Kähler Differentials for Points in IP^{n}

by Gabriel de Dominicis (Genova) and Martin Kreuzer (Regensburg)

Keywords. Kähler differentials, 0-dimensional scheme, Hilbert function, torsion submodule, almost complete intersection

Abstract. If R is the homogeneous coordinate ring of a reduced 0-dimensional subscheme of ℙ^{n}, we study the module of Kähler differentials Ω_{R/K}^{1}. Explicit presentations of it and its torsion submodule are used to describe the module structure. From this we derive many properties of the Hilbert function of Ω_{R/K}^{1}. Finally, this function is computed in a number of special cases, in particular for reduced 0-dimensional almost complete intersections.

Introduction.

Given a set of points 𝕏 ⊆ ℙ^{n} with homogeneous vanishing ideal I_{𝕏} ⊆ A = K[X_{0}, …, X_{n}] and homogeneous coordinate ring R = A/I_{𝕏}, one oftentimes considers the Hilbert function H_{𝕏}(i) = dim_{K}(R_{i}) or the graded Betti numbers of the A-module R. Many geometric properties of the embedding 𝕏 ⊆ ℙ^{n} are encoded in those invariants, or in similar ones derived from them (cf. [GKR], [GM], [Kr1], [O]). Nevertheless, as one expects, not all geometric informations about 𝕏 ⊆ ℙ^{n} can be read off those invariants or I_{𝕏} easily (Ex. 4.2). In this paper, we suggest that it may also be worthwhile to examine the structure and the Hilbert function of the module of Kähler differentials Ω_{R/K}^{1} = J/J^{2}, where J = ker(R ⊗_{K} R ⟶ R). As Prop. 3.9 shows, the Hilbert function of this finitely generated graded R-module is intimately connected to the ideal I_{𝕏}^{(2)} generated by those homogeneous polynomials which together with all of their partial derivatives vanish at the points of 𝕏. Thus the module Ω_{R/K}^{1} presents a way of bringing tools from analysis (derivations, differentials forms) into the study of configurations of points in ℙ^{n}.

Now we describe the content of the paper in more detail. Section 1 outlines the most important aspects of the R-module structure of Ω_{R/K}^{1}. We provide a presentation, in terms of generators and relations (Prop. 1.3), the relations of Ω_{R/K}^{1} with the modules Ω_{R/K[x_{0}]}^{1} and Ω_{(R/(x_{0}))/K}^{1} (assuming 𝕏 ∩ Z(X_{0}) = ∅, cf. Prop. 1.4, Cor. 1.7), and some facts about the module structure of its first syzygy module G = ⟨(\frac{∂F}{∂x_{0}} , …, \frac{∂F}{∂x_{n}})⟩_{F ∈I_{𝕏}} ⊆ R^{n+1} (Prop. 1.8). Then we show in section 2 that most nontrivial aspects of the module structure of Ω_{R/K}^{1} are captured by its torsion submodule T_{R/K} (Prop. 2.3). We give several simple descriptions of T_{R/K} (Prop. 2.2, Cor. 2.4) and represent it explicitly as a quotient module of the syzygy module Syz_{R}^{1}(x_{0}, …, x_{n}) (Prop. 2.6). As a consequence, we see that under a mild assumption on char (K), the module T_{R/K} is generated by the elements of the form x_{i}dx_{j} − x_{j}dx_{i} with 0 ≤ i < j ≤ n (Cor. 2.7).

Section 3 collects our insights into the behaviour of the Hilbert function H_{Ω} of Ω_{R/K}^{1}. Roughly speaking, this function increases (not necessarily monotonically, cf. Exs. 3.3, 3.4) from H_{Ω}(1) = H_{𝕏}(1) to a maximum near σ_{𝕏} + 1 = max{i ∈ ℤ | H_{𝕏}(i) < #𝕏} + 1, then decreases strictly until it finally stabilizes at the number of points of 𝕏. The least integer i such that H_{Ω}(j) equals #𝕏 for all j ≥ i is called the regularity ϱ_{𝕏} of Ω_{R/K}^{1}. We present the relations of H_{Ω} with the Hilbert functions of Ω_{R/K[x_{0}]}^{1} and Ω_{(R/(x_{0}))/K}^{1} (Prop. 3.1), precise versions of the above qualitative remarks (Prop. 3.2), and a sharp upper bound for ϱ_{𝕏} (Prop. 3.5, Ex. 3.3). We also show that Ω_{R/K}^{1} always has torsion elements of degree 2, and in fact in every degree in which we could reasonably expect such elements (Prop. 3.6), thereby strengthening a result of Scheja (cf. [S], 9.8). Furthermore, we characterize the case of maximal regularity ϱ_{𝕏} = 2σ_{𝕏} + 3 (Prop. 3.8) and describe the connection between H_{Ω} and the Hilbert function of I_{𝕏}^{(2)} mentioned above (Prop. 3.9). The squares and symbolic squares of ideals of points have received some attention lately. Although we did not find an obvious connection between the structure of Ω_{R/K}^{1} and of I_{𝕏}^{(2)}, Prop. 3.9 seems to indicate that the module of Kähler differentials could also be used as a tool in the theory of sets of double points Z(I_{𝕏}^{(2)}).

The last two sections are concerned with calculations of Ω_{R/K}^{1} in special cases. In section 4, we deal with sets of points in ℙ^{1} (Prop. 4.1), with complete intersections (Prop. 4.3), and with sets of generic points (Prop. 4.4). The case of reduced 0-dimensional almost complete intersections is treated in section 5. Using a graded version of the Primbasissatz (Prop. 5.1), we construct an explicit presentation of the conormal module I_{𝕏}/I_{𝕏}^{2} (Prop. 5.2) as in [M], and a presentation of the torsion submodule I_{𝕏}^{(2)}/I_{𝕏}^{2} of that conormal module (Prop. 5.3) as in [Ku2]. The calculation of H_{Ω} is reduced to finding the Hilbert function of the kernel D of the Jacobian map. Then D is interpreted as the inverse ideal of the Kähler different of R/K[x_{0}]. In the case of a special almost complete intersection, this Kähler different can be described further (Prop. 5.4), but unfortunately not every reduced 0-dimensional almost complete intersection is special (Ex. 5.5).

Acknowledgements. The authors cordially thank the Departments of Mathematics of Universitá di Genova, Universität Regensburg, and Queen’s University (Kingston) for their hospitality during various stages of the preparation of this paper. They are very grateful to A.V. Geramita and E. Kunz for helpful conversations and encouragements. The calculations underlying most examples and leading to the discovery of several of the propositions were carried out using the computer algebra system CoCoA (cf. [CNR]). A package containing CoCoA-functions used to compute many of the special objects studied in this paper is available from the authors.

1. The module structure of Ω_{R/K}^{1}

First of all we fix the notation which will be used throughout the paper. We work over an algebraically closed field K. By ℙ^{n} we denote the n-dimensional projective space over K. The homogeneous coordinate ring of ℙ^{n} is A = K[X_{0}, …, X_{n}] equipped with its standard grading deg(X_{0}) = ⋯ = deg(X_{n}) = 1. Let 𝕏 be a reduced 0-dimensional subscheme of ℙ^{n}. In other words, let 𝕏 = {P_{1}, …, P_{s}} ⊆ ℙ^{n} be a set of points together with the induced reduced scheme structure. Here s = #𝕏 is the degree of 𝕏 as a projective scheme. If I_{𝕏} denotes the homogeneous ideal of 𝕏 in A, then the graded ring R = A/I_{𝕏} is called the homogeneous coordinate ring of 𝕏. The ring R = ⊕_{i≥0}R_{i} is a standard graded K-algebra, i.e. R_{0} = K and R = K[R_{1}] with dim_{K} R_{1} < ∞. Its homogeneous maximal ideal is denoted by 𝔪 = ⊕_{i>0}R_{i}.

The Hilbert function of 𝕏 (or of R) is defined by H_{𝕏}(i) = H_{R}(i) = dim_{K} R_{i} for all i ∈ ℤ. Analogously, for a finitiely generated graded R-module M, we let H_{M}(i) = dim_{K} M_{i} for all i ∈ ℤ. It is well-known (see e.g. [GKR]) that the growth behaviour of H_{𝕏} can be described as follows. There are intergers α_{𝕏} ≥ 1 and σ_{𝕏} ≥ −1 such that H_{𝕏}(i) = dim_{K} A_{i} = \binom{n+i}{n} for i < α_{𝕏}, such that 1 = H_{𝕏}(0) < H_{𝕏}(1) < ⋯ < H_{𝕏}(σ_{𝕏}) < s, and such that H_{𝕏}(i) = s for i > σ_{𝕏}.

Since K is infinite, we can choose the coordinate system {X_{0}, …, X_{n}} such that 𝕏 ∩ Z(X_{0}) = ∅. Then we can write P_{i} = (1 : p_{i1} : …: p_{in}) with p_{i1}, …, p_{in} ∈ K for i = 1, …, s. Given a polynomial F ∈ A, we shall use the corresponding lower case letter to denote its image in R, and we set f(P_{i}) = F (1, p_{i1}, …, p_{in}) for i = 1, …, s. The ring structure of R is explained by the following results (cf. [GKR], [Kr1]).

1.1. Proposition. a) The image x_{0} of X_{0} in R is not a zerodivisor.

b) The ring R is a 1-dimensional Cohen-Macaulay ring and the monomorphism K[x_{0}] ⟶ R defines a Noether normalization.

c) The graded K[x_{0}]-module R is graded-free of rank s, i.e. it has a homogeneous basis of length s.

d) There is a unique K-basis {f_{1}, …, f_{s}} of R_{σ_{𝕏}+1} such that f_{i}(P_{j}) = δ_{ij} for i, j = 1, …, s. The elements f_{1}, …, f_{s} are called the (normalized) separators of 𝕏.

e) For each i ≥ σ_{𝕏} + 1, the set {x_{0}^{i−σ_{𝕏}−1}f_{1}, …, x_{0}^{i−σ_{𝕏}−1}f_{s}} is a K-basis of R_{i}.

f) If r ∈ R_{i} and c_{1}x_{0}^{j}f_{1} + ⋯ + c_{s}x_{0}^{j}f_{s} ∈ R_{j+σ_{𝕏}+1} for some i, j ≥ 0 and c_{1}, …, c_{s} ∈ K, then

r · (c_{1}x_{0}^{j}f_{1} + ⋯ + c_{s}x_{0}^{j}f_{s}) = c_{1}r(P_{1})x_{0}^{i+j}f_{1} + ⋯ + c_{s}r(P_{s})x_{0}^{i+j}f_{s}.

