Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 37 (1996), No. 1, 149-159.

The Divisor of the Resultant

Günter Scheja and Uwe Storch

Mathematische Fakultät der Universität Tübingen Auf der Morgenstelle, D-72076 Tübingen, Germany

Fakultät für Mathematik der Ruhr-Universität Bochum D-44780 Bochum, Germany

Abstract. Let A be an integrally closed noetherian domain, A[T_{0}, …, T_{n}] a graded polynomial algebra with indeterminates having arbitrary positive (integral) weights and F_{0}, …, F_{n} ∈ A[T] := A[T_{0}, …, T_{n}] homogeneous polynomials of positive degrees, which form a regular sequence. The properties of the subvariety Proj C, C := A[T]/(F_{0}, …, F_{n}), of the weighted projective space Proj A[T] and its canonical image in Spec A are best described in terms of the resultant ideal ℜ which is a principal ideal and can be defined either by a suitable Fitting ideal or a cancelling rule for determinants. By means of duality available for regular sequences the following is proved: The divisor of ℜ equals the divisor defined by the algebra of global sections of Proj C (which is an A–torsion–module). Some applications on the degree of elimination and on the fibres of the elimination mapping are given.

Introduction

In this paper we continue the study [8] of resultants and their use in elimination theory.

The situation will always be the following: The base ring A is an integrally closed noetherian domain. A[T] = A[T_{0}, …, T_{n}] is a graded polynomial algebra where the indeterminates T_{i} have arbitrary positive weights γ_{i} ∈ N^{∗}. We consider a regular sequence F_{0}, …, F_{n} of homogeneous polynomials in A[T] of positive degrees δ_{0}, …, δ_{n} and the closed subvariety Proj C, C := A[T]/(F_{0}, …, F_{n}), they define in the (weighted) projective space Proj A[T]. The (homogeneous) ideal 𝔗 of the Trägheitsformen in C describes the same variety, i.e. Proj C = Proj C̄ with C̄ := C/𝔗, and the zero–set of the elimination ideal 𝔗_{0} = 𝔗 ∩ A ⊆ A is the image of Proj C in Spec A.

Far better than the elimination ideal is the resultant ideal ℜ ⊆ A which is not only a principal ideal with the same zero–set as 𝔗_{0} but also has suitable functorial properties.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1996 Heldermann Verlag, Berlin Furthermore, its divisor div ℜ = χ(C̄_{σ}), σ := (δ_{0} + ⋯ + δ_{n}) − (γ_{0} + ⋯ + γ_{n}), describes precisely the degree of Proj C, which means

div ℜ = χ(Γ(Proj C, O_{Proj C}))

cf. Theorem 2.1.^{1}) For the proof we use Lemma 3.1 from duality theory which, in addition, provides means to compute div ℜ in a way that generalizes long–known techniques used by Sylvester, Hurwitz, Macaulay and others, cf. Theorem 2.2.

In Section 1 we introduce the resultant ideal ℜ and describe its relations to the elimination ideal 𝔗_{0}. The main results and some applications concerning the degree of elimination are given in Section 2. Section 3 supplies the necessary tools from duality theory.

1. The resultant ideal

Let A be an integrally closed noetherian domain. We consider a regular sequence F_{0}, …, F_{n} of homogeneous polynomials of positive degrees δ_{0}, …, δ_{n} in the graded polynomial algebra

A[T] = A[T_{0}, …, T_{n}] , deg T_{i} = γ_{i} ∈ N^{∗}, i = 0, …, n ,

and the graded A–algebra

C = A[T]/(F_{0}, …, F_{n}) .

The zero–ideal in C has the decomposition

0 = 𝔔 ∩ 𝔗

where 𝔔 is the irrelevant primary component of 0 and 𝔗 the ideal of the Trägheitsformen. 𝔔 is nothing else but the A–torsion of C, furthermore 𝔗 = 0 : 𝔔 and 𝔔 = 0 : 𝔗. The graded residue class algebra

C̄ := C/𝔗

has the same projective spectrum as C. The homogeneous parts of degree m in A[T], C, 𝔔, 𝔗, C̄, … will be indicated by attaching m as an index. Of special importance is the degree

σ := (δ_{0} + ⋯ + δ_{n}) − (γ_{0} + ⋯ + γ_{n}) .

