is the fiber W_{Q}/P W_{Q}, and depthW_{Q} = depthT_{P} + depthB by [Mat], (21.C) Corollary 1, p. 154.

But the fibers of T → W are Cohen-Macaulay (in fact, regular), and so

depthW_{Q} ⩾ min {ht P, h + i} + depthB = min {ht P, h + i} +

dim B ⩾ min {ht P + dim B, h + i + dim B} = min {ht Q, h + i + dim B} ⩾

min {ht Q, h + i}, as required.

It remains only to consider the case where Q fails to contains I. Then one of the generators f_{j} of I is a unit in T_{P}, where P is the contraction of Q to T. Again, we pass to T_{P}, and renumber so that f_{r} is a unit. Then the equations g_{μ} that we are killing simply solve for the indeterminates t_{μr} in terms of the others, and the quotient by J may be identified with T_{P} [t_{μj} : 1 ⩽ μ ⩽ h, 1 ⩽ j < r], a polynomial ring over T_{P}. Since this ring is S_{i} so is the polynomial ring, and the result follows.

We shall later need the following fact whose proof is really a variant of the argument that (1) ⇒(3) in Lemma (1.5) above.

Lemma 2.7. Let I be an ideal of pure height h in a Noetherian ring T such that T is S_{i}. Let u_{1}, …, u_{h} be a regular sequence in I such that T/(u_{1}, …, u_{h})T is S_{i}. Let f_{1}, …, f_{r} be generators of I. Let k be an integer, 1 ⩽ k ⩽ h. Let t_{μj} be indeterminates over T, 1 ⩽ μ ⩽ k, 1 ⩽ j ⩽ h and let t′_{μj} also be indeterminates over T, k < μ ⩽ h, 1 ⩽ j ⩽ r. Let W be the ring obtained by adjoining all these independent indeterminates to T. Then the k elements g_{μ} = ∑_{j=1}^{h} u_{j}t_{μj}, 1 ⩽ μ ⩽ k, together with the h−k elements g_{μ} = ∑_{j=1}^{r} f_{j}t′_{μj}, k < μ ⩽ h, give a regular sequence in IW such that the quotient of W by the ideal they generate is S_{i}.

Proof. Consider a prime Q of W that contains the g ’s. The specified depth condition on W_{Q}/(g_{1}, …, g_{h}) is equivalent to the condition that the depth of W_{Q} be at least the lesser of ht Q, h + i.

If Q contains all the elements u_{1}, …, u_{h} this follows because, after we kill this regular sequence, the quotient ring W_{Q}/(u_{1}, …, u_{h}) is S_{i}, since it is flat over T/(u_{1}, …, u_{h}) with regular fibers. If Q fails to contain all the u_{h}, we replace R by its localization at the contraction P of Q to T. We may assume that in each of the first k of the g_{μ}, one of the u_{i} is invertible, permitting us to use g_{μ} = 0 to solve for one of the t_{μj} in terms of the others. But then P ⊉ I as well, and so in each of the last n − k of the g_{μ} at least one of the f_{j} is invertible, permitting us to solve for one of the t_{μj} in terms of the others. This means that the quotient by (g_{1}, …, g_{h}) may be identified with a polynomial ring over T_{P}, and the result follows from the fact that T is S_{i}.

Note that if R is Cohen-Macaulay then so is T, and it is then automatic that every L-presentation is relatively S_{i} for all i.

Our next results are aimed at showing that when S has a relatively S_{i} presentation over R, then every presentation of S over R is relatively S_{i}.

Lemma 2.8. Let R → S be a homomorphism as in (1.0). Let S ≅ T/I be an L-presentation of S as an R-algebra that is relatively S_{i}, and let T_{0} be a polynomial ring in finitely many variables over R contained in T such that T is a localization of T_{0}.

(a) If U is a multiplicative system in T with image V in S then T_{U}/I_{U} is a relatively S_{i} L-presentation of S_{V} over R.

(b) (Descent of localizations.) Assume that the S_{i} locus is open in rings essentially of finite type over R (which holds if R is either a homomorphic image of a Cohen-Macaulay ring or if R is excellent). If T = (T_{0})_{U} for a multiplicative system U in T_{0}, there is an element u ∈ U such that if T_{1} = T_{0}[1/u], and I_{1} = I ∩ T_{1}, then S_{1} = T_{1}/I_{1} is a relatively S_{i} L-presentation.

Proof. Part (a) is clear, since the set of localizations at primes containing I can only become smaller as we localize T. It is worth noting that the condition that I have pure height is needed here: otherwise, the height of I might increase after localization.

To prove part (b), note that localizing at finitely many elements is equivalent to localizing at their product. We may make several choices of u, T_{1}, each time localizing at one more element to get additional conditions to hold. Choose generators f_{1}, …, f_{r} for I and initially choose u so that these will be in T_{1}. Thus, we may assume that I = I_{1}T. Let h = ht I. Introduce hr new indeterminates t_{μj} and define g_{1}, …, g_{h} as in part (3) of Lemma (1.5). Then we know that U^{−1}T_{1}[t_{μj} : μ, j]/(g_{1}, …, g_{h}) is S_{i}, by Lemma (1.5). This means that U meets the defining radical ideal of the non-S_{i} locus in the ring T_{1}[t_{μj} : μ, j]/(g_{1}, …, g_{h}) (our hypothesis guarantees that this locus is closed). Thus, we may choose u_{1} ∈ U meeting that radical ideal. Replacing T_{1} by its localization at u_{1}, we see that we can localize at just one element and get T_{1}[t_{μj} : μ, j]/(g_{1}, …, g_{h}) to be S_{i} (this is a new choice of T_{1}), and so by part (3) of Lemma (1.5), T_{1} ↠ T_{1}/I_{1} is relatively S_{i} for this new choice of T_{1}.

Theorem 2.9 ((independence of presentation)). Let R → S be a homomorphism as in (1.0) and suppose that S has an L-presentation over R that is relatively S_{i}. Then every L-presentation of S over R is relatively S_{i}.