Finally, we let \overline{R} = R/(x_{0}) and observe that \overline{R} is a 0-dimensional graded K-algebra with Hilbert function H\overline{R}(i) = ΔH_{𝕏}(i) = H_{𝕏}(i) − H_{𝕏}(i − 1) for i ∈ ℤ. The topic of this paper is to describe the module structure and Hilbert function of the following R-module.

1.2. Definition. a) In the standard graded K-algebra R⊗_{K} R = ⊕_{i≥0}(⊕_{j+k=i}R_{j} ⊗R_{k}) we have the homogeneous ideal I = ker(R⊗_{K} R ⟶^{μ} R), where μ is the homogeneous Rlinear map of degree zero given by μ(r ⊗ r′) = rr′. The graded R-module Ω_{R/K}^{1} = I/I^{2} is called the module of Kähler differentials of R/K.

b) The homogeneous K-linear map d_{R/K} : R ⟶ Ω_{R/K}^{1} given by r ↦ r⊗1−1⊗r+I^{2} is called the universal derivation of R/K.

More generally, for any homomorphism of graded K-algebras S ⟶ T we can define in the same way the module of Kähler differentials Ω_{T/S}^{1} = J /J^{2}, where J = ker(T ⊗_{S}T ⟶ T), and the universal derivation d_{T/S} : T ⟶ Ω_{T/S}^{1} (t ↦ t⊗1−1⊗t+J^{2}) of T/S (cf. [Ku1]). We shall also be interested in the Hilbert functions of the modules Ω_{R/K[x_{0}]}^{1} and Ω_{\overline{R}/K}^{1}, since they are closely related to the Hilbert function of Ω_{R/K}^{1}. In this section we shall provide some exact sequences connecting those three modules and some insights into their module structure.

For a polynomial F ∈ A and for i ∈ \{0, …, n\}, we shall denote the image of \frac{∂F}{∂X_{i}} in R by \frac{∂F}{∂x_{i}}. Recall that we may present Ω_{R/K}^{1} as follows (cf. [Ku1], 4.13).

1.3. Proposition. There are isomorphisms of graded R-modules

Ω_{R/K}^{1} ≅ ⨁_{i=0}^{n} Ad_{A/K}X_{i}/⟨I_{𝕏}, dI_{𝕏}⟩ ≅ ⨁_{i=0}^{n} Rd_{A/K}X_{i}/\overline{dI_{𝕏}}

where dI_{𝕏} = ⟨\frac{∂F}{∂X_{0}} d_{A/K}X_{0} + ⋯ + \frac{∂F}{∂X_{n}} d_{A/K}X_{n}⟩_{F ∈I_{𝕏}} and where \overline{dI_{𝕏}} is the image of dI_{𝕏} in ⊕_{i=0}^{n}Rd_{A/K}X_{i}. In other words, if we let G = ⟨(\frac{∂F}{∂x_{0}} , …, \frac{∂F}{∂x_{n}})⟩_{F ∈I_{𝕏}} ⊆ R^{n+1}, then there is an exact sequence of graded R-modules

0 ⟶ G(−1) ⟶ R^{n+1}(−1) ⟶ Ω_{R/K}^{1} ⟶ 0

By applying [Ku1], 3.24 to the sequence K ⟶ K[x_{0}] ⟶ R, we get the connection between Ω_{R/K}^{1} and Ω_{R/K[x_{0}]}^{1}.

1.4. Proposition. There is an exact sequence of graded R-modules

R ⊗_{K[x_{0}]} Ω_{K[x_{0}]/K}^{1} ⟶^{α} Ω_{R/K}^{1} ⟶^{β} Ω_{R/K[x_{0}]}^{1} ⟶ 0

where Ω_{K[x_{0}]/K}^{1} ≅ K[x_{0}]dx_{0}, where α is given by α(r ⊗ fdx_{0}) = rfdx_{0}, and where β is given by β(rd_{R/K}r′) = rd_{R/K[x_{0}]}r′.

1.5. Corollary. a) We have Ω_{R/K[x_{0}]}^{1} ≅ Ω_{R/K}^{1}/Rdx_{0}.

b) If {b_{1}, …, b_{s}} is a K[x_{0}]-basis of R (cf. 1.1.c), then {d_{R/K[x_{0}]}b_{1}, …, d_{R/K[x_{0}]}b_{s}} is a system of generators of the K[x_{0}]-module Ω_{R/K[x_{0}]}^{1}.

Proof: Claim a) follows from im α = Rdx_{0}. To prove b), we note that Ω_{R/K[x_{0}]}^{1} is generated by {d_{R/K[x_{0}]}r}_{r∈R} (cf. [Ku1], 1.19), and that for r = f_{1}b_{1} + ⋯ + f_{s}b_{s} with f_{1}, …, f_{s} ∈ K[x_{0}] we get d_{R/K[x_{0}]}r = f_{1}d_{R/K[x_{0}]}b_{1} + ⋯ + f_{s}d_{R/K[x_{0}]}b_{s}. □

The connection between Ω_{R/K}^{1} and Ω_{\overline{R}/K}^{1} is provided by our next proposition, which follows from [Ku1], 4.17.

1.6. Proposition. For any homogeneous ideal I ⊆ R there is an exact sequence of graded R-modules

I/I^{2} ⟶^{δ} Ω_{R/K}^{1}/IΩ_{R/K}^{1} ⟶^{ϵ} Ω_{(R/I)/K}^{1} ⟶ 0

where δ is given by δ(r + I^{2}) = d_{R/K}r + IΩ_{R/K}^{1}, and where ϵ is given by ϵ(rd_{R/K}r′ + IΩ_{R/K}^{1}) = (r + I)d_{(R/I)/K}(r′ + I).

1.7. Corollary. a) We have Ω_{\overline{R}/K}^{1} ≅ Ω_{R/K}^{1}/(x_{0}Ω_{R/K}^{1} +Rdx_{0}) ≅ Ω_{R/K[x_{0}]}^{1}/x_{0}Ω_{R/K[x_{0}]}^{1}.

b) The elements {dx_{0}, …, dx_{n}} form a homogeneous system of generators of the R-module Ω_{R/K}^{1}. If 𝕏 spans ℙ^{n}, this system of generators is minimal.

Proof: Claim a) follows by applying the proposition with I = (x_{0}) and by observing that the image of δ is the module (Rd_{R/K}I +IΩ_{R/K}^{1})/IΩ_{R/K}^{1}. The second isomorphism is an immediate corollary of 1.5.a. In order to show claim b) we apply the proposition with I = 𝔪 and obtain an epimorphism

𝔪/𝔪^{2} ⟶^{δ} Ω_{R/K}^{1}/𝔪Ω_{R/K}^{1} ⟶ Ω_{(R/𝔪)/K}^{1} ≅ Ω_{K/K}^{1} = (0).

If 𝕏 spans ℙ^{n}, the K-linear map δ is also injective, since (I_{𝕏})_{1} = (0) and 1.3 imply dim_{K}(Ω_{R/K}^{1})_{1} = n + 1 = dim_{K}(𝔪/𝔪^{2})_{1}. Now an application of the graded version of Nakayama’s Lemma finishes the proof. □

Finally we describe some properties of the structure of the module G which is by 1.3 up to a shift in degrees the first syzygy module of Ω_{R/K}^{1}.

1.8. Proposition. Let G be the graded R-submodule of R^{n+1} generated by the elements (\frac{∂F}{∂x_{0}} , …, \frac{∂F}{∂x_{n}}) such that F ∈ I_{𝕏}.

a) For each i ∈ ℤ we have G_{i} = \{(\frac{∂F}{∂x_{0}} , …, \frac{∂F}{∂x_{n}}) | F ∈ (I_{𝕏})_{i+1}\}.

b) If {F_{1}, …, F_{m}} is a minimal homogeneous system of generators of I_{𝕏}, and if char (K) > max\{deg(F_{i}) | 1 ≤ i ≤ m\}, then the set \{(\frac{∂F_{i}}{∂x_{0}} , …, \frac{∂F_{i}}{∂x_{n}}) | 1 ≤ i ≤ m\} is a minimal homogeneous system of generators of the R-module G.

Proof: Claim a) follows from the observation that, given homogeneous polynomials F ∈ (I_{𝕏})_{i} and G ∈ A_{j}, we have \overline{G}·(\frac{∂F}{∂x_{0}} , …, \frac{∂F}{∂x_{n}}) = (\frac{∂(F G)}{∂x_{0}}, …, \frac{∂(F G)}{∂x_{n}})−\overline{F} ·(\frac{∂G}{∂x_{0}} , …, \frac{∂G}{∂x_{n}}) = (\frac{∂(F G)}{∂x_{0}}, …, \frac{∂(F G)}{∂x_{n}}). This also shows that the set given in b) generates G. Suppose that for some i ∈ \{1, …, m\} we have a relation (\frac{∂F_{i}}{∂x_{0}} , …, \frac{∂F_{i}}{∂x_{n}}) = ∑_{j≠i} \overline{G}_{j} · (\frac{∂F_{j}}{∂x_{0}} , …, \frac{∂F_{j}}{∂x_{n}}) with homogeneous polynomials G_{j} ∈ A_{deg(F_{i})−deg(F_{j})}. Then the homogeneous polynomial F = F_{i} − ∑_{j≠i} G_{j}F_{j} satisfies \frac{∂F}{∂x_{0}} = ⋯ = \frac{∂F}{∂x_{n}} = 0. Since char (K) > deg(F_{i}), Euler’s relation F = \frac{1}{deg(F)} ∑_{i=0}^{n} X_{i} \frac{∂F}{∂X_{i}} implies F ∈ (X_{0}, …, X_{n}) · I_{𝕏}. Since \{F_{1}, …, F_{m}\} generates I_{𝕏} minimally, we are done. □

2. The torsion submodule of Ω_{R/K}^{1}

For 1-dimensional local rings, the torsion submodule of the module of Kähler differentials has been studied extensively (cf. [B]). In this section, we want to describe the module structure of the torsion submodule of Ω_{R/K}^{1} in our situation. The graded R-module

T_{R/K} = {m ∈ Ω_{R/K}^{1} | rm = 0 for some homogeneous nonzerodivisor r ∈ R}

is called the torsion submodule of Ω_{R/K}^{1}. Our next proposition gives several alternative descriptions of T_{R/K}. Recall that the Euler derivation of R/K is given by e : R ⟶ R (r ⟼ ir for r ∈ R_{i}). Using the universal property of Ω_{R/K}^{1} (cf. [Ku1], sec. 1), this map induces a homogeneous surjective R-linear map φ : Ω_{R/K}^{1} ⟶ 𝔪 satisfying φ(dx_{i}) = x_{i} for i = 0, …, n.

