On the Dedekind different of a Cayley-Bacharach scheme

TL;DR

This paper characterizes Cayley-Bacharach schemes in projective space using the algebraic structure of the Dedekind different, linking geometric properties with algebraic invariants and exploring Gorenstein conditions.

Contribution

It provides new algebraic characterizations of Cayley-Bacharach schemes via Dedekind different and explores properties of almost Gorenstein and nearly Gorenstein schemes.

Findings

Characterization of Cayley-Bacharach property through Dedekind different

Use of Dedekind's formula for the conductor and the complementary module

Analysis of schemes with minimal Dedekind different and Gorenstein properties

Abstract

Given a 0-dimensional scheme $\mathbb{X}$ in a projective space $\mathbb{P}^n_K$ over a field $K$, we characterize the Cayley-Bacharach property of $\mathbb{X}$ in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley-Bacharach schemes by Dedekind's formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.

Equations

$$\ast$$

$$Q^h(R)\cong \prod _{j=1}^s{\mathcal{O}}_{\mathbb{X},p_j}[T_j,T_j^{-1}]\cong R_{x_0}$$

$$\bigg{\{}\begin{array}[]{c}\textrm{maximal}\ p_{j}\textrm{-subschemes}\\ \textrm{of the scheme}\ \mathbb{X}\end{array}\bigg{\}}\longleftrightarrow\bigg{\{}\begin{array}[]{c}\textrm{ideals}\ \langle(0,\dots,0,s_{j},0,\dots,0)\rangle_{\Gamma}\subseteq\Gamma\\ \textrm{with}\ s_{j}\in\mathfrak{G}({\mathcal{O}}_{\mathbb{X},p_{j}})\setminus\{0\}\end{array}\bigg{\}}.$$

$$\tilde{ı}:R⟶Q^h(R)\cong \prod _{j=1}^s{\mathcal{O}}_{\mathbb{X},p_j}[T_j,T_j^{-1}]$$

$$\mu (a):=\min \{i\in \mathbb{N}\mid (0,\dots ,0,a{T}_j^i,0,\dots ,0)\in \tilde{ı}(R)\}$$

$$f_{j{k}_j}^{\ast }:={\tilde{ı}}^{-1}((0,\dots ,0,e_{j{k}_j}{s}_j{T}_j^{\mu (e_{j{k}_j}{s}_j)},0,\dots ,0)).$$

$${\mu }_{\mathbb{Y}/\mathbb{X}}:=\max \{\mathrm{deg}(f_{j{k}_j}^{\ast })\mid k_j=1,\dots ,{\varkappa }_j\}$$

$$\operatorname{ri}(M)=\min\big{\{}\,i\in\mathbb{Z}\mid\operatorname{HF}_{M}(j)=\operatorname{HP}_{M}(j)\ \textrm{for all}\ j\geq i\,\big{\}}$$

$$1={\mathrm{HF}}_{\mathbb{X}}(0)<{\mathrm{HF}}_{\mathbb{X}}(1)<\cdots <{\mathrm{HF}}_{\mathbb{X}}(r_{\mathbb{X}}-1)<\mathrm{deg}(\mathbb{X})$$

$${\mathrm{HF}}_{\mathbb{Y}}(i)=\{\begin{array}{ll}{\mathrm{HF}}_{\mathbb{X}}(i)& \text{if}i<{\alpha }_{\mathbb{Y}/\mathbb{X}},\\ \le {\mathrm{HF}}_{\mathbb{X}}(i)-1& \text{if}{\alpha }_{\mathbb{Y}/\mathbb{X}}\le i<{\mu }_{\mathbb{Y}/\mathbb{X}},\\ {\mathrm{HF}}_{\mathbb{X}}(i)-{\varkappa }_j& \text{if}i\ge {\mu }_{\mathbb{Y}/\mathbb{X}}.\end{array}$$

$$\Delta\operatorname{HF}_{\mathbb{Y}}(i)=\Delta\operatorname{HF}_{\mathbb{X}}(i)-\#\big{\{}\,k\in\{1,\dots,\varkappa_{j}\}\mid\operatorname{deg}(f^{*}_{jk})=i\,\big{\}}.$$

$$\tilde{ı}(f)=(0,\dots ,0,a{s}_j{T}_j^i,0,\dots ,0)\in Q^h(R)$$

$$h={\tilde{ı}}^{-1}((0,\dots ,0,b{a}^{-1}{T}_j^{r_{\mathbb{X}}},0,\dots ,0))\in R_{r_{\mathbb{X}}}.$$

$${\varkappa }_j={\dim }_K⟨g_{j1},\dots ,g_{j{\varkappa }_j}{⟩}_K\le {\dim }_K(I_{\mathbb{Y}/\mathbb{X}}{)}_{{\mu }_{\mathbb{Y}/\mathbb{X}}}\le {\varkappa }_j.$$

$${\mathrm{HF}}_{I_{\mathbb{Y}/\mathbb{X}}}(r_{\mathbb{X}})=\mathrm{deg}(\mathbb{X})-{\mathrm{HF}}_{\mathbb{Y}}(r_{\mathbb{X}})>\mathrm{deg}(\mathbb{X})-\mathrm{deg}(\mathbb{Y})={\varkappa }_j,$$

$$d_{{\alpha }_{\mathbb{Y}/\mathbb{X}}+i}={\mathrm{HF}}_{I_{\mathbb{Y}/\mathbb{X}}}({\alpha }_{\mathbb{Y}/\mathbb{X}}+i)-{\mathrm{HF}}_{I_{\mathbb{Y}/\mathbb{X}}}({\alpha }_{\mathbb{Y}/\mathbb{X}}+i-1)$$

$$\displaystyle\Big{\{}x_{0}^{i}f^{*}_{j1},\dots,x_{0}^{i}f^{*}_{jd_{\alpha_{\mathbb{Y}/\mathbb{X}}}},\dots,$$

$$\displaystyle\hskip 44.10185ptf^{*}_{j{\scriptscriptstyle\sum\limits_{0\leq k<i}d_{\alpha_{\mathbb{Y}/\mathbb{X}}+k}}+1},\dots,f^{*}_{j{\scriptscriptstyle\sum\limits_{0\leq k\leq i}d_{\alpha_{\mathbb{Y}/\mathbb{X}}+k}}}\Big{\}}$$

$$\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{X}}}(i)=\#\big{\{}\,k\in\{1,\dots,\varkappa_{j}\}\mid\operatorname{deg}(f^{*}_{jk})\leq i\,\big{\}}$$

$$\Delta {\mathrm{HF}}_{\mathbb{Y}}(i)$$

$$=({\mathrm{HF}}_{\mathbb{X}}(i)-{\mathrm{HF}}_{I_{\mathbb{Y}/\mathbb{X}}}(i))-({\mathrm{HF}}_{\mathbb{X}}(i-1)-{\mathrm{HF}}_{I_{\mathbb{Y}/\mathbb{X}}}(i-1))$$

$$=\Delta {\mathrm{HF}}_{\mathbb{X}}(i)-({\mathrm{HF}}_{I_{\mathbb{Y}/\mathbb{X}}}(i)-{\mathrm{HF}}_{I_{\mathbb{Y}/\mathbb{X}}}(i-1))$$

$$\displaystyle=\Delta\operatorname{HF}_{\mathbb{X}}(i)-\#\big{\{}\,k\in\{1,\dots,\varkappa_{j}\}\mid\operatorname{deg}(f^{*}_{jk})=i\,\big{\}}.$$

$${\mathrm{HF}}_{\mathbb{Y}}(i)=\{\begin{array}{ll}{\mathrm{HF}}_{\mathbb{X}}(i)& \text{for}i<{\alpha }_{\mathbb{Y}/\mathbb{X}},\\ {\mathrm{HF}}_{\mathbb{X}}(i)-1& \text{for}i\ge {\alpha }_{\mathbb{Y}/\mathbb{X}}\end{array}$$

$$f_{j{k}_j}\cdot f_{j^{\prime }{k}_{j^{\prime }}}\in \{\begin{array}{ll}{x}_0^{r_{\mathbb{X}}}⟨f_{j1},\dots ,f_{j{\varkappa }_j}{⟩}_K& \text{if}j=j^{\prime }\text{and}{\dim }_{\kappa (p_j)}({\mathcal{O}}_{\mathbb{X},p_j})=1,\\ ⟨0⟩& \text{otherwise}.\end{array}$$

$$f\cdot f_{jl}\in (I_{\mathbb{Y}/\mathbb{X}}{)}_{r_{\mathbb{X}}+i}=⟨x_0^i{f}_{j1},\dots ,x_0^i{f}_{j{\varkappa }_j}{⟩}_K$$

$$\{x_0^{i-r_{\mathbb{X}}}{f}_{11},\dots ,x_0^{i-r_{\mathbb{X}}}{f}_{1{\varkappa }_1},\dots ,x_0^{i-r_{\mathbb{X}}}{f}_{s1},\dots ,x_0^{i-r_{\mathbb{X}}}{f}_{s{\varkappa }_s}\}$$

$$\tilde{ı}(x_0^{i-r_{\mathbb{X}}}{f}_{j{k}_j})=(0,\dots ,0,e_{j{k}_j}{T}_j^i,0,\dots ,0)\in Q^h(R)$$

$$\Phi :{\omega }_R(1)$$

$$\phi$$

$${\mathrm{HF}}_{{\mathfrak{C}}_{\mathbb{X}}^{\sigma }}(i)=\mathrm{deg}(\mathbb{X})-{\mathrm{HF}}_{\mathbb{X}}(-i-1)\text{for all}i\in \mathbb{Z}.$$

$${\delta }_{\mathbb{X}}^{\sigma }=({\mathfrak{C}}_{\mathbb{X}}^{\sigma }{)}^{-1}=\{f\in Q^h(R)\mid f\cdot {\mathfrak{C}}_{\mathbb{X}}^{\sigma }\subseteq R\},$$

$$f_{j{k}_j}:={\tilde{ı}}^{-1}((0,\dots ,0,e_{j{k}_j}{T}_j^{r_{\mathbb{X}}},0,\dots ,0))$$

Full text

1. Introduction

Let $K$ be a field, and let $\mathbb{P}^{n}_{K}$ be the $n$-dimensional projective space over $K$. We are interested in studying 0-dimensional subschemes $\mathbb{X}$ of $\mathbb{P}^{n}_{K}$. Classically, the Cayley-Bacharach property of a reduced scheme has been defined to mean that all hypersurfaces of a certain degree which pass through all points of $\mathbb{X}$ but one automatically pass through the last point. Here we generalize this definition to arbitrary 0-dimensional subschemes of $\mathbb{P}^{n}_{K}$ over an arbitrary field $K$. In [4], Geramita et al. used the canonical module $\omega_{R}$ of the homogeneous coordinate ring $R$ of $\mathbb{X}$ to characterize the Cayley-Bacharach property algebraically when $\mathbb{X}$ is reduced and $K$ is algebraically closed. Later, in [11] and [9], this result was generalized to arbitrary 0-dimensional schemes with $K$-rational support.

In this paper we use the Dedekind different to study the Cayley-Bacharach property. The Dedekind different $\delta_{\mathbb{X}}^{\sigma}$ of $R$ is the inverse ideal of its Dedekind complementary module $\mathfrak{C}_{\mathbb{X}}^{\sigma}$ in its homogeneous ring of quotients $Q^{h}(R)$. Here the module $\mathfrak{C}_{\mathbb{X}}^{\sigma}$ is a fractional ideal of $Q^{h}(R)$, which is defined if $\mathbb{X}$ is locally Gorenstein, and $\sigma$ is a fix homogeneous trace map. Theorem 4.5, one of our main results, characterizes Cayley-Bacharach schemes, i.e., schemes having the Cayley-Bacharach property of maximal degree $r_{\mathbb{X}}-1$, in terms of the structure of their Dedekind different $\delta_{\mathbb{X}}^{\sigma}$. Another main result, Theorem 5.7, characterizes Cayley-Bacharach schemes as the ones for which Dedekind’s formula for the conductor and the Dedekind complementary module holds true. Applications include several characterizations of schemes $\mathbb{X}$ with minimal Dedekind different and a characterization of almost Gorenstein scheme $\mathbb{X}$ by the nearly Gorenstein and the Cayley-Bacharach properties.

In the following we describe the contents of the paper in more detail. Section 2 starts by recalling the notion of maximal subschemes, minimal separators and the maximal degree of a minimal separator. We describe the Hilbert function of a maximal subscheme of $\mathbb{X}$, define standard sets of separators, and use them to control the ring structure of $R$ in degrees $\geq r_{\mathbb{X}}$, where $r_{\mathbb{X}}$ is the regularity index of $\mathbb{X}$.

Next, in Section 3, we rework the construction of the Dedekind complementary module $\mathfrak{C}_{\mathbb{X}}^{\sigma}$ from the local case given in [8]. Then we work out explicit descriptions of its homogeneous components and its Hilbert function. As mentioned above, the Dedekind different $\delta_{\mathbb{X}}^{\sigma}$ is defined as the inverse ideal of $\mathfrak{C}_{\mathbb{X}}^{\sigma}$. We provide its Hilbert function, Hilbert polynomial, and a sharp bound for its regularity index. If the containment $\bigoplus_{i\geq 2r_{\mathbb{X}}}R_{i}\subseteq\delta_{\mathbb{X}}^{\sigma}$ is an equality, we say that $\mathbb{X}$ has minimal Dedekind different. For reduced schemes $\mathbb{X}$ in $\mathbb{P}^{2}_{K}$, we show that this condition implies that $\delta_{\mathbb{X}}^{\sigma}$ agrees with the Kähler different of $\mathbb{X}$.

Section 4 starts with the general definition of the Cayley-Bacharach property of degree $d$ (in short, CBP($d$)) and of Cayley-Bacharach schemes. The main result of this section is Theorem 4.5. It shows that a [math]-dimensional locally Gorenstein scheme $\mathbb{X}$ is a Cayley-Bacharach scheme if any only if the Dedekind different $\delta^{\sigma}_{\mathbb{X}}$ satisfies

[TABLE]

for all $p_{j}\in\operatorname{Supp}(\mathbb{X})$ and every maximal $p_{j}$-subscheme $\mathbb{Y}\subseteq\mathbb{X}$. This theorem allows us to detect Cayley-Bacharach schemes by looking at a single homogeneous component of the Dedekind different. Moreover, we can describe the growth of the Hilbert function of the Dedekind different of a Cayley-Bacharach scheme and determine its regularity index (see Proposition 4.8). A property similar to $(\ast)$ allows us to detect the Cayley-Bacharach property of any degree (see Proposition 4.10), but is not equivalent to it in general (see Example 4.11).

In Section 5 we look at the conductor $\mathfrak{F}_{\widetilde{R}/R}$ of $R$ in the ring $\widetilde{R}=\prod_{i=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{i}}[T_{i}]$, where $T_{1},\dots,T_{s}$ are indeterminates. If $\mathbb{X}$ is reduced, this is the classical conductor of $R$ in its integral closure. After showing a chain of inclusions $\mathfrak{F}_{\widetilde{R}/R}^{2}\subseteq\delta_{\mathbb{X}}^{\sigma}\subseteq\mathfrak{F}_{\widetilde{R}/R}$ between the conductor and the Dedekind different, we generalize a result of [4] which characterizes Cayley-Bacharach schemes in terms of their conductors. More precisely, we prove that the Cayley-Bacharach property of degree $d$ is equivalent to $\mathfrak{F}_{\widetilde{R}/R}\subseteq\bigoplus_{i\geq d+1}R_{i}$, and that $\mathbb{X}$ is a Cayley-Bacharach scheme if and only if $\mathfrak{F}_{\widetilde{R}/R}=\bigoplus_{i\geq r_{\mathbb{X}}}R_{i}$ (see Theorem 5.4). A further main result is the generalization of Dedekind’s formula $\mathfrak{F}_{\widetilde{R}/R}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}=\widetilde{R}$ for the conductor and the Dedekind complementary module given in Theorem 5.7. These theorems have a number of applications to schemes with minimal Dedekind different, to locally Gorenstein schemes, and to Cayley-Bacharach schemes (see Proposition 5.5, Corollary 5.9 and Corollary 5.10).

In the last section we use the trace of the Dedekind complementaty module to characterize schemes with minimal Dedekind different by the Cayley-Bacharach property and by $\mathfrak{F}_{\widetilde{R}/R}={\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$. Moreover, we provide a number of contributions to the topics of nearly Gorenstein and almost Gorenstein schemes which have received some attention lately (see [2, 5, 7]). Among others, we prove an analogue of [7, Proposition 6.1] in our setting, which characterizes almost Gorenstein schemes by the nearly Gorenstein property and one value of the Hilbert function of the Dedekind different. Further applications to the case $\Delta_{\mathbb{X}}=\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{X}})-\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{X}}-1)=1$, to Cayley-Bacharach schemes, and to level schemes follow. In particular, we point out that every almost Gorenstein scheme is nearly Gorenstein. In the case $\Delta_{\mathbb{X}}=1$, the converse of this property holds true if $\mathbb{X}$ is a Cayley-Bacharach scheme (see Proposition 6.8). Moreover, we show that $\mathbb{X}$ is almost Gorenstein if and only if it is a Cayley-Bacharach scheme and $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(r_{\mathbb{X}}+1)=\operatorname{HF}_{\mathbb{X}}(1)$ (see Proposition 6.10), and provide a different proof of a result in [5, 10.2-4] when the graded ring has dimension one. In our setting, this result states that a 0-dimensional locally Gorenstein scheme with $\Delta_{\mathbb{X}}\geq 2$ is level and almost Gorenstein if and only if $r_{\mathbb{X}}=1$. Finally, we show that an almost Gorenstein $(2,r_{\mathbb{X}}-1)$-uniform set $\mathbb{X}$ of distinct $K$-rational points with $r_{\mathbb{X}}\geq 2$ satisfies $\Delta_{\mathbb{X}}=1$.

Unless explicitly mentioned otherwise, we use the definitions and notation introduced in the books [14, 15, 16]. All examples in this paper were calculated by using the computer algebra system ApCoCoA (see [1]).

