Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 463-473.

On Mixed Multiplicities of Homogeneous Ideals

Nguyen Duc Hoang

Department of Mathematics Hanoi Pedagogical University, Vietnam

1. Introduction

Let S = ⊕S_{(u,v)} be a standard bigraded R-algebra over an Artinian local ring K = S_{(0,0)}, i.e. S is generated by finitely many forms of degree (1, 0) and (0, 1) over K. The Hilbert function of S is defined as

H(u, v) := ℓ(S_{(u,v)}),

where ℓ denotes the length of the underlying K-module. Van der Waerden [12] proved that if K is a field, then H(u, v) is given by a polynomial

P (u, v) = ∑_{i+j≤dim S−2}a_{ij}\binom{u}{i}\binom{v}{j}

for large u and v, where a_{ij} are integers. This has been extended to the Artinian case by Bhattacharya [1].

A bihomogeneous prime ideal 𝔭 of S is called relevant if 𝔭 does not contain S_{(1,0)} and S_{(0,1)}. Let BiProj(S) denote the set of the relevant bihomogeneous prime ideal of S. The relevant dimension of S is defined as

rdim S := max{dim S/𝔭 | 𝔭 ∈ BiProj(S)}.

As shown by D. Katz, S. Mandal and J. K. Verma [4], deg P (u, v) = rdim S − 2. The numbers a_{ij} with i + j = rdim S − 2 are called the mixed multiplicities of S.

Let (R, 𝔪) be a local ring of positive dimension d and I an ideal of R . We can associate with I the Rees algebra R[It] = ⊕_{i≥0}I^{i}t^{i}. Let M := (𝔪, It) be the maximal graded ideal

of R[It]. The associated graded ring gr_{M}R[It] := ⊕_{n≥0}M^{n}/M^{n+1} has a natural bigrading with

(gr_{M}R[It])_{(u,v)} = 𝔪^{u}I^{v}/𝔪^{u+1}I^{v}.

As shown by Bhattacharya [1], the numerical function dim_{k} 𝔪^{u}I^{v}/𝔪^{u+1}I^{v} is given by a polynomial in u and v for all large values of u and v. Let s be the degree of this polynomial and write the terms of total degree s as

∑_{i+j=s} \frac{a_{ij}}{i!j!}u^{i}v^{j}

where a_{ij} are non negative integers.

Teissier and Risler [9] linked these numbers to the Milnor numbers of general hyperplane sections of complex analytic hypersurfaces with isolated singularities. They called the number a_{ij} a mixed multiplicity of the pair (𝔪, I) and denoted it by e_{ij}(𝔪|I). The multiplicity of the Rees algebra R[It] and of the extended Rees algebra R[It, t^{−1}] can be expressed in terms of the mixed multiplicities as follows:

e(R[It]) = ∑_{i+j=d−1}e_{ij}(𝔪|I)

if I has positive height and, if I ⊊ 𝔪^{2},

e(R[It, t^{−1}] = e(R) + ∑_{i+j=d−1}e_{ij}(𝔪|I).

See [11, Theorem (3.1)], [5, Proof of (3.7)] for more details. Though we have a well developed theory on mixed multiplicities when I is an 𝔪-primary ideal [9], [6], there have been few cases where the mixed multiplicities can be computed in terms of well-known invariants of 𝔪 and I when I is not an 𝔪-primary ideal.

In this paper we study the case R = ⊕_{n≥0}R_{n} is a standard graded algebra over a field

k = R_{0}, 𝔪 = ⊕_{n>0}R_{n} and I a homogeneous ideal of R. Note that we can define the mixed

multiplicities e_{ij}(𝔪|I) as in the local case and that the above formulas for the multiplicities of the Rees algebras can be proved similarly.