The so–called elimination ideal

𝔗_{0} = 𝔗_{0}(F_{0}, …, F_{n}) ⊆ A

of F_{0}, …, F_{n} describes the (closed) image of Proj C = Proj C̄ in Spec A.

In every degree m, there is a canonical isomorphism

𝔗_{m} ⟶^{∼} C_{σ−m}^{∗} = Hom_{A}(C_{σ−m}, A)

1) χ(M) denotes the divisor associated with a finite A–torsion–module M. given by t_{m} ↦ (c_{σ−m} ↦ t_{m}c_{σ−m}/Δ) where t_{m} ∈ 𝔗_{m}, c_{σ−m} ∈ C_{σ−m} and Δ ∈ C_{σ} denotes the residue class in C of the determinant det (A_{ij}), the polynomials A_{ij} ∈ A[T] given by any representations F_{j} = ∑_{i=0}^{n} A_{ij}T_{i}, j = 0, …, n. The element Δ is an A–base of 𝔗_{σ}, and C̄_{σ} = C_{σ}/𝔗_{σ} is a finite A–torsion–module. 𝔗_{0} ≅ C_{σ}^{∗} is a reflexive A–module of rank 1 and therefore a divisorial ideal in A, moreover 𝔗_{0} = Ann_{A} C̄_{σ}.

For details of these constructions and for proofs we refer to [8] or [9]. Furthermore we shall need the following result, cf. Lemma 4.1 in [8] or Lemma 8.2 in [9] :

1.1 div 𝔣_{0}(C̄_{σ}) = χ( C̄_{σ}) = χ(𝔔_{σ}) + χ(A/𝔗_{0}) = div 𝔣_{1}(C_{σ}) + div 𝔗_{0}.

In these formulas 𝔣_{i}(M) denotes the i–th Fitting ideal of a finite A–module M, div 𝔞 = div 𝔞^{∗∗} the divisor defined by an ideal 𝔞 ≠ 0 in A and

χ(M) = div 𝔣_{0}(M) = ∑_{codim 𝔭=1}(length M_{𝔭}) · 𝔭

the divisor associated with a finite A–torsion–module M. Obviously, χ(A/𝔞) = div 𝔞 for every ideal 𝔞 ≠ 0.

Because F_{0}, …, F_{n} is a regular sequence, C_{σ} and therefore C̄_{σ}, too, has a finite free A–resolution (e.g. given by the Koszul complex). The mere existence of such a resolution implies that χ( C̄_{σ}) is a principal divisor, cf. [1], §4, sect.7. The principal ideal in A corresponding to this principal divisor is by definition the resultant ideal

ℜ = ℜ(F_{0}, …, F_{n}) = ℜ_{A}(F_{0}, …, F_{n})

of the polynomials F_{0}, …, F_{n} ∈ A[T]. Because of 𝔗_{0} = Ann_{A} C̄_{σ} and div ℜ = div 𝔣_{0}( C̄_{σ}), the ideals 𝔗_{0} and ℜ describe the same variety in Spec A. By 1.1 the divisorial ideals 𝔗_{0} and ℜ coincide if and only if the divisor χ(𝔔_{σ}) of the torsion submodule 𝔔_{σ} of C_{σ} vanishes, i.e. the support of 𝔔_{σ} has codimension ≥ 2 in Spec A.

The resultant ideal ℜ has better functorial properties than the elimination ideal 𝔗_{0}; in addition, there is a canonical generator for ℜ (uniquely determined up to sign), the so–called resultant

R = R (F_{0}, …, F_{n}) = R_{A}(F_{0}, …, F_{n})

of F_{0}, …, F_{n}, which is compatible with base change A → A′ (as long as A′ is an integrally closed domain and F_{0}, …, F_{n} form a regular sequence in A′[T]). The divisorial ideal 𝔗_{0}, which is completely determined by its divisor

div 𝔗_{0} = div ℜ − div 𝔣_{1}(C_{σ}) ,

does not commute with base changes, even though ℜ (and div ℜ) and 𝔣_{1}(C_{σ}) do commute. In addition, 𝔗_{0} need not be a principal ideal.

