AN INEQUALITY INVOLVING TIGHT CLOSURE AND PARAMETER IDEALS

CĂTĂLIN CIUPERCĂ AND FLORIAN ENESCU

Dedicated to the memory of Professor Nicolae Radu

Abstract. We establish an inequality involving colengths of the tight closure of ideals of systems of parameters in local rings with some mild conditions. As an application, we prove and refine a result by Goto and Nakamura, conjectured by Watanabe and Yoshida, which states that the Hilbert-Samuel multiplicity of a parameter ideal is greater than or equal to the colength of the tight closure of the ideal.

Introduction

In recent years, the Hilbert-Kunz multiplicities have generated constant interest among researchers in commutative algebra. The development of tight closure theory and its connection to Hilbert-Kunz multiplicities has provided a fresh perspective and led to new discoveries. Among them, a characterization of regular local rings in terms of the Hilbert-Kunz multiplicity of the ring at its maximal ideal, due to K.-i. Watanabe and Y. Yoshida, stands out. Under mild conditions, they proved that a local ring (R,𝔪,k) is regular if and only if the Hilbert-Kunz multiplicity at 𝔪 equals 1 [9, Theorem 1.5]. A short and elegant proof of this theorem has also been given by C. Huneke and Y. Yao ([7]). In proving their result, Watanabe and Yoshida were led to a conjecture that ties the Hilbert-Samuel multiplicity of a parameter ideal to the colength of the tight closure of that ideal. Before going further, we would like to give precise definitions to the concepts that appear in our discussion.

Let (R,𝔪) be a local ring of positive characteristic p. If I is an ideal in R, then I^{[q]}= (i^{q} : i ∈ I), where q = p^{e} is a power of the characteristic. Let R^{∘} = R \ ∪P, where P runs over the set of all minimal primes of R. An element x is said to belong to the tight closure of the ideal I if cx^{q} ∈ I^{[q]}for all sufficiently large q = p^{e}. The tight closure of I is denoted by I^{∗}. By a parameter ideal we mean here an ideal generated by a full system of parameters in R. For an 𝔪-primary ideal I, one can consider the Hilbert-Samuel multiplicity and the Hilbert-Kunz multiplicity.

Definition 0.1. Let I be an 𝔪-primary ideal in (R,𝔪).

1. The Hilbert-Samuel multiplicity of R at I is defined by e(I) = e(I,R) :=

lim_{n→∞} d!\frac{λ(R/I^{n})}{n^{d}}. The limit exists and it is positive.

2000 Mathematics Subject Classification: 13D40, 13A35, 13H15.

2. The Hilbert-Kunz multiplicity of R at I is defined by e_{HK}(I) = e_{HK}(I,R) :=

lim_{q→∞} \frac{λ(R/I^{[q]})}{q^{d}}. Monsky has shown that this limit exists and is positive.

It is known that for parameter ideals I, one has e(I) = e_{HK}(I). The following sequence of inequalities is also known to hold:

max\{1, \frac{1}{d!} e(I)\} ≤ e_{HK}(I) ≤ e(I)

for every 𝔪-primary ideal I.

Let Assh(R) = {P ∈ Min(R) : dim(R) = dim(R/P)}. Watanabe and Yoshida have shown that whenever Ass(R̂) = Assh(R̂), the local ring (R,𝔪) is regular if and only if e_{HK}(𝔪) = 1 [9, Theorem 1.5]. Under the same assumption that Ass(R̂) = Assh(R̂), they conjectured that when I is a parameter ideal one has e(I) ≥ λ(R/I^{∗}) and that the equality occurs for one (and hence all) parameter ideals if and only if R is Cohen-Macaulay and F -rational [9, Conjecture 16]. Watanabe and Yoshida have also given an affirmative answer to the conjecture for some particular classes of parameter ideals. The conjecture has been proven by S. Goto and Y. Nakamura under very mild assumptions [2, Theorems 1.1 and 1.2].

Theorem 0.2 (Goto-Nakamura). Let (R,𝔪) be a homomorphic image of a Cohen-Macaulay local ring of characteristic p > 0.

1. Assume that R is equidimensional. Then e(I) ≥ λ(R/I^{∗}) for every parameter ideal I.

2. Assume that Ass(R) = Assh(R). If e(I) = λ(R/I^{∗}) for some parameter ideal I, then R is a Cohen-Macaulay F -rational local ring.