Let x_{1}, …, x_{n} be a sequence of homogeneous elements in R with deg x_{1} ≤ … ≤ deg x_{n}. Let I denote the ideal (x_{1}, …, x_{n}). The multiplicity of the Rees algebra R[It] was computed by Herzog, Trung, and Ulrich [2] when x_{1}, …, x_{n} is a d-sequence and by Trung [10] when x_{1}, …, x_{n} is a subsystem of parameters which is filter-regular. They used a technique which is similar to that of Gröbner bases and which does not involve mixed multiplicities. Using this technique Raghavan and Verma [7] were able to compute the mixed multiplicities e_{ij}(𝔪|I) when x_{1}, …, x_{n} is a d-sequence. However, their method is a bit complicated and can not be applied to study the case I is generated by a subsystem of parameters.

In Section 2 of this paper we will use a simpler argument to compute the mixed multiplicities e_{ij}(𝔪|I) when x_{1}, …, x_{n} is a d-sequence. Let I_{i} = (x_{1}, …, x_{i−1}) : x_{i}, i = 1, …, n, d_{1} = dim R/I_{1} and r = max{i | dim R/I_{i} = d_{1} − i + 1}. We obtain the formula

e_{id_{1}−i−1}(𝔪|I) = \{ \begin{matrix}0 & if 0 ≤ i ≤ d_{1} − r − 1, \\ e(R/I_{d_{1}−i}) & if d_{1} − r ≤ i ≤ d_{1} − 1. \end{matrix}

We point out that this formula is more precise than that of Raghavan and Verma.

In Section 3 we will use the same argument to compute the mixed multiplicities e_{ij}(𝔪|I) when x_{1}, …, x_{n} is a subsystems of homogeneous parameters which is filter-regular with respect to I. Put deg x_{i} = a_{i}. We obtain the formula

e_{id−i−1}(𝔪|I) = \{ \begin{matrix}0 & if 0 ≤ i ≤ d − n − 1, \\ a_{1} … a_{d−i−1}e(R) & if d − n ≤ i ≤ d − 1. \end{matrix}

This formula was posed as a problem in [10]. It is worth to mention that the condition x_{1}, …, x_{n} is a filter-regular sequence with deg x_{1} ≤ … ≤ deg x_{n} is automatically satisfied in a generalized Cohen-Macaulay ring or if I is generated by elements of the same degree.

We do not know whether there is a compact formula for e_{ij}(𝔪|I) in the above cases when the degrees of x_{1}, …, x_{n} are not increasing.

Acknowledgement. The author would like to thank Prof. Ngô Viêt Trung for his guidance.

2. Mixed multiplicities of ideals generated by d-sequences

Let R = ⊕_{n≥0}R_{n} be a standard graded algebra over a field k = R_{0} and 𝔪 = ⊕_{n>0}R_{n}. Let

x_{1}, …, x_{n} be a sequence of homogeneous elements of R and I = (x_{1}, …, x_{n}).

Let A denote the polynomial ring R[T_{1}, …, T_{n}]. If we map T_{i} to x_{i}t, i = 1, …, n, we get a representation of the Rees algebra

R[It] ≅ A/J,

where J is the ideal of A generated by the forms vanishing at x_{1}, …, x_{n}. For all h = (a_{0}, …, a_{n}) ∈ ℕ^{n+1} put

A_{h} := R_{a_{0}}T_{1}^{a_{1}} … T_{n}^{a_{n}}.

Then A = ⊕_{h∈ℕ^{n+1}}A_{h}, that is, A is an ℕ^{n+1}-graded ring. Note that (𝔪, T_{1}, …, T_{n}) is the

maximal graded ideal of A. Define the following degree-lexicographic order on ℕ^{n+1}:

(a_{0}, a_{1}, …, a_{n}) < (b_{0}, b_{1}, …, b_{n})

if the first non-zero component from the left side of

(∑_{i=0}^{n} a_{i} − ∑_{i=0}^{n} b_{i}, a_{0} − b_{0}, …, a_{n} − b_{n})

is negative. Then < is a terms order on ℕ^{n+1}. Set

F_{h}A := ⊕_{h′ ≥h}A_{h′} .

It is clear that F = {F_{h}A}_{h∈ℕ^{n+1}} is a filtration of A. The filtration F imposes a filtration on A/J which we also denote by F.