To give examples, we start from the generic polynomials

F_{0} := XT_{0}^{3} + Y T_{1}^{2} , F_{1} := UT_{0}^{6} + V T_{0}^{3}T_{1}^{2} + W T_{1}^{4}

over A := Z[X, Y, U, V, W], where deg T_{0} := 2, deg T_{1} := 3. Then σ = (6+12)−(2+3) = 13 and

𝔣_{0}( C̄_{σ}) = (X(XW − Y V) + Y^{2}U) , 𝔣_{1}(C_{σ}) = (X, Y) .

Therefore R = R (F_{0}, F_{1}) = X(XW − Y V) + Y^{2}U and ℜ(F_{0}, F_{1}) = 𝔗_{0}(F_{0}, F_{1}). Note that R is a prime element and div 𝔣_{1}(C_{σ}) = 0.

Now we specialize. First we take A := k[x, y, z] = k[X, Y, Z]/(Y^{2} − XZ), k any field, with X ↦ x, Y ↦ y, U ↦ 1, V ↦ y, W ↦ z. Then the polynomials

F_{0} = xT_{0}^{3} + yT_{1}^{2}, F_{1} = T_{0}^{6} + yT_{0}^{3}T_{1}^{2} + zT_{1}^{4}

form a regular sequence in A[T] and

𝔣_{0}( C̄_{σ}) = (y^{2}), 𝔣_{1}(C_{σ}) = (x, y) .

Therefore R = R (F_{0}, F_{1}) = y^{2} and ℜ = ℜ(F_{0}, F_{1}) = (y^{2}), 𝔗_{0} = 𝔗_{0}(F_{0}, F_{1}) = 𝔭 ∩ (z) = 𝔭z = (xz, yz) because of div 𝔗_{0} = div ℜ − div 𝔣_{1}(C_{σ}) = 2𝔭 + 2𝔮 − 𝔭 = 𝔭 + 2𝔮 = 𝔭 + div z with 𝔭 := (x, y), 𝔮 := (z, y). div 𝔗_{0} is not a principal divisor because the ideal 𝔭 is not principal.

Secondly we specialize to A := Z[\sqrt{−5}], X ↦ 2, Y ↦ 1 + \sqrt{−5}, U ↦ 1, V ↦ 1 − \sqrt{−5}, W ↦ 3. Then again the polynomials

F_{0} = 2T_{0}^{3} + (1 + \sqrt{−5})T_{1}^{2}, F_{1} = T_{0}^{6} + (1 − \sqrt{−5})T_{0}^{3}T_{1}^{2} + 3T_{1}^{4}

form a regular sequence in A[T] and

𝔣_{0}( C̄_{σ}) = (2(−2 + \sqrt{−5})), 𝔣_{1}(C_{σ}) = (2, 1 + \sqrt{−5} ) .

Therefore R = 2(−2 + \sqrt{−5}), ℜ = (2(−2 + \sqrt{−5})), 𝔗_{0} = 𝔭 · (−2 + \sqrt{−5}) because of div 𝔗_{0} = div ℜ − div 𝔣_{1}(C_{σ}) = 2𝔭 + 2𝔮 − 𝔭 = 𝔭 + 2𝔮 = 𝔭 + div (−2 + \sqrt{−5}) with 𝔭 := (2, 1 + \sqrt{−5} ), 𝔮 := (3, 1 + \sqrt{−5} ). Again div 𝔗_{0} is not a principal divisor because the ideal 𝔭 is not principal.

Finally, we specialize to A := k[x, y, v, w] = k[X, Y, V, W]/(XW − Y V), k any field (or any integrally closed domain), with X ↦ x, Y ↦ y, U ↦ 1, V ↦ v, W ↦ w. In this case div ℜ = 2 div y = 2𝔭 + 2𝔮, div 𝔣_{1}(C_{σ}) = 𝔭 and div 𝔗_{0} = 𝔭 + 2𝔮, i.e. 𝔗_{0} = 𝔮y, where 𝔭 := (x, y), 𝔮 := (w, y). 𝔮 and therefore 𝔗_{0} is not a principal ideal. Its divisor class has infinite order in the class group Cl(A).

