Some Applications of the Canonical Module

of a 0-dimensional Scheme

Martin Kreuzer

Contents

1. Introduction

2. Applications to hyperplane sections of curves

3. Applications to cohomologically split schemes

4. Applications to pure schemes

1. Introduction

In this note we consider 0-dimensional schemes X together with a fixed embedding into a d-dimensional projective space ℙ^{d} over an algebraically closed field k. In [K1] we studied the canonical module ω_{R} of the projective coordinate ring R = k[X_{0}, … X_{d}]/I_{X} of X ⊆ ℙ^{d}. First of all, we recall some of the basic definitions and results of [K1]. In particular, we are interested in the applications of those results w.r.t. the Hilbert function of X.

In the sequel, we fix our coordinate system such that X ⊆ D_{+}(X_{0}). Thus x_{0} := X_{0} + I_{X} is a linear nonzerodivisor of R and ω_{R} can be described by

ω_{R} ≅ \underline{Hom}_{k[x_{0}]}(R, k[x_{0}])(−1).

Definition. We say that X is a Cayley-Bacharach scheme, if every hypersurface of degree σ_{X} := max{n ∈ ℤ : H_{X}(n) < deg X} containing a subscheme of degree deg X −1 of X contains X.

Theorem 1.1. (cf. [K1], 2.4) The scheme X is a locally Gorenstein Cayley-Bacharach scheme if and only if there exists an element φ ∈ (ω_{R})_{−σ_{X}} such that Ann_{R}(φ) = 0.

Corollary 1.2. If X is a locally Gorenstein Cayley-Bacharach scheme, then

H_{X}(n) + H_{X}(σ_{X} − n) ≤ deg X  for all n ∈ ℤ.

Proof: Since R(σ_{X}) ≅ Rφ ⊆ ω_{R}, we get H_{X}(σ_{X} −n) = dim_{k} R(σ_{X})_{−n} ≤ dim_{k}(ω_{R})_{−n} = deg X − H_{X}(n). For the last equality we used [K1], 1.3.c. □

In section 2 we shall apply Theorem 1.1 to give an algebraic proof of the fact that generic hyperplane sections of certain curves are Cayley-Bacharach schemes.

The uniformity of X required in the definition of a Cayley-Bacharach scheme can be extended as follows.

Definition. Let n ≥ 1. Then X is called n-uniform, if every subscheme Y ⊆ X of degree deg X − n ≤ deg Y ≤ deg X has Hilbert function H_{Y} = min{H_{X}, deg Y }.

It is easy to see (cf. [K1]) that with this definition, Cayley-Bacharach schemes are exactly the 1-uniform schemes, while (deg X − 1)-uniformity agrees with the usual definition of “in uniform position”.

Theorem 1.3. (cf. [K1], 3.1) Let Δ_{X} := H_{X}(σ_{X} + 1) − H_{X}(σ_{X}). The scheme X is Δ_{X}-uniform if and only if the multiplication map

R_{σ_{X}} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0}

is biinjective (i.e. r ∈ R_{σ_{X}}, φ ∈ (ω_{R})_{−σ_{X}}, rφ = 0 implies r = 0 or φ = 0).

Corollary 1.4. If X is Δ_{X}-uniform, then

H_{X}(n) + H_{X}(σ_{X} − n) ≤ deg X − Δ_{X} + 1  for all n ∈ {0, …, σ_{X}}.

Proof: Since x_{0} is a nonzerodivisor of both R and ω_{R}, also R_{n}⊗(ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−σ_{X}+n} is biinjective. By applying the Biinjective Map Lemma (cf. [K1], section 3), we obtain dim_{k}(ω_{R})_{−σ_{X}+n} = deg X − H_{X}(σ_{X} − n) ≥ dim_{k} R_{n} + dim_{k} (ω_{R})_{−σ_{X}} − 1 = H_{X}(n) + Δ_{X} − 1. □

In case Δ_{X} ≥ 2, Corollary 1.4 obviously yields stronger inequalities for the Hilbert function of X than Corollary 1.2. In section 3 we examine intermediate uniformities (between 1-uniform and Δ_{X}-uniform) w.r.t. the question how much uniformity is really needed in order to prove the inequalities in Corollary 1.4.

Finally, in section 4, we discuss the question under which hypothesis it is possible to strengthen the inequalities of Corollary 1.4. An interesting candidate for a set of even stronger inequalities is given by the following notion.