For every polynomial f ∈ A, we denote by f^{∗} the initial term of f, i.e. f^{∗} = f_{h′} if f = ∑_{h∈ℕ^{n+1}}f_{h} and h′ := min{h| f_{h} ≠ 0}. Let J^{∗} denote the ideal of A generated by all

elements f^{∗}, f ∈ J. Then

gr_{F} (A/J) ≅ A/J^{∗}.

The ℕ^{n+1}-graded structure imposes a bigrading on A with

A_{(u,v)} = ⊕_{α_{1}+...+α_{n}=v}A_{(u,α_{1},... ,α_{n})}

for all (u, v) ∈ ℕ^{2}. Since J^{∗} is an ℕ^{n+1}-bigraded ideal of A, J^{∗} is also a bigraded ideal of A. Hence A/J^{∗} is a bigraded algebra over k with respect to the bigrading induced from A.

Now we shall see that the Bhattacharya function of (𝔪, I) coincides with the Hilbert function of A/J^{∗}.

Lemma 2.1. For all (u, v) ∈ ℕ^{2} we have

dim_{k}(𝔪^{u}I^{v}/𝔪^{u+1}I^{v}) = dim_{k}(A/J^{∗})_{(u,v)}.

Proof. We know that

𝔪^{u}I^{v}/𝔪^{u+1}I^{v} = (gr_{M}R[It])_{(u,v)}.

Let 𝔐 = (𝔪, T_{1}, …, T_{n}) be the maximal graded ideal of A. Then

gr_{M}R[It] ≅ gr_{𝔐}(A/J).

The bigrading on gr_{M}R[It] imposes a bigrading on gr_{𝔐}(A/J) with

gr_{𝔐}(A/J)_{(u,v)} =

( ⊕_{{α_{0}≥u \atop α_{1}+...+αn≥v}} A_{(α_{0},... ,α_{n})} + J)/(  ⊕_{{α_{0}≥u \atop α_{1}+...+αn≥v+1}}A_{(α_{0},... ,α_{n})} + ⊕_{{α_{0}≥u+1 \atop α_{1}+...+αn≥v}} A_{(α_{0},... ,α_{n})} + J)

≅ A_{α_{1}+...+α_{n}=v}_{(u,α_{1},... ,α_{n})} + J/J.

Using the filtration F on A/J we can decompose the latter module into a series of graded pieces of the associated ring gr_{F} (A/J) ≅ A/J^{∗} and we obtain

dim_{k} gr_{𝔐}(A/J)_{(u,v)} = ∑_{α_{1}+...+α_{n}=v} dim_{k} (A/J^{∗})_{(u,α_{1},... ,α_{n})}

= dim_{k} ⊕_{α_{1}+...+α_{n}=v}(A/J^{∗})_{(u,α_{1},... ,α_{n})}

= dim_{k} (A/J^{∗})_{(u,v)}. □

According to Lemma 2.1 we can use the Hilbert function of A/J^{∗} to compute the mixed multiplicities e_{i}(𝔪|I). Herzog-Trung-Ulrich [2] computed J^{∗} explicitly when x_{1}, …, x_{n} is a d-sequence of homogeneous elements with increasing degrees. Recall that x_{1}, …, x_{n} is said to be a d-sequence if

(1) x_{i} ∉ (x_{1}, …, x_{i−1}, x_{i+1}, …, x_{n}),

(2) (x_{1}, …, x_{i}) : x_{i+1}x_{k} = (x_{1}, …, x_{i}) : x_{k} for all k ≥ i + 1 and all i ≥ 0.

Lemma 2.2. [2, Lemma 1.2] Let x_{1}, …, x_{n} be a homogeneous d-sequence of R with deg x_{1} ≤ … ≤ deg x_{n}. Then

J^{∗} = (I_{1}T_{1}, …, I_{n}T_{n}),

where I_{j} := (x_{1}, …, x_{j−1}) : x_{j} for 1 ≤ j ≤ n.