In their proof they employ the notion of filter regular sequences. In fact, their proof of part 2 of the Theorem 0.2 is intricate and uses sequences with a property stronger than that of filter regular sequences. Their aim is to reduce the problem to the case of an FLC ring of dimension 2.

Our main result, Theorem 1.1 in Section 1, is an inequality that involves colengths of a certain family of parameter ideals. As an immediate application, we refine part 1 of the result of Goto and Nakamura (as in Remark 1.8). We also provide a short proof of part 2 of their theorem under some mild conditions.

We would like to thank Craig Huneke for comments that allowed us to improve the manuscript. In particular, Remark 1.10 was suggested by him.

1. The main result

The following theorem is the core of this note.

Theorem 1.1. Let (R,𝔪) be an equidimensional local Noetherian ring of characteristic p > 0 which is a homomorphic image of a Cohen Macaulay ring and let x_{1},x_{2},...,x_{d} be a system of parameters. Then for every k_{1},k_{2},...,k_{d} ≥ 1 we have

(1.1.1)  λ(R/(x_{1}^{k_{1}},x_{2}^{k_{2}},...,x_{d}^{k_{d}})^{∗}) ≥ k_{1}k_{2}...k_{d}λ(R/(x_{1},x_{2},...,x_{d})^{∗}).

First, we would like to state the following:

Lemma 1.2. Let (R,𝔪) be a local equidimensional ring which is a homomorphic image of a Cohen-Macaulay ring. Let x_{1},...,x_{d} be a system of parameters in R. Then

(x_{1}^{t+s},x_{2},...,x_{d})^{∗} : x_{1}^{t} ⊂ (x_{1}^{s},x_{2},...,x_{d})^{∗}

for every positive integers t and s.

Proof. The argument is given essentially in [5, (2.3)] and is also implicit in [3, (7.9)]. □

Proof of Theorem 1.1. It is enough to prove that for every system of parameters x_{1},x_{2},...,x_{d} and every k ≥ 1 we have

λ(R/(x_{1}^{k},x_{2},...,x_{d})^{∗}) ≥ kλ(R/(x_{1},x_{2},...,x_{d})^{∗}).

Denote J = (x_{1},x_{2},...,x_{d}) and set m = λ(R/J^{∗}). Take a filtration of J^{∗} ⊆ R,

J^{∗} = 𝔞_{0} ⊆ 𝔞_{1} ⊆ ... ⊆ 𝔞_{m−1} ⊆ 𝔞_{m}= R,

such that λ(𝔞_{i}/𝔞_{i−1} ) = 1 and write 𝔞_{i} = 𝔞_{i−1} + (y_{i}) with y_{i} ∉ a_{i−1} and y_{i}𝔪 ⊆ 𝔞_{i}. For each n ∈ {1,...,k}, let L_{i} = (x_{1}^{n},x_{2},...,x_{d})^{∗}+ x_{1}^{n−1} (y_{1},...,y_{i}). Note that L_{m}= (x_{1}^{n},x_{2},...,x_{d})^{∗}+(x_{1}^{n−1} ), as y_{m}is a unit in R. Then consider the following filtration:

(1.2.1)  (x_{1}^{n},x_{2},...,x_{d})^{∗} = L_{0} ⊆ L_{1} ⊆ ... ⊆ L_{m}⊆ (x_{1}^{n−1} ,x_{2},...,x_{d})^{∗}.

We claim that λ(L_{i}/L_{i−1} ) = 1 for every i ∈ {1,...,m}. Indeed, since y_{i}𝔪 ⊆ 𝔞_{i} = J^{∗}+ (y_{1},...,y_{i}), we have

𝔪x_{1}^{n−1} y_{i} ⊆ x_{1}^{n−1} (J^{∗}+ (y_{1},...,y_{i})) ⊆ (x_{1}^{n},x_{2},...,x_{d})^{∗}+ x_{1}^{n−1} (y_{1},...,y_{i}) = L_{i−1} .