Proof. Step 1. We first consider the case where S is finitely generated over R and we are considering only presentations of S over R, i.e., T is required to be a polynomial ring. Given two presentations T ↠ S, T′ ↠ S, we can compare both with T ⊗_{R} T′ ↠ S. By induction on the number of variables in T′, we reduce to considering a presentation T ↠ S and another T [X] ↠ S, where X is a new indeterminate. Let s be the image of X and let z ∈ T map to s. We may replace X by X − z, and so assume that X maps to 0 in S. Thus, if I = Ker (T ↠ S), the kernel of the second presentation will be IT [X] + XT [X]. The primes of T [X] containing the kernel are in bijective correspondence with the primes of T containing I, and the depth condition is obviously satisfied for T_{P} if and only if it is satisfied for T [X]_{Q}, where Q = P T [X] + XT [X] is the corresponding prime, since T [X]_{Q}/(X) ≅ T_{P}, and X is a nonzerodivisor.

Step 2. We next consider the case where S is finitely generated over R, but can be written S = S_{0}[1/s] where s ∈ S_{0}. Let T_{0} ↠ S_{0} be a presentation of S_{0} with kernel I_{0} and and v ∈ T_{0} an element that maps to s. Then we have a surjection T_{0}[1/v] ↠ S, which is an L-presentation, and we also have a presentation T_{0}[X] ↠ S extending T_{0} → S_{0}, where X maps to 1/s. The kernel of the latter is I_{1} = I_{0}T_{0}[X] + (Xv − 1). Again, there is a bijection between the primes of T_{0}[1/v] containing I_{0} (or primes of T_{0} containing I_{0} and not v) and the primes of T_{0}[X] containing I_{1}, and the depths in the localizations at the latter primes are all one greater: we can see this, because we can start by killing Xv −1, which gives the correspondence. Thus an L-presentation of this type is relatively S_{i} presentable if and only if the corresponding presentation is relatively S_{i} presentable.

Step 3. We now consider the general case. Let T_{0} be a polynomial ring in finitely many variables over R and U a multiplicative system in T_{0} such that U^{−1}T_{0} ↠ S is a relatively S_{i} L-presentation of S over R. By Lemma (1.7) (b) we can choose a localization T_{1} of T_{0} at one element and such that the restriction of T ↠ S gives a surjection T_{1} ↠ S_{1} that is a relatively S_{i} L-presentation, and S is a localization of S_{1}. By part (a), every localization S_{2} of S_{1} at one element has an L-presentation T_{2} ↠ S_{2} that is relatively S_{i} and such that T_{2} is a localization of a polynomial ring at one element. It follows that S is a localization of a finitely generated R-subalgebra S_{1} with the following property: every presentation of S_{1}, and every presentation of any localization S_{1}[1/u], is relatively S_{i}.

Now consider some other L-presentation (U′)^{−1}T′ ↠ S. Then we can choose a localization T′_{1} of T′ at one element of U′ whose image S′_{1} in S contains S_{1}, and we can choose w ∈ S_{1} ⊆ S′_{1} such that S′_{1}[1/w] = S_{1}[1/w], where w is invertible in S. Thus, S′_{1}[1/w] has a presentation that is relatively S_{i}.

Suppose that w = τ/y where τ ∈ T′ and y ∈ U′. Since w is invertible in S, we can choose a ∈ T′ and z ∈ U′ such that aτ/yz = 1 in S, i.e., and then there exists v ∈ U′ such that vaτ = vyz. Then S′_{1} localized at the image ζ of vyz contains 1/w, since that can be represented as a/z, and so S_{2} = S′_{1}[1/ζ] has the property that every presentation is relatively S_{i}, since it is a localization of S_{1}[1/w] at one element, and hence a localization of S_{1} at one element. But T′_{1}[1/vyz] is a localization of T′ at an element of U′ mapping onto S_{2}. From Step 2 we know that this L-presentation is also relatively S_{i} (since the corresponding presentation will be), and then by (1.7) (a), (U′)^{−1}T′ ↠ S is relatively S_{i}.

The following fact will be critical in the proof of the Lipman-Sathaye Jacobian theorem.

Theorem 2.10. Let R → S be as in (1.0). Suppose that S is relatively S_{i} presentable over R. Let s be an element of the total quotient ring of S that is integral over S. Then S[s] is relatively S_{i} presentable over R. Hence, any subring of the integral closure of S module-finite over R is relatively S_{i} presentable over R. Moreover, if T ↠ R is an L-presentation with kernel I of height h such that g_{1}, …, g_{h} is a regular sequence in I with the property that T/(g_{1}, …, g_{h}) is S_{i}, and we extend T ↠ S to T [X] ↠ S[s] by sending X to s, then there is a polynomial F monic in X in the kernel, and T [X]/(g_{1}, …, g_{h}, F) is S_{i}.

Proof. We first establish the last statement. Let B = T/(g_{1}, …, g_{h}) and

C = T [X]/(g_{1}, …, g_{h}, F) ≅ B[X]/(f)

where f is the image of F in B[X] and is still monic in X. Clearly, B[X]/(f) is flat (and even free) over B with zero-dimensional (hence, Cohen-Macaulay) fibers, and so is S_{i}.

The first part now follows. After tensoring with R, S, T R[t_{μj} : μ, j] = R[t] we get such a regular sequence in T [t] such that the quotient is S_{i}, by Lemma (1.5) (3). When we adjoin X the result of the paragraph above together with Lemma (1.5)

(2) shows that we have a relatively S_{i} presentation of S[s].

The following fact is used in Discussion (1.11) to show that there are many case where R → S as in (1.0) is such that S is relatively S_{2} presentable over R, but R is not Cohen-Macaulay.

Proposition 2.11. Let R → S be as in (1.0) (in particular, R satisfies S_{i}) and let R_{1} be a domain flat over R with S_{i} fibers. Suppose that S is relatively S_{i} presentable over R. Then R_{1} → R_{1} ⊗_{R} S satisfies (1.0) and is relatively S_{i} presentable over R_{1}.

Proof. Let T ↠ S be an L-presentation (with kernel I of pure height h)that is relatively S_{i}. After adjoining indeterminates t_{𝔪j} as in (1.5) we obtain a regular sequence g_{1}, …, g_{h} in the expansion of I such that T [t]/(g_{1}, …, g_{h}) is S_{i}. The main point is that R_{1} ⊗_{R} T [t]/(g_{1}, …, g_{h}) is S_{i}. Cf. (1.4).

Discussion 2.12. We conclude this section with an example which shows that the property that S be relatively S_{2} presentable over R is weaker than the assumption that R be Cohen-Macaulay.