In the next section we will put forward some properties of the resultant ideal ℜ(F_{0}, … …, F_{n}) which are related to the projective variety Proj A[T]/(F_{0}, …, F_{n}) described by the polynomials F_{0}, …, F_{n}.

2. Geometric interpretation

As in Section 1 let A denote an integrally closed noetherian domain and F_{0}, …, F_{n} a regular sequence of homogeneous polynomials in the graded polynomial algebra A[T] = A[T_{0}, …, T_{n}] with weighted indeterminates T_{0}, …, T_{n}. Without further notice we take over all the corresponding notations and constructions from Section 1.

The global sections of the projective spectrum of C = A[T]/(F_{0}, …, F_{n}) form a finite A–algebra

B := Γ(Proj C , O_{Proj C})

which is annihilated by the elimination ideal 𝔗_{0} = 𝔗_{0}(F_{0}, …, F_{n}) and thus even an algebra over

Ā := A/𝔗_{0} = C̄_{0} .

The main result is the following theorem:

2.1 Theorem The resultant ℜ = ℜ(F_{0}, …, F_{n}) and the A–algebra B of global sections of O_{Proj C} define the same divisor in Div A:

χ(B) = div ℜ .

Explicitly : For every prime ideal 𝔭 of codimension 1 in A

length B_{𝔭} = length A_{𝔭}/ℜ_{𝔭} = length ( C̄_{σ})_{𝔭} .

Proof. Let 𝔭 be a prime ideal of codimension 1 in A. After localizing by 𝔭 we may assume that A is a discrete valuation ring with maximal ideal 𝔭. In the case C̄ = 0, i.e. 𝔗 = C, we have Proj C = ∅ and ℜ = A.

Now we assume C̄ ≠ 0. Then C̄ is (like C) a graded Macaulay ring of dimension 1 and B is the homogeneous part of degree zero of the total graded quotient ring of C̄ = C/𝔗. By N′ we denote the non–zero submonoid of N of all r such that C̄_{r} contains a non–zero– divisor of C̄.

Trivially, length B ≥ length C̄_{r} for every r ∈ N′ because of C̄_{r}/x_{r} ⊆ B for every non– zero–divisor x_{r} ∈ C̄_{r}. The set of all s ∈ N′ such that length B = length C̄_{r} is a non–empty subset N′′ of N′ with N′ + N′′ ⊆ N′′. Therefore there is a t ∈ N′ with r ∈ N′′ for all r ∈ N′, r ≥ t. Certainly, there is some r ∈ N′′ with r > σ. But for all r ∈ N′ with r > σ

length C̄_{r} = length C̄_{σ} = length A/ℜ ,

which we will prove in the next section, cf. Corollary 3.2.

The last argument in the proof yields the equality length B = length C̄_{r} for all r ∈ N′, r > σ. As is shown in Corollary 3.2, this equality holds even for all r > σ with C_{r} ⊈ 𝔓 for all 𝔓 ∈ Ass_{C} C̄.

In general we set

N_{C} := {r ∈ N : C_{r} ⊈ 𝔓 for all 𝔓 ∈ Ass_{C} C̄ with codim_{A}𝔓_{0} = 1} .

Note that C_{r} = C̄_{r} = 𝔔_{r} for r > σ because of 𝔗_{r} ≅ C_{σ−r}^{∗} = 0 for r > σ. Thus:

2.2 Theorem For all r ∈ N_{C} with r > σ

div ℜ = χ(C_{r}) = div 𝔣_{0}(C_{r}) .

We note that in spite of the (defining) equation div ℜ = χ( C̄_{σ}) the degree σ does not need to be an element of N_{C}. In the simple example F_{0} = T_{0}^{2}, F_{1} = πT_{1} ∈ A[T_{0}, T_{1}] over the discrete valuation ring A with maximal ideal Aπ, deg T_{0} := 2, deg T_{1} := 3, one has σ = 2, N_{C} = 3N. In the classical case γ_{0} = ⋯ = γ_{n} = 1 the monoid N_{C} always coincides with N.

The formula in 2.2 generalizes the well known old computations of resultants in the line of the so–called Macaulay determinants, a very special case (n = 1, γ_{0} = γ_{1} = 1, r = σ+1) being covered by the Sylvester determinant.