Definition. We say that H_{X} is flawless, if ΔH_{X}(σ_{X} +1−n) ≥ ΔH_{X}(n) ≥ ΔH_{X}(n−1) for all 0 ≤ 2n ≤ σ_{X} + 1.

In our discussion of possible hypotheses yielding flawless Hilbert functions we shall arrive at the following question, which at the same time concludes this introduction:

Question. If X is pure (i.e. if ω_{R} is generated by its elements of degree −σ_{X}) and sufficiently uniform (e.g. in uniform position), does this imply that H_{X} is flawless?

2. Applications to hyperplane sections of curves

In this section we let C ⊆ ℙ^{d+1} be an integral curve, and X := C ∩ H ⊆ ℙ^{d} a generic hyperplane section of C. By S we denote the projective coordinate ring of C, and by ℓ ∈ S_{1} the equation of H in S. Then the projective coordinate ring R of X satisfies R = (S/ℓS)/H_{𝔪}^{0}(S/ℓS). Furthermore, we let M := H_{𝔪}^{1}(S) = H_{∗}^{1}(ℙ^{d+1}, J_{C}) denote the Hartshorne-Rao module of C, and consider it as a graded S-module.

Our first result describes the relationship between the canonical modules ω_{R} = H_{𝔪}^{1}(R)^{∗} of R and ω_{S} = H_{𝔪}^{2}(S)^{∗} of S. Here ^{∗} denotes graded dualization w.r.t. k, i.e. \underline{Hom}_{k}(—, k).

Theorem 2.1. (The Fundamental Exact Sequence)

There is an exact sequence of graded R-modules

0 ⟶ (ω_{S}/ℓω_{S})(1) ⟶ ω_{R} ⟶ (M/ℓM)^{∗} ⟶ 0.

Proof: Let T := S/ℓS. Since R = T/H_{𝔪}^{0}(T), we have H_{𝔪}^{1}(R) ≅ H_{𝔪}^{1}(T), and therefore ω_{R} ≅ ω_{T} (cf. [GW]). An application of the long exact sequence of local cohomology modules to

0 ⟶ S(−1) ⟶^{ℓ} S ⟶ T ⟶ 0

yields a homogeneous exact sequence

0 ⟶ H_{𝔪}^{0}(T) ⟶ H_{𝔪}^{1}(S)(−1) ⟶^{ℓ} H_{𝔪}^{1}(S) ⟶ H_{𝔪}^{1}(T) ⟶

⟶ H_{𝔪}^{2}(S)(−1) ⟶^{ℓ} H_{𝔪}^{2}(S) ⟶ 0.

Now we can use M = H_{𝔪}^{1}(S) and apply the functor ^{∗} to obtain

0 ⟶ ω_{S} ⟶^{ℓ} ω_{S}(1) ⟶ ω_{R} ⟶ M^{∗} ⟶^{ℓ} M^{∗}(1) ⟶ H_{𝔪}^{0}(T)^{∗} ⟶ 0.

Since obviously ker(M^{∗} ⟶^{ℓ} M^{∗}(1)) ≅ (M/ℓM)^{∗}, splitting this sequence concludes the proof. □

Using Theorems 1.1 and 2.1 we can now give an algebraic proof of the fact that X is a Cayley-Bacharach scheme (regardless of char k) in the case σ_{X} − 1 = σ_{C} := max{n ∈ ℤ : h^{1}O_{C}(n) ≠ 0}. For this we need a few preparations.

Lemma 2.2. Let N be a finitely generated graded S-module, and let ℓ ∈ S_{1} be a general linear form. Then Ann_{N}(ℓ) := {x ∈ N : ℓx = 0} ⊆ H_{𝔪}^{0}(N).

Proof: From Grothendieck’s standard graded exact sequence we construct a commutative diagram with exact rows

0 ⟶ H_{𝔪}^{0}(N)(−1) ⟶ N(−1) ⟶ Γ_{∗}(Ñ)(−1)
↓^{μ_{ℓ}}  ↓^{μ′_{ℓ}}  ↓^{μ′′_{ℓ}}
0 ⟶  H_{𝔪}^{0}(N)  ⟶  N  ⟶  Γ_{∗}(Ñ)

Since H_{𝔪}^{0}(Γ_{∗}(Ñ)) = 0, we have 𝔪 ∉ Ass (Γ_{∗}(Ñ)), and μ′′_{ℓ} is injective. Hence Ann_{N}(ℓ) = ker μ′_{ℓ} = ker μ_{ℓ} ⊆ H_{𝔪}^{0}(N). □

Lemma 2.3. In the situation described above we have Ass (ω_{S}) = {(0)}.