One way to see this is to let R_{0} be Cohen-Macaulay normal of finite type over an algebraically closed field K, and let B denote a finite type K-algebra that is normal but not Cohen-Macaulay. Take any extension of S_{0} of R_{0} as in (1.0). This will be relatively S_{2}-presentable simply because R_{0} is Cohen-Macaulay. It is easy to see that R = B ⊗_{K} R_{0} → B ⊗_{K} S_{0} = S is still relatively S_{2} presentable (apply (1.10) with R = R_{0} and R_{1} = B ⊗_{K} R_{0}), but R is not Cohen-Macaulay.

3. STATEMENT OF THE THEOREM AND A SKETCH OF THE PROOF

Throughout the rest of this paper R denotes a Noetherian domain with fraction field K, and S denotes an algebra essentially of finite type over R (i.e., a localization at some multiplicative system of a finitely generated R-algebra) such that S is torsion-free and generically étale over R (by which we mean that L = K ⊗_{R} S is a finite product of finite separable algebraic field extensions of K). We shall denote by S′ the integral closure of S in L. We shall see that S′ is module-finite over S if R is regular or excellent (cf. (2.7)).

If A and B are subsets of L we denote by A : _{L}B the set {u ∈ L : uB ⊆ A}. If T is a subring of L and A is a T -module, then so is A : _{L}B.

We shall write J_{S/R} for the Jacobian ideal of S over R. If S is a finitely generated R-algebra, so that we may think of S as R[X_{1}, …, X_{n}]/(f_{1}, …, f_{m}), then J_{S/R} is the ideal of S generated by the images of the size n minors of the Jacobian matrix (∂f_{j}/∂x_{i}) under the surjection R[X] → S. This turns out to be independent of the presentation. Moreover, if s ∈ S, then J_{S_{s}/R} = J_{S/R}S_{s}. From this one sees that if S is essentially of finite type over R and one defines J_{S/R} by choosing a finitely generated subalgebra S_{0} of S such that S = W^{−1}S_{0} for some multiplicative system W of S_{0}, and takes J_{S/R} to be J_{S_{0}/R}S, then J_{S/R} is independent of the choices made. The result we aim to prove is:

Theorem 3.1 (Generalized Lipman-Sathaye Jacobian theorem). Let R be a Noetherian domain with fraction field K. Assume also that the S_{2} locus is open in algebras essentially of finite type over R.^{1} Let S be an extension algebra essentially of finite type over R such that S is torsion-free and generically étale over R. Suppose that for every maximal ideal M of S with contraction P to R, R_{P} is normal, and S_{M} has a relatively S_{2} presentation over R_{P}.^{2} Let L = K ⊗_{R} S and let S′ be the integral closure of S in L. Assume that S′ is module-finite over S. ^{3} Suppose that for every height one prime ideal Q of S′, R_{Q∩R} is regular. Then S′ : _{L}J_{S′/R} ⊆ S : _{L}J_{S/R}.

Overview of the Proof. (2.2) Step 1: The local case suffices Note that it is enough to prove the result when S is replaced by its various localizations at maximal ideals. Thus, we may assume that S is local, although we shall only make this assumption at certain

1This holds automatically if R is excellent or if R is a homomorphic image of a Cohen-Macaulay ring. 2This is automatic if R is normal and Cohen-Macaulay, or if R is normal and S has a relatively S_{2} presentation over R. 3This is automatic if R is regular or excellent: cf. (2.7).

points in the proof. When S is local we may also replace R by its localization at the contraction of the maximal ideal of S, and so there is likewise no loss of generality in assuming that R is local and that R → S is local homomorphism (i.e., the maximal ideal of R maps into that of S). Because of this reduction to the local case, throughout the rest of the argument we assume that R is a normal local domain and, in particular, that R satisfies S_{2}.

(2.3) Step 2: Adjoining indeterminates If T = {t_{λ} : λ ∈ Λ} is a family of indeterminates over a Noetherian ring R we denote by R(T ) or sometimes even R(t) the localization of the polynomial ring R[t_{λ} : λ ∈ Λ] at the multiplicative system consisting of all polynomials whose coefficients generate the unit ideal. This ring is Noetherian even when the family of indeterminates is infinite (cf. [HH7], Remark 2.18, especially the final, parenthetical paragraph) and faithfully flat over R with geometrically regular fibers, since it is a direct limit of smooth R-algebras and flat maps and happens to be Noetherian. To prove Theorem (2.2) it suffices to consider R(t) → R(t) ⊗_{R} S instead of R → S. Note that the normalization of R(t) ⊗_{R} S is R(t) ⊗_{R} S′: it is normal because it is a localization of a polynomial ring over S′. Note that the map R(t) → R(t) ⊗_{R} S is the direct limit of maps R(T_{0}) → R(T_{0}) ⊗_{R} S as T_{0} runs through finite subsets of T .

Note that if R has the property that algebras essentially of finite type over R have open S_{i} locus, then so does R(t). It suffices to see this for algebras of finite type over R(t) and each of these arise from a finite type algebra S_{0} over R(T_{0}) (where T_{0} is finite) by a base change R(T_{0}) → R(t) that is faithfully flat with regular fibers. But then Q in R(t) ⊗_{R(T_{0})} S_{0} with contraction P to S_{0} is such that the localization at Q is S_{i} if and only if (S_{0})_{P} has property S_{i}, i.e., the locus that we want in Spec (R(t) ⊗_{R(T_{0})} S_{0}) is the inverse image of an open set in Spec S_{0} (since S_{0} is essentially of finite type over R).

Note also that if R → S has the property that every height one prime of S′ lies over a prime P in R such that R_{P} is regular, the same is true for R(t) → R(t)⊗_{R} S. Since the latter ring is flat over S, a height one prime there lies over either a height one or height 0 prime of S: in either case, that in turn lies over a prime P_{0} in R such that R_{P_{0}} is regular (if the contraction to S is height 0, the prime lies over 0 in R). Thus, R(t)_{P} will be a Noetherian localization of a polynomial ring over R_{P_{0}}, and, hence, regular.

Thus, we may replace R → S by R(t) → S(t) in proving the theorem, where the set of indeterminates is infinite. This will prove useful in constructing regular sequences in sufficiently general position.

(2.4) Step 3: Presenting S over R Let T denote a localization of R[X_{1}, …, X_{n}] that maps onto S, and let I denote the kernel. Let U denote the complement in T of the set of minimal primes P_{1}, …, P_{r} of of I in T. Since S is reduced, I = ⋂_{i=1}^{r} P_{i}. Since S is a torsion-free R-module, the minimal primes of I do not meet R, and correspond to the minimal primes of I(K ⊗ T). Since killing any of these minimal primes produces an algebraic extension of K, they must correspond to maximal ideals of K[X_{1}, …, X_{n}], and it follows that the P_{i} all have the same height, which must be the same as the number of variables, n. Thus, U^{−1}T is a semilocal regular ring in which each of the maximal ideals M_{i} = P_{i}U^{−1}T is generated by n elements.