2.1. Proposition. a) We have T_{R/K} = {m ∈ Ω_{R/K}^{1} | x_{0}^{i}m = 0 for some i ≥ 0}. b) We have T_{R/K} = H_{𝔪}^{0}(Ω_{R/K}^{1}) = {m ∈ Ω_{R/K}^{1} | 𝔪^{i}m = 0 for some i ≥ 1}.

c) The module T_{R/K} is precisely the kernel of φ, i.e. there is an exact sequence of graded R-modules

0 ⟶ T_{R/K} ⟶^{ı} Ω_{R/K}^{1} ⟶^{φ} 𝔪 ⟶ 0.

d) We have T_{R/K} = {∑_{i=0}^{n} r_{i}dx_{i} ∈ Ω_{R/K}^{1} | ∑_{i=0}^{n} r_{i}x_{i} = 0}.

e) There is an exact sequence of graded R-modules

0 ⟶ T_{R/K} ⟶^{ī} Ω_{R/K[x_{0}]}^{1} ⟶^{φ̄} \overline{𝔪} ⟶ 0

where ī is the composite map T_{R/K} ⟶ Ω_{R/K}^{1} −↠ Ω_{R/K}^{1}/Rdx_{0} ≅ Ω_{R/K[x_{0}]}^{1}, i.e. where ī is given by ī(∑_{i=0}^{n} r_{i}d_{R/K}x_{i}) = ∑_{i=1}^{n} r_{i}d_{R/K[x_{0}]}x_{i}, and where φ̄ is given by φ̄(∑_{i=1}^{n} r_{i}dx_{i}) = ∑_{i=1}^{n} r_{i}x_{i} + (x_{0}).

Proof: “a)” Suppose i ≥ 0, r ∈ R_{i}, and m ∈ Ω_{R/K}^{1} are such that rm = 0 and r is not a zero divisor of R. Then f_{j}r = r(P_{j})x_{0}^{i}f_{j} ≠ 0 (cf. 1.1) and x_{0}^{i}f_{j}m = 0 for j = 1, …, s. Since x_{0}(P_{j}) = 1 for j = 1, …, s, we have f_{1} + ⋯ + f_{s} = x_{0}^{σ_{𝕏}+1}, and therefore x_{0}^{i+σ_{𝕏}+1}m = 0. The converse inclusion holds trivially.

“b)” If i ≥ 0 and m ∈ Ω_{R/K}^{1} are such that x_{0}^{i}m = 0, then Proposition 1.1.e yields 𝔪^{i+σ_{𝕏}+1}m = 𝔪^{σ_{𝕏}+1}x_{0}^{i}m = 0. Again the converse inclusion is obviously true.

“c)” Proposition 1.1.e implies 𝔪 ⟶ 𝔪R_{x_{0}} = R_{x_{0}}. The ring R_{x_{0}} is isomorphic to a product of Laurent polynomial rings (see below). Therefore the localized map (Ω_{R/K}^{1})_{x_{0}} ≅ Ω_{R_{x_{0}} /K} ⟶^{φ_{x_{0}}} 𝔪R_{x_{0}} = R_{x_{0}} is an isomorphism. Thus a) yields ker φ ≅ ker (Ω_{R/K}^{1} ⟶ (Ω_{R/K}^{1})_{x_{0}}) = T_{R/K}.

“d)” follows immediately from c).

“e)” In order to show that īis injective, by 1.5.a we need to show T_{R/K} ∩Rdx_{0} = (0). This follows from c), since φ(rdx_{0}) = rx_{0} = 0 implies r = 0. Since φ̄ is induced by Ω_{R/K}^{1} ⟶^{φ} 𝔪 −↠ \overline{𝔪}, its kernel is the image of ker φ = T_{R/K} in Ω_{R/K[x_{0}]}^{1}. Finally, it is clear that φ̄ is surjective, because φ is surjective. □

Now we give yet another description of T_{R/K}. We recall some basic facts about the integral closure of R in its homogenous ring of quotients Q^{h}(R) (cf. [O], [GKR], [Kr2]). For i = 1, …, s we let 𝔭_{i} ⊆ R be the homogeneous ideal of P_{i} in 𝕏. Then R/𝔭_{i} ≅ K[T_{i}], where T_{i} is an independent variable and corresponds to the image of x_{0} under the canonical epimorphism R −↠ R/𝔭_{i} ≅ K[T_{i}]. The integral closure of R in its homogeneous quotient ring

Q^{h}(R) = \{\frac{r}{r′} | r′ ∈ R homog. nonzerodivisor\} ≅ R_{x_{0}} ≅ K[T_{1}, T_{1}^{−1}] × ⋯ × K[T_{s}, T_{s}^{−1}]

is then given by S = K[T_{1}] × ⋯ × K[T_{s}], and the canonical inclusion R ⟶ S is given by r ↦ (r(P_{1})T_{1}^{i}, …, r(P_{s})T_{s}^{i}) for r ∈ R_{i}. Thus [Ku1], 3.23 yields functorial maps

\begin{matrix}Ω_{R/K}^{1} & ⟶^{Φ} & Ω_{S/K}^{1} ≅ K[T_{1}]dT_{1} × ⋯ × K[T_{s}]dT_{s} & ⟶ & Ω_{Q^{h}(R)/K}^{1} ≅ ∏_{k=1}^{s} K[T_{k}, T_{k}^{−1}]dT_{k} \\ rdx_{i} & ⟼ & (r(P_{1})p_{1i}T_{1}^{j}dT_{1}, …, r(P_{s})p_{si}T_{s}^{j}dT_{s}) \end{matrix}

for r ∈ R_{j}. In this situation we obtain the following characterization of T_{R/K}.

2.2. Corollary. We have T_{R/K} = ker Φ.

Proof: This follows from 2.1.d and the observation ker Φ = {∑_{i=0}^{n} r_{i}dx_{i} ∈ Ω_{R/K}^{1} | ∑_{i=0}^{n} r_{i}x_{i} = 0}. □

Next we collect a few easy consequences of Proposition 2.1. Recall from [Ku1], 10.1 that the i-th Fitting ideal of Ω_{R/K}^{1} is also known as the i-th Kähler different of R/K and denoted by ϑ^{(i)}(R/K).

2.3. Remarks. a) Using 2.1.e, we see that Ω_{R/K[x_{0}]}^{1} and Ω_{\overline{R}/K}^{1} ≅ Ω_{R/K[x_{0}]}^{1}/x_{0}Ω_{R/K[x_{0}]}^{1} are torsion R-modules.

b) Notice that 2.1.e also implies x_{0}Ω_{R/K[x_{0}]}^{1} ⊆ ī(T_{R/K}), since x_{0}\overline{𝔪} = 0.

c) We have {r ∈ R | rdx_{0} = 0} = 0 in Ω_{R/K}^{1}. In particular, Ann_{R}(Ω_{R/K}^{1}) = 0. This follows from 2.1.c, since rdx_{0} ∈ ker φ implies rx_{0} = 0, and therefore r = 0.

d) By [Ku1], 10.4, we have ϑ^{(0)}(R/K) = 0 and ϑ^{(i)}(R/K) ≠ 0 for all i ≥ 1, because one can see for instance from 2.1.c that the R-module Ω_{R/K}^{1} has rank one.

We end this section with an explicit presentation of T_{R/K} by generators and relations.

2.4. Proposition. a) Let Syz_{R}^{1}(x_{0}, …, x_{n}) be the first syzygy module of 𝔪, i.e. let

Syz_{R}^{1}(x_{0}, …, x_{n}) = {(h_{0}, …, h_{n}) ∈ R^{n+1} | ∑_{i=0}^{n}h_{i}x_{i} = 0} ⊆ R^{n+1}

and define G = ⟨(\frac{∂F}{∂x_{0}} , …, \frac{∂F}{∂x_{n}})⟩_{F ∈I_{𝕏}} as in 1.3. Then there is an exact sequence of graded R-modules

0 ⟶ G(−1) ⟶^{γ} Syz_{R}^{1}(x_{0}, …, x_{n})(−1) ⟶^{η} T_{R/K} ⟶ 0

where γ is induced by G ⟶ R^{n+1} and η is given by η((h_{0}, …, h_{n})) = ∑_{i=0}^{n} h_{i}dx_{i}.

b) Let Syz_{A}^{1}(X_{0}, …, X_{n}) ⊆ A^{n+1} be the first syzygy module of 𝔐 = (X_{0}, …, X_{n}). Then there is an exact sequence of graded A-modules

Syz_{A}^{1}(X_{0}, …, X_{n})(−1) ⟶^{λ} Syz_{R}^{1}(x_{0}, …, x_{n})(−1) ⟶^{μ} I_{𝕏}/𝔐I_{𝕏} ⟶ 0

where λ is given by λ((H_{0}, …, H_{n})) = (h_{0}, …, h_{n}) and μ is given by μ((h_{0}, …, h_{n})) = H_{0}X_{0} + ⋯ + H_{n}X_{n} + 𝔐I_{𝕏} for H_{0}, …, H_{n} ∈ A and h_{i} = H_{i} + I_{𝕏}.

Proof: “a)” The map γ is well-defined, since Euler’s formula shows \frac{∂F}{∂x_{0}} x_{0}+⋯+ \frac{∂F}{∂x_{n}} x_{n} = (deg F)F +I_{𝕏} = 0 ∈ R for homogeneous polynomials F ∈ I_{𝕏}. The map η is well-defined and surjective by 2.1.c. Exactness of the sequence follows from 1.3.

“b)” The claim follows by applying the snake lemma to the following commutative diagram with exact rows and columns.

