K\"ahler differential algebras for 0-dimensional schemes

TL;DR

This paper investigates the structure of Kähler differential algebras associated with 0-dimensional schemes in projective space, providing explicit calculations and bounds for their Hilbert functions and polynomials.

Contribution

It offers explicit presentations and detailed analysis of Kähler differential modules for 0-dimensional schemes, including bounds and specific cases like subschemes of P^1 and P^2.

Findings

Explicit Hilbert function values for differential modules

Bounds on Hilbert polynomials and regularity indices

Detailed results for schemes on a conic in P^2

Abstract

Given a 0-dimensional scheme in a projective space $\mathbb{P}^n$ over a field $K$, we study the K\"ahler differential algebra $\Omega_{R/K}$ of its homogeneous coordinate ring $R$. Using explicit presentations of the modules $\Omega^m_{R/K}$ of K\"ahler differential $m$-forms, we determine many values of their Hilbert functions explicitly and bound their Hilbert polynomials and regularity indices. Detailed results are obtained for subschemes of $\mathbb{P}^1$, fat point schemes, and subschemes of $\mathbb{P}^2$ supported on a conic.

Equations

$$\begin{array}[]{ll}\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}:\ 0\ 0\ 3\ 6\ {\bf 4}\ 1\ 0\ 0\cdots,&\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{Y}}/K}}:\ 0\ 0\ 3\ 6\ {\bf 5}\ 1\ 0\ 0\cdots,\\ \operatorname{HF}_{\Omega^{3}_{R_{\mathbb{X}}/K}}:\ 0\ 0\ 0\ 1\ {\bf 0}\ 0\ \cdots,&\operatorname{HF}_{\Omega^{3}_{R_{\mathbb{Y}}/K}}:\ 0\ 0\ 0\ 1\ {\bf 1}\ 0\ \cdots.\end{array}$$

$$0⟶{\Omega }_{R_{\mathbb{X}}/K}^{n+1}⟶{\Omega }_{R_{\mathbb{X}}/K}^n⟶\cdots ⟶{\Omega }_{R_{\mathbb{X}}/K}^2⟶{\Omega }_{R_{\mathbb{X}}/K}^1⟶{\mathfrak{m}}_{\mathbb{X}}⟶0$$

$$\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i)=\dim_{K}(\Omega^{m}_{R_{\mathbb{X}}/K})_{i}\,\quad\,\mbox{for all $i\in\mathbb{Z}$.}$$

$${\Omega }_{R_{\mathbb{X}}/K}^m\cong {\Omega }_{S/K}^m/(I_{\mathbb{X}}{\Omega }_{S/K}^m+d{I}_{\mathbb{X}}{\Omega }_{S/K}^{m-1})$$

$$\left\{F_{j}d{X}_{i_1}\wedge \cdots \wedge d{X}_{i_m}\mid 1\le j\le r,0\le i_1<\cdots <i_m\le n\right\}$$

$$\cup \left\{d{F}_j\wedge d{X}_{j_1}\wedge \cdots \wedge d{X}_{j_{m-1}}\mid 1\le j\le r,0\le j_1<\cdots <j_{m-1}\le n\right\}.$$

$$\Omega^{n+1}_{R_{\mathbb{X}}/K}\cong\big{(}S/\big{\langle}\,\tfrac{\partial F_{j}}{\partial X_{i}}\mid 0\leq i\leq n,1\leq j\leq r\,\big{\rangle}\big{)}(-n-1).$$

$$\frac{\partial F_j}{\partial X_i}d{X}_0\wedge \cdots \wedge d{X}_n=(-1{)}^{i}d{F}_j\wedge d{X}_0\wedge \cdots \wedge \widehat{d{X}_i}\wedge \cdots \wedge d{X}_n\in d{I}_{\mathbb{X}}{\Omega }_{S/K}^n$$

$$dF-(F_{1}d{G}_1+\cdots +F_{r}d{G}_r)$$

$$=G_1\sum _{i=0}^n\frac{\partial F_1}{\partial X_i}d{X}_i+\cdots +G_r\sum _{i=0}^n\frac{\partial F_r}{\partial X_i}d{X}_i$$

$$\cdots \stackrel{\gamma }{⟶}{\Omega }_{R_{\mathbb{X}}/K}^2\stackrel{\gamma }{⟶}{\Omega }_{R_{\mathbb{X}}/K}^1\stackrel{\gamma }{⟶}{\mathfrak{m}}_{\mathbb{X}}⟶0$$

$$\gamma ({\omega }_1\wedge \cdots \wedge {\omega }_m)=\sum _{j=1}^m(-1{)}^{j+1}\gamma ({\omega }_j)\cdot {\omega }_1\wedge \cdots \wedge \widehat{{\omega }_j}\wedge \cdots \wedge {\omega }_m$$

$$\mathcal{K}$$

$$(\gamma \circ d)(\omega )=ifd{x}_{i_1}\wedge \cdots \wedge d{x}_{i_m}-df\wedge \gamma (d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_m})$$

$$(d\circ \gamma )(\omega )$$

$$=df\wedge \gamma (d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_m})+mfd{x}_{i_1}\wedge \cdots \wedge d{x}_{i_m}.$$

$$\begin{array}[]{ll}\operatorname{HF}_{\mathbb{X}}=\operatorname{HF}_{\mathbb{Y}}\!\!\!&:\ 1\ 3\ 5\ 6\ 6\ \cdots\\ \operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}=\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}\!\!\!&:\ 0\ 3\ 8\ 11\ 10\ 7\ 6\ 6\cdots.\end{array}$$

$$\begin{array}[]{ll}\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}:\ 0\ 0\ 3\ 6\ {\bf 4}\ 1\ 0\ 0\cdots,&\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{Y}}/K}}:\ 0\ 0\ 3\ 6\ {\bf 5}\ 1\ 0\ 0\cdots,\\ \operatorname{HF}_{\Omega^{3}_{R_{\mathbb{X}}/K}}:\ 0\ 0\ 0\ 1\ {\bf 0}\ 0\ \cdots,&\operatorname{HF}_{\Omega^{3}_{R_{\mathbb{Y}}/K}}:\ 0\ 0\ 0\ 1\ {\bf 1}\ 0\ \cdots.\end{array}$$

$$\begin{array}[]{ll}\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}&:\ 0\ 2\ 4\ 6\ \cdots\ 2(\mu-2)\ 2(\mu-1)\ 2\mu-1\ 2\mu-2\ \cdots\ 2\mu-s\ 2\mu-s\cdots\\ \operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}&:\ 0\ 0\ 1\ 2\ \cdots\ \mu-2\ \mu-1\ \mu-2\ \mu-3\ \cdots\ \mu-s\ \mu-s\cdots\\ \end{array}$$

$${\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^2}(i)$$

$$={\mathrm{HF}}_{S/⟨G{H}_1,G{H}_2⟩}(i-2)$$

$$={\mathrm{HF}}_{S/⟨G⟩}(i-2)+{\mathrm{HF}}_{⟨G⟩/⟨G{H}_1,G{H}_2⟩}(i-2)$$

$$={\mathrm{HF}}_{S/⟨G⟩}(i-2)+{\mathrm{HF}}_{⟨G⟩}(i-2)-2{\mathrm{HF}}_{⟨G⟩}(i-1-s)+{\mathrm{HF}}_{⟨G⟩}(i-2s)$$

$$={\mathrm{HF}}_S(i-2)-2{\mathrm{HF}}_S(i-1-\mu )+{\mathrm{HF}}_S(i-s-\mu )$$

$$=\left(\genfrac{}{}{0ex}{}{i-1}{1}\right)-2\left(\genfrac{}{}{0ex}{}{i-\mu }{1}\right)+\left(\genfrac{}{}{0ex}{}{i-s-\mu +1}{1}\right)$$

$${\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^2}:0012\cdots \mu -2\mu -1\mu -2\mu -3\cdots \mu -s\mu -s\cdots .$$

$$0⟶{\Omega }_{R_{\mathbb{X}}/K}^2⟶{\Omega }_{R_{\mathbb{X}}/K}^1⟶{\mathfrak{m}}_{\mathbb{X}}⟶0.$$

$${\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^1}:0246\cdots 2(\mu -2)2(\mu -1)2\mu -12\mu -2\cdots 2\mu -s2\mu -s\cdots$$

$$\begin{array}[]{ll}\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}&:\ 0\ 2\ 4\ 6\ 8\ 10\ 11\ 10\ 9\ 9\ \cdots\\ \operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}&:\ 0\ 0\ 1\ 2\ 3\ 4\ 5\ 4\ 3\ 3\ \cdots.\\ \end{array}$$

$${\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^m}(r_{\mathbb{X}}+m)>{\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^m}(r_{\mathbb{X}}+m+1)>\cdots >{\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^m}(\mathrm{ri}({\Omega }_{R_{\mathbb{X}}/K}^m)).$$

$${\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^m}(i)={\mathrm{HF}}_{{\Omega }_{S/K}^m}(i)=\left(\genfrac{}{}{0ex}{}{n+1}{m}\right)\cdot \left(\genfrac{}{}{0ex}{}{n+i-m}{n}\right).$$

$${\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^m}(i)\le {\mathrm{HF}}_{{\Omega }_{S/K}^m/I_{\mathbb{X}}{\Omega }_{S/K}^m}(i)=\left(\genfrac{}{}{0ex}{}{n+1}{m}\right){\mathrm{HF}}_{\mathbb{X}}(i)\le \left(\genfrac{}{}{0ex}{}{n+1}{m}\right)\mathrm{deg}(\mathbb{X})$$

$$({\Omega }_{R_{\mathbb{X}}/K}^m{)}_{i+m}=(R_{\mathbb{X}}{)}_{i}d{x}_0\wedge \cdots \wedge d{x}_{m-1}+\cdots +(R_{\mathbb{X}}{)}_{i}d{x}_{n-m+1}\wedge \cdots \wedge d{x}_n.$$

$${\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^m}(i+m-1)\ge {\mathrm{HF}}_{{\Omega }_{R_{\mathbb{X}}/K}^m}(i+m).$$