Proof: Let 𝔭 ⊆ S be a homogeneous prime of height at least one. From the Graded Duality Theorem (cf. [GW]) we get Ann_{ω_{S}}(𝔭) ≅ \underline{Hom}_{S}(S/𝔭, ω_{S}) ≅ H_{𝔪}^{2}(S/𝔭)^{∗} = (0). Thus 𝔭 ∉ Ass (ω_{S}). □

Theorem 2.4. Let X ⊆ ℙ^{d} be the generic hyperplane section of an integral curve C ⊆ ℙ^{d+1}. If σ_{X} = σ_{C} + 1, then X is a Cayley-Bacharach scheme.

Proof: First of all, we choose a nonzero element Φ ∈ (ω_{S})_{−σ_{C}}. This is possible, since σ_{C} = max{n ∈ ℤ : h^{1}O_{C}(n) ≠ 0} = − min{n ∈ ℤ : (ω_{S})_{n} ≠ 0}. Using Lemma 2.3 we get Ass (SΦ) ⊆ Ass (ω_{S}) = (0), and therefore

dim(S/Ann_{S}(Φ)) = max{dim S/𝔭 : 𝔭 ∈ Ass (SΦ)} = 2.

This implies Ann_{S}(Φ) = 0. Now we construct the following commutative diagram of homogeneous maps with exact rows and columns.

0 ⟶  S(σ_{C})  ⟶^{Φ}  ω_{S}  ⟶  ω_{S}/SΦ  ⟶ 0
↓^{μ_{ℓ}}  ↓^{μ′_{ℓ}}  ↓^{μ′′_{ℓ}}
0 ⟶ S(σ_{C} + 1) ⟶^{Φ}  ω_{S}(1)  ⟶  (ω_{S}/SΦ)(1)  ⟶ 0
↓  ↓  ↓
(S/ℓS)(σ_{X}) ⟶^{μ_{φ}} (ω_{S}/ℓω_{S})(1) ⟶ ω_{S}/(SΦ + ℓω_{S})(1) ⟶ 0
↓  ↓  ↓
0  0  0

Let φ denote the image of Φ in (ω_{S}/ℓω_{S})(1). From Lemma 2.2 we conclude that ker μ′′_{ℓ} = Ann_{ω_{S}/SΦ}(ℓ) ⊆ H_{𝔪}^{0}(ω_{S}/SΦ). Next we apply the Snake Lemma to the above diagram. Since μ′_{ℓ} is injective, we see that ker μ_{φ} ⊆ H_{𝔪}^{0}(S/ℓS). Therefore, if we regard φ via the Fundamental Exact Sequence as an element of ω_{R}, we get Ann_{R}(φ) = 0. Since φ ∈ (ω_{S}/ℓω_{S})(1)_{−σ_{C}−1} ⊆ (ω_{R})_{−σ_{X}}, an application of Theorem 1.1 yields the claim.

□

In case C is arithmetically Cohen-Macaulay, the Fundamental Exact Sequence collapses to an isomorphism ω_{R} ≅ (ω_{S}/ℓω_{S})(1), and the proof of Theorem 2.4 reduces to the one given in [GKR], 5.1. Nontrivial examples of the theorem can be found using arithmetically Buchsbaum curves.

Example. Let C ⊆ ℙ^{d+1} be an integral, arithmetically Buchsbaum curve, and let X = C ∩ H ⊆ ℙ^{d} be a generic hyperplane section of C. It is well-known that the Hartshorne-Rao module M of C is generated in degrees ≤ σ_{C} +2. Since 𝔪M = 0, this implies M_{n} = 0 for all n > σ_{C} + 2. Suppose that also M_{σ_{C}+2} = 0. Then we have (M/ℓM)_{−σ_{C}−2}^{∗} = 0 in the Fundamental Exact Sequence, and therefore −σ_{X} = min{n ∈ ℤ : (ω_{R})_{n} ≠ 0} = min{n ∈ ℤ : (ω_{S}/ℓω_{S})(1)_{n} ≠ 0} = −σ_{C} − 1. Hence Theorem 2.4 can be applied and yields that X is a Cayley-Bacharach scheme.