0  0
↓ ↓
I_{𝕏}^{n+1}(−1) ⟶ I_{𝕏}
↓ ↓
0 ⟶ Syz_{A}^{1}(X_{0}, …, X_{n})(−1) ⟶ A^{n+1}(−1) ⟶ 𝔐 ⟶ 0
↓ ↓ ↓
0 ⟶ Syz_{R}^{1}(x_{0}, …, x_{n})(−1) ⟶ R^{n+1}(−1) ⟶ 𝔪 ⟶ 0
↓ ↓
0  0

Here the map I_{𝕏}^{n+1}(−1) ⟶ I_{𝕏} is induced by e_{i} ↦ X_{i} and its cokernel is I_{𝕏}/𝔐I_{𝕏}. □

2.5. Corollary. If char K = 0 or char K > max{i ∈ ℕ | (I_{𝕏}/𝔐I_{𝕏})_{i} ≠ 0}, then the torsion submodule T_{R/K} of Ω_{R/K}^{1} is generated by {x_{i}dx_{j} − x_{j}dx_{i} | i, j = 0, …, n}. In particular, it is generated by homogeneous elements of degree two.

Proof: From the exact sequence 2.4.b we get that Syz_{R}^{1}(x_{0}, …, x_{n}) is generated by {x_{i}e_{j} − x_{j}e_{i} | i, j = 0, …, n} and by preimages under γ of G_{1} + 𝔐I_{𝕏}, ..., G_{t} + 𝔐I_{𝕏}, where {G_{1}, …, G_{t}} is a minimal homogeneous system of generators of I_{𝕏}. Using the assumption on char K, we can write G_{i} = \frac{1}{deg G_{i}} (\frac{∂G_{i}}{∂X_{0}} X_{0} +⋯+ \frac{∂G_{i}}{∂X_{n}} X_{n}) for i = 1, …, t. Then \frac{1}{deg G_{1}} (\frac{∂G_{1}}{∂x_{0}} , …, \frac{∂G_{1}}{∂x_{n}}), ..., \frac{1}{deg G_{t}} (\frac{∂G_{t}}{∂x_{0}} , …, \frac{∂G_{t}}{∂x_{n}}) are such preimages. Since the exact sequence of 2.4.a shows that they are in the kernel of η, the claim follows from that exact sequence. □

3. The Hilbert function of Ω_{R/K}^{1}

In this section we shall describe some properties of the Hilbert functions of the graded modules Ω_{R/K}^{1}, Ω_{R/K[x_{0}]}^{1}, and Ω_{\overline{R}/K}^{1}. To simplify the notation, we set H_{Ω}(i) = H_{Ω_{R/K}^{1}} (i), H_{Ω_{0}}(i) = H_{Ω_{R/K[x_{0}]}^{1}}(i), H\overline{Ω}(i) = H_{Ω_{\overline{R}/K}^{1}} (i), and H_{T} (i) = H_{T_{R/K}} (i) for all i ∈ ℤ. Using the exact sequences of section 1, we get the following relations between those Hilbert functions.

3.1. Proposition. a) For all i ∈ ℤ we have H_{Ω_{0}}(i) = H_{Ω}(i) − H_{𝕏}(i − 1).

b) For all i ∈ ℤ we have H\overline{Ω}(i) = H_{Ω_{0}}(i) − H_{x_{0}Ω_{0}}(i) = H_{Ω_{0}}(i) − H_{Ω_{0}}(i − 1) + dim_{K}{m ∈ (Ω_{R/K[x_{0}]}^{1})_{i−1} | x_{0}m = 0}.

c) For all i ≥ 1 we have H_{T} (i) = H_{Ω}(i) − H_{𝕏}(i) = H_{Ω_{0}}(i) − H_{𝕏}(i) + H_{𝕏}(i − 1).

d) For all i ∈ ℤ we have H_{Ω}(i) = (n + 1)H_{𝕏}(i − 1) − H_{G}(i − 1) and H_{T} (i) = (n + 1)H_{R}(i − 1) − H_{R}(i) − H_{G}(i − 1).

Proof: Claim a) follows from 1.5.a and 2.3.c. Claim b) follows from 1.7.a and the exact sequence

0 ⟶ \{m ∈ Ω_{R/K[x_{0}]}^{1} | x_{0}m = 0\}(−1) ⟶ Ω_{R/K[x_{0}]}^{1}(−1) ⟶^{x_{0}} Ω_{R/K[x_{0}]}^{1} ⟶ Ω_{\overline{R}/K}^{1} ⟶ 0.

Claim c) follows from 2.1.c, and claim d) is a consequence of 1.3 and c). □

The basic properties of H_{Ω} are summarized by our next proposition. We use the notation α_{𝕏} = min{i ∈ ℤ | (I_{𝕏})_{i} ≠ 0}.

3.2. Proposition. a) For i ≤ 0 we have H_{Ω}(i) = 0.

b) For 1 ≤ i < α_{𝕏} we have H_{Ω}(i) = (n + 1)\binom{n+i−1}{n}.

c) If char K = 0 or char K > α_{𝕏}, then H_{Ω}(α_{𝕏}) = H_{𝕏}(α_{𝕏}) + (α_{𝕏} − 1)\binom{n+α_{𝕏}−1}{n−1}.

d) There is a minimal number ϱ_{𝕏} ≥ 1 such that H_{Ω}(i) = s for all i ≥ ϱ_{𝕏}. We call this number the regularity of Ω_{R/K}^{1}.

e) We have H_{Ω}(σ_{𝕏} +2) ≥ H_{Ω}(σ_{𝕏} +3) ≥ ⋯, and if ϱ_{𝕏} ≥ σ_{𝕏} +2, then H_{Ω}(σ_{𝕏} +2) > H_{Ω}(σ_{𝕏} + 3) > ⋯ > H_{Ω}(ϱ_{𝕏}) = s.

Proof: Claims a) and b) follow from 1.3. To prove c), we consider the exact sequence

0 ⟶ G_{α_{𝕏}−1} ⟶ R_{α_{𝕏}−1}^{n+1} ⟶ (Ω_{R/K}^{1})_{α_{𝕏}} ⟶ 0.

Let {G_{1}, …, G_{r}} be a K-basis of (I_{𝕏})_{α_{𝕏}}. Then the elements dG_{1}, …, dG_{r} form a system of generators of (dI_{𝕏})_{α_{𝕏}}. If we denote the image of \frac{∂G_{i}}{∂X_{j}} in R by \frac{∂G_{i}}{∂x_{j}}, then the set \{(\frac{∂G_{i}}{∂x_{0}} , …, \frac{∂G_{i}}{∂x_{n}})_{i=1,...,r}\} generates G_{α_{𝕏}−1}. We claim that those elements are K-linearly independent.

Suppose there are c_{1}, …, c_{r} ∈ K such that ∑_{i=1}^{r} c_{i}(\frac{∂G_{i}}{∂x_{0}} , …, \frac{∂G_{i}}{∂x_{n}}) = 0. Then G = ∑_{i=1}^{r} c_{i}G_{i} satisfies \frac{∂G}{∂X_{i}} ∈ I_{𝕏} for i = 0, …, n. Since deg G = α_{𝕏} and (I_{𝕏})_{α_{𝕏}−1} = 0, this gives \frac{∂G}{∂X_{0}} = ⋯ = \frac{∂G}{∂X_{n}} = 0. Using our assumption on char K, we conclude G = \frac{1}{α_{𝕏}} (\frac{∂G}{∂X_{0}} X_{0} + ⋯ + \frac{∂G}{∂X_{n}} X_{n}) = 0, and hence c_{1} = ⋯ = c_{r} = 0. In other words, we have shown that H_{G}(α_{𝕏} − 1) = dim_{K}(I_{𝕏})_{α_{𝕏}} = \binom{n+α_{𝕏}}{n} − H_{𝕏}(α_{𝕏}). Therefore we obtain H_{Ω}(α_{𝕏}) = (n + 1)\binom{n+α_{𝕏}−1}{n} − \binom{n+α_{𝕏}}{n} + H_{𝕏}(α_{𝕏}) = H_{𝕏}(α_{𝕏}) + n\binom{n+α_{𝕏}−1}{n} − \binom{n+α_{𝕏}−1}{n} = H_{𝕏}(α_{𝕏}) + (α_{𝕏} − 1)\binom{n+α_{𝕏}−1}{n}.

Now we prove claim d). Clearly x_{0} is not a zero divisor on G, as it isn’t on R and G ⊆ R^{n+1}. If γ denotes the largest degree of a minimal generator of G, every homogeneous element g of G of degree deg g > γ + σ_{𝕏} + 1 is of the form x_{0}g′ with g′ ∈ G_{deg g−1} (cf. Prop. 1.1). Thus the Hilbert function of G stabilizes, and Propositions 1.1 and 1.3 show that also H_{Ω} stabilizes.

Finally we prove claim e). By 1.1.e and 1.7.b, we have (Ω_{R/K}^{1})_{i} = ∑_{j=0}^{n} R_{i−1}dx_{j} = x_{0} ∑_{j=0}^{n} R_{i−2}dx_{j} = x_{0}(Ω_{R/K}^{1})_{i−1} for every i ≥ σ_{𝕏} +3. Thus we have H_{Ω}(i) ≥ H_{Ω}(i−1) for those i.

We also want to show that H_{Ω}(i) = H_{Ω}(i + 1) for some i ≥ σ_{𝕏} + 2 implies H_{Ω}(i) = H_{Ω}(j) for all j ≥ i. In view of 3.1.d, this means that we assume H_{G}(i − 1) = H_{G}(i). As x_{0} ∈ R is not a zero divisor of G, we get G_{i} = x_{0}G_{i−1}. Since I_{𝕏} is generated in degrees ≤ σ_{𝕏} +2 (cf. [GM], 1.1), the R-module G is generated in degrees ≤ σ_{𝕏} +1, and we obtain G_{i+1} = x_{0}G_{i} + ⋯ + x_{n}G_{i} = x_{0}(x_{0}G_{i−1} + ⋯ + x_{n}G_{i−1}) = x_{0}G_{i}. Thus H_{G}(i + 1) = H_{G}(i), and the claim follows by induction. □

The following examples show that H_{Ω}(i) may or may not be monotonic in the range α_{𝕏} ≤ i ≤ σ_{𝕏} + 2.