Now we come to the main result of this section.

Theorem 2.3. Let I be an ideal generated by a homogeneous d-sequence x_{1}, …, x_{n} of R with deg x_{1} ≤ … ≤ deg x_{n}. Let I_{0} = 0, I_{i} = (x_{1}, …, x_{i−1}) : x_{i}, i = 1, …, n and d_{1} = dim R/I_{1}. Then the degree of the Hilbert polynomial of gr_{M}R[It] is d_{1} − 1 and

e_{id_{1}−i−1}(m|I) = \{ \begin{matrix}0 & if 0 ≤ i ≤ d_{1} − r − 1, \\ e(R/I_{d_{1}−i}) & if d_{1} − r ≤ i ≤ d_{1} − 1, \end{matrix}

where r = max{i| dim R/I_{i} = d_{1} − i + 1}.

Proof. We will use an idea from [8, Theorem 3.7] to estimate the coefficients of the terms of the total degree of H_{A/J^{∗}}(u, v). For this we will compute the function

H_{A/J^{∗}}(α_{0}, …, α_{n}) = dim_{k}(A/J^{∗})_{(α_{0},... ,α_{n})}.

Any element f ∈ J^{∗} with deg f = (α_{0}, …, α_{n}) is of the form yT_{1}^{α_{1}} … T_{n}^{α_{n}} with y ∈ (I_{i})_{α_{0}} for some i = 1, …, n with α_{i} ≠ 0. Since I_{1}, …, I_{n} is an increasing sequence of ideals, we get

J_{(α_{0},... ,α_{n})}^{∗} = (I_{m(α_{1},... ,α_{n})})_{α_{0}}T_{1}^{α_{1}} … T_{n}^{α_{n}}

where m(α_{1}, …, α_{n}) := max{i| α_{i} ≠ 0}. Therefore

H_{A/J^{∗}}(α_{0}, …, α_{n}) = dim_{k}(R/I_{m(α_{1},... ,α_{n})})_{α_{0}} = H_{R/I_{m(α_{1},... ,αn)}}(α_{0})

if (α_{1}, …, α_{n}) ≠ 0. From this we get the Hilbert function of A/J^{∗} as a bigraded algebra:

H_{A/J^{∗}}(u, v) = ∑_{α_{1}+...+α_{n}=v}H_{R/I_{m(α_{1},... ,αn)}}(u)

= ∑_{i=1}^{n} \binom{v + i − 2}{i − 1}H_{R/I_{i}}(u),

where the latter equality follows from the fact that the number of vectors (α_{1}, …, α_{n}) with α_{1} + … + α_{n} = v and m(α_{1}, …, α_{n}) = i is given by \binom{v+i−2}{i−1}.

Put d_{i} = dim R/I_{i}. Then

H_{R/I_{i}}(u) = \frac{e(R/I_{i})}{(d_{i} − 1)!}u^{d_{i}−1} + terms of lower degree.

Therefore,

H_{A/J^{∗}}(u, v) = ∑_{i=1}^{n} [\frac{e(R/I_{i})}{(d_{i} − 1)!(i − 1)!}u^{d_{i}−1}v^{i−1} + terms of total degree < d_{i} + i − 2] .

Since x_{i−1} is a non-zerodivisor modulo I_{i−1} we have

d_{i} = dim R/I_{i} ≤ dim R/(I_{i−1}, x_{i−1}) = dim R/I_{i−1} − 1 = d_{i−1} − 1.