Full text

1. Introduction

In the paper [DK], G. de Dominicis and the first author introduced the application of Kähler differential modules to the study of 0-dimensional subschemes $\mathbb{X}$ of a projective space $\mathbb{P}^{n}$ over a field $K$ of characteristic zero. They showed that this graded module over the homogeneous coordinate ring $R_{\mathbb{X}}$ contains numerical and algebraic information which is not readily available from the homogeneous vanishing ideal or from $R_{\mathbb{X}}$. Later, in [KLL], the authors extended and refined these techniques for fat point schemes in $\mathbb{P}^{n}$. Following the classical construction described by E. Kunz in his book [Kun], it is natural to define the Kähler differential algebra $\Omega_{R_{\mathbb{X}}/K}=\bigoplus_{m\in\mathbb{N}}\Omega^{m}_{R_{\mathbb{X}}/K}$ of $\mathbb{X}$ as the exterior algebra of its Kähler differential module $\Omega^{1}_{R_{\mathbb{X}}/K}$. This invites the question whether the Kähler differential algebra contains numerical and algebraic information about $\mathbb{X}$ which is not readily available in $R_{\mathbb{X}}$ or in $\Omega^{1}_{R_{\mathbb{X}}/K}$. Thus the following example provided the initial spark to ignite the curiosity of the authors.

Example 1.1** (See Example 2.6).**

Let $\mathbb{X}$ and $\mathbb{Y}$ be two sets of six reduced $K$-rational points in $\mathbb{P}^{2}$ such that $\mathbb{X}$ is contained in a non-singular conic, and such that $\mathbb{Y}$ consists of three points on a line and three points on another line. Then the Hilbert functions of $R_{\mathbb{X}}$ and $R_{\mathbb{Y}}$ agree, as do the Hilbert functions of $\Omega^{1}_{R_{\mathbb{X}}/K}$ and $\Omega^{1}_{R_{\mathbb{Y}}/K}$. However, the Hilbert functions of $\Omega^{2}_{R_{\mathbb{X}}/K}$ and $\Omega^{2}_{R_{\mathbb{Y}}/K}$ are different, and also the Hilbert functions of $\Omega^{3}_{R_{\mathbb{X}}/K}$ and $\Omega^{3}_{R_{\mathbb{Y}}/K}$ disagree:

[TABLE]

So, the Hilbert functions of the exterior powers of $\Omega^{1}_{R_{\mathbb{X}}/K}$ “know” whether $\mathbb{X}$ is contained in an irreducible or a reducible conic.

This observation motivated the studies underlying this paper. Let us now outline its contents in more detail. In the second section we start by recalling the definitions of the Kähler differential module $\Omega^{1}_{R_{\mathbb{X}}/K}$ and the Kähler differential algebra $\Omega_{R_{\mathbb{X}}/K}=\bigwedge_{R_{\mathbb{X}}}(\Omega^{1}_{R_{\mathbb{X}}/K})$ of a 0-dimensional subscheme $\mathbb{X}$ of $\mathbb{P}^{n}$. As explained in [Kun], we can calculate an explicit presentation of $\Omega^{m}_{R_{\mathbb{X}}/K}=\bigwedge^{m}_{R_{\mathbb{X}}}(\Omega^{1}_{R_{\mathbb{X}}/K})$ for every $m\geq 1$. Moreover, we show that $\Omega^{m}_{R_{\mathbb{X}}/K}=\langle 0\rangle$ for $m>n+1$, provide a simplified presentation for $\Omega^{n+1}_{R_{\mathbb{X}}/K}$, and show that the Koszul complex yields an exact sequence

[TABLE]

where $\mathfrak{m}_{\mathbb{X}}$ is the homogeneous maximal ideal $\mathfrak{m}_{\mathbb{X}}=\langle x_{0},\dots,x_{n}\rangle$ of $R_{\mathbb{X}}$.

In Section 3 we have a brief glance at the case $n=1$, i.e., at 0-dimensional subschemes of a projective line. Unsurprisingly, in this case the Hilbert functions and regularity indices of $\Omega^{1}_{R_{\mathbb{X}}/K}$ and $\Omega^{2}_{R_{\mathbb{X}}/K}$ can be written down explicitly.

In Section 4 we look at the Hilbert function of $\Omega^{m}_{R_{\mathbb{X}}/K}$ in special degrees. We provide explicit values in low degrees, show that the Hilbert polynomial (i.e., the value in high degrees) is constant, and examine monotonicity in intermediate degrees. These insights are accompanied by a bound for the regularity index of $\Omega^{m}_{R_{\mathbb{X}}/K}$ in terms of the regularity index of $\Omega^{1}_{R_{\mathbb{X}}/K}$ in Section 5.

Then, in the next four sections, we look at the modules of Kähler differential $m$-forms for fat point schemes ${\mathbb{W}}$. Such schemes are defined by ideals of the form $I_{\mathbb{W}}=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s}^{m_{s}}$, where the ideals $\wp_{i}$ are the vanishing ideals of distinct $K$-rational points in $\mathbb{P}^{n}$. In Section 6 we prove a regularity bound for $\Omega^{m}_{R_{\mathbb{W}}/K}$ which uses the regularity index of the fattening of ${\mathbb{W}}$, i.e., the scheme $\mathbb{V}$ defined by $I_{\mathbb{V}}=\wp_{1}^{m_{1}+1}\cap\cdots\cap\wp_{s}^{m_{s}+1}$. For fat point schemes, we also give bounds on some specific values of the Hilbert polynomial of $\Omega^{m}_{R_{\mathbb{W}}/K}$. In the reduced case (i.e., when $m_{1}=\cdots=m_{s}=1$), these values are zero for $m\geq 2$, but as soon as one of the exponents $m_{i}$ satisfies $m_{i}\geq 2$, not all of these values are zero anymore. Thus the property of ${\mathbb{W}}$ to be reduced is reflected in the values of the Hilbert polynomials of $\Omega^{m}_{R_{\mathbb{W}}/K}$ (see Cor. 7.5). More generally, Prop. 7.4 provides upper and lower bounds for these Hilbert polynomials.

For the highest non-zero module of Kähler differentials $\Omega^{n+1}_{R_{\mathbb{W}}/K}$, we can sometimes determine its Hilbert polynomial explicitly. More precisely, we have formulas for schemes $\mathbb{W}$ contained in a hyperplane (see Prop. 8.1) and for equimultiple schemes ${\mathbb{W}}$ (see Thm. 8.3). Another case in which we have more detailed information is the module $\Omega^{2}_{R_{\mathbb{W}}/K}$ for an equimultiple fat point scheme ${\mathbb{W}}$ (i.e., a scheme satisfying $m_{1}=\dots=m_{s}$). In this case we can extract the value of the Hilbert polynomial of $\Omega^{2}_{R_{\mathbb{W}}/K}$ from a complex connecting it to its fattening and second fattening (see Prop. 8.5). We end the discussion of Hilbert polynomials with a conjecture for their value for $\Omega^{n+1}_{R_{\mathbb{W}}/K}$.

In Section 9, a rich and detailed set of results describes the Hilbert functions of $\Omega^{m}_{R_{\mathbb{W}}/K}$, where $m=1,2,3$, in the case of a fat point scheme ${\mathbb{W}}$ in $\mathbb{P}^{2}$ supported on a non-singular conic. In this case, the Hilbert function of $\Omega^{1}_{R_{\mathbb{W}}/K}$ can be computed explicitly from the Hilbert functions of suitable fat point schemes (see Thm. 9.2). If ${\mathbb{W}}$ is an equimultiple fat point scheme, we construct a special homogeneous system of generators of $I_{\mathbb{W}}$ in Prop. 9.4 and use it to compute the Hilbert function of $\Omega^{3}_{R_{\mathbb{W}}/K}$ explicitly (see Thm. 9.6 and Prop. 9.7). Consequently, we can use the exact sequence given by the Koszul complex and determine the Hilbert function of $\Omega^{2}_{R_{\mathbb{W}}/K}$ explicitly (see Prop. 9.9).

Finally, in the last section we point out the relation between the Kähler differential algebra $\Omega_{R_{\mathbb{X}}/K}$ and the relative Kähler differential algebra $\Omega_{R_{\mathbb{X}}/K[x_{0}]}$ and use it to deduce many properties of the Hilbert function, the Hilbert polynomial, and the regularity index of $\Omega_{R_{\mathbb{X}}/K[x_{0}]}$ (see Propositions 10.1, 10.2 and 10.3).