We will finish this section with some applications of Theorem 2.1.

Assume the elimination ideal 𝔗_{0} to be prime. Then div ℜ is a multiple of div 𝔗_{0} (note that this implies that the divisor class of 𝔗_{0} in Cl A is necessarily of finite order). Since 𝔗_{0} C̄_{σ} = 0 the length of ( C̄_{σ})_{𝔗_{0}} is simply the rank of C̄_{σ} over the integral domain Ā = A/𝔗_{0}. For the same reasons the length of B_{𝔗_{0}} is the rank [B : Ā] of B over Ā which is called the degree of Proj C over Spec Ā or the degree of elimination. By 2.2 we have:

2.3 Theorem If the elimination ideal 𝔗_{0} is prime, then

div ℜ = (rankĀ C̄_{σ}) · 𝔗_{0} = [B : Ā] · 𝔗_{0} ,

i.e. the degree of elimination is the multiplicity of the resultant divisor div ℜ with respect to the elimination divisor div 𝔗_{0}.

Let us now consider the case that even the ideal 𝔗 of inertia forms is prime, that means that the projective variety defined by the polynomials F_{0}, …, F_{n} is irreducible. Then B is an integral domain and a finite extension of Ā ⊆ B.

2.4 Theorem If 𝔗 is a prime ideal the following conditions are equivalent :

(1) ℜ = 𝔗_{0}, i.e. rankĀ C̄_{σ} = 1.

(2) χ(𝔔_{σ}) = 0.

(3) 𝔔_{σ} = 0, i.e. C_{σ} is torsionfree over A.

(4) Ā and B have the same quotient field, i.e. the projection Proj C = Proj C̄ → Spec Ā is birational.

Proof. (1) and (2) are equivalent by 1.1, and (1) and (4) are equivalent by 2.3. Only the implication (2) ⇒ (3) has yet to be proved. C̄_{σ} = C_{σ}/𝔗_{σ} as an Ā–submodule of the integral domain C̄ = C/𝔗 is torsionfree over Ā. The canonical inclusion 𝔔_{σ} → C̄_{σ} shows that 𝔔_{σ} is also torsionfree as an Ā–module. Therefore χ(𝔔_{σ}) = 0, which means rankĀ 𝔔_{σ} = 0, implies 𝔔_{σ} = 0.

Next we consider a generic situation. For j = 0, …, n let N_{j} be a subset of N_{δ_{j}}(γ) := {ν ∈ N^{n+1} : γ_{0}ν_{0} + ⋯ + γ_{n}ν_{n} = δ_{j}} and F_{j} = ∑_{ν∈N_{j}} U_{jν}T^{ν} the generic polynomial with indeterminate coefficients U_{jν}, ν ∈ N_{j}. We call F_{0}, …, F_{n} strictly admissible if they form a regular sequence in A[T], A := Z[U_{jν}]_{j,ν}, and if, in addition, their ideal 𝔗 of the Trägheitsformen is prime. There is a combinatorial algorithm to decide whether F_{0}, …, F_{n} are strictly admissible, cf. [8], Lemma 2.6, or the full account in [9], Section 4. In this case the degree of elimination is

d := [B: Ā] = [Γ : (Γ_{0} + ⋯ + Γ_{n})] ,

where Γ is the group of syzygies of the weights γ_{0}, …, γ_{n} of the indeterminantes T_{0}, …, T_{n} and Γ_{j} for j = 0, …, n its subgroup generated by the differences ν − μ, ν, μ ∈ N_{j}, cf. [8], Theorem 5.2. By 2.3 we have:

2.5 Theorem For strictly admissible generic polynomials F_{j} = ∑_{ν∈N_{j}} U_{jν}T^{ν}, j = 0, …, n,

ℜ(F_{0}, …, F_{n}) = (𝔗_{0}(F_{0}, …, F_{n}))^{d}, d = [Γ : (Γ_{0} + ⋯ + Γ_{n})] .

As a last application we generalize a result of Chardin about the degree of the fibres of Proj C over Spec A, cf. [3], Proposition 3.