3.3. Example. Let 𝕏 ⊆ ℙ^{2} consist of the following nine points: P_{1} = (1 : 0 : 0), P_{2} = (1 : 0 : 1), P_{3} = (1 : 0 : 2), P_{4} = (1 : 0 : 3), P_{5} = (1 : 0 : 4), P_{6} = (1 : 0 : 5), P_{7} = (1 : 1 : 0), P_{8} = (1 : 2 : 0), and P_{9} = (1 : 1 : 1). Then H_{𝕏} : 1 3 6 7 8 9 9 ⋯, α_{𝕏} = 3, and σ_{𝕏} = 4. The Hilbert function H_{Ω} : 3 9 15 14 13 14 13 12 11 10 9 9 ⋯ satisfies H_{Ω}(σ_{𝕏}) = 14 > H_{Ω}(σ_{𝕏} + 1) = 13 < H_{Ω}(σ_{𝕏} + 2) = 14.

Notice that in this example H_{T} : 3 8 6 5 5 4 3 2 1 is unimodal, i.e. it increases monotonically up to a maximum and then decreases monotonically.

3.4. Example. a) We take four points on a line in ℙ^{4} and three general points off that line, e.g. we let I_{𝕏} ⊆ ℙ^{4} consist of the following seven points: P_{1} = (1 : 0 : 0 : 0 : 0), P_{2} = (1 : 0 : 0 : 0 : 1), P_{3} = (1 : 0 : 0 : 0 : 2), P_{4} = (1 : 0 : 0 : 0 : 3), P_{5} = (1 : 1 : 0 : 0 : 0), P_{6} = (1 : 0 : 1 : 0 : 0), and P_{7} = (1 : 0 : 0 : 1 : 0). Then we have H_{𝕏} : 1 5 6 7 7 ⋯ and α_{𝕏} = σ_{𝕏} = 2 and H_{Ω} : 5 16 9 10 9 8 7 7 · · ·. Again H_{Ω} is not unimodal, and this time not even H_{T} : 10 2 3 2 1 is unimodal.

b) If we take seven generic points in ℙ^{3}, then H_{𝕏} : 1 5 7 7 ⋯, α_{𝕏} = 2, σ_{𝕏} = 1, and H_{Ω} : 5 17 8 7 7 ⋯ is unimodal.

Our next proposition gives an upper bound for the regularity of Ω_{R/K}^{1} Another proof of this result can be derived by using [GGP], 2.4, and Proposition 3.9. A weaker bound is contained in [R]. Example 3.3 shows that our bound is sharp.

3.5. Proposition. The regularity ϱ_{𝕏} = min{i ∈ ℤ | H_{Ω}(j) = s for j ≥ i} of Ω_{R/K}^{1} satisfies ϱ_{𝕏} ≤ 2σ_{𝕏} + 3.

Proof: From 1.1.e and 1.7.b we get that the elements x_{0}^{i−σ_{𝕏}−2}f_{i}dx_{j} such that 1 ≤ i ≤ s and 0 ≤ j ≤ n generate (Ω_{R/K}^{1})_{k} for every k ≥ 2σ_{𝕏} + 3. Using 1.1.f, we calculate

x_{0}^{σ_{𝕏}+1}f_{i}dx_{j} = f_{i}^{2}dx_{j} = d(f_{i}^{2}x_{j}) − x_{j}df_{i}^{2} = d(p_{ij}x_{0}f_{i}^{2}) − x_{j}df_{i}^{2}
= (p_{ij}x_{0} − x_{j})df_{i}^{2} + p_{ij}f_{i}^{2}dx_{0} = (p_{ij}x_{0} − x_{j})2f_{i}df_{i} + p_{ij}f_{i}^{2}dx_{0}
= p_{ij}f_{i}^{2}dx_{0} = p_{ij}x_{0}^{σ_{𝕏}+1}f_{i}dx_{0}.

Thus the elements {x_{0}^{i−σ_{𝕏}−2}f_{j}dx_{0} | 1 ≤ j ≤ s} generate (Ω_{R/K}^{1})_{k} for k ≥ 2σ_{𝕏} +3, and it remains to show that they are K-linearly independent. Suppose we have c_{1}, …, c_{s} ∈ K such that ∑_{j=1}^{s} c_{j}x_{0}^{i−σ_{𝕏}−2}f_{j}dx_{0} = 0, and suppose that for instance c_{1} ≠ 0. Then x_{0}^{i−σ_{𝕏}−2}f_{1}^{2}dx_{0} = −\frac{1}{c_{1}} ∑_{j=2}^{s} c_{j}x_{0}^{i−σ_{𝕏}−2}f_{1}f_{j}dx_{0} = 0 contradicts 2.3.c. □

In the corresponding local situation (e.g. for the ring R_{𝔪}) it is known that the module of Kähler differentials always has torsion, if we have at least two points (cf. [S]). We shall show now that in our situation, the module Ω_{R/K}^{1} has in fact torsion elements of degree two, and in case ϱ_{𝕏} ≥ σ_{𝕏} + 2 in every degree from two up to its regularity.

3.6. Proposition. Suppose 𝕏 consists of s ≥ 2 points. Then we have (T_{R/K})_{2} ≠ 0. Moreover, if char K = 0 or char K > σ_{𝕏} + 2, then ϱ_{𝕏} ≥ σ_{𝕏} + 2 implies (T_{R/K})_{i} ≠ 0 for all i ∈ {2, …, ϱ_{𝕏} − 1}.

Proof: We can assume that 𝕏 spans ℙ^{n}, because otherwise we consider the embedding of 𝕏 in its linear span, and neither H_{𝕏} nor H_{Ω} nor H_{T} change. We start by showing H_{T} (2) ≠ 0.

In case α_{𝕏} ≥ 3, Propositions 3.1.c and 3.2.a yield H_{T} (2) = H_{Ω}(2) − H_{𝕏}(2) = (n + 1)\binom{n+1}{n} − \binom{n+2}{2} = \binom{n+1}{2} > 0. If we have α_{𝕏} = 2, then Proposition 1.3 gives us an exact sequence

0 ⟶ G_{1} ⟶ R_{1}^{n+1} ⟶ (Ω_{R/K}^{1})_{2} ⟶ 0

of K-vector spaces, and Proposition 3.1.c together with the observation dim_{K} G_{1} = dim_{K}⟨(\frac{∂F}{∂x_{0}} , …, \frac{∂F}{∂x_{n}})⟩_{F ∈(I_{𝕏})_{2}} ≤ dim_{K}(I_{𝕏})_{2} implies

H_{T} (2) = H_{Ω}(2) − H_{𝕏}(2) = (n + 1)^{2} − dim_{K} G_{1} − \binom{n + 2}{2} + dim_{K}(I_{𝕏})_{2}

= \binom{n + 1}{2} + dim_{K}(I_{𝕏})_{2} − dim_{K} G_{1} > 0.

For the second claim, we may clearly assume that ϱ_{𝕏} ≥ 3. Then H_{Ω}(ϱ_{𝕏} − 1) ≥ s by 3.1.c, and this cannot be an equality because of the definition of ϱ_{𝕏}. Thus we have (T_{R/K})_{ϱ_{𝕏}−1} ≠ 0. An application of Corollary 2.5 finishes the proof. □

In view of Propositions 3.2.e and 3.6, it seems natural to conjecture that the bound ϱ_{𝕏} ≥ σ_{𝕏} + 2 always holds. If true, that bound would be sharp, as our next example shows.

3.7. Example. Let 𝕏 ⊆ ℙ^{5} consist of nine generic points. Then we have H_{𝕏} : 1 6 9 9 ⋯ and σ_{𝕏} = 1. From H_{Ω} : 6 24 9 9 ⋯ we conclude that ϱ_{𝕏} = 3 = σ_{𝕏} + 2.

Using the separators of 𝕏 (cf. 1.1), we can give an explicit descrition of (T_{R/K})_{2σ_{𝕏}+2} and thus a criterion for maximal regularity ϱ_{𝕏} = 2σ_{𝕏} + 3.

3.8. Proposition. We have (T_{R/K})_{2σ_{𝕏}+2} = ⟨f_{i}df_{j}⟩_{i,j∈{1,...,s},i≠j}.

In particular, for all i, j ∈ {1, …, s} such that i ≠ j, the element f_{i}df_{j} is in the socle of Ω_{R/K}^{1}, and ϱ_{𝕏} equals its maximal value 2σ_{𝕏} + 3 if and only if one of those elements is not zero.

Proof: Let i, j ∈ {1, …, s} such that i ≠ j, and let k ∈ {0, …, n}. Then we have f_{i}df_{j} = d(f_{i}f_{j}) − f_{j}df_{i} = −f_{j}df_{i} and p_{ik}x_{0}f_{i}df_{j} = x_{k}f_{i}df_{j} = −x_{k}f_{j}df_{i} = −p_{jk}x_{0}f_{j}df_{i} = p_{jk}x_{0}f_{i}df_{j}. This is only possible for all k ∈ {0, …, n}, if x_{0}f_{i}df_{j} = 0. By 2.1.a, we get f_{i}df_{j} ∈ T_{R/K}.

Conversely, we see from Proposition 1.1 that {x_{0}^{σ_{𝕏}}f_{i}dx_{j} | 1 ≤ i ≤ s and 0 ≤ j ≤ n} generates (Ω_{R/K}^{1})_{2σ_{𝕏}+2}. We let σ = σ_{𝕏} and calculate

x_{0}^{σ}f_{i}dx_{j} = d(x_{0}^{σ}f_{i}x_{j}) − x_{0}^{σ}x_{j}df_{i} − σx_{0}^{σ−1}x_{j}f_{i}dx_{0}
= p_{ij}d(x_{0}^{σ+1}f_{i}) − ∑_{k=1}^{s} p_{kj}f_{k}df_{i} − σp_{ij}x_{0}^{σ}f_{i}dx_{0}
= p_{ij}x_{0}^{σ}f_{i}dx_{0} + p_{ij}x_{0}^{σ+1}df_{i} − ∑_{k=1}^{s} p_{kj}f_{k}df_{i}
= p_{ij}x_{0}^{σ}f_{i}dx_{0} + ∑_{k≠i}(p_{ij} − p_{kj})f_{k}df_{i}

Therefore also the set {x_{0}^{σ_{𝕏}}f_{i}dx_{0} | 1 ≤ i ≤ s} ∪ {f_{i}df_{j} | 1 ≤ i, j ≤ s and i ≠ j} generates (Ω_{R/K}^{1})_{2σ_{𝕏}+2}. A look at Proposition 2.1.c now finishes the proof. □

Finally, we exhibit an interpretation of H_{Ω} in terms of the Hilbert function of 𝕏 and the scheme of double points 𝕐 concentrated at the support of 𝕏. However, note that there is no clear connection between the module structure of Ω_{R/K}^{1} and the ring structure of the homogeneous coordinate ring of 𝕐.