Throughout the paper we illustrate all results with explicitly computed examples. The necessary calculations were performed using the second author’s package for the computer algebra system ApCoCoA (see [ApC]). Unless explicitly stated otherwise, we adhere to the definitions and notation introduced in [KR1, KR2] and [Kun].

2. Definition and Basic Properties

Throughout this paper we work over a field $K$ of characteristic zero. By $\mathbb{P}^{n}$ we denote the projective $n$-space over $K$. The homogeneous coordinate ring of $\mathbb{P}^{n}$ is $S=K[X_{0},\dots,X_{n}]$. It is equipped with the standard grading $\operatorname{deg}(X_{i})=1$ for $i=0,\dots,n$. Let $\mathbb{X}$ be a 0-dimensional scheme in $\mathbb{P}^{n}$, and let $I_{\mathbb{X}}$ be the (saturated) homogeneous vanishing ideal of $\mathbb{X}$. Then the homogeneous coordinate ring of $\mathbb{X}$ is $R_{\mathbb{X}}=S/I_{\mathbb{X}}$. The ring $R_{\mathbb{X}}=\bigoplus_{i\geq 0}(R_{\mathbb{X}})_{i}$ is a standard graded $K$-algebra. Its enveloping algebra is $R_{\mathbb{X}}\otimes_{K}R_{\mathbb{X}}=\bigoplus_{i\geq 0}(\bigoplus_{j+k=i}(R_{\mathbb{X}})_{j}\otimes(R_{\mathbb{X}})_{k})$. By ${\mathcal{J}}$ we denote the kernel of the homogeneous $R_{\mathbb{X}}$-linear map of degree zero $\mu:\ R_{\mathbb{X}}\otimes_{K}R_{\mathbb{X}}\rightarrow R_{\mathbb{X}}$ given by $\mu(f\otimes g)=fg$. It is well known that $\mathcal{J}$ is the homogeneous ideal of $R_{\mathbb{X}}\otimes_{K}R_{\mathbb{X}}$ generated by $\{x_{i}\otimes 1-1\otimes x_{i}\mid 0\leq i\leq n\}$, where $x_{i}$ is the image of $X_{i}$ in $R_{\mathbb{X}}$ for $i=0,\dots,n$. In this paper we are interested in looking at the algebraic structure and Hilbert function of the following objects.

Definition 2.1**.**

1. (a)

The graded $R_{\mathbb{X}}$-module $\Omega^{1}_{R_{\mathbb{X}}/K}\,=\,{\mathcal{J}}/{\mathcal{J}}^{2}$ is called the module of Kähler differential 1-forms of $R_{\mathbb{X}}/K$, or simply the module of Kähler differentials. 2. (b)

The homogeneous $K$-linear map $d:R_{\mathbb{X}}\rightarrow\Omega^{1}_{R_{\mathbb{X}}/K}$ given by $f\mapsto f\otimes 1-1\otimes f+{\mathcal{J}}^{2}$ is called the universal derivation of $R_{\mathbb{X}}/K$. 3. (c)

The $m$-th exterior power of $\Omega^{1}_{R_{\mathbb{X}}/K}$ over $R_{\mathbb{X}}$ is called the module of Kähler differential $\mathbf{m}$-forms of $R_{\mathbb{X}}/K$ and is denoted by $\Omega^{m}_{R_{\mathbb{X}}/K}$. 4. (d)

The direct sum $\Omega_{R_{\mathbb{X}}/K}:=\bigoplus_{m\in\mathbb{N}}\Omega^{m}_{R_{\mathbb{X}}/K}$ is an $R_{\mathbb{X}}$-algebra. It is called the Kähler differential algebra of $R_{\mathbb{X}}/K$. Here we use $\Omega^{0}_{R_{\mathbb{X}}/K}=R_{\mathbb{X}}$.

More generally, for any graded $K$-algebra $T/S$, we can define the module of Kähler differential $m$-forms $\Omega^{m}_{T/S}$ and the Kähler differential algebra $\Omega_{T/S}$ in analogously (cf. [Kun, Section 2]). The Kähler differential algebra of $\Omega_{R_{\mathbb{X}}/K}$ is in fact a bigraded $K$-algebra whose homogeneous component in degree $(m,d)$ is given by $(\Omega^{m}_{R_{\mathbb{X}}/K})_{d}$. Notice that we have $\operatorname{deg}(dx_{i})=\operatorname{deg}(x_{i})=1$ for $i=0,\dots,n$. For $m\geq 0$, the graded $R_{\mathbb{X}}$-module $\Omega^{m}_{R_{\mathbb{X}}/K}$ is finitely generated and its Hilbert function is defined by

[TABLE]

Note that $\Omega^{0}_{R_{\mathbb{X}}/K}=R_{\mathbb{X}}$ and $\Omega^{1}_{R_{\mathbb{X}}/K}=R_{\mathbb{X}}dx_{0}+\cdots+R_{\mathbb{X}}dx_{n}$. Hence we obtain $\Omega^{m}_{R_{\mathbb{X}}/K}=\langle 0\rangle$ for $m>n+1$ and $\Omega_{R_{\mathbb{X}}/K}=\bigoplus_{m=0}^{n+1}\Omega^{m}_{R_{\mathbb{X}}/K}$. Furthermore, there is a presentation of $\Omega_{R_{\mathbb{X}}/K}$ as $\Omega_{R_{\mathbb{X}}/K}\cong\Omega_{S/K}/\langle I_{\mathbb{X}},dI_{\mathbb{X}}\rangle\Omega_{S/K}$ (cf. [Kun, Proposition 4.12]). From this we deduce the following presentation of the module of Kähler differential $m$-forms.

Proposition 2.2**.**

Let $m\geq 1$ and let $\{F_{1},\dots,F_{r}\}$ be a homogeneous system of generators of $I_{\mathbb{X}}$. The graded $R_{\mathbb{X}}$-module $\Omega^{m}_{R_{\mathbb{X}}/K}$ has a presenation

[TABLE]

where $I_{\mathbb{X}}\Omega^{m}_{S/K}+dI_{\mathbb{X}}\Omega^{m-1}_{S/K}$ is generated by

[TABLE]

In the case $m=n+1$, the presentation of $\Omega^{n+1}_{R_{\mathbb{X}}/K}$ can be described explicitly as follows.

Corollary 2.3**.**

Let $\{F_{1},\dots,F_{r}\}$ be a homogeneous system of generators of $I_{\mathbb{X}}$. There is an isomorphism of graded $R_{\mathbb{X}}$-modules

[TABLE]

Proof.

Note that $\Omega^{n+1}_{S/K}$ is a free $S$-module of rank $1$ with basis $\{dX_{0}\wedge\cdots\wedge dX_{n}\}$, and so $\Omega^{n+1}_{S/K}\cong S(-n-1)$. For $F\in I_{\mathbb{X}}$ and $G\in S$, we have $FdG=d(FG)-GdF\in dI_{\mathbb{X}}$. It follows that $I_{\mathbb{X}}\Omega^{m}_{S/K}\subseteq dI_{\mathbb{X}}\Omega^{m-1}_{S/K}$ for all $m\geq 1$. Let $I=\langle\,\frac{\partial F_{j}}{\partial X_{i}}\mid 0\leq i\leq n,1\leq j\leq r\,\rangle$. We need to show that $dI_{\mathbb{X}}\Omega^{n}_{S/K}=IdX_{0}\wedge\cdots\wedge dX_{n}$. Clearly, we have

[TABLE]

where $\widehat{dX_{i}}$ indicates that $dX_{i}$ is omitted in the wedge product. Hence we get the inclusion $dI_{\mathbb{X}}\Omega^{n}_{S/K}\supseteq IdX_{0}\wedge\cdots\wedge dX_{n}$. For the other inclusion, let $F\in I_{\mathbb{X}}$, and let $\{i_{1},\dots,i_{n}\}\subseteq\{0,\dots,n\}$. Write $F=G_{1}F_{1}+\cdots+G_{r}F_{r}$ with $G_{1},\dots,G_{r}\in S$. Then we have

[TABLE]

and hence $(dF-(F_{1}dG_{1}+\cdots+F_{r}dG_{r}))\wedge dX_{i_{1}}\wedge\cdots\wedge dX_{i_{n}}\in IdX_{0}\wedge\cdots\wedge dX_{n}$. Since the field $K$ has characteristic zero, for $j=1,\dots,r$, Euler’s relation yields that $F_{j}=\frac{1}{\operatorname{deg}(F_{j})}\sum_{i=0}^{n}X_{i}\tfrac{\partial F_{j}}{\partial X_{i}}\in I$. In particular, we have $I_{\mathbb{X}}\subseteq I$. This implies $dF\wedge dX_{i_{1}}\wedge\cdots\wedge dX_{i_{n}}\in IdX_{0}\wedge\cdots\wedge dX_{n}$. Therefore we get the equality $dI_{\mathbb{X}}\Omega^{n}_{S/K}=IdX_{0}\wedge\cdots\wedge dX_{n}$, and the claim follows readily. ∎