2. Separators of Maximal $p_{j}$-Subschemes

Throughout the paper, we work over an arbitrary field $K$. By $\mathbb{P}^{n}_{K}$ we denote the projective $n$-space over $K$. The homogeneous coordinate ring of $\mathbb{P}^{n}_{K}$ is the polynomial ring $P=K[X_{0},\dots,X_{n}]$ equipped with the standard grading. We are interested in studying a 0-dimensional subscheme $\mathbb{X}$ of $\mathbb{P}^{n}_{K}$. Its homogeneous vanishing ideal in $P$ is denoted by $I_{\mathbb{X}}$. The homogeneous coordinate ring of $\mathbb{X}$ is then given by $R:=P/I_{\mathbb{X}}.$ The ring $R$ is a standard graded $K$-algebra. Its homogeneous maximal ideal is denoted by $\mathfrak{m}$.

The set of closed points of $\mathbb{X}$ is called the support of $\mathbb{X}$ and is denoted by $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$. Once and for all, we assume that no point of the support of $\mathbb{X}$ lies on the hyperplane at infinity $\mathcal{Z}(X_{0})$. Consequently, the residue class $x_{0}$ of $X_{0}$ in $R$ is a non-zerodivisor and $R$ is a 1-dimensional Cohen-Macaulay ring. To each point $p_{j}\in\operatorname{Supp}(\mathbb{X})$ we have the associated local ring ${\mathcal{O}}_{\mathbb{X},p_{j}}$. Its maximal ideal is denoted by $\mathfrak{m}_{\mathbb{X},p_{j}}$, and the residue field of $\mathbb{X}$ at $p_{j}$ is denoted by $\kappa(p_{j})$. The degree of $\mathbb{X}$ is defined as $\operatorname{deg}(\mathbb{X})=\sum_{j=1}^{s}\dim_{K}({\mathcal{O}}_{\mathbb{X},p_{j}})$. Furthermore, the homogeneous ring of quotients of $R$, denoted by $Q^{h}(R)$, is defined as the localization of $R$ with respect to the set of all homogeneous non-zerodivisors of $R$. In view of [9, Proposition 3.1], there are isomorphisms of graded $R$-modules

[TABLE]

where $T_{1},\dots,T_{s}$ are indeterminates with $\operatorname{deg}(T_{1})=\cdots=\operatorname{deg}(T_{s})=1$.

The following special class of subschemes of the scheme $\mathbb{X}$ plays an important role in this paper.

Definition 2.1**.**

Let $j\in\{1,\dots,s\}$. A subscheme $\mathbb{Y}\subseteq\mathbb{X}$ is called a $p_{j}$-subscheme if the following conditions are satisfied:

1. (a)

${\mathcal{O}}_{\mathbb{Y},p_{k}}={\mathcal{O}}_{\mathbb{X},p_{k}}$ for $k\neq j$. 2. (b)

The map ${\mathcal{O}}_{\mathbb{X},p_{j}}\twoheadrightarrow{\mathcal{O}}_{\mathbb{Y},p_{j}}$ is an epimorphism.

A $p_{j}$-subscheme $\mathbb{Y}\subseteq\mathbb{X}$ is called maximal if $\operatorname{deg}(\mathbb{Y})=\operatorname{deg}(\mathbb{X})-\dim_{K}\kappa(p_{j})$.

If $\mathbb{X}$ has $K$-rational support (i.e., all closed points of $\mathbb{X}$ are $K$-rational), then a subscheme $\mathbb{Y}\subseteq\mathbb{X}$ of degree $\operatorname{deg}(\mathbb{Y})=\operatorname{deg}(\mathbb{X})-1$ with ${\mathcal{O}}_{\mathbb{Y},p_{j}}\neq{\mathcal{O}}_{\mathbb{X},p_{j}}$ is exactly a maximal $p_{j}$-subscheme of $\mathbb{X}$.

A relationship between maximal $p_{j}$-subschemes of $\mathbb{X}$ and ideals of the product of local rings can be described as follows (cf. [9, Proposition 3.2]).

Proposition 2.2**.**

Let $\Gamma=\prod_{j=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{j}}$, and let $\mathfrak{G}({\mathcal{O}}_{\mathbb{X},p_{j}})={\rm Ann}_{{\mathcal{O}}_{\mathbb{X},p_{j}}}(\mathfrak{m}_{\mathbb{X},p_{j}})$ be the socle of ${\mathcal{O}}_{\mathbb{X},p_{j}}$. There is a 1-1 correspondence

[TABLE]

Let $\mathbb{Y}$ be a maximal $p_{j}$-subscheme of $\mathbb{X}$, let $I_{\mathbb{Y}/\mathbb{X}}$ be the ideal of $\mathbb{Y}$ in $R$, and let $\alpha_{\mathbb{Y}/\mathbb{X}}:=\min\{i\in\mathbb{N}\mid(I_{\mathbb{Y}/\mathbb{X}})_{i}\neq\langle 0\rangle\}$. Furthermore, we let $s_{j}\in\mathfrak{G}({\mathcal{O}}_{\mathbb{X},p_{j}})\setminus\{0\}$ be a socle element corresponding to $\mathbb{Y}$. Then there is a non-zero homogeneous element $f_{\mathbb{Y}}\in(I_{\mathbb{Y}/\mathbb{X}})_{i}$, $i\geq\alpha_{\mathbb{Y}/\mathbb{X}}$, such that $\tilde{\imath}(f_{\mathbb{Y}})=(0,\dots,0,s_{j}T_{j}^{i},0,\dots,0)$. Here the injection

[TABLE]

is the homogeneous map of degree zero given by $\tilde{\imath}(f)=(f_{p_{1}}T_{1}^{i},\dots,f_{p_{s}}T_{s}^{i})$, for $f\in R_{i}$ with $i\geq 0$, where $f_{p_{j}}$ is the germ of $f$ at the point $p_{j}$ of $\operatorname{Supp}(\mathbb{X})$.

Let $\varkappa_{j}:=\dim_{K}\kappa(p_{j})$, and let $\{e_{j1},\dots,e_{j\varkappa_{j}}\}\subseteq{\mathcal{O}}_{\mathbb{X},p_{j}}$ be elements whose residue classes form a $K$-basis of $\kappa(p_{j})$. For $a\in{\mathcal{O}}_{\mathbb{X},p_{j}}$ and for $k_{j}=1,\dots,\varkappa_{j}$, we set

[TABLE]

and

[TABLE]

Definition 2.3**.**

Let $\mathbb{Y}$ be a maximal $p_{j}$-subscheme as above.

1. (a)

The set $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ is called the set of minimal separators of $\mathbb{Y}$ in $\mathbb{X}$ with respect to $s_{j}$ and $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$. 2. (b)

The number

[TABLE]

is called the maximal degree of a minimal separator of $\mathbb{Y}$ in $\mathbb{X}$.

Remark 2.4**.**

Let $\mathbb{Y}$ be a maximal $p_{j}$-subscheme of $\mathbb{X}$.

1. (a)

The maximal degree of a minimal separator of $\mathbb{Y}$ in $\mathbb{X}$ depends neither on the choice of the socle element $s_{j}$ nor on the specific choice of $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$ (see [9, Lemma 3.4]). 2. (b)

Set $U:=\langle(0,\dots,0,s_{j},0,\dots,0)\rangle_{Q^{h}(R)}$. As in the proof of [9, Proposition 4.2], we have $I_{\mathbb{Y}/\mathbb{X}}=\tilde{\imath}^{-1}(U)$ and $\dim_{K}(I_{\mathbb{Y}/\mathbb{X}})_{i}=\dim_{K}U_{i}=\varkappa_{j}$ for $i\gg 0$. In particular, $f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\in I_{\mathbb{Y}/\mathbb{X}}$. 3. (c)

If $\mathbb{X}$ has $K$-rational support, then $\varkappa_{1}=\cdots=\varkappa_{s}=1$ and a minimal separator $f^{*}_{\mathbb{Y}}$ of $\mathbb{Y}$ in $\mathbb{X}$ is nothing but a non-zero element of $(I_{\mathbb{Y}/\mathbb{X}})_{\alpha_{\mathbb{Y}/\mathbb{X}}}$, i.e., $f^{*}_{\mathbb{Y}}$ is a minimal separator of $\mathbb{Y}$ in $\mathbb{X}$ in the sense of [11].

Now we examine the Hilbert function of a maximal $p_{j}$-subscheme of $\mathbb{X}$. Recall that the Hilbert function of a finitely generated graded $R$-module $M$ is a map $\operatorname{HF}_{M}:\mathbb{Z}\rightarrow\mathbb{N}$ given by $\operatorname{HF}_{M}(i)=\dim_{K}(M_{i})$. The unique polynomial $\operatorname{HP}_{M}(z)\in\mathbb{Q}[z]$ for which $\operatorname{HF}_{M}(i)=\operatorname{HP}_{M}(i)$ for all $i\gg 0$ is called the Hilbert polynomial of $M$. The number

[TABLE]

is called the regularity index of $M$ (or of $\operatorname{HF}_{M}$). Whenever $\operatorname{HF}_{M}(i)=\operatorname{HP}_{M}(i)$ for all $i\in\mathbb{Z}$, we let $\operatorname{ri}(M)=-\infty.$ Instead of $\operatorname{HF}_{R}$ we also write $\operatorname{HF}_{\mathbb{X}}$ and call it the Hilbert function of $\mathbb{X}$. Its regularity index is denoted by $r_{\mathbb{X}}.$ Note that $\operatorname{HF}_{\mathbb{X}}(i)=0$ for $i<0$ and

[TABLE]

and $\operatorname{HF}_{\mathbb{X}}(i)=\operatorname{deg}(\mathbb{X})$ for $i\geq r_{\mathbb{X}}$.

Proposition 2.5**.**

Let $\mathbb{Y}\subseteq\mathbb{X}$ be a maximal $p_{j}$-subscheme, let $s_{j}$ be a socle element of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ corresponding to $\mathbb{Y}$, let $\{e_{j1},\dots,e_{j\varkappa_{j}}\}\subseteq{\mathcal{O}}_{\mathbb{X},p_{j}}$ be elements whose residue classes form a $K$-basis of $\kappa(p_{j})$, and let $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ be the set of minimal separators of $\mathbb{Y}$ in $\mathbb{X}$ with respect to $s_{j}$ and $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$. Then the following assertions hold true.

1. (a)

We have $I_{\mathbb{Y}/\mathbb{X}}=\langle f\rangle_{R}^{\mathrm{sat}}$ for every $f\in(I_{\mathbb{Y}/\mathbb{X}})_{i}\setminus\{0\}$ with $i\geq\alpha_{\mathbb{Y}/\mathbb{X}}$, where \langle f\rangle_{R}^{\mathrm{sat}}=\{\,g\in R\mid\mathfrak{m}^{i}g\subseteq\langle f\rangle_{R}\mbox{ for some i\geq 0}\,\} is the saturation of $\langle f\rangle_{R}$. 2. (b)

We have $\alpha_{\mathbb{Y}/\mathbb{X}}\leq\mu_{\mathbb{Y}/\mathbb{X}}\leq r_{\mathbb{X}}$ and the Hilbert function of $\mathbb{Y}$ satisfies

[TABLE] 3. (c)

There is a special choice of a set $\{\,e_{j1},\dots,e_{j\varkappa_{j}}\,\}\subseteq{\mathcal{O}}_{\mathbb{X},p_{j}}$ such that its residue classes form a $K$-basis of $\kappa(p_{j})$, $I_{\mathbb{Y}/\mathbb{X}}\!=\!\langle f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\rangle_{R}$, and for all $i\in\mathbb{Z}$ we have

[TABLE]

Proof.

(a) It is clear that $\langle f\rangle_{R}\subseteq\langle f\rangle_{R}^{\mathrm{sat}}\subseteq I_{\mathbb{Y}/\mathbb{X}}$. For the other inclusion, we use Remark 2.4(b) and write

[TABLE]

for some $a\in{\mathcal{O}}_{\mathbb{Y},p_{j}}\setminus\mathfrak{m}_{\mathbb{X},p_{j}}$. Similarly, for every $g\in(I_{\mathbb{Y}/\mathbb{X}})_{k}$ with $k\geq\alpha_{\mathbb{Y}/\mathbb{X}}$ we have $\tilde{\imath}(g)=(0,\dots,0,bs_{j}T_{j}^{k},0,\dots,0)$ with $b\in{\mathcal{O}}_{\mathbb{X},p_{j}}$. If $b$ is not a unit of ${\mathcal{O}}_{\mathbb{X},p_{j}}$, then $bs_{j}=0$, and so $g=0\in\langle f\rangle_{R}^{\mathrm{sat}}$. Otherwise, since $R_{i+r_{\mathbb{X}}}\cong Q^{h}(R)_{i+r_{\mathbb{X}}}$ for all $i\geq 0$, we let

[TABLE]

Then we have $x_{0}^{r_{\mathbb{X}}+i}\!g=x_{0}^{k}hf\in\!\langle f\rangle_{R}$, and consequently $g\!\in\!\langle f\rangle_{R}^{\mathrm{sat}}$ by [12, Lemma 1.6]. Hence we obtain $I_{\mathbb{Y}/\mathbb{X}}=\langle f\rangle_{R}^{\mathrm{sat}}$.

(b)  Obviously, we have $\alpha_{\mathbb{Y}/\mathbb{X}}\leq\mu_{\mathbb{Y}/\mathbb{X}}$ and $\operatorname{HF}_{\mathbb{Y}}(i)\leq\operatorname{HF}_{\mathbb{X}}(i)-1$ for $\alpha_{\mathbb{Y}/\mathbb{X}}\leq i<\mu_{\mathbb{Y}/\mathbb{X}}$. Now we verify the equality $\operatorname{HF}_{\mathbb{Y}}(i+\mu_{\mathbb{Y}/\mathbb{X}})=\operatorname{HF}_{\mathbb{X}}(i+\mu_{\mathbb{Y}/\mathbb{X}})-\varkappa_{j}$ for all $i\geq 0$. We set $g_{jk_{j}}:=x_{0}^{\mu_{\mathbb{Y}/\mathbb{X}}-\operatorname{deg}(f^{*}_{jk_{j}})}f^{*}_{jk_{j}}\in(I_{\mathbb{Y}/\mathbb{X}})_{\mu_{\mathbb{Y}/\mathbb{X}}}$ for all $k_{j}=1,\dots,\varkappa_{j}$. Then we have $\tilde{\imath}(g_{jk_{j}})=(0,\dots,0,e_{jk_{j}}s_{j}T_{j}^{\mu_{\mathbb{Y}/\mathbb{X}}},0,\dots,0)$. Since $\{e_{j1}s_{j},\dots,e_{j\varkappa_{j}}s_{j}\}$ is $K$-linearly independent, this implies

[TABLE]

So, we get $\dim_{K}(I_{\mathbb{Y}/\mathbb{X}})_{\mu_{\mathbb{Y}/\mathbb{X}}}=\dim_{K}(I_{\mathbb{Y}/\mathbb{X}})_{i+\mu_{\mathbb{Y}/\mathbb{X}}}=\varkappa_{j}$ for all $i\geq 0$. It follows that $\operatorname{HF}_{\mathbb{Y}}(i+\mu_{\mathbb{Y}/\mathbb{X}})=\operatorname{HF}_{\mathbb{X}}(i+\mu_{\mathbb{Y}/\mathbb{X}})-\varkappa_{j}$ for all $i\geq 0$. In particular, $\mu_{\mathbb{Y}/\mathbb{X}}$ is the smallest number $i\in\mathbb{N}$ such that $\operatorname{HF}_{\mathbb{Y}}(i)=\operatorname{HF}_{\mathbb{X}}(i)-\varkappa_{j}$.

Moreover, we see that $\operatorname{HF}_{\mathbb{Y}}(r_{\mathbb{X}})=\operatorname{deg}(\mathbb{Y})$, since otherwise we would have

[TABLE]

which is impossible. Thus $\operatorname{HF}_{\mathbb{Y}}(r_{\mathbb{X}})=\operatorname{deg}(\mathbb{X})-\varkappa_{j}=\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{X}})-\varkappa_{j}$, and hence the inequality $\mu_{\mathbb{Y}/\mathbb{X}}\leq r_{\mathbb{X}}$ holds true.

(c) We may construct the set $\{e_{j1},\dots,e_{j\varkappa_{j}}\}\subseteq{\mathcal{O}}_{\mathbb{X},p_{j}}$ with the desired properties as follows. Let $d_{\alpha_{\mathbb{Y}/\mathbb{X}}}=\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{X}}}(\alpha_{\mathbb{Y}/\mathbb{X}})$ and

[TABLE]

for $i=1,\dots,\mu_{\mathbb{Y}/\mathbb{X}}-\alpha_{\mathbb{Y}/\mathbb{X}}$. Then we have $\varkappa_{j}=d_{\alpha_{\mathbb{Y}/\mathbb{X}}}+d_{\alpha_{\mathbb{Y}/\mathbb{X}}+1}+\cdots+d_{\mu_{\mathbb{Y}/\mathbb{X}}}$. We begin taking a $K$-basis $f^{*}_{j1},\dots,f^{*}_{jd_{\alpha_{\mathbb{Y}/\mathbb{X}}}}$ of $(I_{\mathbb{Y}/\mathbb{X}})_{\alpha_{\mathbb{Y}/\mathbb{X}}}$. For $i=1,\dots,\mu_{\mathbb{Y}/\mathbb{X}}-\alpha_{\mathbb{Y}/\mathbb{X}}$, if $d_{\alpha_{\mathbb{Y}/\mathbb{X}}+i}>0$, we choose $f^{*}_{j{\scriptscriptstyle\sum\limits_{0\leq k<i}d_{\alpha_{\mathbb{Y}/\mathbb{X}}+k}}+1},\dots,f^{*}_{j{\scriptscriptstyle\sum\limits_{0\leq k\leq i}d_{\alpha_{\mathbb{Y}/\mathbb{X}}+k}}}$ such that the set

[TABLE]

forms a $K$-basis of $(I_{\mathbb{Y}/\mathbb{X}})_{\alpha_{\mathbb{Y}/\mathbb{X}}+i}$. Then the ideal $J=\langle f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\rangle_{R}$ is a subideal of $I_{\mathbb{Y}/\mathbb{X}}$ and $\operatorname{HF}_{J}(i)=\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{X}}}(i)$ for all $i\leq\mu_{\mathbb{Y}/\mathbb{X}}$. By (b) we have $\operatorname{HF}_{J}(i)=\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{X}}}(i)=\varkappa_{j}$ for $i\geq\mu_{\mathbb{Y}/\mathbb{X}}$. This implies $I_{\mathbb{Y}/\mathbb{X}}=J=\langle f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\rangle_{R}$. Moreover, it follows from the construction of the set $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ that

[TABLE]

for all $i\in\mathbb{Z}$. Thus we have

[TABLE]

Now let us write $\tilde{\imath}(f^{*}_{jk_{j}})=(0,\dots,0,e_{jk_{j}}s_{j}T_{j}^{\operatorname{deg}(f^{*}_{jk_{j}})},0,\dots,0)$ for $k_{j}=1,\dots,\varkappa_{j}$. Obviously, the set $\{e_{j1}s_{j},\dots,e_{j\varkappa_{j}}s_{j}\}$ is $K$-linearly independent. It remains to show that the residue classes $\{\overline{e}_{j1},\dots,\overline{e}_{j\varkappa_{j}}\}$ form a $K$-basis of $\kappa(p_{j})$. Suppose there are $c_{j1},\dots,c_{j\varkappa_{j}}\in K$ such that $c_{j1}\overline{e}_{j1}+\cdots+c_{j\varkappa_{j}}\overline{e}_{j\varkappa_{j}}=0$. It follows that the element $c_{j1}e_{j1}+\cdots+c_{j\varkappa_{j}}e_{j\varkappa_{j}}$ is contained in $\mathfrak{m}_{\mathbb{X},p_{j}}$. This implies $c_{j1}e_{j1}s_{j}+\cdots+c_{j\varkappa_{j}}e_{j\varkappa_{j}}s_{j}=0$. Since $\{e_{j1}s_{j},\dots,e_{j\varkappa_{j}}s_{j}\}$ is $K$-linearly independent, we deduce $c_{j1}=\cdots=c_{j\varkappa_{j}}=0$. Therefore the set $\{\overline{e}_{j1},\dots,\overline{e}_{j\varkappa_{j}}\}$ is a $K$-basis of $\kappa(p_{j})$, and the conclusion follows. ∎

The set of minimal separators $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ of a maximal $p_{j}$-subscheme $\mathbb{Y}$ in $\mathbb{X}$ as in Proposition 2.5(c) is not necessarily a homogeneous minimal system of generators of $I_{\mathbb{Y}/\mathbb{X}}$, as the following example shows.