3.9. Proposition. a) Let 𝔓_{i} be the prime ideal of P_{i} in A, and let I_{𝕏}^{(2)} = I_{𝕏}^{2}A_{𝔓_{1}} ∩ ⋯ ∩ I_{𝕏}^{2}A_{𝔓_{s}} ∩ A be the symbolic square of I_{𝕏}. Then there is an exact sequence of graded R-modules

0 ⟶ I_{𝕏}^{(2)}/I_{𝕏}^{2} ⟶ I_{𝕏}/I_{𝕏}^{2} ⟶ Ω_{A/K}^{1} ⊗_{A} R ⟶ Ω_{R/K}^{1} ⟶ 0.

b) We have I_{𝕏}^{(2)} = 𝔓_{1}^{2} ∩⋯∩𝔓_{s}^{2}. In other words, I_{𝕏}^{(2)} is the homogeneous vanishing ideal of the 0-dimensional scheme 𝕐 = Z(I_{𝕏}^{(2)}) consisting of double points concentrated at the support of 𝕏.

c) For all i ∈ ℤ we have

H_{Ω}(i) = (n + 1)H_{𝕏}(i − 1) + H_{𝕏}(i) − H_{𝕐}(i)

In particular, the invariant σ_{𝕐} = max{i ∈ ℤ | H_{𝕐}(i) < deg 𝕐 = (n + 1)s} satisfies σ_{𝕐} = max{σ_{𝕏} + 2, ϱ_{𝕏}} − 1.

Proof: “a)” If we localize the homogeneous exact sequence

I_{𝕏}/I_{𝕏}^{2} ⟶^{α} Ω_{A/K}^{1} ⊗_{A} R ⟶ Ω_{R/K}^{1} ⟶ 0

(cf. [Ku1], 4.17) at the element X_{0}, the map α becomes an isomorphism, since 𝕏 ⊆ D_{+}(X_{0}) = 𝔸_{K}^{n} is a local complete intersection (cf. [Ku1], sec. 9). Thus we get ker α = {r + I_{𝕏}^{2} ∈ I_{𝕏}/I_{𝕏}^{2} | X_{0}^{i}r ∈ I_{𝕏}^{2} for some i ≥ 0}. Now every element of ker α lies in I_{𝕏}^{(2)}/I_{𝕏}^{2}, because the image of X_{0} in A_{𝔓_{i}} is a unit for i = 1, …, s. The converse inclusion is clear, since (I_{𝕏}^{(2)})_{i} = (I_{𝕏}^{2})_{i} for i >> 0.

Claim b) follows from I_{𝕏}^{2}A_{𝔓_{i}} = 𝔓_{i}^{2}A_{𝔓_{i}} for i = 1, …, s, and c) follows from a).

□

4. Computation of Ω_{R/K}^{1} in special cases

In this section we provide formulas for Ω_{R/K}^{1}, its torsion submodule, its Hilbert function etc. in some special cases. The case of 0-dimensional almost complete intersections will be treated in the next section. We continue to use the definitions and notations introduced in the previous sections. Our first case is n = 1, i.e. sets of points 𝕏 ⊆ ℙ^{1}.

4.1. Proposition. Let 𝕏 = {P_{1}, …, P_{s}} ⊆ ℙ^{1} such that P_{i} = (1 : p_{i}) for i = 1, …, s. Then Ω_{R/K}^{1} is given by

Ω_{R/K}^{1} = (R^{2}/⟨\frac{∂F}{∂x_{0}} , \frac{∂F}{∂x_{1}} ⟩)(−1),

where \frac{∂F}{∂x_{0}} = − ∑_{i=1}^{s} p_{i} ∏_{j≠i}(x_{1} − p_{j}x_{0}), and where \frac{∂F}{∂x_{1}} = ∑_{i=1}^{s} ∏_{j≠i}(x_{1} − p_{j}x_{0}).

It has Hilbert function H_{Ω} : 2 4 6 ⋯ (2s−2) (2s−1) (2s−2) (2s−3) ⋯ (s+1) s s ⋯ and regularity ϱ_{𝕏} = 2s − 1 = 2σ_{𝕏} + 3. Its torsion submodule T_{R/K} is given by

T_{R/K} = ⟨x_{0}dx_{1} − x_{1}dx_{0}⟩ ≅ (R/(\frac{∂F}{∂x_{0}} , \frac{∂F}{∂x_{1}} ))(−2)

and has Hilbert function H_{T} : 1 2 ⋯ (s−1) (s−2) ⋯ 1.

The proof of this proposition follows easily from 1.3, 2.4 and 3.1. The interested reader may also work out the presentations and Hilbert functions of Ω_{R/K[x_{0}]}^{1} and Ω_{\overline{R}/K}^{1} in this case. We note that Ω_{R/K}^{1} has maximal regularity. This explains the following example.

4.2. Example. Let 𝕏, 𝕐 ⊆ P^{2} be two subsets, the first one consisting of 19 generic points, and the second one consisting of 13 generic points and 6 points on a line. Then both coordinate rings R_{𝕏} and R_{𝕐} have the same Hilbert function H_{𝕏} = H_{𝕐} : 1 3 6 10 15 19 19 ⋯, and in fact even the same graded free resolution over A, namely

0 ⟶ A(−7)^{4} ⟶ A(−5)^{2} ⊕ A(−6)^{3} ⟶ A ⟶ R ⟶ 0.
However, the Hilbert functions H_{Ω_{𝕏}} of Ω_{R_{𝕏}/K}^{1} and H_{Ω_{𝕐}} of Ω_{R_{𝕐}/K}^{1} are
H_{Ω_{𝕏}} : 3 9 18 30 43 48 40 31 21 19 19 ⋯  and
H_{Ω_{𝕐}} : 3 9 18 30 43 48 40 31 21 20 19 ⋯

and differ in degree 10. We see that H_{Ω} contains more information about the geometry of 𝕏 than H_{𝕏} or the resolution of R_{𝕏} alone. The fact that the 6 collinear points 𝕐′ ⊆ 𝕐 force H_{Ω_{𝕐}}(10) to be larger than 19 can be seen as follows. By 4.1, we have H_{Ω_{𝕐′}} : 2 4 6 8 10 11 10 9 8 7 6 6 ⋯, so that (R_{𝕐′}dx_{0})_{10} ⊊ (Ω_{𝕐′})_{10}. Hence (R_{𝕐}dx_{0})_{10} has to be strictly smaller than (Ω_{𝕐})_{10}, too, i.e. we have to have H_{Ω_{𝕐}}(10) > 19.

Another case in which it is possible to describe Ω_{R/K}^{1} completely is the case of 0-dimensional complete intersections.

4.3. Proposition. Let 𝕏 ⊆ ℙ^{n} be a 0-dimensional complete intersection of n hypersurfaces H_{1} = Z(F_{1}), …, H_{n} = Z(F_{n}), and let d_{i} = deg(F_{i}) for i = 1, …, n. Then there is an exact sequence of graded R-modules

0 ⟶ I_{𝕏}/I_{𝕏}^{2} ⟶^{δ} R^{n+1}(−1) ⟶^{ϵ} Ω_{R/K}^{1} ⟶ 0

such that δ(F_{i} + I_{𝕏}^{2}) = (\frac{∂F_{i}}{∂x_{0}} , …, \frac{∂F_{i}}{∂x_{n}}) for i = 1, …, n and ϵ(e_{j}) = dx_{j} for j = 0, …, n. There is also an isomorphism of graded R-modules ı : I_{𝕏}/I_{𝕏}^{2} ⟶^{∼} ⊕_{i=1}^{n}R(−d_{i}) given by ı(F_{i} + I_{𝕏}^{2}) = e_{i} for i = 1, …, n.

In particular, we have H_{Ω}(i) = (n + 1)H_{𝕏}(i − 1) − ∑_{i=1}^{n} H_{𝕏}(i − d_{i}) for all i ∈ ℤ, and ϱ_{𝕏} = σ_{𝕏} + 1 + max{d_{i} | 1 ≤ i ≤ n}.

Proof: From [Ku1], sec. 9, it follows that the map η : ⊕_{i=1}^{n}R(−d_{i}) ⟶ I_{𝕏}/I_{𝕏}^{2} defined by e_{i} ⟼ F_{i} + I_{𝕏}^{2} is an isomorphism. Hence the R-module I_{𝕏}/I_{𝕏}^{2} is torsion-free and the above sequence is exact by 3.9.a. □

In a series of papers leading up to [AH], A. Hirschowitz and J. Alexander succeeded in computing the Hilbert function of a generic set of double points (cf. [AH], Thm. 2). Thus we can apply 3.9.c to determine H_{Ω} for generic sets of points 𝕏 ⊆ ℙ^{n}.

4.4. Corollary. Let 𝕏 ⊆ ℙ^{n} be a generic set of s ≥ n + 1 points.

a) We have H_{𝕏}(i) = min\{\binom{n+i}{n}, s\} and

H_{Ω}(i) = (n + 1)H_{𝕏}(i − 1) + H_{𝕏}(i) − min\{\binom{n+i}{n}, (n + 1)s\}

for all i ∈ ℤ, except if (n, s) ∈ {(2, 5), (3, 9), (4, 7), (4, 14)}.

b) If (n, s) = (2, 5), we have H_{Ω} : 3 8 10 6 5 5 · · ·.

c) If (n, s) = (3, 9), we have H_{Ω} : 4 5 15 25 11 9 9 · · ·.
d) If (n, s) = (4, 7), we have H_{Ω} : 5 17 8 7 7 · · ·.
e) If (n, s) = (4, 14), we have H_{Ω} : 5 24 49 15 14 14 · · ·.