(2.5) Step 4: Special sequences and the modules W_{S/R} Call a sequence g_{1}, …, g_{n} of n elements of I special if it generates each of the M_{i}, is a regular sequence in T, and all of the associated primes of the ideal it generates are of height n. We shall show that special sequences exist (this is really only using that S has a relatively S_{1} presentation over R), and that there are sufficiently many of them that the images of the elements det (∂g_{j}/∂X_{i}) in S with g_{1}, …, g_{n} special generate the Jacobian ideal. Moreover, when g_{1}, …, g_{n} is special the image of det(∂g_{j}/∂X_{i}) in S is not a zerodivisor in S, and so has an inverse in L. Given θ: T → S and a special sequence g_{1}, …, g_{n} we define a map

Φ: \frac{(g_{1}, …, g_{n})T : _{T} I}{(g_{1}, …, g_{n})T} → L

by sending the class of u to \overline{u}/γ, where \overline{u} is the image of u in S and γ is the image of det (∂g_{j}/∂X_{i}) in S. We shall show that Φ is injective. A priori, its image depends on the choice of T → S and on the choice of the special sequence g_{1}, …, g_{n}, but the image turns out to be independent of these choices, and so we have constructed a finitely generated canonically determined S-module W_{S/R} ⊆ L. (2.6) Step 5: The main idea of the argument It will turn out that, quite generally, W_{S/R} ⊆ S : _{L}J_{S/R}. The result then follows from two further observations. The first is that when S is normal, this is an equality. (The idea is to show that W_{S/R} is then S_{2}, which forces Φ with its image restricted to S : _{L}J_{S/R} to be an isomorphism — this can be verified after localizing at height one primes of S. This is the only place in the proof where we use the hypothesis that S has a relatively S_{2} presentation over R.) The second is that when one enlarges S to S_{1} = S[s_{1}] by adjoining one integral fraction s_{1} ∈ L (so that S ⊆ S_{1} ⊆ S′), then W_{S_{1}/R} ⊆ W_{S/R}. Repeated application of this fact yields that W_{S′/R} ⊆ W_{S/R} and then we have

S′ : _{L}J_{S′/R} = W_{S′/R} ⊆ W_{S/R} ⊆ S : _{L}J_{S/R},

and we are done. In the sequel we shall systematically fill in the details of this outline.

We conclude this section with a result, essentially in [LS] but not stated in this generality, which shows that the normalization S′ of S is module-finite over S under very mild hypotheses on R.

Fact 3.2 (finiteness of the normalization). Let S be torsion-free, generically étale, and essentially of finite type over a normal Noetherian domain R. Suppose that the completion of every local ring of R is reduced (which holds if R is either regular or excellent). Then the normalization S′ of S over R is module-finite over S.^{4}

4Sketch of proof. Let S_{0} ∈ S be finite type over R with S = (S_{0})_{W}. Since localization commutes with normalization, we may replace S by S_{0} and assume S is finite type over R. The integral closure of S is the product of the integral closures of the domains obtained by killing a minimal prime of S. Hence, we may assume S is a domain. Each of finitely many generators for S over R satisfies an algebraic equation over R with leading coefficient r_{ν}, say, and so with r = ∏_{ν} r_{ν} ∈ R − {0}, S[1/r] is integral over R[1/r]. The integral closure of the normal domain R[1/r] in the fraction field L of S[1/r] is the same as the normalization of S[1/r], and is module-finite over S[1/r] by Prop. (31.B) of Chapter 12 of [Mat]. We may enlarge S by adjoining finitely many elements of its normalization and so obtain a domain S such that S[1/r] is normal for some r ≠ 0. By Lemma 4

4. BEGINNING THE PROOF: EXISTENCE OF SPECIAL SEQUENCES

Throughout the rest of this paper R denotes a normal Noetherian domain with fraction field K, and S denotes an algebra essentially of finite type over R (i.e., a localization at some multiplicative system of a finitely generated R-algebra) such that S is torsion-free and generically étale over R (by which we mean that L = K ⊗_{R} S is a finite product of finite separable algebraic field extensions of K). We shall denote by S′ the integral closure of S in L, and we shall assume that it is a finite module over S. We also assume that S is relatively S_{1} presentable over R. (We shall not need that it is relatively S_{2} presentable until §4.) Let T ↠ S be an Lpresentation with kernel I of pure height n. Recall that a regular sequence g_{1}, …, g_{n} of length n in I is special if g_{1}, …, g_{n} generate IT_{P} for every minimal prime P of I, and all minimal primes of (g_{1}, …, g_{n})T have height n. We construct many special sequences after replacing R, S, T by their tensor products over R with R(T ), where T is consists of infinitely many new indeterminates as in (2.3). We denote the results of tensoring R, S, T with R(T ) = R_{1} by R_{1}, S_{1}, and T_{1}, respectively. We write I_{1} for the expansion of I to T_{1}, so that S_{1} ≅ T_{1}/I_{1}. In later sections we shall change notation and write R, S, T for the rings denoted R_{1}, S_{1}, T_{1} here.

Lemma 4.1. With hypothesis as in the paragraph above, there are sufficiently many special sequences g_{1}, …, g_{n} in I_{1} that the Jacobian ideal J_{S_{1}/R_{1}} is generated by the images of elements det (∂g_{j}/∂X_{k}) for g_{1}, …, g_{n} special (where the X_{k} are the indeterminates generating the polynomial ring of which T is the localization). Moreover, if T_{1} is local, given any two special sequences there is a finite sequence (which we shall refer to as a chain) of special sequences with one of them as first term and the other as last term with the following property: given any two consecutive terms, one is obtained from the other by letting an invertible matrix over R_{1} act, or else the two consecutive sequences differ in only one term.