Then 𝔪L_{i} = 𝔪(L_{i−1} + x_{1}^{n−1} y_{i}) ⊆ L_{i−1} , hence λ(L_{i}/L_{i−1} ) ≤ 1. On the other hand, we also have L_{i} ≠ L_{i−1} . If not, then

x_{1}^{n−1} y_{i} ∈ L_{i−1} = (x_{1}^{n},x_{2},...,x_{d})^{∗}+ x_{1}^{n−1} (y_{1},...,y_{i−1} ),

so there exist r_{1},...,r_{d} such that

x_{1}^{n−1} (y_{i}− ∑_{j=1}^{i−1} r_{j}y_{j}) ∈ (x_{1}^{n},x_{2},...,x_{d})^{∗}.

Since x_{1},...,x_{d} is a system of parameters, by the previous Lemma it follows that

y_{i}− ∑_{j=1}^{i−1} r_{j}y_{j} ∈ (x_{1}^{n},x_{2},...,x_{d})^{∗} : x_{1}^{n−1} ⊆ (x_{1},x_{2},...,x_{d})^{∗} = J^{∗},

hence y_{i} ∈ J^{∗}+(y_{1},...,y_{i−1} ) = 𝔞_{i−1} . This contradicts the choice of y_{i} and the claim is proved.

From (1.2.1) we then obtain

(1.2.2) λ((x_{1}^{n−1} ,x_{2},...,x_{d})^{∗}/(x_{1}^{n},x_{2},...,x_{d})^{∗}) ≥ m = λ(R/(x_{1},x_{2},...,x_{d})^{∗}),

with equality if and only if L_{m}= (x_{1}^{n−1} ,x_{2},...,x_{d})^{∗}, that is

(1.2.3)  (x_{1}^{n−1} ,x_{2},...,x_{d})^{∗} = (x_{1}^{n},x_{2},...,x_{d})^{∗}+ (x_{1}^{n−1} ).

Finally, as (1.2.2) holds for every n ∈ {1,...,k}, we get

λ(R/(x_{1}^{k},x_{2},...,x_{d})^{∗}) ≥ kλ(R/(x_{1},x_{2},...,x_{d})^{∗}),

and the proof is finished. □

Remark 1.3. Under the same assumptions, if

λ(R/(x_{1}^{k},x_{2},...,x_{d})^{∗}) = kλ(R/(x_{1},x_{2},...,x_{d})^{∗}),

then

(x_{1},x_{2},...,x_{d})^{∗} = (x_{1},x_{2},...,x_{d}) + (x_{1}^{k},x_{2},...,x_{d})^{∗}.

Indeed, by (1.2.2) the first equality implies that

λ((x_{1}^{n−1} ,x_{2},...,x_{d})^{∗}/(x_{1}^{n},x_{2},...,x_{d})^{∗}) = λ(R/(x_{1},x_{2},...,x_{d})^{∗})

for every n ∈ {1,...,k}. By (1.2.3) we then obtain

(1.3.1)  (x_{1}^{n−1} ,x_{2},...,x_{d})^{∗} = (x_{1}^{n},x_{2},...,x_{d})^{∗}+ (x_{1}^{n−1} ) for 1 ≤ n ≤ k,

which by iteration yields

(x_{1},x_{2},...,x_{d})^{∗} = (x_{1},x_{2},...,x_{d}) + (x_{1}^{k},x_{2},...,x_{d})^{∗}.

Corollary 1.4. Let (R,𝔪) be an equidimensional local Noetherian ring of characteristic p > 0 which is a homomorphic image of a Cohen Macaulay ring and let I be a parameter ideal. Then

λ(R/(I^{[p]})^{∗}) ≥ p^{d}λ(R/I^{∗}).

Remark 1.5. For every q = p^{e} set

a_{e} = λ(R/(I^{[q]})^{∗})/q^{d} and b_{e} = λ(R/I^{[q]})/q^{d}.

If (R,𝔪) is equidimensional and homomorphic image of a Cohen-Macaulay ring and I is a parameter ideal, Corollary 1.4 shows that {a_{e}}_{e≥0} is an increasing sequence. On the other hand, it is known that {b_{e}}_{e≥0} is a decreasing sequence whose limit is the Hilbert-Kunz multiplicity e_{HK}(I). In our case I is a parameter ideal, so e_{HK}(I) = e(I). It is also clear that for every e ≥ 0 we have

(1.5.1)  λ(R/I^{∗}) ≤ a_{e} ≤ b_{e} ≤ λ(R/I).