5. Kähler differentials for 0-dimensional almost complete intersections

In this section we shall assume that 𝕏 ⊆ ℙ^{n} is a reduced 0-dimensional almost complete intersection, i.e. that I_{𝕏} is minimally generated by n+1 homogeneous polynomials F_{1}, …, F_{n+1}. For i = 1, …, n + 1, we let d_{i} = deg F_{i}. We shall use the following graded version of the Primbasissatz (cf. [N]) which we prove using arguments in the spirit of [HMV], §4A.

5.1. Proposition. Let I ⊆ A = K[X_{0}, …, X_{n}] be a homogeneous ideal of height h, and let m = μ(I) be its minimal number of generators.

a) There exists a minimal homogeneous system of generators {F_{1}, …, F_{m}} of I such that F_{1}, …, F_{h} is a regular sequence in A.

b) If I is unmixed and 𝔓_{1}, …, 𝔓_{s} ⊆ A are prime ideals containing I such that IA_{𝔓_{i}} = 𝔓_{i}A_{𝔓_{i}} for i = 1, …, s, we can assume additionally that the images of F_{1}, …, F_{h} in A_{𝔓_{i}} form a regular system of parameters of A_{𝔓_{i}} for each i ∈ {1, …, s}.

Proof: If {F_{1}, …, F_{m}} is any minimal homogeneous system of generators of I, and if d_{i} = deg F_{i} for i = 1, …, m, we can assume d_{1} ≥ ⋯ ≥ d_{m}. We shall prove both claims by showing that a generic tuple from

𝔘 = I_{d_{1}} × ⋯ × I_{d_{m}}

satisfies all requirements. Here 𝔘 is viewed as a product of affine spaces over K and equipped with the Zariski topology.

“a)” We note that m ≥ h and proceed by induction on m, the case m = 1 being clear. In case m = h, any minimal homogeneous system of generators of I is a regular sequence. If m > h, we choose a generic element (F_{1}, …, F_{m}) of 𝔘 and consider the ideal J = (F_{2}, …, F_{m}). Since h − 1 ≤ ht(J) ≤ h, there are two cases.

The first case is ht(J) = h. By the induction hypothesis, the elements F_{2}, …, F_{h+1} form a regular sequence of A. It suffices to show that there exists a polynomial G ∈ A_{d_{1}−d_{h+1}} such that F_{1}+GF_{h+1} is not a zero divisor of A/(F_{2}, …, F_{h}). This follows from the fact that for each associated prime 𝔮 of A/(F_{2}, …, F_{h}) there exists a polynomial G ∈ A_{d_{1}−d_{h+1}} such that F_{1} + GF_{h+1} ∉ 𝔮, because in case F_{1} ∉ 𝔮 we can take G = 0 and in case F_{1} ∈ 𝔮 we have F_{h+1} ∉ 𝔮, and therefore F_{h+1} · (X_{0}, …, X_{n})^{d_{1}−d_{h+1}} ⊈ 𝔮. Then a generic element G ∈ A_{d_{1}−d_{h+1}} satisfies F_{1} + GF_{h+1} ∉ 𝔮 for all such primes 𝔮 simultaneously. Thus we obtain a regular sequence F̃_{1}, F_{2}, …, F_{h}, if we replace F_{1} by F̃_{1} = F_{1} + GF_{h+1} or a generic element of I_{d_{1}}.

The second case is ht(J) = h − 1. By the induction hypothesis, the elements F_{2}, …, F_{h} form a regular sequence in A. In the ring B = A/(F_{2}, …, F_{h}), the ideal \overline{I} = (\overline{F}_{1}, …, \overline{F}_{m}) has height one, and therefore contains a homogeneous nonzerodivisor. Since \overline{I} is generated in degrees ≤ d_{1}, there even exists a homogeneous nonzerodivisor of degree d_{1}. Thus a generic element F_{1} of I_{d_{1}} yields a regular sequence F_{1}, …, F_{h}.

“b)” Let i ∈ {1, …, s}. Clearly, the images of F_{1}, …, F_{m} in A_{𝔓_{i}} generate 𝔓_{i}A_{𝔓_{i}}, and we have μ(𝔓_{i}A_{𝔓_{i}}) = h. We have to show that for a generic tuple (F_{1}, …, F_{m}) of 𝔘, the images of F_{1}, …, F_{h} generate 𝔓_{i}A_{𝔓_{i}} minimally. Let 1 ≤ ν_{1} ≤ ⋯ ≤ ν_{h} ≤ m be such that the images of F_{ν_{1}}, …, F_{ν_{h}} generate 𝔓_{i}A_{𝔓_{i}} minimally. We need to show that if ν ∈ {1, …, m} \ {ν_{1}, …, ν_{h}} and ν < ν_{h}, we can replace F_{ν_{h}} by a generic element F_{ν} of I_{d_{ν}} and still have a minimal system of generators {F_{ν}, F_{ν_{1}}, …, F_{ν_{h−1}}} of 𝔓_{i}A_{𝔓_{i}}. For this we only need to find some element F_{ν} ∈ I_{d_{ν}} with the desired property. If G ∈ A_{d_{ν}−d_{ν_{h}}} lies outside 𝔓_{i}, we can take F_{ν} = GF_{ν_{h}}, since the image of G in A_{𝔓_{i}} is a unit. □

In view of this proposition, we shall assume henceforth that F_{1}, …, F_{n} is a regular sequence of A, and that (F_{1}, …, F_{n})A_{𝔓_{i}} = 𝔓_{i}A_{𝔓_{i}} for all homogeneous prime ideals 𝔓_{1}, …, 𝔓_{s} ⊆ A corresponding to the points of 𝕏. This means that the ideal J = (F_{1}, …, F_{n}) defines a 0-dimensional complete intersection containing 𝕏 which is reduced at the points of 𝕏. The graded R-module

ω_{R} = \underline{Ext}_{A}^{n}(R, A)(−n − 1) ≅ \underline{Hom}_{K[x_{0}]}(R, K[x_{0}])(−1)

is called the canonical module of R (cf. [GW], ch. 2). Notice that we can express ω_{R} in the form

ω_{R} ≅ \underline{Hom}_{A/J}(R, A/J)(∑_{i=1}^{n}d_{i} − n − 1) ≅ ((J : I_{𝕏})/J)(∑_{i=1}^{n}d_{i} − n − 1)

(cf. [GW], 2.2.9). In analogy with [M], we shall now construct an explicit presentation of the R-module I_{𝕏}/I_{𝕏}^{2}.

5.2. Proposition. If 𝕏 ⊆ ℙ^{n} is a reduced 0-dimensional almost complete intersection as above, we have an exact sequence of graded R-modules

0 ⟶ ω_{R}(n + 1 − ∑_{i=1}^{n+1}d_{i}) ⟶^{φ} ⊕_{i=1}^{n+1}R(−d_{i}) ⟶^{ψ} I_{𝕏}/I_{𝕏}^{2} ⟶ 0

where ψ is given by ψ(e_{i}) = F_{i} + I_{𝕏}^{2} for i = 1, …, n + 1 and φ is described in the proof. In particular, for all i ∈ ℤ we have

H_{I_{𝕏}/I_{𝕏}^{2}}(i) = ∑_{i=1}^{n+1}H_{𝕏}(i − d_{i}) + H_{𝕏}(∑_{i=1}^{n+1}d_{i} − n − i − 1) − s.

Proof: Considering the above presentation of ω_{R}, we need to define a homogeneous Rlinear map Φ : ((J : I_{𝕏})/J)(−d_{n+1}) ⟶ ⊕_{i=1}^{n+1}R(−d_{i}). For a homogeneous polynomial G ∈ A such that G·I_{𝕏} ⊆ J we write GF_{n+1} = ∑_{i=1}^{n} G_{i}F_{i} with homogeneous polynomials G_{1}, …, G_{n} ∈ A of degree deg(G_{i}) = deg(G) + d_{n+1} − d_{i} for i = 1, …, n. Then we let Φ(G + J) = (G_{1} + I_{𝕏}, …, G_{n} + I_{𝕏}, −G + I_{𝕏}). Let us show that this yields a well-defined map Φ.

For a homogeneous polynomial H ∈ A such that H · I_{𝕏} ⊆ J and G + J = H + J we get an analogous representation HF_{n+1} = H_{1}F_{1} + ⋯ + H_{n}F_{n}. Then we have G − H = ∑_{i=1}^{n} c_{i}F_{i} with homogeneous polynomials c_{i} ∈ A, and ∑_{i=1}^{n}(G_{i} − H_{i})F_{i} = (G − H)F_{n+1} = ∑_{i=1}^{n} c_{i}F_{i}F_{n+1}. Since F_{1}, …, F_{n} is a regular sequence, the relation ∑_{i=1}^{n}(G_{i}−H_{i}−c_{i}F_{n+1})F_{i} = 0 implies G_{i}−H_{i}−c_{i}F_{n+1} ∈ J, i.e. we have G_{i}+I_{𝕏} = H_{i}+I_{𝕏} for i = 1, …, n.

It is easy to check that the map Φ is R-linear and homogeneous. Next we prove that Φ is injective. Let G ∈ A be represented as above and suppose that Φ(G + J) = (G_{1} + I_{𝕏}, …, G_{n} + I_{𝕏}, −G + I_{𝕏}) = 0. Then we have G = H_{1}F_{1} + ⋯ + H_{n+1}F_{n+1} for some homogeneous polynomials H_{1}, …, H_{n+1}. We want to show H_{n+1}F_{n+1} ∈ J. Since H_{n+1}F_{n+1}^{2} = GF_{n+1} − H_{1}F_{1}F_{n+1} − ⋯ − H_{n}F_{n}F_{n+1} = ∑_{i=1}^{n}(G_{i} − H_{i}F_{n+1})F_{i} ∈ J, we need to show J : (F_{n+1}^{2}) = J : (F_{n+1}). From the fact that Z(J) is reduced at the points of 𝕏 we get (J : I_{𝕏}) ∩ I_{𝕏} = J. Thus H_{n+1}F_{n+1} ∈ (J : I_{𝕏}) ∩ I_{𝕏} implies the claim.