Example 2.6**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{2}_{\mathbb{Q}}$ be the [math]-dimensional reduced complete intersection with $I_{\mathbb{X}}=\langle X_{2},X_{0}^{5}X_{1}-\frac{11}{6}X_{0}^{4}X_{1}^{2}+2X_{0}^{3}X_{1}^{3}-2X_{0}^{2}X_{1}^{4}+X_{0}X_{1}^{5}-\frac{1}{6}X_{1}^{6}\rangle$. Then $\mathbb{X}$ contains the set of $\mathbb{Q}$-rational points $\mathbb{Y}=\{(1:0:0),(1:1:0),(1:2:0),(1:3:0)\}$ which is a maximal $p$-subscheme, where $p$ is the closed point corresponding to the homogeneous prime ideal $\mathfrak{P}=\langle X_{1}^{2}+X_{0}^{2},X_{2}\rangle$. We see that $\operatorname{deg}(\mathbb{Y})=4=\operatorname{deg}(\mathbb{X})-2$, and two minimal separators of $\mathbb{Y}$ in $\mathbb{X}$ are $f^{*}_{1}=x_{0}^{3}x_{1}-\frac{11}{6}x_{0}^{2}x_{1}^{2}+x_{0}x_{1}^{3}-\frac{1}{6}x_{1}^{4}$ and $f^{*}_{2}=x_{1}f^{*}_{1}$. Moreover, the equality of the first difference function of $\operatorname{HF}_{\mathbb{Y}}$ in Proposition 2.5(c) holds true, while $I_{\mathbb{Y}/\mathbb{X}}=\langle f^{*}_{1}\rangle_{R}$.

When $\mathbb{Y}\subseteq\mathbb{X}$ is a $p_{j}$-subscheme of degree $\operatorname{deg}(\mathbb{Y})=\operatorname{deg}(\mathbb{X})-1$, we have $\alpha_{\mathbb{Y}/\mathbb{X}}=\mu_{\mathbb{Y}/\mathbb{X}}$ and the Hilbert function of $\mathbb{Y}$ is given by

[TABLE]

(see also [12, Lemma 1.7]). Furthermore, if $\mathbb{X}=\{p_{1},\dots,p_{s}\}$ is a set of distinct $K$-rational points in $\mathbb{P}^{n}_{K}$, we write $p_{j}=(1:p_{j1}:...:p_{jn})$ with $p_{jk}\in K$, and for $f\in R$ we set $f(p_{j}):=F(1,p_{j1},\dots,p_{jn})$ where $F$ is any representative of $f$ in $P$. Then a separator of $\mathbb{X}\setminus\{p_{j}\}$ in $\mathbb{X}$ is an element $f\in R_{r_{\mathbb{X}}}$ such that $f(p_{j})\neq 0$ and $f(p_{k})=0$ for $k\neq j$. In general setting we introduce the following definition.

Definition 2.7**.**

In the setting of Proposition 2.5, we let $f_{jk_{j}}=x_{0}^{r_{\mathbb{X}}-\mu(e_{jk_{j}}s_{j})}f^{*}_{jk_{j}}$ for $k_{j}=1,\dots,\varkappa_{j}$. The set $\{f_{j1},\dots,f_{j\varkappa_{j}}\}$ is called the standard set of separators of $\mathbb{Y}$ in $\mathbb{X}$ with respect to $s_{j}$ and $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$.

Some basic properties of standard sets of separators of a maximal $p_{j}$-subscheme are summarized in the following lemma which generalizes some results in [12, Lemmas 1.9 and 1.10].

Lemma 2.8**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional scheme, let $f\in R_{i}$ with $i\geq 0$, let $\mathbb{Y}$ be a maximal $p_{j}$-subscheme of $\mathbb{X}$, and let $\{f_{j1},\dots,f_{j\varkappa_{j}}\}\subseteq R_{r_{\mathbb{X}}}$ be a standard set of separators of $\mathbb{Y}$ in $\mathbb{X}$.

1. (a)

We have $f\cdot f_{jl}=\sum_{k_{j}=1}^{\varkappa_{j}}c_{jk_{j}l}x_{0}^{i}f_{jk_{j}}$ for some $c_{j1l},\dots,c_{j\varkappa_{j}l}\in K$ and $l\in\{1,\dots,\varkappa_{j}\}$. 2. (b)

If $f\cdot f_{jl}=0$ for some $l\in\{1,\dots,\varkappa_{j}\}$, then $f\cdot f_{j\lambda}=0$ for all $\lambda\in\{1,\dots,\varkappa_{j}\}$. Moreover, $f\cdot f_{jl}\neq 0$ if and only if $f_{p_{j}}\notin\mathfrak{m}_{\mathbb{X},p_{j}}$. 3. (c)

Let $\mathbb{Y}^{\prime}$ be a maximal $p_{j^{\prime}}$-subscheme of $\mathbb{X}$, and let $\{f_{j^{\prime}1},\dots,f_{j^{\prime}\varkappa_{j^{\prime}}}\}\subseteq R_{r_{\mathbb{X}}}$ be a standard set of separators of $\mathbb{Y}^{\prime}$ in $\mathbb{X}$. Then we have

[TABLE]

Proof.

Claim (a) is a consequence of the fact that

[TABLE]

for $l=1,\dots,\varkappa_{j}$. Claims (b) and (c) follow by using the injection $\tilde{\imath}$ and the fact that $(f_{jk_{j}})_{p_{j}}$ is a socle element of $\mathfrak{G}({\mathcal{O}}_{\mathbb{X},p_{j}})$ for $k_{j}=1,\dots,\varkappa_{j}$. ∎

In the case that the scheme $\mathbb{X}$ is reduced and $i\geq r_{\mathbb{X}}$, we can use standard sets of separators of $\mathbb{X}$ to describe a $K$-basis of the vector space $R_{i}$ as follows (see [4, Proposition 1.13(a)] for the case of sets of distinct $K$-rational points).

Corollary 2.9**.**

Let  $\mathbb{X}\,\subseteq\,\mathbb{P}^{n}_{K}$  be a reduced [math]-dimensional scheme with support $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$, let $\{f_{j1},\dots,f_{j\varkappa_{j}}\}\subseteq R_{r_{\mathbb{X}}}$ be a standard set of separators of $\mathbb{X}\setminus\{p_{j}\}$ in $\mathbb{X}$ for $j=1,\dots,s$. Then the set

[TABLE]

is a $K$-basis of $R_{i}$ for every $i\geq r_{\mathbb{X}}$.

Proof.

Since the scheme $\mathbb{X}$ is reduced, we have ${\mathcal{O}}_{\mathbb{X},p_{j}}=\kappa(p_{j})=\mathfrak{G}({\mathcal{O}}_{\mathbb{X},p_{j}})$ for $j=1,\dots,s$. Let $i\geq r_{\mathbb{X}}$. We write

[TABLE]

for $j=1,\dots,s$ and $k_{j}=1,\dots,\varkappa_{j}$, where $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$ is a $K$-basis of ${\mathcal{O}}_{\mathbb{X},p_{j}}$. Then the set $\{\,\tilde{\imath}(x_{0}^{i-r_{\mathbb{X}}}f_{11}),\dots,\tilde{\imath}(x_{0}^{i-r_{\mathbb{X}}}f_{s\varkappa_{s}})\,\}$ is $K$-linearly independent, and so it forms a $K$-basis of $Q^{h}(R)_{i}$. Since $i\geq r_{\mathbb{X}}$, the restriction $\tilde{\imath}|_{R_{i}}:R_{i}\rightarrow Q^{h}(R)_{i}$ is an isomorphism of $K$-vector spaces, it follows that $\{\,x_{0}^{i-r_{\mathbb{X}}}f_{11},\dots,x_{0}^{i-r_{\mathbb{X}}}f_{s\varkappa_{s}}\,\}$ is a $K$-basis of $R_{i}$. ∎

3. Dedekind Differents of 0-Dimensional Schemes

In this section we define and examine the Dedekind complementary module and the Dedekind different for a 0-dimensional scheme $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$. For this we need to restrict our attention to a special class of [math]-dimensional schemes, namely locally Gorenstein schemes. Here we say that $\mathbb{X}$ is locally Gorenstein if the local ring ${\mathcal{O}}_{\mathbb{X},p_{j}}$ is a Gorenstein ring for every point $p_{j}\in\operatorname{Supp}(\mathbb{X})$.

Recall that the graded $R$-module $\omega_{R}=\underline{\operatorname{Hom}}_{K[x_{0}]}(R,K[x_{0}])(-1)$ is called the canonical module of $R$. It is a finitely generated graded $R$-module with Hilbert function $\operatorname{HF}_{\omega_{R}}(i)=\operatorname{deg}(\mathbb{X})-\operatorname{HF}_{\mathbb{X}}(-i)$ for all $i\in\mathbb{Z}$ (see [11, Proposition 1.3]).

In the following we assume that $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ is a 0-dimensional locally Gorenstein scheme and let $L_{0}=K[x_{0},x_{0}^{-1}]$. In this case one can embed the canonical module $\omega_{R}$ of $R$ as a fractional ideal into its homogeneous ring of quotients $Q^{h}(R)$ (see [8] or [16, Appendix G]). Explicitly, this construction is based on the existence of a homogeneous trace map of the graded algebra $Q^{h}(R)/L_{0}$. Recall that a homogeneous trace map of a finite graded algebra $T/S$ is a homogeneous $T$-basis of the graded module $\underline{\operatorname{Hom}}_{S}(T,S)$. For further information on (canonical, homogeneous) trace maps we refer to [16, Appendix F].

The following proposition indicates that the graded algebra $Q^{h}(R)/L_{0}$ has a homogeneous trace map of degree zero, which is shown in [9, Proposition 3.3].

Proposition 3.1**.**

The following statements hold true.

1. (a)

The algebra $Q^{h}(R)/L_{0}$ has a homogeneous trace map $\sigma$ of degree zero. 2. (b)

The map $\Sigma:Q^{h}(R)\rightarrow\underline{\operatorname{Hom}}_{L_{0}}(Q^{h}(R),L_{0})$ given by $\Sigma(1)=\sigma$ is an isomorphism of graded $Q^{h}(R)$-modules.

Now let $\sigma$ be a fixed homogeneous trace map of degree zero of $Q^{h}(R)/L_{0}$. Note that $\sigma\in\underline{\operatorname{Hom}}_{L_{0}}(Q^{h}(R),L_{0})$ satisfies $\underline{\operatorname{Hom}}_{L_{0}}(Q^{h}(R),L_{0})=Q^{h}(R)\cdot\sigma$. Furthermore, there is an injective homomorphism of graded $R$-modules

[TABLE]

The image of $\Phi$ is a homogeneous fractional $R$-ideal $\mathfrak{C}^{\sigma}_{\mathbb{X}}$ of $Q^{h}(R)$. It is also a finitely generated graded $R$-module and

[TABLE]

Definition 3.2**.**

The $R$-module $\mathfrak{C}^{\sigma}_{\mathbb{X}}$ is called the Dedekind complementary module of $\mathbb{X}$ (or of $R/K[x_{0}]$) with respect to $\sigma$. Its inverse,

[TABLE]

is called the Dedekind different of $\mathbb{X}$ (or of $R/K[x_{0}]$) with respect to $\sigma$.

When $\mathbb{X}$ is a finite set of distinct $K$-rational points of $\mathbb{P}^{n}_{K}$, we also denote the Dedekind complementary module (respectively, the Dedekind different) with respect to the canonical trace map by $\mathfrak{C}_{\mathbb{X}}$ (respectively, $\delta_{\mathbb{X}}$).

A system of generators of $\mathfrak{C}^{\sigma}_{\mathbb{X}}$ can be computed as follows.

Remark 3.3**.**

Let $<_{\tau}$ be a degree-compatible term ordering on the set of terms $\mathbb{T}^{n}$ of $K[X_{1},\dots,X_{n}]$, and let $d=\operatorname{deg}(\mathbb{X})$. Then $\mathbb{T}^{n}\setminus\mathrm{LT}_{\tau}(I_{\mathbb{X}}^{\mathrm{deh}})=\{\,T^{\prime}_{1},\dots,T^{\prime}_{d}\,\}$ with $T^{\prime}_{j}=X_{1}^{\alpha_{j1}}\cdots X_{n}^{\alpha_{jn}}$ and $\alpha_{j}=(\alpha_{j1},\dots,\alpha_{jn})\in\mathbb{N}^{n}$ for $j=1,\dots,d$. W.l.o.g. we assume that $T^{\prime}_{1}<_{\tau}\cdots<_{\tau}T^{\prime}_{m}$. Let $t_{j}=T^{\prime}_{j}+I_{\mathbb{X}}\in R$ and set $\operatorname{deg}(t_{j}):=\operatorname{deg}(T^{\prime}_{j})=n_{j}$ for $j=1,\dots,d$. Then $n_{1}\leq\cdots\leq n_{d}\leq r_{\mathbb{X}}$ and the set $\{\,t_{1},\dots,t_{d}\,\}$ is a $K[x_{0}]$-basis of $R$ (cf. [15, Theorem 4.3.22]). Let $\{t_{1}^{*},\dots,t_{d}^{*}\}$ be the dual basis of $\{t_{1},\dots,t_{d}\}$, and let $g_{j}=\Phi(t_{j}^{*})$ for $j=1,\dots,d$. We get $\mathfrak{C}^{\sigma}_{\mathbb{X}}=\langle g_{1},\dots,g_{d}\rangle_{K[x_{0}]}\subseteq Q^{h}(R)$.

Now we want to take a closer look at each homogeneous component of the Dedekind complementary module of $\mathbb{X}$. For this we use the following notation. Let $\nu_{j}:=\dim_{K}({\mathcal{O}}_{\mathbb{X},p_{j}})$ and let $\{e_{j1},\dots,e_{j\nu_{j}}\}$ be a $K$-basis of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ for $j=1,\dots,s$. Using the injection $\tilde{\imath}:R\hookrightarrow Q^{h}(R)$, we set

[TABLE]

for $k_{j}=1,\dots,\nu_{j}$. It is easy to see that $R_{r_{\mathbb{X}}}=\langle f_{11},\dots,f_{1\nu_{1}},\dots,f_{s1},\dots,f_{s\nu_{s}}\rangle_{K}$. Since $\mathbb{X}$ is locally Gorenstein, ${\mathcal{O}}_{\mathbb{X},p_{j}}/K$ has a trace map $\overline{\sigma}_{j}\in\operatorname{Hom}_{K}({\mathcal{O}}_{\mathbb{X},p_{j}},K)$. Also, there is a $K$-basis $\{e^{\prime}_{j1},\dots,e^{\prime}_{j\nu_{j}}\}$ of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ such that

[TABLE]

for all $k_{j},k^{\prime}_{j}=1,\dots,\nu_{j}$. The $K$-basis $\{e^{\prime}_{j1},\dots,e^{\prime}_{j\nu_{j}}\}$ is known as a dual basis of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ to the $K$-basis $\{e_{j1},\dots,e_{j\nu_{j}}\}$ w.r.t. $\overline{\sigma}_{j}$. Moreover, these maps $\overline{\sigma}_{j}$ induce a homogeneous trace map $\sigma$ of degree zero of $Q^{h}(R)/L_{0}$.

A description of the Dedekind complementary module of $\mathbb{X}$ is given by our next proposition.

Proposition 3.4**.**

Using the above notation, let $\Phi$ be the monomorphism of graded $R$-modules defined by (3.1), let $i\geq 0$, and let $\varphi\in(\omega_{R})_{i-r_{\mathbb{X}}+1}$. We write $\varphi(f_{jk_{j}})=c_{jk_{j}}x_{0}^{i}$ with $c_{jk_{j}}\in K$. Then we have

[TABLE]

In particular, $\Phi(\varphi)$ can be identified with the element $x_{0}^{i-2r_{\mathbb{X}}}(\sum_{j=1}^{s}\sum_{k_{j}=1}^{\nu_{j}}c_{jk_{j}}\widetilde{f}_{jk_{j}})$ of $R_{x_{0}}\cong Q^{h}(R)$, where $\widetilde{f}_{jk_{j}}=\tilde{\imath}^{-1}((0,\dots,0,e^{\prime}_{jk_{j}}T_{j}^{r_{\mathbb{X}}},0,\dots,0))\in R_{r_{\mathbb{X}}}$ for all $j=1,\dots,s$ and for all $k_{j}=1,\dots,\nu_{j}$.