Now let $\mathfrak{m}_{\mathbb{X}}=\langle x_{0},\dots,x_{n}\rangle$ be the homogeneous maximal ideal of $R_{\mathbb{X}}$, and let $e:R_{\mathbb{X}}\rightarrow R_{\mathbb{X}}$ be the Euler derivation of $R_{\mathbb{X}}/K$ given by $f\mapsto i\cdot f$ for $f\in(R_{\mathbb{X}})_{i}$. By universal property of $\Omega^{1}_{R_{\mathbb{X}}/K}$ (cf. [Kun, Section 1]), there is a unique homogeneous $R_{\mathbb{X}}$-linear map $\gamma:\Omega^{1}_{R_{\mathbb{X}}/K}\rightarrow R_{\mathbb{X}}$ such that $e=\gamma\circ d$. In particular, we have $\gamma(dx_{i})=x_{i}$ for all $i=0,\dots,n$ and $\gamma(df)=\operatorname{deg}(f)\cdot f$ for every homogeneous element $f\in R_{\mathbb{X}}\setminus\{0\}$. The Koszul complex of $\gamma$ is the complex

[TABLE]

where $\gamma:\Omega^{m}_{R_{\mathbb{X}}/K}\rightarrow\Omega^{m-1}_{R_{\mathbb{X}}/K}$ is a homogeneous $R_{\mathbb{X}}$-linear map defined by

[TABLE]

for all $\omega_{1},\dots,\omega_{m}\in\Omega^{1}_{R_{\mathbb{X}}/K}$, and where $\gamma(\omega\wedge\omega^{\prime})=\gamma(\omega)\wedge\omega^{\prime}+(-1)^{m}\omega\wedge\gamma(\omega^{\prime})$ for $\omega\in\Omega^{m}_{R_{\mathbb{X}}/K}$ and $\omega^{\prime}\in\Omega^{k}_{R_{\mathbb{X}}/K}$ (cf. [BH, 1.6.1-2]). In our setting, this complex is an exact sequence, as the following proposition shows.

Proposition 2.4**.**

The Koszul complex

[TABLE]

is an exact sequence of graded $R_{\mathbb{X}}$-modules.

Proof.

Let $1\leq m\leq n+1$, let $i\geq 0$, and let $\omega=fdx_{i_{1}}\wedge\cdots\wedge dx_{i_{m}}\in\Omega^{m}_{R_{\mathbb{X}}/K}$ with $f\in(R_{\mathbb{X}})_{i}$ and $0\leq i_{1}<\cdots<i_{m}\leq n$. Then we have

[TABLE]

and

[TABLE]

This implies $(\gamma\circ d+d\circ\gamma)(\omega)=(m+i)\omega$. Hence $(\gamma\circ d+d\circ\gamma)(\omega)=\operatorname{deg}(\omega)\omega$ for every homogeneous element $\omega\in\Omega^{m}_{R_{\mathbb{X}}/K}$. Now suppose that $\omega\in\Omega^{m}_{R_{\mathbb{X}}/K}\setminus\{0\}$ is a homogeneous element with $\gamma(\omega)=0$. Set $\widetilde{\omega}=\frac{1}{\operatorname{deg}(\omega)}d\omega\in\Omega^{m+1}_{R_{\mathbb{X}}/K}$. We get $\gamma(\widetilde{\omega})=\omega$, and the proof is complete. ∎

Obviously, the ring $R_{\mathbb{X}}$ is Noetherian and the graded $R_{\mathbb{X}}$-module $\Omega^{m}_{R_{\mathbb{X}}/K}$ is finitely generated, and so the Hilbert polynomial of $\Omega^{m}_{R_{\mathbb{X}}/K}$ exists (cf. [KR2, 5.1.21]) and is denoted by $\operatorname{HP}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(z)$. The number \operatorname{ri}(\Omega^{m}_{R_{\mathbb{X}}/K})=\min\{i\in\mathbb{Z}\mid\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(j)=\operatorname{HP}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(j)\mbox{\ for all j\geq i}\} is called the regularity index of $\Omega^{m}_{R_{\mathbb{X}}/K}$. In the following, we denote the Hilbert function of $R_{\mathbb{X}}$ by $\operatorname{HF}_{\mathbb{X}}$ and its regularity index by $r_{\mathbb{X}}$. As a consequence of the exact sequence ($\mathcal{K}$), we have the following bound for $\operatorname{ri}(\Omega^{n+1}_{R_{\mathbb{X}}/K})$.

Corollary 2.5**.**

We have $\operatorname{ri}(\Omega^{n+1}_{R_{\mathbb{X}}/K})\leq\max\{r_{\mathbb{X}},\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K}),\dots,\operatorname{ri}(\Omega^{n}_{R_{\mathbb{X}}/K})\}$.

Let us examine the Hilbert functions of the modules of Kähler differential $m$-forms and their regularity indices in a concrete case.

Example 2.6**.**

Let $\mathbb{X}$ and $\mathbb{Y}$ be two sets of six reduced $K$-rational points in $\mathbb{P}^{2}$ such that $\mathbb{X}$ is contained in a non-singular conic and $\mathbb{Y}$ lies on the union of two lines and no 5 points of $\mathbb{Y}$ are collinear. Then the Hilbert functions of $\mathbb{X}$ and $\mathbb{Y}$ agree, as do the Hilbert functions of $\Omega^{1}_{R_{\mathbb{X}}/K}$ and $\Omega^{1}_{R_{\mathbb{Y}}/K}$, namely

[TABLE]

It is clear that $r_{\mathbb{X}}=r_{\mathbb{Y}}=3$ and $\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})=\operatorname{ri}(\Omega^{1}_{R_{\mathbb{Y}}/K})=6$. We also have $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i)=\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{Y}}/K}}(i)=0$ for $m=1,2,3$ and $i\leq 0$.

However, the Hilbert functions of $\Omega^{2}_{R_{\mathbb{X}}/K}$ and $\Omega^{2}_{R_{\mathbb{Y}}/K}$ are different, and also the Hilbert functions of $\Omega^{3}_{R_{\mathbb{X}}/K}$ and $\Omega^{3}_{R_{\mathbb{Y}}/K}$ disagree:

[TABLE]

In addition, we have $\operatorname{ri}(\Omega^{2}_{R_{\mathbb{X}}/K})=\operatorname{ri}(\Omega^{2}_{R_{\mathbb{Y}}/K})\!=6$, $\operatorname{ri}(\Omega^{3}_{R_{\mathbb{X}}/K})\!=4$, and $\operatorname{ri}(\Omega^{3}_{R_{\mathbb{Y}}/K})\!=5$. In this case, the inequality of regularity indices in Corollary 2.5 is a strict inequality. Moreover, the Hilbert functions of the exterior powers $\Omega^{2}_{R_{\mathbb{X}}/K}$ and $\Omega^{3}_{R_{\mathbb{X}}/K}$ distinguish a set $\mathbb{X}$ of six points on an irreducible conic from a set $\mathbb{Y}$ of six points on a reducible conic.

3. Kähler Differential Algebras

for Subschemes of $\mathbb{P}^{1}$

In this section we consider the easiest case, namely 0-dimensional subschemes $\mathbb{X}$ of $\mathbb{P}^{1}$. It is well known that Hilbert functions do not change under base field extensions (for instance, see [KR2, 5.1.20]). Thus, in order to compute the Hilbert function of the Kähler differential algebra for the [math]-dimensional scheme $\mathbb{X}$ of $\mathbb{P}^{1}$, we may assume that the field $K$ is algebraically closed. In this case the homogeneous vanishing ideal $I_{\mathbb{X}}$ is a principal ideal generated by a homogeneous polynomial $F\in S=K[X_{0},X_{1}]$. Moreover, after a suitable change of coordinates, we may also assume that $F$ is of the form $F=\prod_{i=1}^{s}(X_{1}-a_{i}X_{0})^{m_{i}}$ where $s\geq 1$, $m_{1},\dots,m_{s}\geq 1$ and $a_{1},\dots,a_{s}\in K$ such that $a_{i}\neq a_{j}$ for $i\neq j$.

In [Rob, Section 4], L.G. Roberts gave a formula for the Hilbert function of $\Omega_{R_{\mathbb{X}}/K}$ when $m_{1}=m_{2}=\cdots=m_{s}=1$. Now we extend his result to arbitrary exponents $m_{1},\dots,m_{s}\geq 1$ as follows.

Proposition 3.1**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{1}$ be a [math]-dimensional scheme, and let $I_{\mathbb{X}}=\langle F\rangle$, where $F=\prod_{i=1}^{s}(X_{1}-a_{i}X_{0})^{m_{i}}$ for some $s$, $m_{1},\dots,m_{s}\geq 1$, and $a_{i}\in K$ with $a_{i}\neq a_{j}$ for $i\neq j$, and let $\mu=\sum_{i=1}^{s}m_{i}$. Then the Hilbert functions of the Kähler differential modules of $R_{\mathbb{X}}/K$ are given by

[TABLE]

In particular, we have $\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})=\operatorname{ri}(\Omega^{2}_{R_{\mathbb{X}}/K})=\mu+s-1$.

Proof.