Proof. Pick generators f_{1}, …, f_{r} for I. Let [t_{μj}] be an r×r matrix of indeterminates from the infinite set T . The elements g_{μ} = ∑_{j=1}^{r} f_{j}t_{μj} give a new set of generators for I_{1}, because the determinant of [t_{μj}] is invertible in R_{1}. By Lemma (1.5) applied with i = 1, any n of these generators form a regular sequence W = T [t_{μj} : μj] such that the ideal they generate has no embedded primes (the last follows from the fact that the quotient is S_{1}). Because the map W → T_{1} is flat with Cohen-Macaulay fibers, the sequence retains these properties. Next, we want check that any n of these generate after localizing at a minimal prime P of I_{1}. Since they generate I_{1},

of Ch. 12 of [Mat], S has finite normalization if S_{Q} does for every maximal ideal Q of S. Choose s ∈ S with L = K[s]. Let R_{1} = R[s] and P = Q ∩ R_{1}. Then S_{Q} is a localization of (R_{1})_{P} [S], and S is generated over (R_{1})_{P} by elements of its fraction field. By a result of [Rees], if the completion of a local domain B is reduced, then the normalization of any algebra C finitely generated over B by fractions is module-finite over C. Thus, it suffices to show that the completion of (R_{1})_{P} is reduced. We may replace R by its localization at the contraction of P, and so we may assume that (R, m) is local with reduced completion. The completion of (R_{1})_{P} is one of the local rings of the completion of R_{1} with respect to m. Thus, it suffices if this completion of R_{1} is reduced. But this is R_{1} ⊗_{R} R̂ ⊆ L ⊗_{R} R̂ ≅ L ⊗_{K} (K ⊗_{R} R̂), and the result follows because K ⊗_{R} R̂ is reduced and L/K is separable.

it is certainly true that we can choose n of them to generate I_{1}(T_{1})_{P}. Given the symmetric roles played by the indeterminates, it must be true that any n generate. Since we may use all n elements subsets of this set of generators of I_{1}, we obtain enough elements to generate the Jacobian ideal.

Now suppose that we have two special sequences in I_{1}. They only involve finitely many of the indeterminates in T . After replacing R by R(T_{0}) for a suitable finite subset T_{0} of T , and making corresponding changes in S, T, we may assume without loss of generality that the two special sequences are in T.

We choose n^{2} + nr new indeterminates from T to construct two n × n matrices α, β of indeterminates from T as well as an n × r matrix γ. We replace one special sequence by letting α act on it: call it a_{1}, …, a_{n}. We replace the other by letting β act on it: call the result b_{1}, …, b_{n}. Also let γ act on the f_{1}, …, f_{r} to form n linear combinations, say g_{1}, …, g_{n}, of the f_{1}, …, f_{r} with indeterminate coefficients. If we replace the terms in a_{1}, …, a_{n} one at a time by the f ’s, we make a chain from a_{1}, …, a_{n} to f_{1}, …, f_{n}. We can then reverse the process to make a chain from f_{1}, …, f_{n} to b_{1}, …, b_{n}, changing only one term at a time. The fact that each of the sequences formed is a regular sequence generating an ideal with no embedded primes is a consequence of Lemma (1.6). The proof that these sequence all generate I_{1}(T_{1})_{P} = P (T_{1})P for each minimal prime P of I_{1} reduces to studying the vector space V = P (T_{1})_{P} /P^{2}(T_{1})_{P}. One needs to know that given two sets of generators for this vector space, one can take general linear combinations of the first set (i.e., with indeterminate coefficients), say v_{1}, …, v_{k} of them, together with general linear combinations of the second, and if the number of these is dim V, they will be a basis. The point is that the elements v_{1}, …, v_{k} form part of a basis. To decide whether the remaining elements complete the basis one may pass to V/Span {v_{1}, …, v_{k}}. Since we are working with elements that span this space, this is clear.

5. THE MAP Φ AND THE MODULES W_{S/R}

Our next main goal is to construct the maps Φ mentioned briefly in (2.5). Throughout this section we assume that R is a normal Noetherian domain, with fraction field K, that S is a torsion-free generically étale R-algebra with total quotient ring L (by hypothesis, a finite product of separable field extensions of K), that T is a localization of the polynomial ring R[X_{1}, …, X_{n}] that maps onto S, with kernel I, and that the minimal primes of I in T are P_{1}, …, P_{r}. We also assume that S is relatively S_{1} presentable over R. However, in parts (c), (d) and (f) of Theorem (4.4), we shall need the stronger hypothesis that S is relatively S_{2} presentable over R. Furthermore, we assume that R → S has arisen from R_{0} → S_{0} satisfying the same conditions by tensoring with R = R_{0}(T ) for some infinite family T of indeterminates, so that the results of Lemma (3.1) are applicable.

Lemma 5.1. Let g_{1}, …, g_{n} ∈ I. Let X denote X_{1}, …, X_{n}. Then the images of the g_{j} in R[X]_{P_{ν}} generate P_{ν}R[X]_{P_{ν}} if and only if det (∂g_{j}/∂X_{i}) ∉ P_{ν}. (The same holds if we only assume that g_{1}, …, g_{n} ∈ P_{ν}.) Hence, if g_{1}, …, g_{n} is a special sequence in I, then the image γ of det (∂g_{j}/∂X_{i}) is not a zerodivisor in S, and so represents an invertible element of the total quotient ring L of S.

Proof. The last statement clearly follows from the first, and we prove the first under the assumption that g_{1}, …, g_{n} ∈ P = P_{ν}. Let M be the maximal ideal corresponding to P in K[X]. Then P R[X]_{P} ≅ K[X]_{M}, and so this is really a statement about when elements of M generate M in K[X]. For the rest of this proof we shall write L (instead of, say, L_{ν}) for the residue field R[X]_{P}, which is the same as the residue field of K[X]_{M}. It is critical in what follows that L is separable over K. Consider the universal K-derivation d : K[X] → Ω_{K[X]/K}, the module of Kähler differentials, which is the free module K[X]-module generated by the elements dX_{1}, …, dX_{n}. Of course, if f ∈ K[X] then df = ∑_{j=1}^{n}(∂f/∂x_{j}) dx_{j}. The restriction of d to M gives a K-linear map M → Ω_{K[X]/K}, and by the defining property of a derivation it sends M^{2} → MΩ_{K[X]/K}. Thus, there is an induced map of K-vector spaces

δ: M/M^{2} → L ⊗_{K[X]} Ω_{K[X]/K}.