As before let A be an integrally closed domain. For the moment, the homogeneous polynomials F_{0}, …, F_{n} ∈ A[T_{0}, …, T_{n}] are not supposed to form a regular sequence. Nevertheless the resultant R = R_{A}(F_{0}, …, F_{n}) is defined as the element one gets by specializing the resultant of the generic homogeneous polynomials of the same degrees. (If these generic polynomials do not form a regular sequence — which can happen when some weights are > 1 — then R = 0 by definition.) As explained in Section 1, if F_{0}, …, F_{n} form a regular sequence in A[T] this resultant generates the resultant ideal ℜ = ℜ(F_{0}, …, F_{n}).

For every prime ideal 𝔭 ⊆ A of codimension 1 we denote by v_{𝔭}(R) the multiplicity of 𝔭 in the principal ideal AR and by X_{𝔭} = Proj (C_{𝔭}/𝔭C_{𝔭}) the fibre of X := Proj C, C = A[T]/(F_{0}, …, F_{n}), over 𝔭 ∈ Spec A. If this fibre is finite,

deg X_{𝔭} = dim_{κ(𝔭)}Γ(X_{𝔭}, O_{X_{𝔭}})

denotes its degree. As above, B = Γ(X, O_{X}), which is finite over A in every case. With these notations we have:

2.6 Theorem If for the prime ideal 𝔭 ⊆ A of codimension 1 the fibre X_{𝔭} is finite then

deg X_{𝔭} ≤ v_{𝔭}(R) .

Equality holds if and only if 𝔭B_{𝔭} = 0.

Proof. We may assume right away that A is a discrete valuation ring with maximal ideal 𝔭 = Aπ. The assumption on the fibre is equivalent with the following condition: codim (π, F_{0}, …, F_{n}) ≥ n + 1.

First we consider the case that F_{0}, …, F_{n} is not a regular sequence. Then there is a homogeneous prime ideal 𝔓 of codimension ≤ n containing F_{0}, …, F_{n}. Therefore π ∉ 𝔓 and F_{0}, …, F_{n} generate an ideal of codimension ≤ n in K[T] where K is the quotient field of A. Then R = 0, K ⊗_{A} B ≠ 0 and 𝔭B ≠ 0, which settles the theorem in this case (because v_{𝔭}(0) = ∞).

Now let us assume that F_{0}, …, F_{n} is a regular sequence. Then X ≅ Spec B is affine and finite over Spec A which implies Γ(X_{𝔭}, O_{X_{𝔭}}) = B/𝔭B. With Theorem 2.1 one gets deg X_{𝔭} = length B/𝔭B ≤ length B = v_{𝔭}(R), and equality holds if and only if 𝔭B = 0.

3. Supplements in duality

In this section the base ring A is a Dedekind domain. Again F_{0}, …, F_{n} is a regular sequence of homogeneous polynomials of positive degrees in A[T] = A[T_{0}, …, T_{n}] and C = A[T]/(F_{0}, …, F_{n}). We adopt freely notations and constructions from Section 1. Among other things we prove a result we needed for Theorems 2.1 and 2.2.

3.1 Lemma Assume A to be a Dedekind domain. Let C̄ = C/𝔗, Ā = A/𝔗_{0} = C̄_{0} and ωĀ := 𝔗_{0}^{−1}/A. For every homogeneous element z ∈ C of positive degree r, which is a non–zero–divisor in C̄ (which is true if and only if C/zC is a finite A–algebra), there is an isomorphism

𝔔/z𝔔 ≅ HomĀ (C̄/zC̄, ωĀ)(−σ − r)

of graded C–modules.

𝔔 denotes as in Section 1 the irrelevant component of the zero–ideal in C. Recall that for any graded module V over a graded ring the shifted module V (m), m ∈ Z, is the graded module having as homogeneous part of degree i ∈ Z the subgroup V (m)_{i} = V_{m+i}.

Proof of 3.1. Let A[Z] denote the graded polynomial algebra over A in the single indeterminate Z of degree r. Then C and C̄ are finite graded A[Z]–algebras by Z ↦ z and

HomĀ (C̄/zC̄ , ωĀ) = Hom_{Ā[Z]}(C̄/zC̄ , ωĀ) = Hom_{Ā[Z]}(C̄ , ωĀ) .