Remark 1.6. Assume that (R,𝔪) has a test element and let I be an 𝔪-primary ideal in R. Then lim_{q→∞} λ((I^{[q]})^{∗}/I^{[q]}))/q^{d} = 0. Indeed, fix c a test element. Then (I^{[q]})^{∗} ⊆ (I^{[q]}: c) so it is enough to show that lim_{q→∞} λ((I^{[q]}: c)/I^{[q]}))/q^{d} = 0. But λ((I^{[q]}: c)/I^{[q]})) = λ(R/(c,I^{[q]})) and lim_{q→∞} λ((R/(c,I^{[q]}))/q^{d} = 0 because R/cR is (d − 1)-dimensional and lim_{q→∞} λ((R/(c,I^{[q]}))/q^{d−1} represents the Hilbert-Kunz multiplicity of the image of I in R/cR.

This also shows that lim_{q→∞} λ(R/(I^{[q]})^{∗})/q^{d} exists and equals e_{HK}(I) for an 𝔪-primary ideal I. In particular, for a parameter ideal I, lim_{q→∞} λ(R/(I^{[q]})^{∗})/q^{d} = e(I), since e(I) = e_{HK}(I). So, according to our main result, in this case the sequence a_{e} = λ(R/(I^{[q]})^{∗})/q^{d} is increasing and its limit equals e(I).

As an immediate consequence of the Remark 1.5, we obtain the following result proved by Goto and Nakamura [2, Theorem 1.1]. It had been conjectured and proved in some special cases by Watanabe and Yoshida [9].

Corollary 1.7 (Goto–Nakamura). Let (R,𝔪) be an equidimensional local Noetherian ring of characteristic p > 0 which is a homomorphic image of a Cohen Macaulay ring and let I be a parameter ideal. Then

e(I) ≥ λ(R/I^{∗}).

Proof. From (1.5.1) we get λ(R/I^{∗}) ≤ lim_{e→∞}b_{e} = e_{HK}(I) = e(I). □

Remark 1.8. In fact, under the conditions of the above Corollary, one has that

e(I) ≥ λ(R/(I^{[q]})^{∗})/q^{d} ≥ λ(R/I^{∗}).

Proof. On one hand, q^{d}e(I) = e(I^{[q]}) ≥ λ(R/(I^{[q]})^{∗}). On the other hand, we have that λ(R/(I^{[q]})^{∗})/q^{d} ≥ λ(R/I^{∗}), according to the Theorem 1.1 □

As mentioned in the introduction, Watanabe and Yoshida [9] also conjectured that if R is unmixed (Ass(R̂) = Assh(R̂)) and e(I) = λ(R/I^{∗}) for some parameter ideal I, then R is a Cohen-Macaulay F-rational ring. If R is a homomorphic image of a Cohen-Macaulay ring and Ass(R) = Assh(R), Goto and Nakamura [2, Theorem 1.2] proved that the conjecture holds true. Using some considerations on the line of the argument employed in Theorem 1.1, we are able to give a much shorter proof of this result in the case when either R̂is reduced or R has a parameter test element.

Corollary 1.9 (Goto–Nakamura). Let (R,𝔪) be an equidimensional local Noetherian ring of characteristic p > 0 which is a homomorphic image of a Cohen Macaulay ring. Assume that either R̂is reduced or that R has a (parameter) test element and Ass(R) = Assh(R). If e(I) = λ(R/I^{∗}) for some parameter ideal I, then R is a Cohen-Macaulay F-rational ring.

Proof. Let I = (x_{1},...,x_{d}). By (1.5.1), for every e ≥ 0 we have

λ(R/I^{∗}) ≤ a_{e} ≤ limb_{n} = e_{HK}(I) = e(I),

hence

λ(R/I^{∗}) = λ(R/(I^{[q]})^{∗})/q^{d} for every q = p^{e}.

On the other hand, by Proposition 1.1, for any i ∈ {1,...,d}and all q = p^{e} we have

λ(R/(I^{[q]})^{∗}) ≥ q^{d−i} λ(R/(x_{1}^{q},...,x_{i}^{q},x_{i+1} ...,x_{d})^{∗}) ≥ q^{d}λ(R/I^{∗}) = λ(R/(I^{[q]})^{∗}),

which implies that

λ(R/(x_{1}^{q},...,x_{i}^{q},x_{i+1} ,...,x_{d})^{∗}) = qλ(R/(x_{1}^{q},...,x_{i−1}^{q} ,x_{i},...,x_{d})^{∗}).