Both modules are L-vector spaces and it follows from the defining property of a derivation that δ is actually L-linear. Since K[X]_{M} is regular of dimension n, M/M^{2} is an n-dimensional vector space over L. The key point is that under the hypothesis that L is separable over K, the map δ is an isomorphism of L-vector spaces. This is well known, but we give a very short proof. The map δ sends the elements represented by generators g_{1}, …, g_{n} for M to the elements represented by the dg_{j}, and so it has a matrix which is the image of the matrix (∂g_{j}/∂x_{i}) after mapping the entries to L. Thus, δ is an isomorphism if and only if the Jacobian determinant det (∂g_{j}/∂x_{i}) has nonzero image in L. But this determinant generates J_{L/K}, and so δ is an isomorphism if and only if the Jacobian ideal of L over K is L. But we may use any presentation of L over K to calculate J_{L/K}, and so we may instead use L ≅ K[Z]/f(Z) where Z here represents just one variable and where f is a single separable polynomial. The Jacobian determinant is then the value of f′(Z) in L, which is not zero by virtue of the separability.

Thus, δ is an L-isomorphism. Moreover, we have already seen that if g_{1}, …, g_{n} are generators of M then the Jacobian determinant is not 0 in L. But the converse is also clear, because if g_{1}, …, g_{n} are any elements of M, they generate M if and only if their images in M/M^{2} span this vector space over L, by Nakayama’s lemma, and this will be the case if and only if their further images in L ⊗_{K[X]} Ω_{K[X]/K} span that vector space over L, since δ is an isomorphism, i.e., if and only if the images of the dg_{j} span. But this is equivalent to the assertion that the images of the columns of the matrix (∂g_{j}/∂x_{i}), after the entries are mapped to L, span an n-dimensional space, i.e., to the nonvanishing of det (∂g_{j}/∂x_{i}) in L, which is, of course, equivalent to its not being in P.

(4.2) The definition of θ and W_{S/R} We continue the conventions in the first paragraph of this section, but because we shall let both S and its presentation vary we shall write θ for the map T → S and we shall denote by g a special sequence g_{1}, …, g_{n} in I. We may then temporarily define

Φ_{θ,g}: \frac{(g_{1}, …, g_{n})T : _{T} I)}{(g_{1}, …, g_{n})T} → L

by sending the class of u to \overline{u}/γ where \overline{u} is the image of u in L, and γ is the image of det (∂g_{j}/∂x_{i}) in L: the element γ is invertible in L by Lemma (4.1). We shall often write Φ when θ and g are understood. We shall soon show that the image of Φ is contained in S : _{L}J_{S/R}. Once this is established we shall change the definition of Φ very slightly by restricting its range to be S : _{L}J_{S/R} ⊆ L.

We note that ((g_{1}, …, g_{n})T : _{T} I)/(g_{1}, …, g_{n})T ≅ Hom_{T} (T/I, T/(g_{1}, …, g_{n})T). We shall denote the image of Φ_{θ,g} in L by W_{S/R}(θ, g). However, we shall see just

below that it is independent of the choices of θ and g, and once we know this we shall simply write it as W_{S/R} ⊆ L.

Lemma 5.2. The map Φ_{θ,g} is injective, and its image in L is independent of the choice of g, and of the choice of θ. Its image is contained in S : _{L}J_{S/R}.

Proof. The domain of Φ may be thought of as the submodule of T/(g_{1}, …, g_{n})T consisting of elements killed by I. Since g_{1}, …, g_{n} is a special sequence, the associated primes of g_{1}, …, g_{n} are all minimal, and it suffices to show that the map is injective after localizing at each minimal prime of (g_{1}, …, g_{n})T. If the prime does not contain I the domain of Φ becomes 0 after localization and there is nothing to prove. If the prime contains I it is one of the P_{ν}, and after localization the map becomes the composition of the isomorphism T_{P_{ν}} /P_{ν}T_{P_{ν}} ≅ L_{P_{ν}} with the automorphism of L_{P_{ν}} induced by multiplication by the image of γ, which is nonzero.

To prove for a fixed presentation that the map is independent of the choice of special sequence suppose that we have two special sequences that yield maps with different images. We can preserve the fact that the images are different while localizing at a suitable prime or even maximal ideal of T: S is replaced by its localization at a corresponding prime. Thus, there is no loss of generality in assuming that T and S are local. The sequences in question remain special as we localize. But we may now apply Lemma (3.1) to conclude the existence of a finite chain of special sequences joining the two that we are comparing such that any two consecutive sequences differ either in at most one spot, or by the action of an invertible matrix over R. Thus, we need only make the comparison when the two sequences differ in just one term, and since the sequences are permutable we may assume without loss of generality that one of them is g_{1}, …, g_{n} and the other is h_{1}, …, h_{n} where h_{j} = g_{j} for j ⩾ 2. We set up an isomorphism

σ: \frac{(g_{1}, …, g_{n})T : _{T} I}{(g_{1}, …, g_{n})T} ≅ \frac{(h_{1}, …, h_{n})T : _{T} I}{(h_{1}, …, h_{n})T}

as follows. If u is an element of (g_{1}, …, g_{n})T : _{T} I then since uI ⊆ (g_{1}, …, g_{n})T we may write uh_{1} = vg_{1} + w, where w ∈ (g_{2}, …, g_{n})T = (h_{2}, …, h_{n})T. We map the class of u to the class of v. Note that any such v is automatically in (h_{1}, …, h_{n})T : _{T} I. (If a ∈ I then avg_{1} + aw = auh_{1}. Now, au = bg_{1} + w′ where w′ ∈ (g_{2}, …, g_{n})T and so avg_{1} + aw = bg_{1}h_{1} + w′h_{1} and so g_{1}(av − bh_{1}) = w′h_{1} − aw ∈ (g_{2}, …, g_{n})T. Since g_{1} is not a zerodivisor on (g_{2}, …, g_{n})T, we have that av − bh_{1} ∈ (g_{2}, …, g_{n})T, and so av ∈ (h_{1}, g_{2}, …, g_{n})T = (h_{1}, …, h_{n})T, as required, for all a ∈ I.) Next note that the choice of v given u is unique modulo (g_{2}, …, g_{n})T. Thus, we have defined a map from (g_{1}, …, g_{n})T : _{T} I to ((h_{1}, …, h_{n})T : _{T} I)/(h_{1}, …, h_{n})T. It is easy to see that this map kills (g_{1}, …, g_{n})T and so we have a map as required. It is also clear that if we similarly define a map

\frac{(h_{1}, …, h_{n})T : _{T} I}{(h_{1}, …, h_{n})T} → \frac{(g_{1}, …, g_{n})T : _{T} I}{(g_{1}, …, g_{n})T}

it will be an inverse for the map already constructed.