Let $G=\prod_{i=1}^{s}(X_{1}-a_{i}X_{0})^{m_{i}-1}$, let $H_{1}=\sum_{i=1}^{s}m_{i}a_{i}\prod_{j\neq i}(X_{1}-a_{j}X_{0})$, and let $H_{2}=\sum_{i=1}^{s}m_{i}\prod_{j\neq i}(X_{1}-a_{j}X_{0})$. Note that $\operatorname{deg}(G)=\sum_{i=1}^{s}(m_{i}-1)$ and $\operatorname{deg}(H_{1})=\operatorname{deg}(H_{2})=s-1$. We verify that $\gcd(H_{1},H_{2})=1$. Suppose for a contradiction that $\gcd(H_{1},H_{2})=H$ with $\operatorname{deg}(H)\geq 1$. Euler’s relation $\mu F=G(-X_{0}H_{1}+X_{1}H_{2})$ implies $\mu\prod_{i=1}^{s}(X_{1}-a_{i}X_{0})=-X_{0}H_{1}+X_{1}H_{2}$. So, $H$ is a divisor of $\prod_{i=1}^{s}(X_{1}-a_{i}X_{0})$. There exists an index $i\in\{1,\dots,s\}$ such that $(X_{1}-a_{i}X_{0})\mid H$, but $(X_{1}-a_{i}X_{0})\nmid H_{2}$, a contradiction. Hence we obtain $\gcd(H_{1},H_{2})=1$. Thus the sequence $\{H_{1},H_{2}\}$ is an $S$-regular sequence. Consequently, this sequence is also a regular sequence for the principal ideal $\langle G\rangle$ which is regarded as a graded $S$-module. So, for $i\in\mathbb{Z}$, we have

[TABLE]

Thus the Hilbert function of $\Omega^{2}_{R_{\mathbb{X}}/K}$ is

[TABLE]

Moreover, it is clear that $\operatorname{HF}_{\mathfrak{m}_{\mathbb{X}}}:\ 0\ 2\ 3\ 4\ \cdots\ \mu-1\ \mu\ \mu\cdots$. By Proposition 2.4, we have the exact sequence of graded $R_{\mathbb{X}}$-modules

[TABLE]

Hence the Hilbert function of $\Omega^{1}_{R_{\mathbb{X}}/K}$ satisfies $\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}(i)=\operatorname{HF}_{\mathfrak{m}_{\mathbb{X}}}(i)+\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}(i)$ for all $i\in\mathbb{Z}$. More precisely, it is of the form

[TABLE]

as claimed. ∎

Let us apply this proposition in an explicit example.

Example 3.2**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{1}$ be the 0-dimensional scheme with the homogeneous vanishing ideal $I_{\mathbb{X}}=\langle X_{1}(X_{1}-X_{0})^{2}(X_{1}-2X_{0})^{3}\rangle$. Clearly, we have $s=3$ and $\mu=6$ and $\operatorname{HF}_{\mathfrak{m}_{\mathbb{X}}}:\ 0\ 2\ 3\ 4\ 5\ 6\ 6\cdots$. An application of Proposition 3.1 yields

[TABLE]

In this case we have $\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})=\operatorname{ri}(\Omega^{2}_{R_{\mathbb{X}}/K})=\mu+s-1=8$.

4. Special Values of the Hilbert Function of $\Omega^{m}_{R_{\mathbb{X}}/K}$

In this section we describe the values of the Hilbert function of the module of Kähler differential $m$-forms for a [math]-dimensional scheme $\mathbb{X}$ at some special degrees. From now on, the coordinates $\{X_{0},\dots,X_{n}\}$ of $\mathbb{P}^{n}$ are always chosen such that no point of $\mathbb{X}$ lies on the hyperplane $\mathcal{Z}^{+}(X_{0})$. By the choice of the coordinates, $x_{0}$ is a non-zerodivisor of $R_{\mathbb{X}}$. Moreover, $x_{0}$ is also a non-zerodivisor for any nontrivial graded submodule of a graded free $R_{\mathbb{X}}$-module, as the following lemma shows.

Lemma 4.1**.**

Let $d\geq 1$, let $\delta_{1},\dots,\delta_{d}\in\mathbb{Z}$, and let $V$ be a non-trivial graded submodule of the graded free $R_{\mathbb{X}}$-module $\bigoplus_{j=1}^{d}R_{\mathbb{X}}(-\delta_{j})$. Then $x_{0}$ is not a zerodivisor for $V$, i.e., if $x_{0}\cdot v=0$ for some $v\in V$ then $v=0$.

Proof.

Let $\{e_{1},\dots,e_{d}\}$ be the canonical $R_{\mathbb{X}}$-basis of $\bigoplus_{j=1}^{d}R_{\mathbb{X}}(-\delta_{j})$, and let $i\in\mathbb{Z}$. Then every homogeneous element $v\in V_{i}$ has a representation $v=g_{1}e_{1}+\cdots+g_{d}e_{d}$ for some homogeneous elements $g_{1},\dots,g_{d}\in R_{\mathbb{X}}$, where $\operatorname{deg}(g_{j})=\operatorname{deg}(v)-\delta_{j}$ for $j=1,\dots,d$. Suppose that $x_{0}\cdot v=0$. This implies that $x_{0}g_{1}e_{1}+\cdots+x_{0}g_{d}e_{d}=0$, and so $x_{0}g_{1}=\cdots=x_{0}g_{d}=0$ in $R_{\mathbb{X}}$. Since $x_{0}$ is a non-zerodivisor for $R_{\mathbb{X}}$, we have $g_{1}=\cdots=g_{d}=0$, and hence $v=0$. Thus the claim follows. ∎

The number $\alpha_{\mathbb{X}}=\min\{i\in\mathbb{N}|(I_{\mathbb{X}})_{i}\neq 0\}$ is called the initial degree of $I_{\mathbb{X}}$. Using this notation, some basic properties of $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}$ can be summarized as follows.

Proposition 4.2**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional scheme, and let $1\leq m\leq n+1$.

1. (a)

For $i<m$, we have $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i)=0$. 2. (b)

For $m\leq i<\alpha_{\mathbb{X}}+m-1$, we have $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i)=\binom{n+1}{m}\cdot\binom{n+i-m}{n}$. 3. (c)

The Hilbert polynomial of $\Omega^{m}_{R_{\mathbb{X}}/K}$ is constant. 4. (d)

We have $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(r_{\mathbb{X}}+m)\geq\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(r_{\mathbb{X}}+m+1)\geq\cdots$, and if $\operatorname{ri}(\Omega^{m}_{R_{\mathbb{X}}/K})\geq r_{\mathbb{X}}+m$ then

[TABLE]

Proof.

(a) Obivously, every non-zero homogeneous element $\omega$ of $\Omega^{m}_{R_{\mathbb{X}}/K}$ has degree $\operatorname{deg}(\omega)\geq m$, and hence $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i)=0$ for all $i<m$.

(b) Let $m\leq i<\alpha_{\mathbb{X}}+m-1$. Notice that $I_{\mathbb{X}}\Omega^{m}_{S/K}\subseteq dI_{\mathbb{X}}\Omega^{m-1}_{S/K}$. Also, we have $(dI_{\mathbb{X}}\Omega^{m-1}_{S/K})_{i}=\langle 0\rangle$ for all $i<\alpha_{\mathbb{X}}+m-1$, since a non-zero homogeneous element of $dI_{\mathbb{X}}\Omega^{m-1}_{S/K}$ is always of the form $\sum_{k}dF_{k}\wedge\omega_{k}$, where $F_{k}\in(I_{\mathbb{X}})_{\geq\alpha_{\mathbb{X}}}$ and $\omega_{k}\in(\Omega^{m-1}_{S/K})_{\geq m-1}$. By Proposition 2.2, for all $i<\alpha_{\mathbb{X}}+m-1$, we obtain

[TABLE]

(c) It follows from Proposition 2.2 that

[TABLE]

for all $i\in\mathbb{Z}$. Hence the Hilbert polynomial of $\Omega^{m}_{R_{\mathbb{X}}/K}$ is a constant polynomial.

(d) The graded $R_{\mathbb{X}}$-module $\Omega^{m}_{R_{\mathbb{X}}/K}$ has the following form:

[TABLE]

Observe that $(R_{\mathbb{X}})_{i}=x_{0}(R_{\mathbb{X}})_{i-1}$ if $i>r_{\mathbb{X}}$. Thus $(\Omega^{m}_{R_{\mathbb{X}}/K})_{i+m}=x_{0}(\Omega^{m}_{R_{\mathbb{X}}/K})_{i+m-1}$ for all $i>r_{\mathbb{X}}$. So, for all $i>r_{\mathbb{X}}$, we have the inequality

[TABLE]

Now let $\mathcal{G}=\langle(\frac{\partial F}{\partial x_{0}},\dots,\frac{\partial F}{\partial x_{n}})\mid F\in I_{\mathbb{X}}\rangle_{R_{\mathbb{X}}}$. By [DK, Proposition 1.3] and [SS, X.83], there is an exact sequence of graded $R_{\mathbb{X}}$-modules

[TABLE]

Suppose $i\geq r_{\mathbb{X}}$ satisfies $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i+m)=\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i+m+1)$. Then it follows from the above exact sequence that

[TABLE]

For every $j\geq r_{\mathbb{X}}$, $\operatorname{HF}_{\mathbb{X}}(j)=\operatorname{deg}(\mathbb{X})$, and so $\operatorname{HF}_{\bigwedge_{R_{\mathbb{X}}}^{m}(R_{\mathbb{X}}^{n+1})}(j)=\operatorname{HF}_{\bigwedge_{R_{\mathbb{X}}}^{m}(R_{\mathbb{X}}^{n+1})}(j+1)$. Consequently, we have

[TABLE]

In addition, Lemma 4.1 shows that $x_{0}$ is a non-zerodivisor for the graded $R_{\mathbb{X}}$-submodule $\mathcal{G}\wedge_{R_{\mathbb{X}}}\bigwedge^{m-1}_{R_{\mathbb{X}}}(R_{\mathbb{X}}^{n+1})$ of the graded-free $R_{\mathbb{X}}$-module $\bigwedge^{m}_{R_{\mathbb{X}}}(R_{\mathbb{X}}^{n+1})$. This implies

[TABLE]

In view of [GM, Proposition 1.1], the ideal $I_{\mathbb{X}}$ can be generated by homogeneous polynomials of degrees $\leq r_{\mathbb{X}}+1$. So, the graded $R_{\mathbb{X}}$-module $\mathcal{G}\wedge_{R_{\mathbb{X}}}\bigwedge^{m-1}_{R_{\mathbb{X}}}(R_{\mathbb{X}}^{n+1})$ is generated in degrees $\leq r_{\mathbb{X}}$. Thus we obtain

[TABLE]

Altogether, we have $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i+m+1)=\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i+m+2)$, and the claim follows by induction. ∎

The following example shows that $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K}}(i)$ may or may not be monotonic in the range $\alpha_{\mathbb{X}}+m\leq i\leq r_{\mathbb{X}}+m$.