Proof.

We set $\epsilon_{jk_{j}}:=(0,\dots,0,e_{jk_{j}},0,\dots,0)\in\prod_{l=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{l}}$ for $j=1,\dots,s$ and $k_{j}=1,\dots,\nu_{j}$. It is not difficult to see that the set $\{\epsilon_{11},\dots,\epsilon_{1\nu_{1}},\dots,\epsilon_{s1},\dots,\epsilon_{s\nu_{s}}\}$ is a $L_{0}$-basis of $Q^{h}(R)$. So, the mapping $\varphi\otimes\mathrm{id}_{L_{0}}:Q^{h}(R)\cong R\otimes_{K[x_{0}]}L_{0}\rightarrow L_{0}$ satisfies

[TABLE]

for $j=1,\dots,s$ and $k_{j}=1,\dots,\nu_{j}$. Thus we have $(\varphi\otimes\mathrm{id}_{L_{0}})(\epsilon_{jk_{j}})=c_{jk_{j}}x_{0}^{i-r_{\mathbb{X}}}$ for all $j=1,\dots,s$ and $k_{j}=1,\dots,\nu_{j}$. On the other hand, we see that

[TABLE]

This implies that we have

[TABLE]

in $\underline{\operatorname{Hom}}_{L_{0}}(Q^{h}(R),L_{0})$. Hence we get

[TABLE]

In addition, we observe that

[TABLE]

Therefore the claim follows. ∎

Next we collect from [9, Proposition 3.7] the following basic properties of the Dedekind different of $\mathbb{X}$.

Proposition 3.5**.**

Let $\sigma$ be a trace map of $Q^{h}(R)/L_{0}$.

1. (a)

The Dedekind different $\delta^{\sigma}_{\mathbb{X}}$ is a homogeneous ideal of $R$ and $x_{0}^{2r_{\mathbb{X}}}\in\delta^{\sigma}_{\mathbb{X}}$. 2. (b)

The Hilbert function of $\delta^{\sigma}_{\mathbb{X}}$ satisfies $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)=0$ for $i<0$, $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)=\operatorname{deg}(\mathbb{X})$ for $i\geq 2r_{\mathbb{X}}$, and

[TABLE] 3. (c)

The regularity index of $\delta^{\sigma}_{\mathbb{X}}$ satisfies $r_{\mathbb{X}}\leq\operatorname{ri}(\delta^{\sigma}_{\mathbb{X}})\leq 2r_{\mathbb{X}}$.

The upper bound for the regularity index of the Dedekind different given in this proposition is attained for a finite set of distinct $K$-rational points, as the next corollary shows.

Corollary 3.6**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{s}\}\subseteq\mathbb{P}^{n}_{K}$ be a set of $s$ distinct $K$-rational points. Then we have $\operatorname{HP}_{\delta_{\mathbb{X}}}(z)=s$ and $\operatorname{ri}(\delta_{\mathbb{X}})=2r_{\mathbb{X}}$.

Proof.

If $n=1$, then $\mathbb{X}$ is a complete intersection, and so $\operatorname{HF}_{\delta_{\mathbb{X}}}(i)=\operatorname{HF}_{\mathbb{X}}(i-s+1)$ for all $i\in\mathbb{Z}$. In particular, we have $\operatorname{ri}(\delta_{\mathbb{X}})=2r_{\mathbb{X}}=(n+1)(s-1)$.

Now suppose that $n\geq 2$. For $j\in\{1,\dots,s\}$, let $f_{j}\in R_{r_{\mathbb{X}}}$ be the separator of $\mathbb{X}\setminus\{p_{j}\}$ in $\mathbb{X}$ with $f(p_{j})=1$ and $f(p_{k})=0$ for $k\neq j$, and let $\overline{f}_{j}$ denote the image of $f_{j}$ in $\overline{R}:=R/\langle x_{0}\rangle$. Set $\Delta_{\mathbb{X}}:=\dim_{K}\overline{R}_{r_{\mathbb{X}}}=\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{X}})-\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{X}}-1)$. Note that $\Delta_{\mathbb{X}}\geq 1$. Since $\{\overline{f}_{1},\dots,\overline{f}_{s}\}$ generates the $K$-vector space $\overline{R}_{r_{\mathbb{X}}}$, we can renumber $\{p_{1},\dots,p_{s}\}$ in such a way that $\{\overline{f}_{1},\dots,\overline{f}_{\Delta_{\mathbb{X}}}\}$ is a $K$-basis of $\overline{R}_{r_{\mathbb{X}}}$. Because $\overline{f}_{i}\neq 0$ for every $i\in\{1,\dots,\Delta_{\mathbb{X}}\}$, this implies $f_{1},\dots,f_{\Delta_{\mathbb{X}}}\notin x_{0}R_{r_{\mathbb{X}}-1}$. For $j=1,\dots,s-\Delta_{\mathbb{X}}$, we write

[TABLE]

where $\beta_{j1},\dots,\beta_{j\Delta_{\mathbb{X}}}\in K$. By [13, Corollary 1.10], the elements

[TABLE]

such that $1\leq j\leq\Delta_{\mathbb{X}}$ form a $K$-basis of $(\mathfrak{C}_{\mathbb{X}})_{-r_{\mathbb{X}}}$.

Now suppose for a contradiction that $\operatorname{HF}_{\delta_{\mathbb{X}}}(2r_{\mathbb{X}}-1)=s$. This implies that $(\delta_{\mathbb{X}})_{2r_{\mathbb{X}}-1}=R_{2r_{\mathbb{X}}-1}$. In particular, we have $x_{0}^{r_{\mathbb{X}}-1}f_{1}\in(\delta_{\mathbb{X}})_{2r_{\mathbb{X}}-1}$. Using Lemma 2.8, we also have

[TABLE]

It follows that $f_{1}\in x_{0}R_{r_{\mathbb{X}}-1}$, a contradiction. Thus we must have $\operatorname{HF}_{\delta_{\mathbb{X}}}(2r_{\mathbb{X}}-1)<s$, and hence $\operatorname{ri}(\delta_{\mathbb{X}})=2r_{\mathbb{X}}$. ∎

In view of the preceding proposition, for a 0-dimensional locally Gorenstein scheme $\mathbb{X}$ the inclusion $\bigoplus_{i\geq 2r_{\mathbb{X}}}R_{i}\subseteq\delta_{\mathbb{X}}^{\sigma}$ always holds true. When this inclusion becomes an equality, we use the following name.

Definition 3.7**.**

We say that $\mathbb{X}$ has minimal Dedekind different if its Dedekind different satisfies $\delta_{\mathbb{X}}^{\sigma}=\bigoplus_{i\geq 2r_{\mathbb{X}}}R_{i}$.

Recall that the Kähler different $\vartheta_{\mathbb{X}}$ of $\mathbb{X}$ is the homogeneous ideal of $R$ generated by all $n$-minors of the Jacobian matrix $\left(\frac{\partial F_{j}}{\partial x_{i}}\right)_{\begin{subarray}{l}i=1,\dots,n\\ j=1,\dots,r\end{subarray}}$, where $\{F_{1},\dots,F_{r}\}$ is a homogeneous system of generators of $I_{\mathbb{X}}$. For finite sets of distinct $K$-rational points in $\mathbb{P}^{2}_{K}$ which have minimal Dedekind different, the Dedekind and Kähler differents agree, as the following corollary shows.

Corollary 3.8**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{s}\}\subseteq\mathbb{P}^{2}_{K}$ be a set of $s$ distinct $K$-rational points. If $\mathbb{X}$ has minimal Dedekind different, then $\delta_{\mathbb{X}}=\vartheta_{\mathbb{X}}$.

Proof.

By [9, Proposition 3.8], we have $\vartheta_{\mathbb{X}}\subseteq\delta_{\mathbb{X}}$. Because $\mathbb{X}$ has minimal Dedekind different, we have $\operatorname{HP}_{\vartheta_{\mathbb{X}}}(2r_{\mathbb{X}}-1)=\operatorname{HF}_{\delta_{\mathbb{X}}}(2r_{\mathbb{X}}-1)=0$. Moreover, it follows from [10, Theorem 2.5] and $n=2$ that $\operatorname{ri}(\vartheta_{\mathbb{X}})\leq nr_{\mathbb{X}}=2r_{\mathbb{X}}$ and $\operatorname{HF}_{\vartheta_{\mathbb{X}}}(i)=s$ for all $i\geq 2r_{\mathbb{X}}$. Thus we obtain $\delta_{\mathbb{X}}=\vartheta_{\mathbb{X}}=\bigoplus_{i\geq 2r_{\mathbb{X}}}R_{i}$. ∎

Example 3.9**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{6}\}\subseteq\mathbb{P}^{2}_{\mathbb{Q}}$ be the set of six points given by $p_{1}=(1:0:0)$, $p_{2}=(1:2:0)$, $p_{3}=(1:2:1)$, $p_{4}=(1:0:2)$, $p_{5}=(1:1:2)$, and $p_{6}=(1:2:2)$. We sketch $\mathbb{X}$ in the affine plane $D_{+}(X_{0})=\mathbb{A}^{2}_{\mathbb{Q}}$ as follows:

[TABLE]

Then $\mathbb{X}$ has the Hilbert function $\operatorname{HF}_{\mathbb{X}}:\ 1\ 3\ 6\ 6\cdots$ and the regularity index $r_{\mathbb{X}}=2$. Moreover, the Dedekind different is given by

[TABLE]

Thus the scheme $\mathbb{X}$ has minimal Dedekind different, and Corollary 3.8 yields that $\delta_{\mathbb{X}}=\vartheta_{\mathbb{X}}=\bigoplus_{i\geq 4}R_{i}$.

Notice that the Dedekind and Kähler differents do not always agree, e.g. when $\mathbb{X}$ is a non-reduced complete intersection in $\mathbb{P}^{2}_{K}$ (see [9, Example 3.9]). However, for finite sets of distinct points in $\mathbb{P}^{2}_{K}$ we propose the following conjecture.

Conjecture 3.10**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{s}\}\subseteq\mathbb{P}^{2}_{K}$ be a set of $s$ distinct $K$-rational points. Then we have $\delta_{\mathbb{X}}=\vartheta_{\mathbb{X}}$.

Recall that a 0-dimensional scheme $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ is an almost complete intersection if $I_{\mathbb{X}}$ is minimally generated by $n+1$ homogeneous polynomials in $P$. The above conjecture holds true when the set $\mathbb{X}$ is an almost complete intersection. This follows from [20, Satz 4], because in this case the Hilbert-Burch Theorem (cf. [19, Theorem 24.2]) implies that $\mathbb{X}$ is also a special almost complete intersection (see [20, Definition 1]). Note that Corollary 3.8 and Conjecture 3.10 are not true in $\mathbb{P}^{3}_{K}$.

Example 3.11**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{9}\}\subseteq\mathbb{P}^{3}_{\mathbb{Q}}$ be the set of nine points given by $p_{1}=(1:0:0:0)$, $p_{2}=(1:1:0:0)$, $p_{3}=(1:1:1:0)$, $p_{4}=(1:1:-1:1)$, $p_{5}=(1:-1:1:1)$, $p_{6}=(1:-2:1:0)$, $p_{7}=(1:-2:2:0)$, $p_{8}=(1:-1:2:1)$, and $p_{9}=(1:0:2:0)$. We have $\operatorname{HF}_{\mathbb{X}}:\ 1\ 4\ 9\ 9\cdots$ and $r_{\mathbb{X}}=2$. In this case the Hilbert functions of the Kähler and Dedekind differents are given by

[TABLE]

It follows that $\delta_{\mathbb{X}}=\bigoplus_{i\geq 2r_{\mathbb{X}}}R_{i}$, and so $\mathbb{X}$ has minimal Dedekind different. However, we have $\vartheta_{\mathbb{X}}=\bigoplus_{i\geq 2r_{\mathbb{X}}+1}R_{i}\subsetneq\delta_{\mathbb{X}}$.

4. The Cayley-Bacharach Property

In this section we relate the algebraic structure of the Dedekind different to the Cayley-Bacharach property of a 0-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^{n}_{K}$. First we use the notion of the maximal degree of a minimal separator introduced in Section 1 to define the degree of a point in $\mathbb{X}$.

Definition 4.1**.**

For every $p_{j}\in\operatorname{Supp}(\mathbb{X})$, the degree of $p_{j}$ in $\mathbb{X}$ is defined as

[TABLE]

Obviously, we have $\operatorname{deg}_{\mathbb{X}}(p_{j})\leq r_{\mathbb{X}}$ for all $j=1,\dots,s$. In case all points of $\operatorname{Supp}(\mathbb{X})$ have degree greater than some natural number $d$, we have the following notion.

Definition 4.2**.**

Let $d\geq 0$, let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional scheme, and let $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$. We say that $\mathbb{X}$ has the Cayley-Bacharach property of degree $d$ (in short, $\mathbb{X}$ has CBP($d$)) if every point $p_{j}\in\operatorname{Supp}(\mathbb{X})$ has degree $\operatorname{deg}_{\mathbb{X}}(p_{j})\geq d+1$. In the case that $\mathbb{X}$ has CBP($r_{\mathbb{X}}-1$) we also say that $\mathbb{X}$ is a Cayley-Bacharach scheme.

If $\mathbb{X}$ has CBP($d$), then $\mathbb{X}$ has CBP($d-1$), and every [math]-dimensional scheme $\mathbb{X}$ with $\operatorname{deg}(\mathbb{X})\geq 2$ has CBP([math]). Moreover, the number $r_{\mathbb{X}}-1$ is the largest degree $d\geq 0$ such that $\mathbb{X}$ can have CBP($d$). So, it suffices to consider the Cayley-Bacharach property in degree $d\in\{0,\dots,r_{\mathbb{X}}-1\}$.

The following proposition gives a characterization of Cayley-Bacharach property using standard sets of separators of $\mathbb{X}$.

Proposition 4.3**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional scheme, let $0\leq d\leq r_{\mathbb{X}}-1$, let $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$, and let $\varkappa_{j}=\dim\kappa(p_{j})$. Then the following statements are equivalent.

1. (a)

The scheme $\mathbb{X}$ has CBP($d$). 2. (b)

If $\mathbb{Y}\subseteq\mathbb{X}$ is a maximal $p_{j}$-subscheme and $\{f_{j1},\dots,f_{j\varkappa_{j}}\}$ is a standard set of separators of $\mathbb{Y}$ in $\mathbb{X}$, then there exists $k_{j}\in\{1\dots,\varkappa_{j}\}$ such that $x_{0}^{r_{\mathbb{X}}-d}\nmid f_{jk_{j}}$. 3. (c)

For all $p_{j}\in\operatorname{Supp}(\mathbb{X})$, every maximal $p_{j}$-subscheme $\mathbb{Y}\subseteq\mathbb{X}$ satisfies

[TABLE]

Proof.

Let $\mathbb{Y}$ be a maximal $p_{j}$-subscheme of $\mathbb{X}$ and $\{f_{j1},\dots,f_{j\varkappa_{j}}\}$ a standard set of separators of $\mathbb{Y}$ in $\mathbb{X}$. If we write $f_{jk_{j}}=x_{0}^{r_{\mathbb{X}}-\operatorname{deg}(f^{*}_{jk_{j}})}f^{*}_{jk_{j}}$ with $f^{*}_{jk_{j}}\in R_{\operatorname{deg}(f^{*}_{jk_{j}})}\setminus x_{0}R_{\operatorname{deg}(f^{*}_{jk_{j}})-1}$ for $k_{j}=1,\dots,\varkappa_{j}$, then the set $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ is a set of minimal separators of $\mathbb{Y}$ in $\mathbb{X}$. Hence the equivalence of (a) and (b) follows.

Now we prove the equivalence of (a) and (c). We always have $\dim_{K}(I_{\mathbb{Y}/\mathbb{X}})_{i}\leq\varkappa_{j}$ for $i\geq 0$. Moreover, we see that $\dim_{K}(I_{\mathbb{Y}/\mathbb{X}})_{d}=\varkappa_{j}$ if and only if $\operatorname{deg}(f^{*}_{jk_{j}})\leq d$ for all $k_{j}=1,\dots,\varkappa_{j}$. This is equivalent to $\operatorname{deg}_{\mathbb{X}}(p_{j})\leq d$. Thus the claim follows. ∎

Let us apply the proposition to a concrete case.

Example 4.4**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{2}_{\mathbb{Q}}$ be the 0-dimensional scheme of degree $8$ with support $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{6}\}$, where $p_{1}=(1:0:0)$, $p_{2}=(1:1:0)$, $p_{3}=(1:0:1)$, $p_{4}=(1:1:1)$, $p_{5}$ corresponds to $\mathfrak{P}_{5}=\langle X_{1}^{2}+3X_{0}^{2},X_{2}\rangle$, and $p_{6}$ corresponds to $\mathfrak{P}_{6}=\langle X_{1}-2X_{0},2X_{0}^{2}+X_{2}^{2}\rangle$. We have $\varkappa_{1}=\cdots=\varkappa_{4}=1$ and $\varkappa_{5}=\varkappa_{6}=2$. The Hilbert functions of $\mathbb{X}$ and its subschemes are

[TABLE]

We see that $\dim_{K}(I_{\mathbb{X}\setminus\{p_{j}\}/\mathbb{X}})_{r_{\mathbb{X}}-1}=\dim_{K}(I_{\mathbb{X}\setminus\{p_{j}\}/\mathbb{X}})_{2}=0<\varkappa_{j}$ for $j=1,\dots,5$ and $\dim_{K}(I_{\mathbb{X}\setminus\{p_{6}\}/\mathbb{X}})_{r_{\mathbb{X}}-1}=1<2=\varkappa_{6}$. Consequently, the scheme $\mathbb{X}$ is a Cayley-Bacharach scheme by Proposition 4.3.

Next we consider the subscheme $\mathbb{Y}=\mathbb{X}\setminus\{p_{4}\}$ of $\mathbb{X}$. We have $\operatorname{HF}_{\mathbb{Y}}:\ 1\ 3\ 6\ 7\ 7\cdots$ and $r_{\mathbb{Y}}=3$. The Hilbert functions of subschemes of $\mathbb{Y}$ are given by

[TABLE]

It follows that $\mathbb{Y}$ has CBP($d$) for $d=0,1$. But $\dim_{K}(I_{\mathbb{Y}\setminus\{p_{3}\}/\mathbb{Y}})_{r_{\mathbb{Y}}-1}=1=\nu_{3}$ and $\dim_{K}(I_{\mathbb{Y}\setminus\{p_{6}\}/\mathbb{Y}})_{r_{\mathbb{Y}}-1}=2=\nu_{6}$. Therefore Proposition 4.3 yields that the scheme $\mathbb{Y}$ is not a Cayley-Bacharach scheme.