Corresponding to the exact sequence 0 → ωĀ[Z] ⟶^{Z} ωĀ[Z] → ωĀ → 0 there is the exact sequence

0 → Hom_{Ā[Z]}(C̄ , ωĀ [Z]) ⟶^{Z} Hom_{Ā[Z]}(C̄ , ωĀ [Z]) → Hom_{Ā[Z]}(C̄ , ωĀ) → Ext_{Ā[Z]}^{1}(C̄)

where Ext_{Ā[Z]}^{1}(C̄) := Ext_{Ā[Z]}^{1}(C̄ , ωĀ[Z]) = 0, however, which is proved as follows: From the exact sequence 0 → C̄ ⟶^{z} C̄ → C̄/zC̄ → 0 we get an exact sequence

Ext_{Ā[Z]}^{1}(C̄) ⟶^{z} Ext_{Ā[Z]}^{1}(C̄) → Ext_{Ā[Z]}^{2}(C̄/zC̄ , ωĀ[Z]) .

The Ext^{2}– group is zero because ωĀ[Z] is an Ā[Z]–module of injective dimension 1. Hence multiplication by z on the finite graded Ā[Z]–module Ext_{Ā[Z]}^{1}(C̄) is surjective, which implies that the module vanishes.

It suffices to show now that

Hom_{Ā[Z]}(C̄ , ωĀ[Z]) = Hom_{A[Z]}(C̄ , ωĀ[Z]) ≅ 𝔔(σ + r)

as graded C–modules.

First we prove that the homomorphism C → C̄ induces an isomorphism

Hom_{A[Z]}(C̄ , ωĀ[Z]) = Hom_{A[Z]}(C, ωĀ[Z])

by showing that Hom_{A[Z]}(𝔗, ωĀ[Z]) = 0. For this we may assume that A is a discrete valuation ring with maximal ideal Aπ. Then we consider the exact sequence

Hom_{A[Z]}(𝔗, A[Z]) → Hom_{A[Z]}(𝔗, Ā[Z]) → Ext_{A[Z]}^{1}(𝔗, A[Z])⟶^{π^{s}}Ext_{A[Z]}^{1}(𝔗, A[Z])

belonging to 0 → A[Z] ⟶^{π^{s}} A[Z] → Ā[Z] → 0, where π^{s} is a generator of 𝔗_{0}. For trivial reasons, Hom_{A[Z]}(𝔗, A[Z]) = 0. Therefore it remains to show that Ext_{A[Z]}^{1}(𝔗, A[Z]) is a torsionsfree A–module. But this follows from an isomorphism

Ext_{A[Z]}^{1}(𝔗, A[Z]) = Hom_{A}(𝔗, A)

of modules. For the construction of such an isomorphism see [7], Lemma 4.1; one has to use that 𝔗 is a finite projective A–module.

Next we prove that Hom_{A[Z]}(C, ωĀ[Z]) is isomorphic to the 𝔗_{0}–torsion of the module Ext_{A[Z]}^{1}(C, A[Z]). For this we consider the exact sequence

Hom_{A[Z]}(C, 𝔗_{0}^{−1}[Z]) → Hom_{A[Z]}(C, ωĀ[Z]) → Ext_{A[Z]}^{1}(C, A[Z]) → Ext_{A[Z]}^{1}(C, 𝔗_{0}^{−1}[Z])

belonging to 0 → A[Z] → 𝔗_{0}^{−1}[Z] → ωĀ[Z] → 0. Again Hom_{A[Z]}(C, 𝔗_{0}^{−1}[Z]) = 0 for trivial reasons. Thus Hom_{A[Z]}(C, ωĀ[Z]) is the kernel of

Ext_{A[Z]}^{1}(C, A[Z]) → Ext_{A[Z]}^{1}(C, 𝔗_{0}^{−1}[Z]) ,

which is the 𝔗_{0}–torsion of Ext_{A[Z]}^{1}(C, A[Z]) as can be verified locally with respect to A.

Because 𝔔 is the 𝔗_{0}–torsion of C, it finally suffices to show that there is an isomorphism

Ext_{A[Z]}^{1}(C, A[Z]) ≅ C(σ + r)

of graded C–modules.