Example 4.3**.**

Let $K=\mathbb{Q}$, and let $\mathbb{X}\subseteq\mathbb{P}^{2}$ be the set of nine points $\mathbb{X}=\{(1:1:0),$ $(1:1:1),(1:1:2),(1:1:3),(1:1:4),(1:1:5),(1:0:1),(1:2:1),(1:2:2)\}$. Notice that $\mathbb{X}$ contains six points on a line and three non-collinear points off that line. It is clear that $\operatorname{HF}_{\mathbb{X}}:\ 1\ 3\ 6\ 7\ 8\ 9\ 9\cdots$, $\alpha_{\mathbb{X}}=3$, and $r_{\mathbb{X}}=5$. The Hilbert functions of the Kähler differential modules of $R_{\mathbb{X}}/K$ are given by

[TABLE]

We see that $\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}(\alpha_{\mathbb{X}}+1)=14>13=\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}(\alpha_{\mathbb{X}}+2)$ and $\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}(\alpha_{\mathbb{X}}+2)=$ $13<14=\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}(r_{\mathbb{X}}+1)$. So, $\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{X}}/K}}(i)$ is not monotonic in the range $\alpha_{\mathbb{X}}+1\leq i\leq r_{\mathbb{X}}+1$. Similarly, $\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}(i)$ is not monotonic in the range $\alpha_{\mathbb{X}}+2\leq i\leq r_{\mathbb{X}}+2$.

Next we consider the set $\mathbb{Y}=\mathbb{X}\cup\{(1:0:2)\}$. We have $\operatorname{HF}_{\mathbb{Y}}:\ 1\ 3\ 6\ 8\ 9\ 10\ 10\cdots$, $\alpha_{\mathbb{Y}}=3$, and $r_{\mathbb{Y}}=5$. The Hilbert functions of the Kähler differential modules of $R_{\mathbb{Y}}/K$ are

[TABLE]

Hence $\operatorname{HF}_{\Omega^{1}_{R_{\mathbb{Y}}/K}}(i)$ is monotonic in the range $\alpha_{\mathbb{Y}}+1\leq i\leq r_{\mathbb{Y}}+1$, and $\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{Y}}/K}}(i)$ is also monotonic in the range $\alpha_{\mathbb{Y}}+2\leq i\leq r_{\mathbb{Y}}+2$.

5. Bounds for the Regularity Index of $\Omega^{m}_{R_{\mathbb{X}}/K}$

In this section we give an upper bound for the regularity index of the module of Kähler differential $m$-forms $\Omega^{m}_{R_{\mathbb{X}}/K}$ for a 0-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^{n}$. To do this, we need the following lemmas.

Lemma 5.1**.**

Let $d\geq 1$, let $\delta_{1},\dots,\delta_{d}\in\mathbb{Z}$ such that $\delta_{1}\leq\cdots\leq\delta_{d}$, let $W=\bigoplus_{j=1}^{d}R_{\mathbb{X}}(-\delta_{j})$ be the graded free $R_{\mathbb{X}}$-module, and let $V$ be a non-trivial graded submodule of $W$. Then, for $1\leq m\leq d$, we have

[TABLE]

Proof.

First we note that the Hilbert polynomial of $W$ is $\operatorname{HP}_{W}(z)=d\cdot\operatorname{deg}(\mathbb{X})$ and that $\operatorname{ri}(W)=r_{\mathbb{X}}+\delta_{d}$. This shows that the Hilbert polynomial of $V$ is a constant polynomial $\operatorname{HP}_{V}(z)=u\leq d\cdot\operatorname{deg}(\mathbb{X})$. Let $r=\operatorname{ri}(V)$, and let $v_{1},\dots,v_{u}$ be a $K$-basis of $V_{r}$. By Lemma 4.1, the elements $\{x_{0}^{i}v_{1},\dots,x_{0}^{i}v_{u}\}$ form a $K$-basis of the $K$-vector space $V_{r+i}$ for all $i\in\mathbb{N}$. We let $\{e_{1},\dots,e_{d}\}$ be the canonical $R_{\mathbb{X}}$-basis of $W$, we let $t=\binom{d}{m}$, and we let $\{\varepsilon_{1},\dots,\varepsilon_{t}\}$ be a basis of the graded free $R_{\mathbb{X}}$-module $\bigwedge_{R_{\mathbb{X}}}^{m}(W)$ w.r.t. $\{e_{1},\dots,e_{d}\}$. We set $\delta=\delta_{d-m+1}+\dots+\delta_{d}$, and let

[TABLE]

Let $\varrho=\dim_{K}N$, and let $w_{1},\dots,w_{\varrho}$ be a $K$-basis of $N$. It is not difficult to verify that $N=\langle w_{1},\dots,w_{\varrho}\rangle_{K}=(V\wedge_{R_{\mathbb{X}}}{\textstyle\bigwedge_{R_{\mathbb{X}}}^{m}}(W))_{\delta+r}$. Moreover, for any $i\geq 0$, the set $\{x_{0}^{i}w_{1},\dots,x_{0}^{i}w_{\varrho}\}$ is $K$-linearly independent. Indeed, assume that there are elements $a_{1},\dots,a_{\varrho}\in K$ such that $\sum_{j=1}^{\varrho}x_{0}^{i}a_{j}w_{j}=0$. Since $x_{0}$ is a non-zerodivisor for $V\wedge_{R_{\mathbb{X}}}\bigwedge_{R_{\mathbb{X}}}^{m}(W)$ by Lemma 4.1, we get $\sum_{j=1}^{\varrho}a_{j}w_{j}=0$, and hence $a_{1}=\cdots=a_{\varrho}=0$.

Now it suffices to prove that the set $\{x_{0}^{i}w_{1},\dots,x_{0}^{i}w_{\varrho}\}$ generates the $K$-vector space $(V\wedge_{R_{\mathbb{X}}}\bigwedge_{R_{\mathbb{X}}}^{m}(W))_{\delta+r+i}$ for all $i\geq 0$. Let $w\in(V\wedge_{R_{\mathbb{X}}}\bigwedge_{R_{\mathbb{X}}}^{m}(W))_{\delta+r+i}$ be a non-zero homogeneous element. Then $w=\sum_{j,k}\widetilde{v}_{j}\wedge h_{k}\varepsilon_{k}=\sum_{j,k}h_{k}\widetilde{v}_{j}\wedge\varepsilon_{k}$ for some homogeneous elements $\widetilde{v}_{j}\in V$ and $h_{k}\in R_{\mathbb{X}}$ such that $\operatorname{deg}(\widetilde{v}_{j})+\operatorname{deg}(h_{k})=\delta+r+i-\operatorname{deg}(\varepsilon_{k})$ for all $j,k$. Note that $\operatorname{deg}(h_{k}\widetilde{v}_{j})=\delta+r+i-\operatorname{deg}(\varepsilon_{k})\geq r+i$. Also, we have

[TABLE]

So, there are $b_{jk1},\dots,b_{jku}\in K$ such that $h_{k}\widetilde{v}_{j}=\sum_{l=1}^{u}b_{jkl}x_{0}^{\delta+i-\operatorname{deg}(\varepsilon_{k})}v_{l}$. This implies

[TABLE]

for some $c_{jklq}\in K$. Thus we get $w\in\langle x_{0}^{i}w_{1},\dots,x_{0}^{i}w_{\varrho}\rangle_{K}$, and consequently $\operatorname{HF}_{V\wedge_{R_{\mathbb{X}}}\bigwedge_{R_{\mathbb{X}}}^{m}(W)}(i)=\varrho$ for all $i\geq\delta+r$. Therefore $\operatorname{ri}(V\wedge_{R_{\mathbb{X}}}\bigwedge_{R_{\mathbb{X}}}^{m}(W))\leq\operatorname{ri}(V)+\delta$, as we wanted to show. ∎

Lemma 5.2**.**

Let $V$ be a graded $R_{\mathbb{X}}$-module generated by the set of homogeneous elements $\{v_{1},\dots,v_{d}\}$ for some $d\geq 1$. Let $\delta_{j}=\operatorname{deg}(v_{j})$ for $j=1,\dots,d$, and let $m\geq 1$. Assume that $\delta_{1}\leq\cdots\leq\delta_{d}$, and set $\delta=\delta_{d-m+1}+\cdots+\delta_{d}$ if $m\leq d$. Then the regularity index of $\bigwedge_{R_{\mathbb{X}}}^{m}(V)$ satisfies $\operatorname{ri}(\bigwedge_{R_{\mathbb{X}}}^{m}(V))=-\infty$ if $m>d$ and

[TABLE]

if $1\leq m\leq d$. In particular, if $1\leq m\leq d$ and $\delta_{1}=\cdots=\delta_{d}=t$ then we have $\operatorname{ri}({\textstyle\bigwedge_{R_{\mathbb{X}}}^{m}}(V))\leq\max\{\,r_{\mathbb{X}}+mt,\operatorname{ri}(V)+(m-1)t\,\}$.