At this point we are ready to characterize Cayley-Bacharach schemes in terms of their Dedekind differents.

Theorem 4.5**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional locally Gorenstein scheme and let $\sigma$ be a homogeneous trace map of degree zero of $Q^{h}(R)/L_{0}$. Then $\mathbb{X}$ is a Cayley-Bacharach scheme if and only if, for all $p_{j}\in\operatorname{Supp}(\mathbb{X})$, every maximal $p_{j}$-subscheme $\mathbb{Y}\subseteq\mathbb{X}$ satisfies

[TABLE]

Proof.

Suppose that $\mathbb{X}$ is a Cayley-Bacharach scheme. By [9, Proposition 3.2], for every $j\in\{1,\dots,s\}$, we find an element $g^{*}_{j}\in(\mathfrak{C}^{\sigma}_{\mathbb{X}})_{-r_{\mathbb{X}}}$ such that $g^{*}_{j}=x_{0}^{-2r_{\mathbb{X}}}\widetilde{g}^{*}_{j}$ with $\widetilde{g}^{*}_{j}\in R_{r_{\mathbb{X}}}\setminus\{0\}$ and $(\widetilde{g}^{*}_{j})_{p_{j}}\in{\mathcal{O}}_{\mathbb{X},p_{j}}\setminus\mathfrak{m}_{\mathbb{X},p_{j}}$. We assume for a contradiction that there is a maximal $p_{j}$-subscheme $\mathbb{Y}_{j}\subseteq\mathbb{X}$ such that

[TABLE]

For such $j$, let $s_{j}$ be the socle element in $\mathcal{O}_{\mathbb{X},p_{j}}$ corresponding to the scheme $\mathbb{Y}_{j}$, let $\{e_{j1},\dots,e_{j\varkappa_{j}}\}\subseteq\mathcal{O}_{\mathbb{X},p_{j}}$ be elements whose residue classes form a $K$-basis of $\kappa(p_{j})$, and let $\{f_{j1},\dots,f_{j\varkappa_{j}}\}$ be the standard set of separators of $\mathbb{Y}_{j}$ in $\mathbb{X}$ w.r.t. $s_{j}$ and $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$. We want to show that $x_{0}\mid f_{jk}$ for $k=1,\dots,\varkappa_{j}$. It suffices to show $x_{0}\mid f_{j1}$, since the other cases follow similarly. We write

[TABLE]

and put

[TABLE]

Then $0\neq x_{0}^{r_{\mathbb{X}}-1}f\in x_{0}^{r_{\mathbb{X}}-1}(I_{\mathbb{Y}/\mathbb{X}})_{r_{\mathbb{X}}}$ and $f\widetilde{g}^{*}_{j}=x_{0}^{r_{\mathbb{X}}}f_{j1}$, especially, $x_{0}^{r_{\mathbb{X}}-1}f\in(\delta^{\sigma}_{\mathbb{X}})_{2r_{\mathbb{X}}-1}$. Also, we observe that

[TABLE]

So, it follows from the inclusion $\mathfrak{C}^{\sigma}_{\mathbb{X}}\cdot\delta^{\sigma}_{\mathbb{X}}\subseteq R$ that $x_{0}^{-1}f_{j1}\in R_{r_{\mathbb{X}}-1}\setminus\{0\}$. This implies $f_{j1}\in x_{0}R_{r_{\mathbb{X}}-1}$ or $x_{0}\mid f_{j1}$. Therefore Proposition 4.3 yields that $\mathbb{X}$ is not a Cayley-Bacharach scheme, a contradiction.

Conversely, suppose that $\mathbb{X}$ is not a Cayley-Bacharach scheme. Then there is a maximal $p_{j}$-subscheme $\mathbb{Y}_{j}\subseteq\mathbb{X}$ such that $\operatorname{deg}(f^{*}_{jk_{j}})\leq r_{\mathbb{X}}-1$ for all $k_{j}=1,\dots,\varkappa_{j}$. Notice that $f_{jk_{j}}=x_{0}^{r_{\mathbb{X}}-\operatorname{deg}(f^{*}_{jk_{j}})}f^{*}_{jk_{j}}$ in $x_{0}^{r_{\mathbb{X}}-\operatorname{deg}(f^{*}_{jk_{j}})}R_{\operatorname{deg}(f^{*}_{jk_{j}})}$ for all $k_{j}=1,\dots,\varkappa_{j}$. As in Remark 3.3, we may write $\mathfrak{C}^{\sigma}_{\mathbb{X}}=\langle g_{1},\dots,g_{\operatorname{deg}(\mathbb{X})}\rangle_{K[x_{0}]}$, where $g_{k}=x_{0}^{-2r_{\mathbb{X}}}\widetilde{g}_{k}$ with $\widetilde{g}_{k}\in R_{2r_{\mathbb{X}}-n_{k}}$ for $k=1,\dots,\operatorname{deg}(\mathbb{X})$ and $n_{k}\leq r_{\mathbb{X}}$. By Lemma 2.8, there are $c_{j1},\dots,c_{j\varkappa_{j}}\in K$ such that $f_{j1}\cdot\widetilde{g}_{k}=\sum_{k_{j}=1}^{\varkappa_{j}}c_{jk_{j}}x_{0}^{2r_{\mathbb{X}}-n_{k}}f_{jk_{j}}$. We calculate

[TABLE]

This implies $x_{0}^{r_{\mathbb{X}}-1}f_{j1}g_{k}\in R_{2r_{\mathbb{X}}-n_{k}-1}$ for every $k\in\{1,\dots,\operatorname{deg}(\mathbb{X})\}$. Hence the element $x_{0}^{r_{\mathbb{X}}-1}f_{j1}$ is contained in $(\delta^{\sigma}_{\mathbb{X}})_{2r_{\mathbb{X}}-1}$. Similarly, we can show that $x_{0}^{r_{\mathbb{X}}-1}f_{jk_{j}}$ is a homogeneous element of degree $2r_{\mathbb{X}}-1$ of $\delta^{\sigma}_{\mathbb{X}}$ for all $k_{j}=2,\dots,\varkappa_{j}$. Therefore we obtain

[TABLE]

in contradiction to the assumption that $x_{0}^{r_{\mathbb{X}}-1}(I_{\mathbb{Y}_{j}/\mathbb{X}})_{r_{\mathbb{X}}}\nsubseteq(\delta^{\sigma}_{\mathbb{X}})_{2r_{\mathbb{X}}-1}$. ∎

The following corollary is an immediate consequence of Theorem 4.5.

Corollary 4.6**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional locally Gorenstein scheme.

1. (a)

If $\mathbb{X}$ has $K$-rational support then it is a Cayley-Bacharach scheme if and only if for every subscheme $\mathbb{Y}\subseteq\mathbb{X}$ of degree $\operatorname{deg}(\mathbb{Y})=\operatorname{deg}(\mathbb{X})-1$ and for every separator $f_{\mathbb{Y}}$ of $\mathbb{Y}$ in $\mathbb{X}$ we have $x_{0}^{r_{\mathbb{X}}-1}f_{\mathbb{Y}}\notin(\delta^{\sigma}_{\mathbb{X}})_{2r_{\mathbb{X}}-1}$. 2. (b)

If $\mathbb{X}$ has minimal Dedekind different then it is a Cayley-Bacharach scheme.

Let us apply the corollary to some explicit cases.

Example 4.7**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{6}\}\subseteq\mathbb{P}^{2}_{\mathbb{Q}}$ be the set of six points given in Example 3.9. We know that $\mathbb{X}$ has minimal Dedekind different. Therefore Corollary 4.6(b) yields that $\mathbb{X}$ is a Cayley-Bacharach scheme. Similarly, the set of nine points in $\mathbb{P}^{3}_{\mathbb{Q}}$ given in Example 3.9 is also a Cayley-Bacharach scheme.

Next we consider the 0-dimensional scheme $\mathbb{Y}\subseteq\mathbb{P}^{2}_{\mathbb{Q}}$ of degree $6$ with support $\operatorname{Supp}(\mathbb{Y})=\{p_{1},\dots,p_{5}\}$, where $p_{1}=(1:0:0)$, $p_{2}=(1:1:0)$, $p_{3}=(1:0:1)$, $p_{4}=(1:1:1)$, and $p_{5}$ corresponds to $\mathfrak{P}_{5}=\langle X_{1}-2X_{0},2X_{0}^{2}+X_{2}^{2}\rangle$. The Hilbert function of $\mathbb{Y}$ is $\operatorname{HF}_{\mathbb{Y}}:\ 1\ 3\ 6\ 6\ \cdots$ and $r_{\mathbb{Y}}=2$. In this case the Hilbert function of the Dedekind different is given by

[TABLE]

It follows that $\mathbb{Y}$ has minimal Dedekind different, and so it is a Cayley-Bacharach scheme by Corollary 4.6(b).

For a Cayley-Bacharach scheme $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$, the Hilbert function of the Dedekind different is described in our next proposition.

Proposition 4.8**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a 0-dimensional locally Gorenstein Cayley-Bacharach scheme and let $\sigma$ be a homogeneous trace map of degree zero of $Q^{h}(R)/\!L_{0}$. Then the Hilbert function of $\delta_{\mathbb{X}}^{\sigma}$ satisfies $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(i)=0$ for $i<r_{\mathbb{X}}$, $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(i)=\operatorname{deg}(\mathbb{X})$ for $i\geq 2r_{\mathbb{X}}$ and

[TABLE]

In this case, the regularity index of $\delta_{\mathbb{X}}^{\sigma}$ is exactly $2r_{\mathbb{X}}$.

Proof.

Since the scheme $\mathbb{X}$ is a Cayley-Bacharach scheme, there are homogeneous elements $g_{1}^{*},\dots,g_{s}^{*}$ in $(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-r_{\mathbb{X}}}$ such that $g^{*}_{j}=x_{0}^{-2r_{\mathbb{X}}}\widetilde{g}^{*}_{j}$ with $\widetilde{g}^{*}_{j}\in R_{r_{\mathbb{X}}}$ and $(\widetilde{g}^{*}_{j})_{p_{j}}\in{\mathcal{O}}_{\mathbb{X},p_{j}}\setminus\mathfrak{m}_{\mathbb{X},p_{j}}$ by [9, Proposition 3.2]. Let $h\in(\delta_{\mathbb{X}}^{\sigma})_{i}$ with $i<r_{\mathbb{X}}$. Then we have

[TABLE]

for $j=1,\dots,s$. This implies $h\widetilde{g}^{*}_{j}=0$, in particular, $h_{p_{j}}\cdot(\widetilde{g}^{*}_{j})_{p_{j}}=0$ in ${\mathcal{O}}_{\mathbb{X},p_{j}}$ for all $j\in\{1,\dots,s\}$. Since $(\widetilde{g}^{*}_{j})_{p_{j}}$ is a unit of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ for $j=1,\dots,s$, we have to get $h_{p_{j}}=0$ for all $j=1,\dots,s$. In other words, we have $\tilde{\imath}(h)=0$, and so $h=0$ (as $\tilde{\imath}$ is an injection). Subsequently, we get $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(i)=0$ for $i<r_{\mathbb{X}}$.

Now, according to Proposition 3.5, we only need to show that $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(2r_{\mathbb{X}}-1)<\operatorname{deg}(\mathbb{X})$, i.e., $(\delta_{\mathbb{X}}^{\sigma})_{2r_{\mathbb{X}}-1}\subsetneq R_{2r_{\mathbb{X}}-1}$. But this follows from Theorem 4.5, since otherwise we would have $x_{0}^{r_{\mathbb{X}}-1}(I_{\mathbb{Y}/\mathbb{X}})_{r_{\mathbb{X}}}\subseteq(\delta_{\mathbb{X}}^{\sigma})_{2r_{\mathbb{X}}-1}$ for every maximal $p_{j}$-subscheme $\mathbb{Y}\subseteq\mathbb{X}$, and thus $\mathbb{X}$ would not be a Cayley-Bacharach scheme. ∎

Remark 4.9**.**

The upper bound for the regularity index of the Dedekind different given in Proposition 3.5 is attained for 0-dimensional locally Gorenstein Cayley-Bacharach schemes. Moreover, a 0-dimensional locally Gorenstein Cayley-Bacharach scheme $\mathbb{X}$ satisfies $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}})>0$ if and only if $\mathbb{X}$ is arithmetically Gorenstein (see [9, Proposition 4.8]).

Proposition 4.10**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional locally Gorenstein scheme, let $0\leq d\leq r_{\mathbb{X}}-1$, and let $\sigma$ be a homogeneous trace map of degree zero of $Q^{h}(R)/\!L_{0}$. If for every $p_{j}\in\operatorname{Supp}(\mathbb{X})$ the maximal $p_{j}$-subscheme $\mathbb{Y}_{j}\subseteq\mathbb{X}$ satisfies

[TABLE]

then $\mathbb{X}$ has CBP($d$). In particular, if $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}}+d)=0$ then $\mathbb{X}$ has CBP($d$).

Proof.

Suppose for contradiction that $\mathbb{X}$ does not have CBP($d$). There are a maximal $p_{j}$-subscheme $\mathbb{Y}_{j}\subseteq\mathbb{X}$ and a set of minimal separators $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ of $\mathbb{Y}_{j}$ in $\mathbb{X}$ such that $\operatorname{deg}(f^{*}_{jk_{j}})\leq d$ for $k_{j}=1,\dots,\varkappa_{j}$. Set $f_{jk_{j}}:=x_{0}^{r_{\mathbb{X}}-\operatorname{deg}(f_{jk_{j}}^{*})}f^{*}_{jk_{j}}$ for $k_{j}\in\{1\dots,\varkappa_{j}\}$. Then the set $\{f_{j1},\dots,f_{j\varkappa_{j}}\}$ is a standard set of separators of $\mathbb{Y}_{j}$ in $\mathbb{X}$. We write $\mathfrak{C}_{\mathbb{X}}^{\sigma}=\langle g_{1},\dots,g_{\operatorname{deg}(\mathbb{X})}\rangle_{K[x_{0}]}$, where $g_{k}=x_{0}^{-r_{\mathbb{X}}-n_{k}}\widetilde{g}_{k}$ with $\widetilde{g}_{k}\in R_{r_{\mathbb{X}}}$ and $n_{k}\leq r_{\mathbb{X}}$ for $k=1,\dots,\operatorname{deg}(\mathbb{X})$ (see Remark 3.3). We have

[TABLE]

for some $c_{j1},\dots,c_{j\varkappa_{j}}\in K$. Since $r_{\mathbb{X}}-n_{l}\geq 0$ and $d-\operatorname{deg}(f_{jk_{j}}^{*})\geq 0$, this implies that $(x_{0}^{d}f_{jk})\cdot(x_{0}^{-r_{\mathbb{X}}-n_{l}}\widetilde{g}_{l})\in R_{r_{\mathbb{X}}+d-n_{l}}$ for all $l=1,\dots,\operatorname{deg}(\mathbb{X})$. Consequently, the element $x_{0}^{d}f_{jk}$ is contained in $(\delta_{\mathbb{X}}^{\sigma})_{r_{\mathbb{X}}+d}$ for all $k=1,\dots,\varkappa_{j}$. Therefore we get the inclusion $x_{0}^{d}(I_{\mathbb{Y}/\mathbb{X}})_{r_{\mathbb{X}}}\subseteq(\delta_{\mathbb{X}}^{\sigma})_{r_{\mathbb{X}}+d}$, in contradiction to our assumption. ∎

The following example shows that the converse of Proposition 4.10 is not true in the general case (except for the case $d=r_{\mathbb{X}}-1$).

Example 4.11**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{2}_{\mathbb{Q}}$ be the set consisting of the points $p_{1}=(1:0:0)$, $p_{2}=(1:1:0)$, $p_{3}=(1:2:0)$, $p_{4}=(1:3:1)$, $p_{5}=(1:4:0)$, $p_{6}=(1:5:0)$, $p_{7}=(1:6:1)$, and $p_{8}=(1:1:1)$. It is easy to see that $\operatorname{HF}_{\mathbb{X}}:1\ 3\ 5\ 7\ 8\ 8\ \cdots$ and $r_{\mathbb{X}}=4$. The Dedekind different is computed by

[TABLE]

and its Hilbert function is $\operatorname{HF}_{\delta_{\mathbb{X}}}:\ 0\ 0\ 0\ 0\ 1\ 3\ 5\ 7\ 8\ 8\ \cdots$. Clearly, $\mathbb{X}$ is not arithmetically Gorenstein and $\operatorname{HF}_{\delta_{\mathbb{X}}}(r_{\mathbb{X}})\neq 0$. Hence $\mathbb{X}$ is not a Cayley-Bacharach scheme by Remark 4.9. Also, we can check that $\mathbb{X}$ has CBP($d$) for $0\leq d\leq 2$. Now the subscheme $\mathbb{Y}_{4}:=\mathbb{X}\setminus\{p_{4}\}$ has a separator of the form $f_{4}=x_{0}x_{1}^{2}x_{2}-7x_{0}x_{1}x_{2}^{2}+6x_{0}x_{2}^{3}$. It is not difficult to verify that $x_{0}^{r_{\mathbb{X}}-2}f_{4}\in(\delta_{\mathbb{X}})_{2r_{\mathbb{X}}-2}$. Thus $\mathbb{X}$ has CBP($2$), but $x_{0}^{2}(I_{\mathbb{Y}_{4}/\mathbb{X}})_{r_{\mathbb{X}}}\subseteq(\delta_{\mathbb{X}})_{r_{\mathbb{X}}+2}$.

5. Dedekind’s Formula

In previous sections, we mainly considered the Dedekind different to study the Cayley-Bacharach property of 0-dimensional locally Gorenstein schemes. This different is a subideal of the conductor of $R$ in the ring $\prod_{j=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{j}}[T_{j}]$. In [4], Geramita et al. characterized a finite set of points to be a Cayley-Bacharach scheme in terms of the conductor and showed that Dedekind’s formula for the conductor and the complementary module always holds for finite sets of points. In this section we generalize these results substantially. We work over an arbitrary base field $K$, and let $\mathbb{X}$ be an arbitrary 0-dimensional subscheme of $\mathbb{P}^{n}_{K}$. Let the support of $\mathbb{X}$ be given by $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$.