3. Applications to cohomologically split schemes

For this section, we shall assume that X ⊆ ℙ^{d} is a reduced 0-dimensional subscheme. In the introduction we saw that Δ_{X}-uniform schemes X satisfy the inequalities of Corollary 1.4, while for 1-uniform schemes we obtained only weaker inequalities. The following example demonstrates that the stronger set of inequalities actually fails for some 1-uniform schemes.

Example. Let L_{1}, L_{2} ⊆ ℙ^{3} be two skew lines, and let {P_{1}, …, P_{4}} ⊆ L_{1} as well as {P_{5}, …, P_{8}} ⊆ L_{2} be two subsets of four distinct points each. Then X := {P_{1}, …, P_{8}} ⊆ ℙ^{3} is a Cayley-Bacharach scheme, but H_{X} : 1 4 6 8 8 ⋯ does not satisfy the inequalities of Corollary 1.4.

Intuitively, the failure of those inequalities comes from the fact that X “splits” into two disjoint subsets having high individual uniformities, but little to do with each other. Algebraically, we can formalize this as follows.

Write X = {P_{1}, …, P_{s}}, where s = deg X. For each i ∈ {1, …, s} the ideal of X \ {P_{i}} in X defines a unique 1-dimensional subvectorspace V_{i} of R_{σ_{X}+1}.

Definition. We say that X splits cohomologically, if we can decompose X = Y ∪ Z such that Y ≠ ∅, Z ≠ ∅, Y ∩ Z = ∅, and the only elements r ∈ ∑_{P_{i}∈Y} V_{i}, r′ ∈ ∑_{P_{i}∈Z} V_{i}

with r + r′ ∈ x_{0}R_{σ_{X}} are r = r′ = 0.

In order to make this an intrinsic definition, we have to show that it does not depend on the choice of the linear nonzerodivisor x_{0} ∈ R_{1}. At the same time, we want to explain the choice of the name.

Remark. The local cohomology module H_{𝔪}^{1}(R) is a graded R-module whose grading satisfies σ_{X} = max{n ∈ ℤ : H_{𝔪}^{1}(R)_{n} ≠ 0}. Each subscheme Y_{i} := X \ {P_{i}} corresponds to a 1-dimensional subspace c(Y_{i}) ⊆ H_{𝔪}^{1}(R)_{σ_{X}} in the following way: if f_{i} ∈ V_{i} \ {0}, then the generalized fraction \left[{f_{i} \atop x_{0}}\right] defines c(Y_{i}). Using the rules for computing with generalized fractions, the above definition translates into the following statement:

X splits cohomologically if and only if X = Y ∪ Z with Y ≠ ∅, Z ≠ ∅,

Y ∩ Z = ∅, and ∑_{P_{i}∈Y} c(Y_{i}) ∩ ∑_{P_{i}∈Z} c(Y_{i}) = (0).

Since c(Y_{i}) clearly does not depend on the choice of x_{0}, neither does the notion of cohomological splitting.

The reader may verify that the example given above constitutes a cohomologically split scheme. Our next theorem characterizes cohomologically non-split schemes in terms of their canonical module. This will allow us to show that they enjoy a uniformity which is indeed intermediate between 1-uniformity and Δ_{X}-uniformity.

Theorem 3.1. Let X = {P_{1}, …, P_{s}} ⊆ ℙ^{d} be a reduced 0-dimensional subscheme. The following conditions are equivalent.

a) X does not split cohomologically.

b) The multiplication map R_{σ_{X}} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} is nondegenerate and surjective.

The proof of this theorem relies on a careful study of the canonical ideal of X and is contained in [K2]. Here we are interested in the following application.

Corollary 3.2. For reduced 0-dimensional subschemes X ⊆ ℙ^{d}, the following implications hold.