To complete the proof of the independence of the image from the choice of special sequence we note that the following diagram commutes:

\frac{(g_{1}, …, g_{n})T : _{T} I}{(g_{1}, …, g_{n})T} ⟶^{σ} \frac{(h_{1}, …, h_{n})T : _{T} I}{(h_{1}, …, h_{n})T}
Φ_{θ,g} ↓  ↓^{Φ_{θ,h}}
L  ⟶_{1_{L}}  L

To see this, one simply needs to see that if

(∗) uh_{1} − vg_{1} = ∑_{j=2}^{n} t_{j}g_{j}

in T, then \overline{u}/γ = \overline{v}/η in cL, where γ, η are the respective images of det (∂g_{j}/∂X_{i}) and det (∂h_{j}/∂X_{i}) in L, i.e., that u det (∂h_{j}/∂X_{i}) ≡ v det (∂g_{j}/∂X_{i}) modulo I. By differentiating (∗) with respect to each X_{j} in turn and using the fact that all the g_{j} and h_{j} are in I, we see that

u∇h_{1} − v∇g_{1} ≡ ∑_{j=2}^{n} t_{j}∇g_{j} modulo I

which implies (∗).

For a given special sequence g it is obvious from the definition of Φ_{θ,g} that γ multiplies the image of Φ_{θ,g} into S ⊆ L. Since the image is independent of the choice of special sequence, and since by Lemma (3.1) as the special sequence varies the values of γ generate J_{S/R}, it follows that the image is contained in S : _{L}J_{S/R}.

It remains only to prove that the image of Φ_{T,g} is independent of the choice of θ: T → S as well. We first consider the case of a finitely generated R-algebra S. The choice of a presentation is equivalent to the choice of a finite set of generators for S over R. We can compare the results from each of two different presentations with the result from their union, and so it suffices to see what happens when we enlarge a set of generators. By induction, it suffices to show that the image does not change when we enlarge a set of generators by one element, and so we may assume that we have θ : T = R[X_{1}, …, X_{n}] ↠ S and an extension of θ, θ′: T [X_{n+1}] ↠ S by sending X_{n+1} to s. Let T′ = T [X_{n+1}]. We can choose an element F ∈ T such that F maps to s in S, and it follows easily that the kernel I′ of θ′ is I + (X_{n+1} − F). It also follows easily that if g = g_{1}, …, g_{n} is special in I then g′ = g_{1}, …, g_{n+1} with g_{n+1} = X_{n+1} − F is a special sequence in I′. The larger (size n + 1) Jacobian matrix has the same determinant γ as the size n Jacobian matrix of g_{1}, …, g_{n} with respect to X_{1}, …, X_{n}, and it is easy to check that there is an isomorphism

τ: \frac{(g_{1}, …, g_{n})T : _{T} I}{(g_{1}, …, g_{n})T} ≅ \frac{(g_{1}, …, g_{n+1})T′ : _{T′}I′}{(g_{1}, …, g_{n+1})T′}

which is induced by the inclusion (g_{1}, …, g_{n})T : _{T} I ⊆ (g_{1}, …, g_{n+1})T′ : _{T′}I′. Since the Jacobian determinants are the same we have a commutative diagram

\frac{(g_{1}, …, g_{n})T : _{T} I}{(g_{1}, …, g_{n})T} ⟶^{τ} \frac{(g_{1}, …, g_{n+1})T′ : _{T′}I′}{(g_{1}, …, g_{n+1})T′}
Φ_{θ,g} ↓  ↓^{Φ_{θ′,g′}}
L  ⟶_{1_{L}}  L

and this yields that the images are the same.

We have now justified the notation W_{S/R} when S is finitely generated over R. We leave it to the reader to verify that if s is a nonzerodivisor in S, then W_{S[s^{−1}]/R} = (W_{S/R})_{s}, and that W_{S/R}(θ) is independent of θ when S is essentially of finite type over R.

Proposition 5.3. Proposition (4.4) Let S be generically étale, torsion-free and essentially of finite type over the Noetherian normal domain R. Assume that S is relatively S_{1} presentable over R. Let W = W_{S/R}.

(a) For any multiplicative system U in S, W_{U^{−1}S/R} = U^{−1}W.

(b) W is torsion-free over S.

For the remaining parts, assume that S is relatively S_{2} presentable over R (automatic in part (e), since R is regular).

(c) For every prime ideal P of S, if u, v is part of a system of parameters for S_{P} then it is a regular sequence on W_{P}. (Thus, W is S_{2}.)

(d) If W ⊆ W′ ⊆ L and W_{P} = W′_{P} for all height one primes of S and for all minimal primes of S that are also maximal ideals, then W = W′.

(e) If R → S is a local homomorphism of regular local rings then J_{S/R} is principal and W = S : _{L}J_{S/R}.

(f) If S is normal and R_{P} is regular for every prime ideal P of R lying under a height one prime ideal Q of S, then W = S : _{L}J_{S/R}.

Proof. Part (a) is essentially the last part of (4.3), while (b) is evident from the fact that W ⊆ L, by definition.

To prove (c) note that by (a) we may assume that S is local and that u, v is part of a system of parameters. We may choose a presentation θ: T ↠ S and think of W as ≅ ((g_{1}, …, g_{n})T : _{T} I)/(g_{1}, …, g_{n})T, where the sequence g_{1}, …, g_{n} is not only special, but has the additional property that T/(g_{1}, …, g_{n}) is S_{2}. Let u_{0}, v_{0} ∈ T be representatives of u, v. Then u_{0} + I cannot be contained in the union of the associated primes of (g_{1}, …, g_{n}) (these are the same as the minimal primes), or else it will be contained in one of them by [Kap], Theorem 124. Since this will contain I, it will be a minimal prime of I, and contradicts the statement that u is part of a system of parameters in S = T/I. Thus, we can replace u_{0} by an element u_{1} representing u such that g_{1}, …, g_{n}, u_{1} is a regular sequence in T. Similarly, v_{0}+I cannot be contained in the union of the associated primes of (g_{1}, …, g_{n}, u_{1})T, or else it is contained in one of them, say Q. Thinking modulo I, we see that, since Q contains u_{0}, v_{0}, it has height at least two more than I. But then Q has depth at least n + 2, a contradiction, since it is supposedly an associated prime of an ideal generated by a regular sequence of length n + 1. Thus, we may choose u_{1}, v_{1} in T representing u, v respectively and such that g_{1}, …, g_{n}, u_{1}, v_{1} is a regular sequence. Clearly, u_{1}, v_{1} form a regular sequence on T/(g_{1}, …, g_{n})T. We claim they also form a regular sequence on the set of elements killed by I. It is clear that u_{1} remains not a zerodivisor on this set. Suppose that v_{1}z = u_{1}y where z, y are killed by I. Then z = u_{1}x, y = −v_{1}x where, a priori, x ∈ T/(g_{1}, …, g_{n})T. But Iz = 0 and so Iu_{1}x = 0, and since u_{1} is not a zerodivisor on T/(g_{1}, …, g_{n})T, it follows that Ix = 0 as well.