C has the representation

C = C[Z]/(Z − z) = A[Z, T]/(F_{0}, …, F_{n}, Z − z)

where F_{0}, …, F_{n}, Z − z is a regular sequence in A[Z, T]. Furthermore, C has homological dimension 1 over A[Z] because C is finite over A[Z] and is a Macaulay ring of dimension 1. Thus there is a canonical isomorphism

Ext_{A[Z]}^{1}(C, A[Z]) ≅ Ext_{A[Z,T]}^{n+2}(C, A[Z, T])(−γ_{0} − ⋯ − γ_{n})

(γ_{i} being the degree of T_{i}) of graded C–modules, cf. again [7], Lemma 4.1. From the Koszul resolution of C over A[Z, T] one gets the isomorphism

Ext_{A[Z,T]}^{n+2}(C, A[Z, T]) ≅ C(δ_{0} + ⋯ + δ_{n} + r)

(δ_{i} being the degree of F_{i}, r the degree of Z − z) of graded C–modules. The composition of both isomorphisms yields an isomorphism of Ext_{A[Z]}^{1}(C, A[Z]) onto C(σ + r) we were looking for, because −γ_{0} − ⋯ − γ_{n} + δ_{0} + ⋯ + δ_{n} + r is simply σ + r.

Let us recall the definition of the monoid N_{C} (given before Theorem 2.2) which for a Dedekind domain reads as follows:

N_{C} = {r ∈ N : C_{r} ⊈ 𝔓 for all 𝔓 ∈ Ass_{C} C̄} .

3.2 Corollary Assume A to be a Dedekind domain. For every r ∈ N_{C} and s ∈ Z

χ(𝔔_{r+s}) − χ(𝔔_{s}) = χ( C̄_{σ−s}) − χ( C̄_{σ−s−r}) .

In particular,

χ( C̄_{σ}) = χ(𝔔_{r}) (= χ(C_{r}) = χ( C̄_{r}))

for every r ∈ N_{C} with r > σ.

Proof. We may assume that A is a discrete valuation ring and, using a suitable flat extension which preserves lengths, that its residue field is infinite. Then N_{C} is the monoid of all r ∈ N such that C_{r} contains an element z which is a non–zero–divisor in C̄. Now the corollary follows from 3.1 simply by the fact that dualizing by ωĀ ≅ Ā preserves lengths.

As another simple consequence of 3.1 we get:

3.3 Corollary Assume A to be a Dedekind domain : Let z ∈ C_{r} be a homogeneous element of positive degree r, which is a non–zero–divisor in C. Then (C/zC)_{m} = 0 for every m ≥ σ + r.

There is a canonical way to formulate the duality decribed in Lemma 3.1 (using the mere existence of such a duality): We do this by constructing a canonical Ā–bilinear homogeneous form

𝔔/z𝔔 × (C̄/zC̄)(σ + r) ⟶ ωĀ

which defines a perfect duality.

The isomorphism 𝔗_{0} ⟶^{∼} C_{σ}^{∗} = (C_{σ}/𝔔_{σ})^{∗} induces canonical isomorphisms C_{σ}/𝔔_{σ} =

(C_{σ}/𝔔_{σ})^{∗∗}⟶^{∼} 𝔗_{0}^{∗} = 𝔗_{0}^{−1} and C̄_{σ}/𝔔_{σ} = C_{σ}/(𝔔_{σ} + AΔ) ⟶^{∼} 𝔗_{0}^{−1}/A = ωĀ. So we may

replace ωĀ by C̄_{σ}/𝔔_{σ}. The value of the bilinear form on ([q], [c]), q ∈ 𝔔_{m}, c ∈ C_{σ+r−m}, is

⟨[q], [c]⟩ := [\frac{qc}{z}] ∈ C̄_{σ}/𝔔_{σ} .

Note that qc ∈ C_{σ+r} can be written in the form qc = zu with u ∈ C_{σ} by 3.3. It suffices to show that this form is non–degenerate over Ā because the modules involved have finite lengths. But non–degeneracy follows from the fact that there is an isomorphism 𝔔/z𝔔 ≅ HomĀ( C̄/zC̄, C̄_{σ}/𝔔_{σ})(−σ − r) of graded C–modules, cf. 3.1.

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Received: July 15, 1995