From this it follows that d_{i} < d_{i−1} < … < d_{1}. Hence d_{1} − 1 is the total degree of H_{A/J^{∗}}(u, v). By the assumption, d_{i} = d_{1} − i + 1 if 1 ≤ i ≤ r, and d_{i} < d_{1} − i + 1 if r < i ≤ n. Hence from the above formula for H_{A/J^{∗}}(u, v) we obtain

e_{id_{1}−i−1}(m|I) = \{ \begin{matrix}0 & if 0 ≤ i ≤ d_{1} − r − 1, \\ e(R/I_{d_{1}−i}) & if d_{1} − r ≤ i ≤ d_{1} − 1. \end{matrix} □

Remark. Raghavan and Verma [7] already computed the bigraded Hilbert series gr_{𝔐}R[It]. From this they get the formula

e_{ij}(𝔪|I) = e_{i}(R/I_{j+1}) − e_{i}(R/I_{j+2}) (i + j = s),

where for a standard graded algebra B over a field the symbol e_{i}(B) denotes the i-th coefficient of the Hilbert polynomial P_{B}(u) of B, i.e. P_{B}(u) = ∑_{i≥0}e_{i}(B)\binom{u+i}{i}. This formula is not explicit as our formula in Theorem 2.3.

Examples 2.4. It is known that the sequence x_{1}, …, x_{n} is a d-sequence in the following cases (see [3]). Hence we can use Theorem 2.3 to compute the mixed multiplicities.

(1) Regular sequence. Let I be generated by an R-sequence x_{1}, …, x_{n} of homogeneous elements with deg x_{i} = a_{i}, a_{1} ≤ … ≤ a_{n}. Since e(R/I_{i}) = a_{1} … a_{i−1}e(R), we have

e_{id−i−1}(m|I) = a_{1} … a_{i−1}e(R), 1 ≤ i ≤ n.

(2) Subsystem of parameters of Buchsbaum rings. Let R be a graded Buchsbaum ring and I an ideal of R generated by a subsequence x_{1} …, x_{n} of a homogeneous system of parameters of R with deg x_{i} = a_{i}, a_{1} ≤ … ≤ a_{n}. By [2, Corollary 1.5] e(R/I_{i}) = a_{1} … a_{i−1}e(R). Hence

e_{id−i−1}(m|I) = a_{1} … a_{i−1}e(R), 1 ≤ i ≤ n.

(3) Almost complete intersection. Let R be a Gorenstein ring and I = (x_{1}, …, x_{n}) a homogeneous almost complete intersection of R of height n − 1 > 0 which satisfies the following conditions:

(i) x_{1}, …, x_{n−1} is a regular sequence,

(ii) a_{1} ≤ … ≤ a_{n}, a_{i} = deg x_{i},

(iii) R/I is Cohen-Macaulay,

(iv) IR_{P} = (x_{1}, …, x_{n−1})_{P} for all minimal prime ideals P of I. Note that e(R/I_{i}) = a_{1} … a_{i−1}e(R) for i = 1, …, n − 1, and e(R/I_{n}) = a_{1} … a_{n−1} e(R) − e(R/I) because (x_{1}, …, x_{n−1}) = I_{n} ∩ I. Then we obtain

e_{id−i−1}(m|I) = \{ \begin{matrix}a_{1} … a_{i−1}e(R) & if 1 ≤ i ≤ n − 1, \\ a_{1} … a_{n−1}e(R) − e(R/I) & if i = n. \end{matrix}

3. Mixed multiplicities of subsystems of parameters

Let R= ⊕_{n≥0}R_{n} be a standard graded algebra over a field k, 𝔪= ⊕_{n>0}R_{n} and I =(x_{1}, …, x_{n})

a homogeneous ideal of R. Assume that R[It] ≅ A/J, where A = R[T_{1}, …, T_{n}]. As we have seen in Section 2, A has a natural ℕ^{n+1}-graded structure. The degree lexicographical order on ℕ^{n+1} induces a filtration F on R[It]. We may write

gr_{F} R[It] ≅ A/J^{∗},

where J^{∗} is the ideal generated by the initial elements of J. Moreover, A/J^{∗} is a bigraded algebra with respect to the bigrading induced from A. By Lemma 2.1, the Bhattacharya function ℓ(𝔪^{u}I^{v}/𝔪^{u+1}I^{v}) coincides with the Hilbert function of A/J^{∗}.