Part (d) is a standard consequence of what we proved in part (c), but we give a short argument. If W ≠ W′ we can localize at a minimal prime of the support of W′/W and preserve the counterexample. By hypothesis, this prime cannot have height one (nor height 0, since if a height 0 prime is not maximal then we can localize at it in two steps: first localize at a height one prime that contains it). Thus, we may assume that S is local of height two or more, and that W′/W is a nonzero module of finite length. It follows that we can choose an element x ∈ W′ − W and part of a system of parameters u, v for S such that uz and vz are in W. The relations v(uz) = u(vz) over W together with part (c) show that uz ∈ uW, and it follows that z ∈ W after all, a contradiction.

To prove (e) note that when R is regular so is T, and so T → S will be a surjection of local rings. The kernel of such a surjection must be generated by part of a minimal set of generators for the maximal ideal of T. It follows that I is a prime and we have I = (g_{1}, …, g_{n})T is itself generated by a suitable special sequence. Then J_{S/R} is generated by γ = det (∂g_{j}/∂X_{i}), and ((g_{1}, …, g_{n})T : _{T} I)/(g_{1}, …, g_{n})T = (I : _{T} I)/I = T/I = S and Φ sends 1 to \frac{1}{γ}, so that W = S \frac{1}{γ}, and one sees that S : _{L}J_{S/R} = S : _{L}γS = W, as claimed.

To prove (f) it suffices by (d) to consider the problem after localizing at a height one or zero prime Q of S, and, without affecting the issue, one may also localize R at its contraction. If the prime of S has height 0, so does its contraction to R, and both rings become regular after localization. If the prime of S has height one, then, again, both rings become regular after localization, S because it is normal and R by hypothesis. In either case the result follows from part (e).

6. A CRITICAL LEMMA AND THE FINAL STEP OF THE PROOF

The following result of Lipman and Sathaye is critical in establishing that W_{S/R} decreases as S is increased by adjoining integral fractions.

Lemma 6.1 (Lipman-Sathaye). Let T be a commutative ring, Y an indeterminate, and J an ideal of T [Y] such that J contains a monic polynomial h in Y of degree d, and such that J also contains an element of the form αY − β where α, β ∈ T are such that J : _{T [Y]}αT [Y] = J, i.e., such that α is not a zerodivisor modulo J. Let G ⊆ T be an ideal of T with G ⊆ J. Then for every element v ∈ T [Y] such that vJ ⊆ (h, G)T [Y] there is a u ∈ T such that u(J ∩ T) ⊆ G and such that

v ≡ u \frac{∂h}{∂Y} modulo J.

Proof. See Lemma (3.17) on p. 216 of [LS].

We now use this to prove:

Theorem 6.2. If S_{1} is obtained from S by adjoining finitely many integral fractions of L, then W_{S_{1}/R} ⊆ W_{S/R}.

Proof. By induction on the number of fractions adjoined, it is obviously sufficient to prove this when S_{1} = S[λ], where λ is a single element of L. Choose a presentation θ: T → S and a special sequence g_{1}, …, g_{n} in the kernel I. Let Y be a new indeterminate and extend θ to a map T [Y] ↠ S[λ] by sending Y to λ. Since λ is integral over S there is a monic polynomial h = h(Y) ∈ T [Y] of degree say, d, in the kernel J of T [Y] ↠ S[λ]. If λ ∈ S there is nothing to prove so that we may assume that d ⩾ 2. Since λ is in L we may also choose α and β in T with α not a zerodivisor on I such that αY − β is in the kernel. Consider the image of h(Y) in S[Y]. There will be a certain subset of the minimal primes of S such that the image of λ is a multiple root of the image of h modulo those primes. If that set of primes is empty, we shall not alter h. If it is not empty choose an element of S that is not in any of those minimal primes but that is in the others, and represent it by an element t ∈ T. Then h(Y) + t(αY − β) has the property that its image modulo any minimal prime of S has the image of λ as a simple root, and so we may assume, using this polynomial in place of the original choice of h, that h is a monic polynomial of degree d ⩾ 2 such that image of λ modulo every minimal prime of L is a simple root of the image of h.

Because h is monic in Y, the sequence g_{1}, …, g_{n}, h is a regular sequence,

and the Jacobian determinant with respect to X_{1}, …, X_{n}, Y is γ \frac{∂h}{∂Y}, where γ is

det (∂g_{j}/∂X_{i}). Our choice of h implies that \frac{∂h}{∂Y} has image that is not in any minimal prime of L, and it follows, using Lemma (1.9), that g_{1}, …, g_{n}, h is a special sequence in J and can be used to calculate W_{S[λ]/R}. Let v ∈ (g_{1}, …, g_{n}, h)T [Y] : _{T [Y]}J. We may now apply Lemma (5.1) with this T, Y, J, v, α, β and h, while taking G = (g_{1}, …, g_{n})T. Note that J ∩ T = I. Now, v gives rise to a typical element, the

image of v/(γ \frac{∂h}{∂Y}) in L, in W_{S[λ]/R}, and we want to show that this element is in

W_{S/R}. Pick u as in Lemma (5.1). Then u ∈ (g_{1}, …, g_{n})T : _{T} I and since v ≡ u \frac{∂h}{∂y}

modulo J, this image is the same as the image of (u \frac{∂h}{∂y})/(γ \frac{∂h}{∂y}) = u/γ, and so is

in W_{S/R}, as required.

The proof of the main theorem. Theorem (5.2) and Proposition (4.4f) justify the argument given much earlier in (2.6), and the proof of Theorem (2.1) is now complete.

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Melvin Hochster hochster@math.lsa.umich.edu

Department of Mathematics University of Michigan Ann Arbor, MI 48109–1109 USA