X is Δ_{X}-uniform ⇓ X does not split cohomologically ⇓ X is a Cayley-Bacharach scheme (i.e. 1-uniform)

Proof: The second implication immediately follows from [K1], Theorem 2.6, which says that X is a Cayley-Bacharach scheme if and only if the multiplication map μ : R_{σ_{X}} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} is nondegenerate. For the first implication, we only have to make the additional observation that the multiplication map μ is surjective, if it is biinjective (which it is by 1.3), since the Biinjective Map Lemma (cf. [K1], section 3) yields dim_{k} im μ ≥ dim_{k} R_{σ_{X}} + dim_{k} (ω_{R})_{−σ_{X}} − 1 = s − 1 = dim_{k} (ω_{R})_{0}. □

It is interesting to note that cohomological non-splitting rules out all known examples of Cayley-Bacharach schemes violating the inequalities of 1.4. Therefore we are led to putting forward the following

Question. If X ⊆ ℙ^{d} is a reduced 0-dimensional subscheme which does not split cohomologically, does it follow that H_{X}(n) + H_{X}(σ_{X} − n) ≤ deg X − Δ_{X} + 1 for all n ∈ {1, …, σ_{X}} ?

Although we have not yet been able to decide this question, we shall end this section with a partial result. Recall that a generic basis {φ_{1}, …, φ_{Δ_{X}} } of (ω_{R})_{−σ_{X}} satisfies Ann_{R}(φ_{i}) = (0) for i = 1, …, Δ_{X}, if X is a Cayley-Bacharach scheme (cf. [GKR]). Let us say that X has property (∗), if (ω_{R})_{0} has a decompostion of the form

(ω_{R})_{0} = R_{σ_{X}} φ_{1} ⊕ kx_{0}^{σ_{X}} φ_{2} ⊕ … ⊕ kx_{0}^{σ_{X}} φ_{Δ_{X}} .

Theorem 3.3. Let X ⊆ ℙ^{d} be a reduced 0-dimensional subscheme.

a) If X has property (∗), then H_{X}(n) + H_{X}(σ_{X} − n) ≤ deg X − Δ_{X} + 1 for all n ∈ {0, …, σ_{X}}.

b) If X has property (∗), then X does not split cohomologically.

Proof: “a)” Since x_{0} is not a zerodivisor of ω_{R}, it follows from property (∗) that (ω_{R})_{−σ_{X}+n} contains R_{n}φ_{1} ⊕ kx_{0}^{n}φ_{2} ⊕ … ⊕ kx_{0}^{n}φ_{Δ_{X}} for all n ∈ {0, …, σ_{X}}. Hence deg X − H_{X}(σ_{X} − n) = dim_{k}(ω_{R})_{−σ_{X}+n} ≥ H_{X}(n) + Δ_{X} − 1.

“b)” Property (∗) yields that the multiplication map R_{σ_{X}} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} is nondegenerate and surjective. Then we can apply Theorem 3.1. □

Since in [K2] we also show that Δ_{X}-uniform schemes have property (∗), we have found another condition of uniformity, which is intermediate between Δ_{X}-uniformity and cohomological non-splitting, and which at the same time implies the inequalities of Corollary 1.4. Therefore the above question really asks:

Question. Do cohomologically non-split schemes have property (∗) ?

4. Applications to pure schemes

After having examined the inequalities of Corollary 1.4, we venture further and ask under which circumstances it is possible to prove even stronger inequalities for the Hilbert function of a 0-dimensional scheme X ⊆ ℙ^{d}. The inequalities describing flawless Hilbert functions appear to be natural candidates because of a result of T. Hibi which we are now going to explain.

From the presentation R ≅ k[X_{0}, …, X_{d}]/I_{X} we get \overline{R} ≅ k[X_{1}, …, X_{d}]/J_{X}, where \overline{R} := R/(x_{0}) and J_{X} := I_{X} mod X_{0}. Choose a degree-compatible ordering τ of the monomials of k[X_{1}, …, X_{d}], and form the ideal of leading terms Lt_{τ} (J_{X}) := (Lt_{τ} f)_{f∈J_{X}}. Let M be the set of all monomials of k[X_{1}, …, X_{d}] which are not in Lt_{τ} (J_{X}). It is wellknown that the elements of M form an order ideal of monomials (i.e. if m ∈ M and m′ | m, then m′ ∈ M), and that the Hilbert function of X satisfies ΔH_{X}(n) = #{m ∈ M : deg m = n} for all n ∈ ℤ. We say that X is combinatorically pure, if we can choose x_{0} and τ in such a way that all maximal elements of M have the same degree (namely σ_{X} + 1).

Theorem 4.1. (cf. [H]) If X is combinatorically pure, then H_{X} is flawless.

In the present situation this result is not entirely satisfactory, because the choices involved in the above definition make it hard to check in practice whether a given X is combinatorically pure. As a more algebraic notion, we propose the following.