Applying successively Remark 1.3, we obtain

(x_{1},x_{2},...,x_{d})^{∗}= (x_{1},x_{2},...,x_{d}) + (x_{1}^{q},x_{2},...,x_{d})^{∗}
= (x_{1},x_{2},...,x_{d}) + (x_{1}^{q},x_{2}^{q},x_{3}...,x_{d})^{∗}
...
= (x_{1},x_{2},...,x_{d}) + (x_{1}^{q},x_{2}^{q},...,x_{d}^{q})^{∗},

so I^{∗} = I + (I^{[q]})^{∗} for all q = p^{e}. In particular, I^{∗} ⊆ I + \overline{𝔪^{q}} for all q = p^{e}, or equivalently, I^{∗} ⊆ I + \overline{𝔪^{n}} for all n ≥ 1.

First we consider the case when R̂ is reduced. Passing to the completion R̂, we get I^{∗}R̂⊆ IR̂+ \overline{𝔪^{n}R̂} for all n ≥ 1. Since R̂ is reduced, ⋂ _{n} \overline{𝔪^{n}R̂}= 0 and by Chevalley’s Lemma, for each k ≥ 1 there exists n_{k} with \overline{𝔪^{n_{k}}R̂}⊆ 𝔪^{k}R̂. This implies that I^{∗}R̂⊆ IR̂+ 𝔪^{k}R̂ for all k, hence I^{∗}R̂⊆ IR̂, or equivalently, I^{∗} = I. As I is a parameter ideal, this shows that R is a Cohen-Macaulay F-rational ring.

Now, assume that R admits a (parameter) test element c ∈ R^{∘}. Then (I^{[q]})^{∗} ⊆ (I^{[q]}: c). This shows that c(I^{[q]})^{∗} ⊆ Rc ∩ I^{[q]}. By the Artin-Rees Lemma, one can find k such that Rc ∩ I^{[q]}⊆ cI^{q−k} for all sufficiently large q In conclusion, (I^{[q]})^{∗} ⊆ (0 : c) + I^{q−k}. In the first part of proof, we have seen that equality stated in the hypothesis implies that I^{∗} = I + (I^{[q]})^{∗} for all q = p^{e}. Hence, I^{∗} ⊆ I + (0 : c) + I^{q−k} and taking the intersection over all sufficiently large q we get that I^{∗} ⊆ I + (0 : c). One can notice now that whenever Ass(R) = Assh(R), c ∈ R^{∘} implies that c is a nonzerodivisor on R, so (0 : c) = 0. In conclusion, I is tightly closed and therefore R is F -rational and Cohen-Macaulay. □

Remark 1.10. The assumption that R admits a parameter test element that appears in one of the cases of the above Corollary is a mild condition which is generally easy to test in practice. In fact, if one employs the notion of limit closure instead of tight closure, the assumption can be removed when R is a homomorphic image of a Gorenstein ring and Ass(R) = Assh(R).

For any parameter ideal I generated by a system of parameters x_{1},...,x_{d}, one can define the limit closure of I = (x_{1},...,x_{d}) by I^{lim}:= ⋃ _{t}(x_{1}^{t+1},...,x_{d}^{t+1}) : (x_{1}··· x_{d})^{t}. In general, I ⊆ I^{lim} ⊆ I^{∗} if R is equidimensional and homomorphic image of a Cohen-Macaulay ring. (For more details on the limit closure we refer the reader to [5, 6].) The arguments presented in the paper use only the “colon-capturing” part of the tight closure and in fact work for the limit closure too. Hence, one can obtain similar inequalities as in our Theorem that are valid for the limit closure operation.