Definition 5.1**.**

Let $\widetilde{R}=\prod_{j=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{j}}[T_{j}]$, and let $\mathfrak{F}_{\widetilde{R}/R}$ be the ideal defined as

[TABLE]

The ideal $\mathfrak{F}_{\widetilde{R}/R}$ is called the conductor of $R$ in $\widetilde{R}$.

When the scheme $\mathbb{X}$ is reduced, the ring $\widetilde{R}$ is the integral closure of $R$ in its full quotient ring, and hence $\mathfrak{F}_{\widetilde{R}/R}$ is the conductor of $R$ in its integral closure in the traditional sense. Furthermore, $\mathfrak{F}_{\widetilde{R}/R}$ is an ideal of both $R$ and $\widetilde{R}$. We recall from [12, Proposition 2.9] the following description of the conductor of $R$ in $\widetilde{R}$.

Proposition 5.2**.**

For $j\in\{1,\dots,s\}$ and $a\in{\mathcal{O}}_{\mathbb{X},p_{j}}$, let $\mu(a)=\min\{i\in\mathbb{N}\mid(0,\dots,0,aT_{j}^{i},0,\dots,0)\in\tilde{\imath}(R)\}$, where $\tilde{\imath}$ is the injection from $R$ to $Q^{h}(R)$, and let $\nu(a)=\max\{\,\mu(ab)\mid b\in{\mathcal{O}}_{\mathbb{X},p_{j}}\setminus\{0\}\,\}$. Then, as an ideal of $R$, we have

[TABLE]

where $f_{a}$ is the preimage of $(0,\dots,0,aT_{j}^{\nu(a)},0,\dots,0)$ under the injection $\tilde{\imath}$.

Some relations between the Dedekind different and the conductor are given by the next proposition.

Proposition 5.3**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional locally Gorenstein scheme, and let $\sigma$ be a homogeneous trace map of degree zero of $Q^{h}(R)/\!L_{0}$. Then we have

[TABLE]

Proof.

We know that $Q^{h}(R)=\prod_{j=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{j}}[T_{j},T_{j}^{-1}]$ and $(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{i}=Q^{h}(R)_{i}=\widetilde{R}_{i}$ for all $i\geq 0$. This implies $\widetilde{R}\subseteq\mathfrak{C}_{\mathbb{X}}^{\sigma}$. Thus we get

[TABLE]

Since $\mathbb{X}$ is a locally Gorenstein scheme, we have $\operatorname{Hom}_{K}({\mathcal{O}}_{\mathbb{X},p_{j}},K)\cong{\mathcal{O}}_{\mathbb{X},p_{j}}$ for all $j=1,\dots,s$. This implies the isomorphism $\widetilde{R}\cong\underline{\operatorname{Hom}}_{K[x_{0}]}(\widetilde{R},K[x_{0}])$. Hence we get $\operatorname{HF}_{\underline{\operatorname{Hom}}_{K[x_{0}]}(\widetilde{R},K[x_{0}])}(i)=\operatorname{deg}(\mathbb{X})$ if and only if $i\geq 0$. Let $f\in(\mathfrak{F}_{\widetilde{R}/R})_{i}$, let $g\in(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{k}$, and let $\varphi\in(\omega_{R})_{k+1}$ such that $g=\Phi(\varphi)$ where $\Phi$ was defined by (3.1). Observe that $(f\cdot\varphi)(\widetilde{R})=\varphi(f\widetilde{R})\subseteq\varphi(R)\subseteq K[x_{0}]$. This yields $f\cdot\varphi\in\underline{\operatorname{Hom}}_{K[x_{0}]}(\widetilde{R},K[x_{0}])_{i+k}$. If $f\cdot\varphi\neq 0$, then $\operatorname{deg}(f\cdot\varphi)=i+k\geq 0$. Thus we have $f\cdot g=f\cdot\Phi(\varphi)=\Phi(f\cdot\varphi)\in\widetilde{R}$, and hence we get the inclusion $\mathfrak{F}_{\widetilde{R}/R}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}\subseteq\widetilde{R}$. Now we see that $\mathfrak{F}_{\widetilde{R}/R}^{2}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}\subseteq\mathfrak{F}_{\widetilde{R}/R}\widetilde{R}\subseteq R$. This yields the inclusion $\mathfrak{F}_{\widetilde{R}/R}^{2}\subseteq\delta_{\mathbb{X}}^{\sigma}$. Altogether, the claim follows. ∎

The Cayley-Bacharach property of a 0-dimensional scheme can be characterized in terms of the conductor of $R$ in $\widetilde{R}$, as the following theorem shows.

Theorem 5.4**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional scheme, and let $0\leq d\leq r_{\mathbb{X}}-1$. Then $\mathbb{X}$ has CBP($d$) if and only if $\mathfrak{F}_{\widetilde{R}/R}\subseteq\bigoplus_{i\geq d+1}R_{i}$. In particular, $\mathbb{X}$ is a Cayley-Bacharach scheme if and only if $\mathfrak{F}_{\widetilde{R}/R}=\bigoplus_{i\geq r_{\mathbb{X}}}R_{i}$.

Proof.

Suppose that $\mathbb{X}$ has CBP($d$), but $\mathfrak{F}_{\widetilde{R}/R}\nsubseteq\bigoplus_{i\geq d+1}R_{i}$. It follows from Proposition 5.2 that there are a non-zero element $a\in{\mathcal{O}}_{\mathbb{X},p_{j}}$ and a homogeneous element $f_{a}\in\mathfrak{F}_{\widetilde{R}/R}\setminus\{0\}$ such that $\tilde{\imath}(f_{a})=(0,\dots,0,aT_{j}^{\nu(a)},0,\dots,0)$ and $\nu(a)\leq d$. So, we can find an element $b\in{\mathcal{O}}_{\mathbb{X},p_{j}}$ with $s_{j}:=ab\in\mathfrak{G}({\mathcal{O}}_{\mathbb{X},p_{j}})\setminus\{0\}$. By Proposition 2.2, there is a maximal $p_{j}$-subscheme $\mathbb{Y}$ of $\mathbb{X}$ associated to the socle element $s_{j}$. We want to prove that $\mu_{\mathbb{Y}/\mathbb{X}}\leq d$. Let $\varkappa_{j}=\dim_{K}\kappa(p_{j})$, let $\{e_{j1},\dots,e_{j\varkappa_{j}}\}\subseteq{\mathcal{O}}_{\mathbb{X},p_{j}}$ be such that their residue classes form a $K$-basis of $\kappa(p_{j})$, and let $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ be the set of minimal separators of $\mathbb{Y}$ in $\mathbb{X}$ w.r.t. $s_{j}$ and $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$. Notice that $\tilde{\imath}(f^{*}_{jk_{j}})=(0,\dots,0,e_{jk_{j}}s_{j}T_{j}^{\mu(e_{jk_{j}}s_{j})},0,\dots,0)$ and $\operatorname{deg}(f^{*}_{jk_{j}})=\mu(e_{jk_{j}}s_{j})$ for $k_{j}=1,\dots,\varkappa_{j}$. Clearly, we have

[TABLE]

This implies $\mu_{\mathbb{Y}/\mathbb{X}}\leq\nu(s_{j})$. Moreover, we also see that

[TABLE]

This yields $\mu_{\mathbb{Y}/\mathbb{X}}\leq\nu(s_{j})\leq d$. Thus we get $\operatorname{deg}_{\mathbb{X}}(p_{j})\leq d$, and hence $\mathbb{X}$ does not have CBP($d$), a contradiction.

Conversely, suppose that $\mathfrak{F}_{\widetilde{R}/R}\subseteq\bigoplus_{i\geq d+1}R_{i}$. Let $\mathbb{Y}\subseteq\mathbb{X}$ be a maximal $p_{j}$-subscheme, and let $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ be the set of minimal separators of $\mathbb{Y}$ in $\mathbb{X}$ w.r.t. $s_{j}$ and $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$. As above, we always have

[TABLE]

Also, it is easy to check that $\mu(a+b)\leq\max\{\mu(a),\mu(b)\}$ for all $a,b\in{\mathcal{O}}_{\mathbb{X},p_{j}}\setminus\{0\}$. Let $a\in{\mathcal{O}}_{\mathbb{X},p_{j}}$ be such that $as_{j}\neq 0$. Then we have $a\notin\mathfrak{m}_{\mathbb{X},p_{j}}$ and we may write $a=c_{j1}e_{j1}+\cdots+c_{j\varkappa_{j}}e_{j\varkappa_{j}}\ ({\rm mod}\ \mathfrak{m}_{\mathbb{X},p_{j}})$ for $c_{j1},\dots,c_{j\varkappa_{j}}\in K$, not all equal to zero. We deduce $as_{j}=c_{j1}e_{j1}s_{j}+\cdots+c_{j\varkappa_{j}}e_{j\varkappa_{j}}s_{j}$. Hence we have

[TABLE]

This implies $\nu(s_{j})=\max\{\mu(e_{jk}s_{j})\mid k=1,\dots,\varkappa_{j}\}$. Without loss of generality, we may assume that $\nu(s_{j})=\operatorname{deg}(f^{*}_{j1})=\mu(e_{j1}s_{j})$. Thus we have $\nu(s_{j})=\nu(e_{j1}s_{j})$ and $f^{*}_{j1}\in\mathfrak{F}_{\widetilde{R}/R}$. Since $\mathfrak{F}_{\widetilde{R}/R}\subseteq\bigoplus_{i\geq d+1}R_{i}$, it follows that $\nu(s_{j})=\operatorname{deg}(f^{*}_{j1})\geq d+1$. From this we conclude that $\operatorname{deg}_{\mathbb{X}}(p_{j})\geq d+1$ for all $j=1,\dots,s$. In other words, the scheme $\mathbb{X}$ has CBP($d$).

Moreover, if we identify $R$ with its image under $\tilde{\imath}$, we have $R_{i}=\widetilde{R}_{i}$ for all $i\geq r_{\mathbb{X}}$. Thus the ideal $\bigoplus_{i\geq r_{\mathbb{X}}}R_{i}$ is an ideal of both $R$ and $\widetilde{R}$, and it is contained in the conductor $\mathfrak{F}_{\widetilde{R}/R}$. Hence the additional claim follows. ∎

The inclusion $\mathfrak{F}_{\widetilde{R}/R}^{2}\subseteq\delta_{\mathbb{X}}^{\sigma}$ in Proposition 5.3 can be an equality in the following case. In this case the converse of Corollary 4.6(b) holds true.

Proposition 5.5**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional locally Gorenstein scheme, and let $\sigma$ be a homogeneous trace map of degree zero of $Q^{h}(R)/L_{0}$. Then the scheme $\mathbb{X}$ has minimal Dedekind different if and only if $\mathbb{X}$ is a Cayley-Bacharach scheme and $\mathfrak{F}_{\widetilde{R}/R}^{2}=\delta_{\mathbb{X}}^{\sigma}$.

Proof.

Suppose that the scheme $\mathbb{X}$ has minimal Dedekind different. Then the Dedekind different satisfies $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(2r_{\mathbb{X}}-1)=0$. By Corollary 4.6(b), the scheme $\mathbb{X}$ is a Cayley-Bacharach scheme. So, Theorem 5.4 yields that $\mathfrak{F}_{\widetilde{R}/R}=\bigoplus_{i\geq r_{\mathbb{X}}}R_{i}$. Hence we have

[TABLE]

Conversely, if $\mathbb{X}$ is a Cayley-Bacharach scheme and $\mathfrak{F}_{\widetilde{R}/R}^{2}=\delta_{\mathbb{X}}^{\sigma}$, then Theorem 5.4 implies the equality $\delta_{\mathbb{X}}^{\sigma}=\bigoplus_{i\geq 2r_{\mathbb{X}}}R_{i}$. Thus $\mathbb{X}$ has minimal Dedekind different. ∎

Example 5.6**.**

Let $\mathbb{X}$ and $\mathbb{Y}$ be the two 0-dimensional reduced schemes given in Example 4.7. Both $\mathbb{X}$ and $\mathbb{Y}$ have minimal Dedekind different. Thus the Dedekind different equals to the square of the conductor for these 0-dimensional schemes by the preceding proposition.

Our next theorem presents a generalization of Dedekind’s formula for the conductor $\mathfrak{F}_{\widetilde{R}/R}$ and the Dedekind complementary module $\mathfrak{C}_{\mathbb{X}}^{\sigma}$. We use the notation $\nu_{j}=\dim_{K}{\mathcal{O}}_{\mathbb{X},p_{j}}$ for all $j=1,\dots,s$.

Theorem 5.7**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional locally Gorenstein scheme with support $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$, and let $\sigma$ be a homogeneous trace map of degree zero of $Q^{h}(R)/L_{0}$. Further, we let $I_{j}$ be the homogeneous vanishing ideal of $\mathbb{X}$ at $p_{j}$, and we let $\mathbb{Y}_{j}$ be the subscheme of $\mathbb{X}$ defined by $I_{\mathbb{Y}_{j}}=\bigcap_{k\neq j}I_{k}$ for $j=1,\dots,s$. Then the formula

[TABLE]

holds true if one of the following conditions is satisfied:

1. (a)

The scheme $\mathbb{X}$ is a Cayley-Bacharach scheme. 2. (b)

For all $j\in\{1,\dots,s\}$, the Hilbert function of $\mathbb{Y}_{j}$ is of the form

[TABLE]

Proof.

As in the proof of Proposition 5.3, we have the inclusion $\mathfrak{F}_{\widetilde{R}/R}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}\subseteq\widetilde{R}$. Now we prove the reverse inclusion if (a) or (b) is satisfied.

(a) For every $j\in\{1,\dots,s\}$, we let $\{\,e_{j1},\dots,e_{j\nu_{j}}\,\}$ be a $K$-basis of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ and set $\epsilon_{jk_{j}}:=(0,\dots,0,e_{jk_{j}},0,\dots,0)\in\widetilde{R}$, where $k_{j}\in\{1,\dots,\nu_{j}\}$. Then the elements $\{\epsilon_{11},\dots,\epsilon_{s\nu_{s}}\}$ form a $K[x_{0}]$-basis of $\widetilde{R}$. Thus it is enough to show that $\epsilon_{11},\dots,\epsilon_{s\nu_{s}}$ are contained in $\mathfrak{F}_{\widetilde{R}/R}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}$. Since $\mathbb{X}$ is a Cayley-Bacharach scheme, for $j=1,\dots,s$ we find $g^{*}_{j}\in(\mathfrak{C}_{R/K[x_{0}]})_{-r_{\mathbb{X}}}$ such that $g^{*}_{j}=x_{0}^{-2r_{\mathbb{X}}}\widetilde{g}^{*}_{j}$, where $\widetilde{g}^{*}_{j}\in R_{r_{\mathbb{X}}}$ and $(\widetilde{g}^{*}_{j})_{p_{j}}$ is a unit of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ (cf. [9, Proposition 3.2]). By identifying $R$ with its image in $Q^{h}(R)$ under $\tilde{\imath}$, the element $h_{jk_{j}}:=(0,\dots,0,(\widetilde{g}^{*}_{j})^{-1}_{p_{j}}e_{jk_{j}}T_{j}^{r_{\mathbb{X}}},0,\dots,0)$ is contained in $R_{r_{\mathbb{X}}}\setminus\{0\}$ for all $j\in\{1,\dots,s\}$ and $k_{j}\in\{1,\dots,\nu_{j}\}$. We see that

[TABLE]

By Theorem 5.4, we have $\mathfrak{F}_{\widetilde{R}/R}=\bigoplus_{i\geq r_{\mathbb{X}}}R_{i}$. This implies $h_{11},\dots,h_{s\nu_{s}}\in\mathfrak{F}_{\widetilde{R}/R}$. Therefore we obtain $\epsilon_{11},\dots,\epsilon_{s\nu_{s}}\in\mathfrak{F}_{\widetilde{R}/R}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}$, as desired.

(b) In a similar fashion, we proceed to show that $\epsilon_{11},\dots,\epsilon_{s\nu_{s}}\in\mathfrak{F}_{\widetilde{R}/R}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}$. For $j=1,\dots,s$, let $\overline{\sigma}_{j}$ denote the trace map of the algebra ${\mathcal{O}}_{\mathbb{X},p_{j}}/K$ (associated to $\sigma$), and let $\{e^{\prime}_{j1},\dots,e^{\prime}_{j\nu_{j}}\}$ be the $K$-basis of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ which is dual to the $K$-basis $\{e_{j1},\dots,e_{j\nu_{j}}\}$ w.r.t. $\overline{\sigma}_{j}$. W.l.o.g., we may assume that $e^{\prime}_{j1}$ is a unit of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ for all $j\in\{1,\dots,s\}$. Note that the subscheme $\mathbb{Y}_{j}$ has degree $\operatorname{deg}(\mathbb{Y}_{j})=\operatorname{deg}(\mathbb{X})-\nu_{j}$ for all $j=1,\dots,s$. It follows from the assumption that $\alpha_{\mathbb{Y}_{j}/\mathbb{X}}=\mu(e_{j1})=\cdots=\mu(e_{j\nu_{j}})$. Then we have $I_{\mathbb{Y}_{j}/\mathbb{X}}=\langle\,f^{*}_{j1},\dots,f^{*}_{j\nu_{j}}\,\rangle$, where

[TABLE]

for $k_{j}=1,\dots,\nu_{j}$. The set $\{(0,\dots,0,aT_{j}^{\alpha_{\mathbb{Y}_{j}/\mathbb{X}}},0,\dots,0)\,\mid\,a\in{\mathcal{O}}_{\mathbb{X},p_{j}}\}$ is the image of $(I_{\mathbb{Y}_{j}/\mathbb{X}})_{\alpha_{\mathbb{Y}_{j}/\mathbb{X}}}$ in $\widetilde{R}$. This implies $\nu(a)=\mu(a)=\alpha_{\mathbb{Y}_{j}/\mathbb{X}}$ for every non-zero element $a\in{\mathcal{O}}_{\mathbb{X},p_{j}}$. Thus Proposition 5.2 yields that $I_{\mathbb{Y}_{j}/\mathbb{X}}\subseteq\mathfrak{F}_{\widetilde{R}/R}$.