Definition. We say that X is pure, if ω_{R} is generated by its elements of minimal degree, i.e. by (ω_{R})_{−σ_{X}}.

First of all, let us show that this notion generalizes the above notion of combinatorical purety. For that and for other purposes it is useful to characterize pure schemes in a number of different ways.

Theorem 4.2. The following conditions are equivalent.

a) X is combinatorically pure.

b) The canonical module ω\overline{R} of \overline{R} is generated by its elements of minimal degree (which is −σ_{X} − 1).

c) For all m, n ≥ 0 with m+n ≤ σ_{X} +1, the multiplication map \overline{R}_{m}⊗\overline{R}_{n} ⟶ \overline{R}_{m+n} is nondegenerate.

d) The socle 𝔖(\overline{R}) := \{r ∈ \overline{R} : 𝔪r = 0\} is precisely 𝔖(\overline{R}) = \overline{R}_{σ_{X}+1}.

Proof: The equivalence of a) and b) follows from the facts that x_{0} is a nonzerodivisor of ω_{R} and that ω\overline{R} ≅ (ω_{R}/x_{0}ω_{R})(1) by applying the graded version of Nakayama’s Lemma. Clearly, b) is equivalent with the multiplication maps \overline{R}_{n} ⊗ (ω\overline{R})_{−σ_{X}−1} ⟶ (ω\overline{R})_{−σ_{X}−1+n} being surjective for n = 0, …, σ_{X} + 1. Using ω\overline{R} ≅ \overline{R}^{∗}, a standard dualization argument shows that it is also equivalent with the multiplication maps \overline{R}_{n} ⊗ \overline{R}_{σ_{X}+1−n} ⟶ \overline{R}_{σ_{X}+1} being nondegenerate for n = 0, …, σ_{X} + 1. This in turn is easily seen to be equivalent with c) and d). □

The proof of the following corollary is based on an observation of L. Robbiano.

Corollary 4.3. If X is combinatorically pure, then X is pure.

Proof: If r ∈ 𝔖(\overline{R}), then Lt_{τ} (r) ∈ 𝔖(gr_{τ} (\overline{R})), because Lt_{τ} (x_{i}) · Lt_{τ} (r) is either zero or it equals Lt_{τ} (x_{i}r) = Lt_{τ} (0) = 0. Now 𝔖(gr_{τ} (\overline{R})) = gr_{τ} (\overline{R})_{σ_{X}+1} because of our assumption about the maximal elements of M. Hence r ∈ \overline{R}_{σ_{X}+1}. The other inclusion is clear, so that 4.2.d yields the claim. □

In other words, Theorem 4.2.d means that \overline{R} is a level ring. Using the other conditions of that theorem, we can derive some implications for the Hilbert functions of pure schemes which have been announced first in [S].

Corollary 4.4. Let X ⊆ ℙ^{d} be pure.

a) For all m, n ≥ 0 with m+n ≤ σ_{X} +1 we habe ΔH_{X}(m) ≤ ΔH_{X}(n)·ΔH_{X}(m+n).

b) The function f(n) := ΔH_{X}(σ_{X} + 1 − n) is the sum of Δ_{X} Hilbert functions of quotient rings of \overline{R}.

c) If Y ⊆ X is a subscheme with deg Y = H_{X}(n) for some n ∈ {0, …, σ_{X} + 1}, and if H_{Y} = min{H_{X}, deg Y }, then Y is pure again. In particular, H_{Y} satisfies the above restrictions, too.

Proof: “a)” follows from 4.2.c, since that condition implies that the canonical map \overline{R}_{m} ⟶ Hom_{k}(\overline{R}_{n}, \overline{R}_{m+n}) is injective.

“b)” follows from 4.2.b by induction on Δ_{X}. If φ_{1}, …, φ_{Δ_{X}} ∈ (ω\overline{R})_{−σ_{X}−1} is a minimal set of generators of ω\overline{R}, then ω\overline{R}/\overline{R}φ_{1} is generated by the images of φ_{2}, …, φ_{Δ_{X}}, while \overline{R}φ_{1} ≅ (\overline{R}/Ann\overline{R}(φ_{1}))(σ_{X} +1) has (up to a shift) the Hilbert function of a quotient ring of \overline{R}. Now note that f(n) is nothing but the Hilbert function of ω\overline{R} shifted by σ_{X}+1.