Returning to the second part of Corollary, whenever R is a homomorphic image of a Gorenstein ring one can prove directly the existence of an element c that multiplies (I^{[q]})^{lim} into I^{[q]}. Indeed, if one denotes 𝔞_{i} = Ann_{R}(H_{𝔪}^{i}(R)), it is known that the ideal 𝔠 = 𝔞_{0}··· 𝔞_{d−1} kills all the modules (x_{1}^{t},...,x_{k}^{t}) : x_{k+1}^{t}/(x_{1}^{t},...,x_{k}^{t}) for all positive integers k ≤ d − 1 and all positive integers t. Then, as in [8, p. 208], one can show that 𝔠^{d} multiplies (I^{[q]})^{lim}into I^{[q]}. Since dimR/𝔞_{i} ≤ i for all i [1, 8.1.1(b)] and Ass(R) = Assh(R), the ideal 𝔠^{d} contains a nonzerodivisor c which, therefore, has the above stated property.

Going back to the Corollary, the inequalities e(I) ≥ λ(R/I^{lim}) ≥ λ(R/I^{∗}) coupled with the hypothesis e(I) = λ(R/I^{∗}) give that I^{lim}= I^{∗}. As in the proof of the Corollary, one can now conclude that I^{lim} = I + (I^{[q]})^{lim} for all q = p^{e}. The existence of a nonzerodivisor c as above allows us to continue the proof in similar fashion and conclude that I = I^{lim}. So, I = I^{lim} = I^{∗} and therefore R is F -rational and Cohen-Macaulay.

The reader should also note that one good property of the limit closure is that it commutes with the completion at the maximal ideal. The same is true for the multiplicity of a parameter ideal. Consequently, if one chooses to pass to the completion first, the completion is a homomorphic image of a regular ring and the proof outlined above will work similarly if one assumes now that R is a homomorphic image of a Cohen-Macaulay ring and Ass(R̂) = Assh(R̂).

The previous results can be easily extended to the class of arbitrary ideals primary to the maximal ideal.

Corollary 1.11. Let (R,𝔪,k) be an equidimensional local Noetherian ring of characteristic p > 0 which is a homomorphic image of a Cohen Macaulay ring. Assume that either R and R̂have a common test element, or k is infinite.

1. For every 𝔪-primary ideal I, we have

e(I) ≥ λ(R/I^{∗}).

2. If e(I) = λ(R/I^{∗}) for some 𝔪-primary ideal I, then R is a Cohen-Macaulay Frational ring.

Proof. If k is finite and R and R̂have a common test element, we can enlarge the residue field of (R,𝔪,k) such that k is an infinite field, by passing to R[X]_{𝔪R[x]}. (The tight closure of I commutes with this base change by [4, Theorem 7.16].) Hence we can assume that the residue field of R is infinite. This allows us to consider a reduction J of I such that J is a parameter ideal. Then e(I) = e(J) ≥ λ(R/J^{∗}) ≥ λ(R/I^{∗}), where the last inequality holds because J ⊆ I and hence J^{∗} ⊆ I^{∗}.

To prove part 2, notice that the equality for I implies that e(J) ≥ λ(R/J^{∗}), for J chosen as above. Hence R is Cohen-Macaulay and F -rational by Theorem 0.2. □

References

[1] W. Bruns and J. Herzog, Cohen-Macaulay rings (Cambridge University Press, Cambridge, 1993).

[2] S. Goto and Y. Nakamura, ‘Multiplicity and tight closures of parameters’, J. Algebra 244 (2001), no. 1, 302–311.

[3] M. Hochster and C. Huneke, ‘Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116.

[4] M. Hochster and C. Huneke, ‘F -regularity, test elements, and smooth base change’, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62.

[5] C. Huneke, ‘Tight closure, parameter ideals and geometry’, Six lectures on commutative algebra, Progr. Math. 166 (Birkhäuser, Basel, 1998), pp. 187–239.

[6] C. Huneke and K. E. Smith, Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math. 484 (1997), 127–152.

[7] C. Huneke and Y. Yao, ‘Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular’, Proc. Amer. Math. Soc. 130 (2002), no. 3, 661–665.

[8] J. R. Strooker, Homological questions in local algebra, London Mathematical Society Lecture Note Series 145 (Cambridge University Press, Cambridge, 1990).

[9] K.-i. Watanabe and K. Yoshida, ‘Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J Algebra 230 (2000), no. 1, 295–317.

Department of Mathematics, University of California, Riverside, CA 92521 USA

E-mail address: ciuperca@math.ucr.edu

Department of Mathematics, University of Utah, Salt Lake City, UT 84112 USA and The Institute of Mathematics of the Romanian Academy, Bucharest, Romania

E-mail address: enescu@math.utah.edu