Obviously, we have $f^{*}_{jk_{j}}\notin\langle x_{0}\rangle$ and its image $\overline{f}^{*}_{jk_{j}}$ in $\overline{R}=R/\langle x_{0}\rangle$ is a non-zero element for $k_{j}=1,\dots,\nu_{j}$. If there exist elements $a_{j1},\dots,a_{j\nu_{j}}\in K$, not all equal to zero, such that $\sum_{k_{j}=1}^{\nu_{j}}a_{jk_{j}}\overline{f}^{*}_{jk_{j}}=0$, then $f=\sum_{k_{j}=1}^{\nu_{j}}a_{jk_{j}}f^{*}_{jk_{j}}$ is contained in $(I_{\mathbb{Y}_{j}/\mathbb{X}})_{\alpha_{\mathbb{Y}_{j}/\mathbb{X}}}\setminus\{0\}$, and we get $\overline{f}=0$. This means $f=x_{0}h\in x_{0}R_{\alpha_{\mathbb{Y}_{j}/\mathbb{X}}-1}$ for some $h\in R_{\alpha_{\mathbb{Y}_{j}/\mathbb{X}}-1}\setminus\{0\}$. Since the ideal $I_{\mathbb{Y}_{j}/\mathbb{X}}$ is saturated, [11, Lemma 1.2] implies $h\in I_{\mathbb{Y}_{j}/\mathbb{X}}\setminus\{0\}$, a contradiction. Thus we have shown that the set $\{\overline{f}^{*}_{j1},\dots,\overline{f}^{*}_{j\nu_{j}}\}$ is $K$-linearly independent.

Consequently, there is a homogeneous $K$-linear map $\overline{\varphi}_{j1}:\overline{R}\rightarrow K$ of degree $-\alpha_{\mathbb{Y}_{j}/\mathbb{X}}$ with $\overline{\varphi}_{j1}(\overline{f}^{*}_{j1})\neq 0$ and $\overline{\varphi}_{j1}(\overline{f}^{*}_{jk_{j}})=0$ for $k_{j}=2,\dots,\nu_{j}$. Using the epimorphism $\omega_{R}(1)\twoheadrightarrow\underline{\operatorname{Hom}}_{K}(\overline{R},K)$, we can lift $\overline{\varphi}_{j1}$ to obtain a homogeneous element $\varphi_{j1}\in(\omega_{R})_{-\alpha_{\mathbb{Y}_{j}/\mathbb{X}}+1}$ with $\varphi_{j1}(f^{*}_{j1})\neq 0$ and $\varphi_{j1}(f^{*}_{jk_{j}})=0$ for $k_{j}=2,\dots,\nu_{j}$. Clearly, the set $\{x_{0}^{r_{\mathbb{X}}-\mu(e_{11})}f^{*}_{11},\dots,x_{0}^{r_{\mathbb{X}}-\mu(e_{s\nu_{s}})}f^{*}_{s\nu_{s}}\}$ forms a $K$-basis of the $K$-vector space $R_{r_{\mathbb{X}}}$. We write $\varphi_{j1}(x_{0}^{r_{\mathbb{X}}-\mu(e_{j^{\prime}k_{j^{\prime}}})}f^{*}_{j^{\prime}k_{j^{\prime}}})=c_{j^{\prime}k_{j^{\prime}}}x_{0}^{r_{\mathbb{X}}-\alpha_{\mathbb{Y}_{j}/\mathbb{X}}}$ for all $j^{\prime}=1,\dots,s$ and $k_{j^{\prime}}=1,\dots,\nu_{j^{\prime}}$. By Proposition 3.4, we have

[TABLE]

Since $e^{\prime}_{j1}$ is a unit of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ and $c_{j1}\in K\setminus\{0\}$, for $k_{j}=1,\dots,\nu_{j}$ we set

[TABLE]

Then $h_{j1},\dots,h_{j\nu_{j}}\in I_{\mathbb{Y}_{j}/\mathbb{X}}\subseteq\mathfrak{F}_{\widetilde{R}/R}$. In $\widetilde{R}$, we have

[TABLE]

since $c_{j2}=\cdots=c_{j\nu_{j}}=0$. Thus we obtain $\epsilon_{jk_{j}}\in\mathfrak{F}_{\widetilde{R}/R}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}$, as was to be shown. ∎

When we specialize to the case of sets of points, the condition (b) of Theorem 5.7 is satisfied. Therefore we recover the following result of A.V. Geramita et al. (see [4, Proposition 3.15]).

Corollary 5.8**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{s}\}\subseteq\mathbb{P}^{n}_{K}$ be a set of $s$ distinct $K$-rational points. Then we have $\mathfrak{F}_{\widetilde{R}/R}\cdot\mathfrak{C}_{\mathbb{X}}=\widetilde{R}$.

We end this section with some straightforward consequences of the theorem.

Corollary 5.9**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional locally Gorenstein scheme, let $0\leq d\leq r_{\mathbb{X}}-1$, and let $\sigma$ be a homogeneous trace map of degree zero of $Q^{h}(R)/L_{0}$. If $\mathbb{X}$ has CBP($d$), then $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(d)=0$.

Proof.

If $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(d)\neq 0$, then there exists a non-zero homogeneous element $h$ in $(\delta^{\sigma}_{\mathbb{X}})_{d}$. Proposition 5.3 yields that $h\in(\mathfrak{F}_{\widetilde{R}/R})_{d}$. By Theorem 5.4, the scheme $\mathbb{X}$ does not have CBP($d$), a contradiction. ∎

Corollary 5.10**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{s}\}\subseteq\mathbb{P}^{n}_{K}$ be a set of $s$ distinct $K$-rational points, and for $j=1,\dots,s$ let $f_{j}$ be the separator of $\mathbb{X}\setminus\{p_{j}\}$ in $\mathbb{X}$ such that $f_{j}(p_{j})=1$ and $f_{j}(p_{k})=0$ for $k\neq j$. Then $\mathbb{X}$ is a Cayley-Bacharach scheme if and only if $x_{0}^{r_{\mathbb{X}}-2}f_{j}\notin(\delta_{\mathbb{X}})_{2r_{\mathbb{X}}-2}$ for all $j=1,\dots,s$.

Proof.

It is clear that $x_{0}^{r_{\mathbb{X}}-1}f_{j}\in(\delta_{\mathbb{X}})_{2r_{\mathbb{X}}-1}$ if $x_{0}^{r_{\mathbb{X}}-2}f_{j}\in(\delta_{\mathbb{X}})_{2r_{\mathbb{X}}-2}$. By Corollary 4.6(a), we get $x_{0}^{r_{\mathbb{X}}-2}f_{j}\notin(\delta_{\mathbb{X}})_{2r_{\mathbb{X}}-2}$ for every $j\in\{1,\dots,s\}$ if $\mathbb{X}$ is a Cayley-Bacharach scheme. Conversely, if $\mathbb{X}$ is not a Cayley-Bacharach scheme, we find a minimal separator $f_{j}^{*}\in R$ such that $d_{j}=\operatorname{deg}(f^{*}_{j})\leq r_{\mathbb{X}}-1$ and $f_{j}^{*}(p_{j})=1$. Notice that $f_{j}^{*}\in\mathfrak{F}_{\widetilde{R}/R}$. By Proposition 5.3, we get $(f_{j}^{*})^{2}\in\mathfrak{F}_{\widetilde{R}/R}^{2}\subseteq\delta_{\mathbb{X}}$. Moreover, we have $x_{0}^{d_{j}}f_{j}^{*}=(f_{j}^{*})^{2}$ and $f_{j}=x_{0}^{r_{\mathbb{X}}-d_{j}}f_{j}^{*}\in R_{r_{\mathbb{X}}}$. This implies that $x_{0}^{r_{\mathbb{X}}-2}f_{j}=x_{0}^{2r_{\mathbb{X}}-2d_{j}-2}(x_{0}^{d_{j}}f_{j}^{*})\in(\delta_{\mathbb{X}})_{2r_{\mathbb{X}}-2}$. Therefore the proof is complete. ∎

6. The Trace of the Dedekind Complementary Module

In this section we let $\mathbb{X}$ be a 0-dimensional locally Gorenstein scheme in $\mathbb{P}^{n}_{K}$, let $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$, and let $\sigma$ be a fixed homogeneous trace map of degree zero of the graded algebra $Q^{h}(R)/L_{0}$.

Definition 6.1**.**

The trace of the Dedekind complementary module $\mathfrak{C}_{\mathbb{X}}^{\sigma}$, denoted ${\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$, is the sum of the ideals $\phi(\mathfrak{C}_{\mathbb{X}}^{\sigma})$ with $\phi\in\underline{\operatorname{Hom}}_{R}(\mathfrak{C}_{\mathbb{X}}^{\sigma},R)$, i.e.,

[TABLE]

The following remark collects some basic properties of ${\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$. For the general theory of traces of modules we refer to [7, 17].

Remark 6.2**.**

Notice that we have $\omega_{R}(1)\cong\mathfrak{C}_{\mathbb{X}}^{\sigma}$, and so ${\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})={\rm tr}(\omega_{R}(1))$. Moreover, there is an isomorphism of graded $R$-modules

[TABLE]

given by $h\mapsto\mu_{h}$, where $\mu_{h}:\mathfrak{C}_{\mathbb{X}}^{\sigma}\rightarrow R$ is the multiplication by $h$. This implies that

[TABLE]

In particular, the scheme $\mathbb{X}$ is arithmetically Gorenstein if and only if ${\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})=R$.

The relation between the trace ${\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$ and the conductor of $R$ in the graded ring $\widetilde{R}=\prod_{j=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{j}}[T_{j}]$ is given by the following proposition.

Proposition 6.3**.**

Let $\mathfrak{F}_{\widetilde{R}/R}$ be the conductor of $R$ in $\widetilde{R}$.

1. (a)

If $\mathbb{X}$ is a Cayley-Bacharach scheme, then $\mathfrak{F}_{\widetilde{R}/R}\subseteq{\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$. 2. (b)

The scheme $\mathbb{X}$ is a Cayley-Bacharach scheme such that $\mathfrak{F}_{\widetilde{R}/R}={\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$ if and only if $\mathbb{X}$ has minimal Dedekind different.

Proof.

Suppose that $\mathbb{X}$ is a Cayley-Bacharach scheme. Then Theorem 5.4 yields $\mathfrak{F}_{\widetilde{R}/R}=\bigoplus_{i\geq r_{\mathbb{X}}}R_{i}$. Furthermore, by [9, Proposition 3.2], for every $j\in\{1,\dots,s\}$, we find an element $x_{0}^{-2r_{\mathbb{X}}}\widetilde{g}^{*}_{j}\in(\mathfrak{C}^{\sigma}_{\mathbb{X}})_{-r_{\mathbb{X}}}\setminus\{0\}$ such that $\widetilde{g}^{*}_{j}\in R_{r_{\mathbb{X}}}$ and $(\widetilde{g}^{*}_{j})_{p_{j}}$ is a unit of ${\mathcal{O}}_{\mathbb{X},p_{j}}$. It is also clear that $\bigoplus_{i\geq 2r_{\mathbb{X}}}R_{i}\subseteq\delta_{\mathbb{X}}^{\sigma}$. Hence we have $R_{r_{\mathbb{X}}}\subseteq\delta_{\mathbb{X}}^{\sigma}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}={\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$, and claim (a) follows.

Now we prove (b). Assume that $\mathbb{X}$ is a Cayley-Bacharach scheme such that $\mathfrak{F}_{\widetilde{R}/R}={\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$. For a contradiction suppose that $\mathbb{X}$ does not have minimal Dedekind different. This implies $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(2r_{\mathbb{X}}-1)\neq 0$. Let $h\in(\delta_{\mathbb{X}}^{\sigma})_{2r_{\mathbb{X}}-1}\setminus\{0\}$. Then there is an index $j\in\{1,\dots,s\}$ such that $h_{p_{j}}\neq 0$ in ${\mathcal{O}}_{\mathbb{X},p_{j}}$. Let $\widetilde{g}^{*}_{j}\in R_{r_{\mathbb{X}}}$ be given as in the proof of (a). Then we have $(h\widetilde{g}^{*}_{j})_{p_{j}}\neq 0$ in ${\mathcal{O}}_{\mathbb{X},p_{j}}$. It follows that $0\neq hx_{0}^{-2r_{\mathbb{X}}}\widetilde{g}^{*}_{j}\in({\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma}))_{r_{\mathbb{X}}-1}$. But ${\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})=\mathfrak{F}_{\widetilde{R}/R}=\bigoplus_{i\geq r_{\mathbb{X}}}R_{i}$, which is impossible.

Conversely, suppose that the scheme $\mathbb{X}$ has minimal Dedekind different. Then $\mathbb{X}$ is a Cayley-Bacharach scheme by Corollary 4.6(b). Moreover, the Dedekind different satisfies $\delta_{\mathbb{X}}^{\sigma}=\bigoplus_{i\geq 2r_{\mathbb{X}}}R_{i}$. It follows that $({\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma}))_{r_{\mathbb{X}}-1}=(\delta_{\mathbb{X}}^{\sigma})_{2r_{\mathbb{X}}-1}(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-r_{\mathbb{X}}}=\langle 0\rangle$. Therefore the equality $\mathfrak{F}_{\widetilde{R}/R}={\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$ follows from claim (a). ∎

Example 6.4**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{9}\}\subseteq\mathbb{P}^{3}_{\mathbb{Q}}$ be the set of nine points given in Example 3.11. We saw that $\operatorname{HF}_{\mathbb{X}}:\ 1\ 4\ 9\ 9\cdots$ and $r_{\mathbb{X}}=2$. Moreover, $\mathbb{X}$ has minimal Dedekind different, and so it is a Cayley-Bacharach scheme. In addition, we have ${\rm tr}(\mathfrak{C}_{\mathbb{X}})=\bigoplus_{i\geq 2}R_{i}$ by Proposition 6.3(b).

In view of the theory of nearly and almost Gorenstein rings given in the papers [2, 5, 7], we introduce the following two special classes of 0-dimensional schemes in $\mathbb{P}^{n}_{K}$. Note that $\mathfrak{m}$ denotes the homogeneous maximal ideal of $R$.

Definition 6.5**.**

Let $\mathbb{X}$ be a 0-dimensional locally Gorenstein scheme in $\mathbb{P}^{n}_{K}$.

1. (a)

The scheme $\mathbb{X}$ is called a nearly Gorenstein scheme if $\mathfrak{m}\subseteq{\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$. 2. (b)

The scheme $\mathbb{X}$ is called an almost Gorenstein scheme if there is an exact sequence of graded $R$-modules

[TABLE]

with $\mathfrak{m}\cdot C=\langle 0\rangle$.

Note that every arithmetically Gorenstein scheme $\mathbb{X}$ is nearly Gorenstein and almost Gorenstein, and that $\mathbb{X}$ is a Cayley-Bacharach scheme if it is an almost Gorenstein scheme (since there exists an element $g\in(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-r_{\mathbb{X}}}$ with ${\rm Ann}_{R}(g)=\langle 0\rangle$).

In our setting, the class of almost Gorenstein schemes is smaller than that of nearly Gorenstein schemes. The following proof of this property mimics the proof of [7, Proposition 6.1] for local rings.

Proposition 6.6**.**

If $\mathbb{X}$ is an almost Gorenstein scheme, then it is a nearly Gorenstein scheme and $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}}+1)=\operatorname{HF}_{\mathbb{X}}(1)$.

Proof.

If $\mathbb{X}$ is arithmetically Gorenstein, we have $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}}+1)=\operatorname{HF}_{\mathbb{X}}(1)$ by [9, Proposition 5.8]. So, we may assume that $\mathbb{X}$ is not arithmetically Gorenstein. Then $C\neq\langle 0\rangle$ and $\mathfrak{m}\cdot C=\langle 0\rangle$, and so $\underline{\operatorname{Hom}}_{R}(C,R)=\langle 0\rangle$. By applying the functor $\underline{\operatorname{Hom}}_{R}(-,R)$ to the homogeneous exact sequence

[TABLE]

we get the exact sequence

[TABLE]

Here the map $\theta^{*}:\delta_{\mathbb{X}}^{\sigma}(r_{\mathbb{X}})\rightarrow R$ is given by $h\mapsto h\theta(1)$ and $\operatorname{deg}(\theta(1))=-r_{\mathbb{X}}$. Also, we have $\mathfrak{m}\cdot{\rm\underline{Ext}}^{1}_{R}(C,R)=\langle 0\rangle$. This implies $\mathfrak{m}\subseteq\delta_{\mathbb{X}}^{\sigma}\cdot\theta(1)\subseteq\delta_{\mathbb{X}}^{\sigma}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}={\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$, and so $\mathbb{X}$ is a nearly Gorenstein scheme. Moreover, Remark 4.9 yields $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}})=0$, and so we have $\mathfrak{m}=\delta_{\mathbb{X}}^{\sigma}\cdot\theta(1)$. Consequently, we get $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}}+1)=\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}\cdot\theta(1)}(1)=\operatorname{HF}_{\mathfrak{m}}(1)=\operatorname{HF}_{\mathbb{X}}(1)$, since ${\rm Ann}_{R}(\theta(1))=\langle 0\rangle$. ∎

Notice that every nearly Gorenstein scheme $\mathbb{X}$ satisfies $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}}+1)\neq 0$, because otherwise we would have $\mathfrak{m}_{1}\nsubseteq({\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma}))_{1}=\langle 0\rangle$. Hence this implies the following corollary.

Corollary 6.7**.**

If $\mathbb{X}$ has minimal Dedekind different and $r_{\mathbb{X}}\geq 2$, then it is not a nearly Gorenstein scheme.

It is natural to ask: If $\mathbb{X}$ is a nearly Gorenstein scheme, when is $\mathbb{X}$ an almost Gorenstein scheme? In the case that $\Delta_{\mathbb{X}}=\operatorname{deg}(\mathbb{X})-\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{X}}-1)=1$, we have the following answer to this question.

Proposition 6.8**.**

Let $\mathbb{X}$ be a 0-dimensional locally Gorenstein scheme in $\mathbb{P}^{n}_{K}$ such that $\Delta_{\mathbb{X}}=1$. Then the following conditions are equivalent.

1. (a)

$\mathbb{X}$* is an almost Gorenstein scheme.* 2. (b)

$\mathbb{X}$* is a nearly Gorenstein Cayley-Bacharach scheme.*

Proof.