Since ψ ∘ Φ(G + J) = ∑_{i=1}^{n} G_{i}F_{i} − GF_{n+1} + I_{𝕏}^{2} = 0, it remains to show ker ψ ⊆ im Φ. Suppose there are homogeneous polynomials H_{1}, …, H_{n+1} ∈ A such that ψ(H_{1} + I_{𝕏}, …, H_{n+1}+I_{𝕏}) = ∑_{i=1}^{n+1} H_{i}F_{i}+I_{𝕏}^{2} = 0. The equation I_{𝕏}^{2} = I_{𝕏}J +(F_{n+1}^{2}) implies that we find polynomials G_{ij}, G ∈ A such that ∑_{i=1}^{n+1} H_{i}F_{i} = ∑_{j=1}^{n} ∑_{i=1}^{n+1} G_{ij}F_{i}F_{j} + GF_{n+1}^{2}. Then we conclude from ∑_{j=1}^{n}(−H_{j} + ∑_{i=1}^{n} G_{ij}F_{i})F_{j} + (GF_{n+1} − H_{n+1})F_{n+1} = 0 that Φ(GF_{n+1} − H_{n+1} + J) = (H_{1} − ∑_{i=1}^{n+1} G_{i1}F_{i} + I_{𝕏}, …, H_{n} − ∑_{i=1}^{n+1} G_{in}F_{i} + I_{𝕏}) = (H_{1} + I_{𝕏}, …, H_{n} + I_{𝕏}). □

Recall that in section 2 we introduced the notation ϑ^{(i)}(R/K) for the i-th Fitting ideal of Ω_{R/K}^{1}. In the sequel, we shall denote the i-th Fitting ideal of Ω_{R/K[x_{0}]}^{1} by ϑ^{(i)}(R/K[x_{0}]). Since we know the Hilbert function of I_{𝕏}/I_{𝕏}^{2} from 5.2, all that is needed to determine the Hilbert function of Ω_{R/K}^{1} using 3.9.a is a description of the torsion submodule I_{𝕏}^{(2)}/I_{𝕏}^{2} of I_{𝕏}/I_{𝕏}^{2}. Our next proposition provides several such descriptions.

5.3. Proposition. Let 𝕏 ⊆ ℙ^{n} be a reduced 0-dimensional almost complete intersection as above. Then there is an exact sequence of graded R-modules of the form

0 ⟶ ω_{R}(n + 1 − ∑_{i=1}^{n+1}d_{i}) ⟶^{λ} D ⟶^{μ} I_{𝕏}^{(2)}/I_{𝕏}^{2} ⟶ 0

where the finitely generated graded R-module D can be described in one of the following ways.

a) The module D is the kernel of the R-linear map j : ⊕_{i=1}^{n+1}R(−d_{i}) ⟶ ⊕_{i=1}^{n}R(−1) given by e_{i} ⟼ (\frac{∂F_{i}}{∂x_{1}} , …, \frac{∂F_{i}}{∂x_{n}}) for i = 1, …, n + 1.

b) The module D is the kernel of the R-linear map j_{0} : ⊕_{i=1}^{n+1}R(−d_{i}) ⟶ ⊕_{i=0}^{n+1}R(−1) given by e_{i} ⟼ (\frac{∂F_{i}}{∂x_{0}} , …, \frac{∂F_{i}}{∂x_{n}}) for i = 1, …, n + 1.

c) The module D is ϑ^{(0)}(R/K[x_{0}])^{−1}(− ∑_{i=1}^{n+1} d_{i}), where ϑ^{(0)}(R/K[x_{0}])^{−1} denotes the R-submodule of R_{x_{0}} generated by all elements \frac{r}{x_{0}^{i}} such that r ·ϑ^{(0)}(R/K[x_{0}]) ⊆ x_{0}^{i}R.

d) If char (K) > max{d_{1}, …, d_{n+1}}, then the module D is also the second syzygy module of Ω_{R/K}^{1}.

In particular, for all i ∈ ℤ we have

H_{Ω}(i) = (n + 1)H_{𝕏}(i − 1) − ∑_{i=1}^{n+1}H_{𝕏}(i − d_{i}) + H_{D}(i) and

H_{I_{𝕏}^{(2)}/I_{𝕏}^{2}}(i) = H_{D}(I) + H_{𝕏}(−i) − s.

Proof: Using 3.9.a and 5.2, we can construct the following commutative diagram with exact rows and columns.

0  0
↓  ↓
0 ⟶ ω_{R}(n+1− ∑_{i=1}^{n+1} d_{i}) ⟶^{id} ω_{R}(n+1− ∑_{i=1}^{n+1} d_{i}) ⟶  0  ⟶ 0
↓  ↓^{φ}  ↓  ↓
0 ⟶  D  ⟶ ⊕_{i=1}^{n+1} R(−d_{i}) ⟶^{j_{0}} ⊕_{i=0}^{n} R(−1) ⟶ Ω_{R/K}^{1} ⟶ 0
↓  ↓^{ψ}  ↓≅  ∥
0 ⟶  I_{𝕏}^{(2)}/I_{𝕏}^{2}  ⟶  I_{𝕏}/I_{𝕏}^{2}  ⟶ R ⊗_{A} Ω_{A/K}^{1} ⟶ Ω_{R/K}^{1} ⟶ 0
↓  ↓  ↓  ↓
0  0  0  0

From this diagram we get a) and the additional claim. Since an element (a_{1}, …, a_{n+1}) ∈ ⊕_{i=1}^{n+1}R(−d_{i}) is in ker(j_{0}) if and only if a_{1} \frac{∂F_{1}}{∂x_{i}} + ⋯ + a_{n+1} \frac{∂F_{n+1}}{∂x_{i}} = 0 for i = 0, …, n, it is clear that ker(j_{0}) ⊆ ker(j). Conversely, if we assume that (a_{1}, …, a_{n+1}) ∈ ker(j), i.e. that those equations hold for i = 1, …, n, then Euler’s relation, applied to the homogeneous polynomial ∑_{j=1}^{n+1} a_{j}F_{j}, shows that also a_{1} \frac{∂F_{1}}{∂x_{0}} + ⋯ + a_{n+1} \frac{∂F_{n+1}}{∂x_{0}} = 0.

The proof of c) proceeds in analogy with the proof of [Ku2], Lemma 1. If we denote the maximal minors of the matrix (\frac{∂F_{i}}{∂x_{j}})_{i∈\{1,...,n+1\},j∈\{1,...,n\}} by Δ_{1}, …, Δ_{n+1}, then Cramer’s rule yields D ≅ {r ∈ R_{x_{0}} | rΔ_{1}, …, rΔ_{n+1} ∈ R} · (Δ_{1}, …, Δ_{n+1}) ⊆ R_{x_{0}}^{n+1}. Thus the map D ⟶ R_{x_{0}}(− ∑_{i=1}^{n+1} d_{i}) given by r · (Δ_{1}, …, Δ_{n+1}) ⟼ r identifies D with ϑ^{(0)}(R/K[x_{0}])^{−1}(− ∑_{i=1}^{n+1} d_{i}). Claim d) follows from 1.8 and the exact sequence in the middle row of the above diagram. □

Since the task of finding the Hilbert function of Ω_{R/K}^{1} has been reduced to finding the Hilbert function of ϑ^{(0)}(R/K[x_{0}])^{−1}, we would like to point out the following result of R. Waldi (cf. [W]). Recall that a reduced 0-dimensional almost complete intersection 𝕏 ⊆ ℙ^{n} is called a special almost complete intersection, if there exist homogeneous polynomials G_{1}, …, G_{n} ∈ A and a homogeneous n × n-matrix 𝔄 = (a_{ij}) over A such that {(G_{1}, …, G_{n}) · 𝔄} ∪ {det(𝔄)} is a homogeneous set of polynomials which generates I_{𝕏}.

5.4. Proposition. Let 𝕏 ⊆ ℙ^{n} be a reduced 0-dimensional almost complete intersection.

a) If 𝕏 is a special almost complete intersection, then we have ϑ^{(0)}(R/K[x_{0}]) ≅ ϑ_{𝔇}(R/K[x_{0}]) ≅ ℭ_{R/K[x_{0}]}^{−1}, where ϑ_{𝔇}(R/K[x_{0}]) is the Dedekind different and ℭ_{R/K[x_{0}]} is the Dedekind complementary module of R/K[x_{0}] (cf. [Ku1], App. G).

b) For all i ∈ ℤ, we have

\begin{aligned}H_{D}(i) & = H_{ϑ^{(0)}(R/K[x_{0}])^{−1}}(i − ∑_{j=1}^{n+1}d_{j}) ≥ H_{((ℭ_{R/K[x_{0}]})^{−1})^{−1}}(i − ∑_{j=1}^{n+1}d_{j}) \\ & ≥ H_{ℭ_{R/K[x_{0}]}}(i − ∑_{j=1}^{n+1}d_{j}) = s − H_{𝕏}(∑_{j=1}^{n+1}d_{j} − i − 1). \end{aligned}

Moreover, if 𝕏 is a special almost complete intersection, the first inequality is an equality.

Proof: Claim a) follows from [W], Satz 4, and b) from [Ku1], 10.18. □

Finally, we give an example of a reduced 0-dimensional almost complete intersection 𝕏 ⊆ ℙ^{n} which is not special, and we compute H_{Ω}, H_{D}, and H_{I_{𝕏}/I_{𝕏}^{2}} in this case (see also [W], Beispiel 1).

5.5. Example. Let 𝕏 ⊆ ℙ^{4} consists of 10 generic points. Then we have H_{𝕏} : 1 5 10 10 ⋯, the ideal I_{𝕏} is minimally generated by 5 quadratic forms, and the Cohen-Macaulay type of R is 5. Thus 𝕏 is an almost complete intersection, but ht(I_{𝕏}) = 4 < 5 = type(R) implies that 𝕏 is not special (cf. [W], Bem. 1). It is not difficult to calculate the following Hilbert functions in degrees i ≥ 1.

\begin{aligned}H_{Ω} & : 5 20 25 10 10 ⋯ \\ H_{D} & : 0 0 0 10 10 ⋯ \\ H_{I_{𝕏}/I_{𝕏}^{2}} & : 0 5 25 45 41 40 40 ⋯ \end{aligned}

References

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Address of the authors:

Gabriel de Dominicis  Martin Kreuzer
Dipartimento di Matematica Fakultät für Mathematik
Universitá di Genova  Universität Regensburg
Via Dodecaneso 35  Postfach 397
I-16146 Genova, Italia  D-93040 Regensburg, Deutschland
dedo@mordred.dima.unige.it  martin.kreuzer@mathematik.uni-regensburg.de