“c)” follows from 4.2.c and 4.2.d, since R/(I_{Y/X} + (x_{0})) ≅ ⨁_{i=0}^{n} \overline{R}_{i} has its socle

concentrated in degree n. □

In view of Corollary 4.3 it would be natural to conjecture that any pure scheme has a flawless Hilbert function. Unfortunately, this seems not to be true, since there are examples of 0-dimensional standard graded Gorenstein rings \overline{R} (i.e. level rings of type one) whose Hilbert functions are not unimodal. Here H_{X} (or H\overline{R}) is called unimodal, if there exists an n ∈ {0, …, σ_{X} + 1} such that ΔH_{X}(0) ≤ … ≤ ΔH_{X}(n) ≥ … ≥ ΔH_{X}(σ_{X} + 1). Thus we would like to pose the following more modest Question. If X is pure and sufficiently uniform (e.g. in uniform position), does this imply that H_{X} is flawless?

Let us show how one can use the uniformity assumption in a discussion of this question.

Remark. Suppose we can show that any Δ_{X}-uniform, pure scheme satisfies ΔH_{X}(n) ≤ ΔH_{X}(σ_{X} + 1 − n) for 0 ≤ n ≤ σ_{X} + 1. Then any pure scheme in uniform position has a flawless Hilbert function. Namely, we can show ΔH_{X}(n−1) ≤ ΔH_{X}(n) in the specified range by considering a subscheme Y ⊆ X of degree deg Y = H_{X}(2n), and by applying 4.4.c and the assumption to it.

Moreover, the above uniformity assumption seems natural, because any pure scheme has at least some uniformity.

Theorem 4.5. If X is pure, then X is a Cayley-Bacharach scheme.

Proof: Let Y ⊆ X be a subscheme of degree deg Y = deg X −1, and let I_{Y/X} ⊆ R be its defining ideal. Choose f ∈ I_{Y/X} \ {0} of minimal degree. Then (I_{Y/X})_{deg f+1} = kx_{0}f implies that the image of f in \overline{R} is an element of the socle of \overline{R}. Hence deg f = σ_{X} + 1, i.e. X is a Cayley-Bacharach scheme (cf. [GKR]). □

Finally, we would like to point out an interesting consequence, if the above question has a positive answer.

Theorem 4.6. Let char k = 0. Suppose any pure scheme in uniform position has a flawless Hilbert function. Then the Hilbert function of any n-dimensional standard graded Gorenstein domain S has a unimodal difference function Δ^{n}H_{S}.

Proof: Since the case dim S = 1, S = k[X] is trivial, we can assume n ≥ 2. Let d+1 = embdim (S), and let V ⊆ ℙ^{d} be the projective variety with projective coordinate ring S. Since dim V = n − 1, we can take n − 1 generic hyperplane sections and arrive at a set of points X ⊆ ℙ^{d−n−1} which is in uniform position by the Uniform Position Lemma. Notice that H_{X} = Δ^{n−1}H_{S} and X is arithmetically Gorenstein, hence pure. By assumption, H_{X} is flawless, and therefore Δ^{n}H_{S} = ΔH_{X} is symmetrical and unimodal.

□

References

[GKR]A.V. Geramita, M. Kreuzer and L. Robbiano, Cayley-Bacharach schemes and

their canonical modules, Trans. Amer. Math. Soc. (to appear)

[GW] S. Goto and K. Watanabe, On graded rings I, J. Math. Soc. Japan 30 (1978), 179–213

[H] T. Hibi, What can be said about pure O-sequences?, J. Comb. Theory 50 (1989), 319–322

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Acknowledgements. Part of this research was done while the author was staying at Queen’s University, Kingston. He wishes to thank A.V. Geramita for his hospitality and encouragement. Furthermore, the author is indebted to L. Robbiano and L.G. Roberts for stimulating discussions, and to E. Kunz for the opportunity to lecture about a preliminary version of this paper in his seminar. Last but not least, cordial thanks to the organizers of the Ravello conference for providing such a stimulating and warm atmosphere.

AMS classification: 13C05, 14C99, 14N05

Keywords: canonical module, 0-dimensional scheme, Cayley-Bacharach property, Hilbert function

Address of the author:

Fakultät für Mathematik

Universität Regensburg

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D - W 8400 Regensburg

Deutschland