It suffices to prove the implication “(b)$\Rightarrow$(a)”. Suppose that $\mathbb{X}$ is a nearly Gorenstein Cayley-Bacharach scheme. We may assume that $\mathbb{X}$ is not arithmetically Gorenstein. By [9, Proposition 5.8], we have $(\delta_{\mathbb{X}}^{\sigma})_{r_{\mathbb{X}}}=\langle 0\rangle$. Since $\mathbb{X}$ is nearly Gorenstein, we have $\mathfrak{m}={\rm tr}(\mathfrak{C}_{\mathbb{X}}^{\sigma})$. This implies

[TABLE]

Since $\mathbb{X}$ is a Cayley-Bacharach scheme and $\Delta_{\mathbb{X}}=1$, [9, Proposition 4.12] shows that there exists an element $g\in(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-r_{\mathbb{X}}}$ such that $(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-r_{\mathbb{X}}}=\langle g\rangle_{K}$ and ${\rm Ann}_{R}(g)=\langle 0\rangle$. Hence we have $(\delta_{\mathbb{X}}^{\sigma})_{r_{\mathbb{X}}+1}\cdot g=\mathfrak{m}_{1}$. Consider the exact sequence of graded $R$-modules

[TABLE]

where $\theta:R\rightarrow\mathfrak{C}_{\mathbb{X}}^{\sigma}(-r_{\mathbb{X}})$ is the injection given by $1\mapsto g$ and $C=\mathfrak{C}_{\mathbb{X}}^{\sigma}(-r_{\mathbb{X}})/\langle g\rangle_{R}$. Now we want to show that $\mathfrak{m}\cdot C=0$. Clearly, $\mathfrak{m}\cdot C=0$ if and only if $\mathfrak{m}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}(-r_{\mathbb{X}})=\mathfrak{m}\cdot g$. This is equivalent to $\mathfrak{m}_{1}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}(-r_{\mathbb{X}})\subseteq\mathfrak{m}\cdot g$. Let $i\geq 0$, $g^{\prime}\in(\mathfrak{C}_{\mathbb{X}}^{\sigma}(-r_{\mathbb{X}}))_{i}$, and $\ell\in\mathfrak{m}_{1}$. Set $\ell=h\cdot g$ with $h\in(\delta_{\mathbb{X}}^{\sigma})_{r_{\mathbb{X}}+1}$, since $(\delta_{\mathbb{X}}^{\sigma})_{r_{\mathbb{X}}+1}\cdot g=\mathfrak{m}_{1}$. We have

[TABLE]

Since $h\cdot g^{\prime}\in R_{1}$, we get $\ell\cdot g^{\prime}\in\mathfrak{m}\cdot g$. It follows that $\mathfrak{m}_{1}\cdot\mathfrak{C}_{\mathbb{X}}^{\sigma}(-r_{\mathbb{X}})\subseteq\mathfrak{m}\cdot g$, and hence $\mathfrak{m}\cdot C=0$, as desired. ∎

Let us apply this proposition to an explicit example.

Example 6.9**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{7}\}\subseteq\mathbb{P}^{2}_{\mathbb{Q}}$ be the set of seven points given by $p_{1}=(1:0:0)$ $p_{2}=(1:1:0)$, $p_{3}=(1:0:1)$, $p_{4}=(1:1:1)$, $p_{5}=(1:0:2)$, $p_{6}=(1:2:1)$, and $p_{7}=(1:2:2)$. Sketch $\mathbb{X}$ in the affine plane $D_{+}(X_{0})$ as follows:

[TABLE]

The Hilbert function of $\mathbb{X}$ is $\operatorname{HF}_{\mathbb{X}}:\ 1\ 3\ 6\ 7\ 7\cdots$ and $r_{\mathbb{X}}=3$. We also have $\Delta_{\mathbb{X}}=1$ and the scheme $\mathbb{X}$ is a Cayley-Bacharach scheme. A calculation yields

[TABLE]

Since $\operatorname{HF}_{\delta_{\mathbb{X}}}(r_{\mathbb{X}})=\operatorname{HF}_{\delta_{\mathbb{X}}}(3)=0$, the scheme $\mathbb{X}$ is not arithmetically Gorenstein. Furthermore, we have

[TABLE]

Hence $\mathbb{X}$ is a nearly Gorenstein scheme. An application of Proposition 6.8 implies that $\mathbb{X}$ is an almost Gorenstein scheme. In this case we do not have $\mathfrak{C}_{\mathbb{X}}=\langle(\mathfrak{C}_{\mathbb{X}})_{-r_{\mathbb{X}}}\rangle_{R}$, since

[TABLE]

Moreover, if we let $p^{\prime}_{7}=(1:2:0)$ and $\mathbb{Y}=\{p_{1},\dots,p_{6},p^{\prime}_{7}\}\subseteq\mathbb{P}^{2}_{\mathbb{Q}}$, then the set $\mathbb{Y}$ satisfies $\operatorname{HF}_{\mathbb{Y}}=\operatorname{HF}_{\mathbb{X}}$, but it is not an almost Gorenstein scheme, since it is not a Cayley-Bacharach scheme.

When $\mathbb{X}$ is a Cayley-Bacharach scheme, the following proposition provides a necessary and sufficient condition for $\mathbb{X}$ to be almost Gorenstein.

Proposition 6.10**.**

Let $K$ be an infinite field, and let $\mathbb{X}$ be a 0-dimensional locally Gorenstein scheme in $\mathbb{P}^{n}_{K}$. Suppose that $\mathbb{X}$ is a Cayley-Bacharach scheme.

1. (a)

We have $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(i)\leq\operatorname{HF}_{\mathbb{X}}(i-r_{\mathbb{X}})$ for all $i\in\mathbb{Z}$. In particular, the scheme $\mathbb{X}$ is arithmetically Gorenstein if and only if $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(i)=\operatorname{HF}_{\mathbb{X}}(i-r_{\mathbb{X}})$ for all $i\in\mathbb{Z}$. 2. (b)

$\mathbb{X}$* is an almost Gorenstein scheme if and only if $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}}+1)=\operatorname{HF}_{\mathbb{X}}(1)$.*

Proof.

Since $\mathbb{X}$ is a Cayley-Bacharach scheme, we have $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(i)=\operatorname{HF}_{\mathbb{X}}(i-r_{\mathbb{X}})$ for $i<r_{\mathbb{X}}$ or $i\geq 2r_{\mathbb{X}}$ by Proposition 4.8. Hence it suffices to consider the case $r_{\mathbb{X}}\leq i<2r_{\mathbb{X}}$. Note that $K$ is infinite. By [9, Remark 4.13], there exists a homogeneous element $g\in(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-r_{\mathbb{X}}}$ such that ${\rm Ann}_{R}(g)=\langle 0\rangle$. Then we have $g\cdot(\delta_{\mathbb{X}}^{\sigma})_{i}\subseteq R_{i-r_{\mathbb{X}}}$. This implies $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(i)\leq\operatorname{HF}_{\mathbb{X}}(i-r_{\mathbb{X}})$ for $r_{\mathbb{X}}\leq i<2r_{\mathbb{X}}$. Moreover, the additional claim of (a) follows from Remark 4.9.

To prove (b), according to Proposition 6.6 and (a) we only need to prove that $\mathbb{X}$ is almost Gorenstein if $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}})=0$ and $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}}+1)=\operatorname{HF}_{\mathbb{X}}(1)$. In this case we have $(\delta_{\mathbb{X}}^{\sigma})_{r_{\mathbb{X}}+1}\cdot g=\mathfrak{m}_{1}$, where $g\in(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-r_{\mathbb{X}}}$ is given as above. A similar argument as in the proof of Proposition 6.8 implies that $\mathbb{X}$ is an almost Gorenstein scheme. ∎

Recall that a 0-dimensional scheme $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ is called level if the socle of the Artinian local ring $\overline{R}=R/\langle x_{0}\rangle$ equals $\overline{R}_{r_{\mathbb{X}}}$. According to [12, Satz 11.6], the scheme $\mathbb{X}$ is level if and only if the canonical module $\omega_{R}$ is generated by homogeneous elements of degree $-r_{\mathbb{X}}+1$. It is also known that $\mathbb{X}$ is a Cayley-Bacharach scheme if it is level (see [3, Proposition 6.1]). Furthermore, Example 6.9 also shows that an almost Gorenstein scheme may not be a level scheme.

Proposition 6.11**.**

Let $K$ be an infinite field, and let $\mathbb{X}$ be a 0-dimensional locally Gorenstein scheme in $\mathbb{P}^{n}_{K}$.

1. (a)

If $r_{\mathbb{X}}=1$ then $\mathbb{X}$ is an almost Gorenstein level scheme. 2. (b)

If $r_{\mathbb{X}}=1$ and $\operatorname{deg}(\mathbb{X})>2$ then $\mathbb{X}$ has minimal Dedekind different. 3. (c)

If $\mathbb{X}$ is level and $\min\{\Delta_{\mathbb{X}},r_{\mathbb{X}}\}\geq 2$, then $\mathbb{X}$ is not an almost Gorenstein scheme.

Proof.

(a) Suppose that $r_{\mathbb{X}}=1$ and $\mathbb{X}$ is not arithmetically Gorenstein. It is clear that $\mathbb{X}$ is a Cayley-Bacharach scheme. Since $K$ is infinite, [9, Remark 4.13] yields an element $g\in(\mathfrak{C}_{\mathbb{X}}^{\sigma}(-1))_{0}$ such that ${\rm Ann}_{R}(g)=\langle 0\rangle$. We see that $\dim_{K}(\mathfrak{m}_{1}\cdot g)=\operatorname{deg}(\mathbb{X})=\dim_{K}(\mathfrak{C}_{\mathbb{X}}^{\sigma}(-1))_{1}$. This implies $\mathfrak{m}_{1}\cdot g=(\mathfrak{C}_{\mathbb{X}}^{\sigma}(-1))_{1}$. Hence $\mathbb{X}$ is a level scheme. Furthermore, we have $\delta_{\mathbb{X}}^{\sigma}=\bigoplus_{i\geq 2}R_{i}$ by [9, Proposition 5.8], and thus $\operatorname{HF}_{\delta_{\mathbb{X}}^{\sigma}}(r_{\mathbb{X}}+1)=\operatorname{HF}_{\mathbb{X}}(1)$. Consequently, Proposition 6.10 shows that $\mathbb{X}$ is an almost Gorenstein scheme.

(b) Since $\operatorname{deg}(\mathbb{X})>2$, we have $\Delta_{\mathbb{X}}=\operatorname{deg}(\mathbb{X})-1\geq 2$. So, $\mathbb{X}$ is not an arithmetically Gorenstein scheme. As above, the Dedekind different satisfies $\delta_{\mathbb{X}}^{\sigma}=\bigoplus_{i\geq 2}R_{i}$. Hence $\mathbb{X}$ has minimal Dedekind different.

(c) Let us write

[TABLE]

where $c_{i}=\operatorname{deg}(\mathbb{X})-h_{r_{\mathbb{X}}-i-1}$ for $i=1,\dots,r_{\mathbb{X}}-1$ and $c_{0}=\operatorname{deg}(\mathbb{X})-1$. Suppose that $\mathbb{X}$ is an almost Gorenstein level scheme with $\min\{\Delta_{\mathbb{X}},r_{\mathbb{X}}\}\geq 2$. We choose an exact sequence

[TABLE]

of graded $R$-modules so that $\mathfrak{m}\cdot C=\langle 0\rangle$. Set $g=\theta(1)$. For $i\geq 1$, we have

[TABLE]

Since $\mathbb{X}$ is level, we have $\mathfrak{C}_{\mathbb{X}}^{\sigma}=\langle(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-r_{\mathbb{X}}}\rangle_{R}$ by [12, Satz 11.6]. This implies

[TABLE]

Therefore the Hilbert function of $\mathfrak{C}_{\mathbb{X}}^{\sigma}(-r_{\mathbb{X}})$ has the form

[TABLE]

It follows that $\operatorname{deg}(\mathbb{X})-1=c_{r_{\mathbb{X}}-1}=h_{r_{\mathbb{X}}-1}=\operatorname{deg}(\mathbb{X})-\Delta_{\mathbb{X}}$. Because $\Delta_{\mathbb{X}}>1$, we have $\operatorname{deg}(\mathbb{X})-1\neq\operatorname{deg}(\mathbb{X})-\Delta_{\mathbb{X}}$, a contradiction. ∎

Our next corollary is an immediate consequence of this proposition. This result also follows from [5, Lemma 10.2 and Theorem 10.4].

Corollary 6.12**.**

Let $K$ be an infinite field, and let $\mathbb{X}$ be a 0-dimensional locally Gorenstein scheme in $\mathbb{P}^{n}_{K}$ such that $\Delta_{\mathbb{X}}\geq 2$. Then $\mathbb{X}$ is an almost Gorenstein level scheme if and only if $r_{\mathbb{X}}=1$.

Finally, we are interested in the question: if $\mathbb{X}$ is an almost Gorenstein scheme with $r_{\mathbb{X}}\geq 2$, then does $\Delta_{\mathbb{X}}=1$ hold? When $\mathbb{X}$ is a set of $s$ distinct $K$-rational points in uniform position, [6, Theorem 4.7] provides an affirmative answer to this question with the help of the Biinjective Map Lemma (cf. [11]). Recall that a set of $s$ distinct $K$-rational points $\mathbb{X}$ is called ${\bf(i,j)}$-uniform if every subscheme $\mathbb{Y}\subseteq\mathbb{X}$ of degree $\operatorname{deg}(\mathbb{X})-i$ satisfies $\operatorname{HF}_{\mathbb{Y}}(j)=\operatorname{HF}_{\mathbb{X}}(j)$. Notice that $\mathbb{X}$ is a Cayley-Bacharach scheme if and only if it is $(1,r_{\mathbb{X}}-1)$-uniform, and if $\mathbb{X}$ is $(i,j)$-uniform then it is also $(i-1,j)$-uniform and $(i,j-1)$-uniform. For further information about the uniformity of $\mathbb{X}$ see [3, 11]. The following proposition shows that the above question also has an affirmative answer when $\mathbb{X}$ is $(2,r_{\mathbb{X}}-1)$-uniform.

Proposition 6.13**.**

Let $\mathbb{X}=\{p_{1},\dots,p_{s}\}\subseteq\mathbb{P}^{n}_{K}$ be a $(2,r_{\mathbb{X}}-1)$-uniform set of $s$ distinct $K$-rational points. Suppose that $\mathbb{X}$ is an almost Gorenstein scheme and $r_{\mathbb{X}}\geq 2$. Then we have $\Delta_{\mathbb{X}}=1$.

Proof.

Suppose for a contradiction that $\Delta_{\mathbb{X}}>1$. Since $\mathbb{X}$ is an almost Gorenstein scheme, we choose an exact sequence

[TABLE]

of graded $R$-modules so that $\mathfrak{m}\cdot C=\langle 0\rangle$, and set $g=\theta(1)$. We write $g=x_{0}^{-2r_{\mathbb{X}}}\widetilde{g}$ with $\widetilde{g}\in R_{r_{\mathbb{X}}}$. Then $\widetilde{g}(p_{j})\neq 0$ for all $j=1,\dots,s$. For each $f\in R_{i}$ with $i\geq 0$, we define the value $\eta(f):=\#\{\,j\,\mid\,1\leq j\leq s,f(p_{j})=0\,\}$. Clearly, we have $\eta(\ell\widetilde{g})=\eta(\ell)$ for all $\ell\in R_{1}$. Now we let $\ell_{0}\in R_{1}$ be a non-zero element such that $\eta(\ell_{0})=\max\{\,\eta(\ell)\,\mid\,\ell\in R_{1}\setminus\{0\}\,\}$. Since $r_{\mathbb{X}}\geq 2$ and $\mathbb{X}$ is a Cayley-Bacharach scheme, there exist at least two points $p_{j_{1}},p_{j_{2}}\in\mathbb{X}$ such that $\ell_{0}(p_{j_{1}})\neq 0$ and $\ell_{0}(p_{j_{2}})\neq 0$. Let $f_{j}\in R_{r_{\mathbb{X}}}$ be the separator of $\mathbb{X}\setminus\{p_{j}\}$ in $\mathbb{X}$ with $f_{j}(p_{j})=1$ and $f_{j}(p_{k})=0$ for $k\neq j$. Since $\mathbb{X}$ is $(2,r_{\mathbb{X}}-1)$-uniform, [11, Proposition 3.4] yields that $\{\overline{f}_{j_{1}},\overline{f}_{j_{2}}\}$ is linearly independent in $\overline{R}_{r_{\mathbb{X}}}$. Let $\Sigma=\{j_{1},\dots,j_{\Delta_{\mathbb{X}}}\}$ be a subset of $\{1,\dots,s\}$ such that $\overline{f}_{j_{1}},\overline{f}_{j_{2}},\cdots,\overline{f}_{j_{\Delta_{\mathbb{X}}}}$ form a $K$-basis of $\overline{R}_{r_{\mathbb{X}}}$. By [13, Corollary 1.10], there exist elements $g_{j_{1}},g_{j_{2}}\in(\mathfrak{C}_{\mathbb{X}})_{-r_{\mathbb{X}}}$ of the form $g_{j_{l}}=x_{0}^{-2r_{\mathbb{X}}}(f_{j_{l}}+\sum_{k\notin\Sigma}\beta_{kj_{l}}f_{k})$ for $l=1,2$, where $\beta_{kj_{l}}\in K$. Letting $\widetilde{g}_{j_{1}}=f_{j_{1}}+\sum_{k\notin\Sigma}\beta_{kj_{1}}f_{k}$, we have $\ell_{0}\widetilde{g}_{j_{1}}\neq 0$ and $\eta(\ell_{0}\widetilde{g}_{j_{1}})\geq\eta(\ell_{0})+1$. Thus we get $\eta(\ell_{0}\widetilde{g}_{j_{1}})>\eta(\ell\widetilde{g})$ for all $\ell\in R_{1}\setminus\{0\}$. Since $x_{0}$ is a non-zerodivisor of $R$, this implies that $0\neq\ell_{0}g_{j_{1}}\notin\mathfrak{m}_{1}\cdot g$. In particular, we have $\mathfrak{m}\cdot C\neq\langle 0\rangle$, a contradiction. ∎

Acknowledgments.

This paper is partially based on the third author’s dissertation [18]. The authors thank Ernst Kunz for his encouragement to elaborate some results presented here. The second author would also like to acknowledge her financial support from OeAD.