We shall see that the ideal J^{∗} can be estimated if I is generated by a filter-regular sequence. Recall that a sequence x_{1}, …, x_{n} of elements of R is called filter-regular with respect to I if x_{i} ∉ P for all associated prime ideals P ⊉ I of (x_{1}, …, x_{i−1}), i = 1, …, n (see e.g. [10]). For i = 1, …, n we set

J_{i} := ∪_{m=1}^{∞}(x_{1}, …, x_{i−1}) : I^{m}.

Note that J_{i} is equal to the intersection of all primary components of (x_{1}, …, x_{i−1}) whose associated prime ideals do not contain I.

Lemma 3.1. [10, Lemma 3.1] Let I be generated by a filter-regular sequence x_{1}, …, x_{n} with respect to I with deg x_{1} ≤ … ≤ deg x_{n}. Let P := (J_{1}T_{1}, …, J_{n}T_{n}). Then

J^{∗} ⊆ P.

Set I_{i} := (x_{1}, …, x_{i−1})R, i = 1, …, n, and

L := (I_{2}T_{2}, …, I_{n}T_{n}).

Since L is the ideal generated by the initial forms of the relations x_{i}T_{j} − x_{j}T_{i} we have

L ⊆ J^{∗}.

If I is generated by a subsystem of parameters with increasing degrees which is filterregular, we can use Lemma 2.1 to show that the mixed multiplicities of A/J^{∗} and A/L are the same. Note that every subsystem of parameters of R is filter-regular if R is a generalized Cohen-Macaulay ring.

Proposition 3.2. Let I be a homogeneous ideal generated by a subsystem of parameters x_{1}, …, x_{n} which is a filter-regular sequence with deg x_{1} ≤ … ≤ deg x_{n}. Then the mixed multiplicities of A/J^{∗} and A/L are equal.

To prove Proposition 3.2 we shall need the following observation on the additivity of mixed multiplicities.

Lemma 3.3. Let S be a standard bigraded algebra with 0 = Q_{1} ∩ … ∩ Q_{s} ∩ Q, where Q_{1}, …, Q_{s} are the relevant primary components of highest dimension. Then

e_{j}(S) = ∑_{i=1}^{s} e_{j}(S/Q_{i}).

Proof. We use induction on s. If s = 1, from the exact sequence

0 ⟶ S = S/Q_{1} ∩ Q ⟶ S/Q_{1} ⊕ S/Q ⟶ S/Q_{1} + Q ⟶ 0

we get

H_{S}(u, v) = H_{S/Q_{1}}(u, v) + H_{S/Q}(u, v) − H_{S/Q_{1}+Q}(u, v).

Since rdim S/Q_{1} > rdim S/Q ≥ rdim S/Q_{1} + Q,

e_{j}(S) = e_{j}(S/Q_{1}).

If s > 1, put P = Q_{2} ∩ … ∩ Q_{s} ∩ Q. From the exact sequence

0 ⟶ S = S/Q_{1} ∩ P ⟶ S/Q_{1} ⊕ S/P ⟶ S/Q_{1} + P ⟶ 0

we get

H_{S}(u, v) = H_{S/Q_{1}}(u, v) + H_{S/P} (u, v) − H_{S/Q_{1}+P} (u, v).

Since the associated prime ideals of P are not contained in the associated prime ideal of Q_{1}, rdim S/Q_{1} + P < rdim S/Q_{1} = rdim S/P = rdim S. Hence

e_{j}(S) = e_{j}(S/Q_{1}) + e_{j}(S/P).

By induction we may assume that

e_{j}(S/P) = ∑_{i=2}^{s} e_{j}(S/Q_{i}).

Therefore,

e_{j}(S) = ∑_{i=1}^{s} e_{j}(S/Q_{i}).  □

Proof of Proposition 3.2. By Lemma 3.3 we only need to show that the relevant primary components of highest dimension of J^{∗} and L are equal. The ideal L has the decomposition

L = ∩_{i=1}^{n}(I_{i}, T_{i+1}, …, T_{n}).