Proof.

If $m>d$, then $\bigwedge_{R_{\mathbb{X}}}^{m}(V)=\langle 0\rangle$, and hence $\operatorname{ri}(\bigwedge_{R_{\mathbb{X}}}^{m}(V))=-\infty$. Now we assume that $1\leq m\leq d$. Obviously, the $R_{\mathbb{X}}$-linear map $\alpha:W=\bigoplus_{j=1}^{d}R_{\mathbb{X}}(-\delta_{j})\rightarrow V$ given by $e_{j}\mapsto v_{j}$ is a homogeneous $R_{\mathbb{X}}$-epimorphism of degree zero. Set $\mathcal{G}=\operatorname{Ker}(\alpha)$. According to [SS, X.83], there is an exact sequence of graded $R_{\mathbb{X}}$-modules

[TABLE]

Thus an application of Lemma 5.1 yields that

[TABLE]

Here the last inequality follows from the fact that $\operatorname{ri}(\mathcal{G})\leq\max\{r_{\mathbb{X}}+\delta_{d},\operatorname{ri}(V)\}$. ∎

Now we are able to give an upper bound for the regularity index of the module of Kähler differential $m$-forms $\Omega^{m}_{R_{\mathbb{X}}/K}$.

Proposition 5.3**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}$ be a [math]-dimensional scheme, and let $1\leq m\leq n+1$. The regularity index of the module of Kähler differential $m$-forms $\Omega^{m}_{R_{\mathbb{X}}/K}$ satisfies

[TABLE]

Proof.

We set $\mathcal{G}=\langle(\frac{\partial F}{\partial x_{0}},\dots,\frac{\partial F}{\partial x_{n}})\in R_{\mathbb{X}}^{n+1}\mid F\in I_{\mathbb{X}}\rangle$. By [DK, Proposition 1.3], we have the short exact sequence of graded $R_{\mathbb{X}}$-modules

[TABLE]

Applying Lemma 5.2 to the graded $R_{\mathbb{X}}$-module $\Omega^{1}_{R_{\mathbb{X}}/K}$ which is generated by the set $\{dx_{0},\dots,dx_{n}\}$, we get $\operatorname{ri}(\Omega^{m}_{R_{\mathbb{X}}/K})\leq\max\{r_{\mathbb{X}}+m,\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})+m-1\},$ as we wished. ∎

Remark 5.4**.**

We have $\operatorname{ri}(\Omega^{n+1}_{R_{\mathbb{X}}/K})\leq\max\{r_{\mathbb{X}}+n,\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})+n-1\}.$ Indeed, the exact sequence $(\mathcal{K})$ of graded $R_{\mathbb{X}}$-modules yields

[TABLE]

Moreover, if we set $\varrho_{m}=\max\{r_{\mathbb{X}}+m,\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})+m-1\}\}$ for $m\geq 1$, then we get the upper bound for the regularity index of $\Omega^{m}_{R_{\mathbb{X}}/K}$ as $\operatorname{ri}(\Omega^{m}_{R_{\mathbb{X}}/K})\leq\min\{\varrho_{n},\varrho_{m}\}$.

6. Bounds for $\operatorname{ri}(\Omega^{m}_{R_{{\mathbb{W}}}/K})$

for a Fat Point Scheme ${\mathbb{W}}$

Let $s\geq 1$, and let $\mathbb{X}=\{P_{1},\dots,P_{s}\}$ be a set of $s$ distinct $K$-rational points in $\mathbb{P}^{n}$. For $i=1,\dots,s$, we let $\wp_{i}$ be the associated prime ideal of $P_{i}$ in $S$.

Definition 6.1**.**

Given a sequence of positive integers $m_{1},\dots,m_{s}$, the intersection $I_{\mathbb{W}}:=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s}^{m_{s}}$ is a saturated homogeneous ideal in $S$ and is therefore the vanishing ideal of a [math]-dimensional subscheme ${\mathbb{W}}$ of $\mathbb{P}^{n}$.

1. (a)

The scheme ${\mathbb{W}}$, denoted by ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$, is called a fat point scheme in $\mathbb{P}^{n}$. The homogeneous vanishing ideal of ${\mathbb{W}}$ is $I_{\mathbb{W}}$. The number $m_{j}$ is called the multiplicity of the point $P_{j}$ for $j=1,\dots,s$. 2. (b)

If $m_{1}=\cdots=m_{s}=\nu$, we denote ${\mathbb{W}}$ also by $\nu\mathbb{X}$ and call it an equimultiple fat point scheme. 3. (c)

For $i\geq 1$, the fat point scheme ${\mathbb{W}}^{(i)}=(m_{1}+i)P_{1}+\cdots+(m_{s}+i)P_{s}$ is called the $i$-th fattening of ${\mathbb{W}}$. We simple say the fattening of ${\mathbb{W}}$ instead of the first fattening of ${\mathbb{W}}$.

The regularity index of the module of Kähler differential $m$-forms for fat point schemes can be bounded as follows.

Proposition 6.2**.**

Let ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a fat point scheme in $\mathbb{P}^{n}$, and let $\mathbb{V}={\mathbb{W}}^{(1)}$ be the fattening of ${\mathbb{W}}$.

1. (a)

For $1\leq m\leq n+1$, we have

[TABLE] 2. (b)

If $m_{1}\leq\cdots\leq m_{s}$ and if $\operatorname{Supp}({\mathbb{W}})=\{P_{1},\dots,P_{s}\}$ is in general position, then we have

[TABLE]

for $1\leq m\leq n+1$.

Proof.

Claim (a) follows from Remark 5.4 and [KLL, Corollary 1.9(iii)]. Moreover, if $\operatorname{Supp}({\mathbb{W}})$ is in general position, then [CTV, Theorem 6] implies that

[TABLE]

Thus claim (b) follows from (a). ∎

The following example shows that the upper bounds for the regularity index of $\Omega^{m}_{R_{{\mathbb{W}}}/K}$ given in Proposition 6.2 are sharp.

Example 6.3**.**

Let $K=\mathbb{Q}$, and let ${\mathbb{W}}$ be the fat point scheme

[TABLE]

where $P_{1}=(1:9:0:0)$, $P_{2}=(1:6:0:1)$, $P_{3}=(1:2:3:3)$, $P_{4}=(1:9:3:5)$, $P_{5}=(1:3:0:4)$, $P_{6}=(1:0:1:3)$, $P_{7}=(1:0:2:0)$, and $P_{8}=(1:3:0:10)$. Let $\mathbb{V}$ be the fat point scheme $\mathbb{V}=2P_{1}+3P_{2}+2P_{3}+2P_{4}+3P_{5}+3P_{6}+3P_{7}+2P_{8}$. We have $r_{{\mathbb{W}}}=3$ and $r_{\mathbb{V}}=5$, and so $\max\{r_{{\mathbb{W}}}+m,r_{\mathbb{V}}+m-1\}=m+4$ for $m=1,\dots,4$. In this case the regularity index of $\Omega^{m}_{R_{{\mathbb{W}}}/K}$ is $m+4$ for $m=1,\dots,3$ and $\operatorname{ri}(\Omega^{4}_{R_{{\mathbb{W}}}/K})=7$. Thus the bound for the regularity index in Proposition 6.2(a) is sharp.

Next let $\mathbb{Y}$ be the scheme $\mathbb{Y}=P_{4}+P_{5}+P_{6}+P_{7}+P_{8}$ in $\mathbb{P}^{3}$. Then $\mathbb{Y}$ is in general position. For $m=1,2,3$, the regularity index of $\Omega^{m}_{R_{2\mathbb{Y}}/K}$ is $4+m$. Thus, for $m=1,2,3$, we have

[TABLE]

In addition, for $m=4$, we have

[TABLE]

and hence the bound in Proposition 6.2(b) is also sharp.

7. Bounds for the Hilbert Polynomial

of $\Omega^{m}_{R_{{\mathbb{W}}}/K}$ for a Fat Point Scheme ${\mathbb{W}}$

First we determine the Hilbert polynomial of the module of Kähler differential $m$-forms for a set of $s$ distinct $K$-rational points $\mathbb{X}=\{P_{1},\dots,P_{s}\}$ in $\mathbb{P}^{n}$. Notice that all points of $\mathbb{X}$ are assumed to lie outside the hyperplane $\mathcal{Z}^{+}(X_{0})$, so we may write $P_{j}=(1:p_{j1}:\dots:p_{jn})$ with $p_{j1},\dots,p_{jn}\in K$ for $j=1,\dots,s$. Furthermore, for every element $f\in R_{\mathbb{X}}$ and $j\in\{1,\dots,s\}$, we also write $f(P_{j})=F(P_{j})$, where $F$ is any representative of $f$ in $S$.