It is clear that every relevant primary component of highest dimension of L must be of the form (𝔮, T_{i+1}, …, T_{n}) for some primary component 𝔮 of I_{i} with

dim R/𝔮 = dim R/I_{i} = dim R − i + 1, i = 1, …, n.

Let 𝔭 denote the associated prime ideal of 𝔮. Then I ⊈ 𝔭 because dim R/I < dim R/𝔭. Therefore

J_{i}R_{𝔭} = (∪_{m=1}^{∞}I_{i} : I^{m})R_{𝔭} = I_{i}R_{𝔭}.

From this we deduce that 𝔮 ⊇ J_{i}. On the other hand, the ideal P = (J_{1}T_{1}, …, J_{n}T_{n}) has the following decomposition

P = ∩_{i=1}^{n}(J_{i}, T_{i+1}, …, T_{n}) ∩ (T_{1}, …, T_{n}).

So P is contained in all relevant primary components of highest dimension of L. But L ⊆ J^{∗} ⊆ P by Lemma 3.1. Therefore, the relevant primary components of highest dimension of L and J^{∗} must be equal. □

Now we will compute the mixed multiplicities of A/L and therefore the mixed multiplicities e_{ij}(𝔪|I).

Lemma 3.4. [10, Lemma 1.6] Let x be a homogeneous filter-regular element with respect to an ideal I of R with ht I ≥ 2, set a := deg x. Then

e(R/(x)) = ae(R).

Theorem 3.5. Let I be a homogeneous ideal of R generated by a subsystem of parameters x_{1}, …, x_{n} which is a filter-regular sequence with respect to I with deg x_{1} = a_{1} ≤ … ≤ deg x_{n} = a_{n}. Then

e_{id−i−1}(𝔪|I) = \{ \begin{matrix}0 & if 0 ≤ i ≤ d − n − 1, \\ a_{1} … a_{d−i−1}e(R) & if d − n ≤ i ≤ d − 1. \end{matrix}

Proof. By Lemma 2.1 and Lemma 3.2, e_{ij}(A/J^{∗}) = e_{ij}(A/L). Therefore we only need to compute the mixed multiplicities of A/L. As in the proof of Theorem 2.3 we have

H_{A/L}(u, v) = ∑_{α_{1}+...+α_{n}=v}H_{R/I_{m(α_{1},... ,αn)}}(u) = ∑_{i=1}^{n} \binom{v + i − 2}{i − 1}H_{R/I_{i}}(u).

Since dim R/I_{i} = d − i + 1,

H_{R/I_{i}}(u) = \frac{e(R/I_{i})}{(d − i)!}u^{d−i} + terms of lower degree.

Using Lemma 3.4 we can easily show that

e(R/I_{i}) = a_{1} … a_{i−1}e(R).

Thus

H_{A/L}(u, v) = ∑_{i=1}^{n} \frac{a_{1} … a_{i−1}e(R)}{(d − i)!(i − 1)!}u^{d−i}v^{i−1} + terms of total degree < d − 1.

From this it follows show that

e_{id−i−1}(𝔪|I) = \{ \begin{matrix}0 & if 0 ≤ i ≤ d − n − 1, \\ a_{1} … a_{d−i−1}e(R) & if d − n ≤ i ≤ d − 1. \end{matrix} □

Remark. The formula of Theorem 3.5 was posed as a problem in [10, Remark of Th. 3.3].

Using the characterization of the multiplicity of the Rees algebra R[It] and the extended Rees algebra R[It, t^{−1}] we immediately obtain the following result which was proved in [10] by a different method.

Corollary 3.6. [10, Corollary 3.6 and Corollary 4.4] Let I be as in Theorem 3.5. Then

e(R[It]) = (1 + ∑_{i=1}^{n−1} a_{1} … a_{i})e(R),

e(R[It, t^{−1}]) = (1 + ∑_{i=l}^{n−1} a_{1} … a_{i})e(R),

where l is the largest integer for which a_{l} = 1 (l = 0 and a_{1} … a_{l} = 1 if a_{i} > 1 for all i = 1, …, n).

References

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Received December 7, 1999