Recall that an element $f_{j}\in(R_{\mathbb{X}})_{r_{\mathbb{X}}}$ is the normal separator of $\mathbb{X}\setminus\{P_{j}\}$ in $\mathbb{X}$ if $f_{j}(P_{j})=1$ and $f_{j}(P_{k})=0$ for $k\neq j$. The set $\{\,x_{0}^{i-r_{\mathbb{X}}}f_{1},\dots,x_{0}^{i-r_{\mathbb{X}}}f_{s}\,\}$ is a $K$-basis of $(R_{\mathbb{X}})_{i}$ for all $i\geq r_{\mathbb{X}}$. For more details about separators of a 0-dimensional scheme in $\mathbb{P}^{n}$ see [GKR, GMT, Kre1, Kre2].

The following proposition gives a description of the Hilbert polynomial of the module of Kähler differential $m$-forms $\Omega^{m}_{R_{\mathbb{X}}/K}$ for every $1\leq m\leq n+1$.

Proposition 7.1**.**

Let $\mathbb{X}=\{P_{1},\dots,P_{s}\}\subseteq\mathbb{P}^{n}$ be a set of $s$ distinct $K$-rational points, and let $1\leq m\leq n+1$. We have

[TABLE]

In particular, the regularity index of $\Omega^{m}_{R_{\mathbb{X}}/K}$ satisfies $\operatorname{ri}(\Omega^{m}_{R_{\mathbb{X}}/K})\leq 2r_{\mathbb{X}}+m$.

Proof.

For $m=1$, we have $\operatorname{HP}_{\Omega^{1}_{R_{\mathbb{X}}/K}}(z)=\operatorname{deg}(\mathbb{X})$ and $\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})\leq 2r_{\mathbb{X}}+1$ (see [DK, Proposition 3.5]). Assume that $m\geq 2$. We see that $\Omega^{m}_{R_{\mathbb{X}}/K}$ is a graded $R_{\mathbb{X}}$-module generated by the set of $\binom{n+1}{m}$ elements

[TABLE]

For $j\in\{1,\dots,s\}$, let $f_{j}$ be the normal separator of $\mathbb{X}\setminus\{P_{j}\}$ in $\mathbb{X}$. Since the set $\{x_{0}^{i-r_{\mathbb{X}}}f_{1},\dots,x_{0}^{i-r_{\mathbb{X}}}f_{s}\}$ is a $K$-basis of the $K$-vector space $(R_{\mathbb{X}})_{i}$ for $i\geq r_{\mathbb{X}}$, the set

[TABLE]

is a system of generators of the $K$-vector space $(\Omega^{m}_{R_{\mathbb{X}}/K})_{k}$ for all $k\geq r_{\mathbb{X}}+m$. Note that $f_{j}^{2}=f_{j}(P_{j})x_{0}^{r_{\mathbb{X}}}f_{i}=x_{0}^{r_{\mathbb{X}}}f_{i}$ and $x_{i}f_{j}=p_{ji}x_{0}f_{j}$ (see, e.g., [GKR, Proposition 1.13]). Therefore we get

[TABLE]

Since $m\geq 2$, we may use the same method as above to get the equality

[TABLE]

This implies $x_{0}^{r_{\mathbb{X}}}f_{j}dx_{i_{1}}\wedge\cdots\wedge dx_{i_{m}}=0$ for $j=1,\dots,s$ and $\{i_{1},\dots,i_{m}\}\subseteq\{0,\dots,n\}$. Thus we obtain $(\Omega^{m}_{R_{\mathbb{X}}/K})_{k}=\langle 0\rangle$ for all $k\geq 2r_{\mathbb{X}}+m$, and the claim follows. ∎

By combining Corollary 2.5 and Proposition 7.1, we obtain upper bounds for the regularity indices of modules of the Kähler differential $m$-forms $\Omega^{m}_{R_{\mathbb{X}}/K}$ as follows.

Corollary 7.2**.**

In the setting of Proposition 7.1, we have

[TABLE]

The preceding bounds for the regularity indices of $\Omega^{m}_{R_{\mathbb{X}}/K}$ are sharp, as our next example shows.

Example 7.3**.**

Let $K=\mathbb{Q}$, and let $\mathbb{X}\subseteq\mathbb{P}^{3}$ be the set of four $K$-rational points $\mathbb{X}=\{P_{1},P_{2},P_{3},P_{4}\}$, where $P_{1}=(1:9:0:0)$, $P_{2}=(1:6:0:1)$, $P_{3}=(1:2:3:3)$, and $P_{4}=(1:9:3:5)$. It is clear that $\operatorname{HF}_{\mathbb{X}}:1\ 4\ 4\cdots$ and $r_{\mathbb{X}}=1$. Moreover, we have

[TABLE]

It follows that $\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})=2r_{\mathbb{X}}+m=2r_{\mathbb{X}}+1=3$, $\operatorname{ri}(\Omega^{2}_{R_{\mathbb{X}}/K})=2r_{\mathbb{X}}+m=2r_{\mathbb{X}}+2=4$, and $\operatorname{ri}(\Omega^{3}_{R_{\mathbb{X}}/K})=\operatorname{ri}(\Omega^{4}_{R_{\mathbb{X}}/K})=\min\{2r_{\mathbb{X}}+m,2r_{\mathbb{X}}+n\}=2r_{\mathbb{X}}+n=5$. Hence we obtain the equality $\operatorname{ri}(\Omega^{m}_{R_{\mathbb{X}}/K})=\min\{2r_{\mathbb{X}}+m,2r_{\mathbb{X}}+n\}$ for $m=1,\dots,4$. Consequently, the bounds in Corollary 7.2 are sharp.

Now we give bounds for the Hilbert polynomial of the module of Kähler differential $m$-forms for a non-reduced fat point scheme.

Proposition 7.4**.**

Let ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a fat point scheme in $\mathbb{P}^{n}$ such that $m_{i}\geq 2$ for some $i\in\{1,\dots,s\}$, and let $1\leq m\leq n+1$. The Hilbert polynomial of $\Omega^{m}_{R_{{\mathbb{W}}}/K}$ is a constant polynomial which is bounded by

[TABLE]

Proof.

Let $\mathbb{Y}$ be the subscheme $\mathbb{Y}=(m_{1}-1)P_{1}+\cdots+(m_{s}-1)P_{s}$ of ${\mathbb{W}}$. Since we have $dI_{{\mathbb{W}}}\subseteq I_{\mathbb{Y}}\Omega^{1}_{S/K}$, this implies $dI_{{\mathbb{W}}}\Omega^{m-1}_{S/K}\subseteq I_{\mathbb{Y}}\Omega^{m}_{S/K}$. Obviously, we have the inclusion $I_{{\mathbb{W}}}\subseteq I_{\mathbb{Y}}$, and so $I_{{\mathbb{W}}}\Omega^{m}_{S/K}\subseteq I_{\mathbb{Y}}\Omega^{m}_{S/K}$. From this we deduce

[TABLE]

By Proposition 2.2, the Hilbert function of $\Omega^{m}_{R_{{\mathbb{W}}/K}}$ satisfies

[TABLE]

for all $i\in\mathbb{Z}$. Also, we see that $\operatorname{HP}_{\Omega^{m}_{S/K}/I_{\mathbb{Y}}\Omega^{m}_{S/K}}(z)=\sum_{i=1}^{s}\binom{n+1}{m}\binom{m_{i}+n-2}{n}>0$ since $m_{i}\geq 2$ for some $i\in\{1,\dots,s\}$. Hence we get the stated lower bound for the Hilbert polynomial of $\Omega^{m}_{R_{{\mathbb{W}}}/K}$. In particular, we have $\operatorname{HP}_{\Omega^{m}_{R_{{\mathbb{W}}}/K}}(z)>0$.

Furthermore, Proposition 4.2 shows that $\operatorname{HP}_{\Omega^{m}_{R_{{\mathbb{W}}}/K}}(z)$ is a constant polynomial. Now we find an upper bound for $\operatorname{HP}_{\Omega^{m}_{R_{{\mathbb{W}}}/K}}(z)$. Clearly, the $R_{{\mathbb{W}}}$-module $\Omega^{m}_{R_{{\mathbb{W}}}/K}$ is generated by the set $\{\,dx_{i_{1}}\wedge\cdots\wedge dx_{i_{m}}\mid 0\leq i_{1}<\cdots<i_{m}\leq n\,\}$ consisting of $\binom{n+1}{m}$ elements. This implies $\operatorname{HF}_{\Omega^{m}_{R_{{\mathbb{W}}}/K}}(i)\leq\binom{n+1}{m}\operatorname{HF}_{{\mathbb{W}}}(i-m)$ for all $i\geq 0$. Hence we get $\operatorname{HP}_{\Omega^{m}_{R_{{\mathbb{W}}}/K}}(z)\leq\binom{n+1}{m}\sum_{i=1}^{s}\binom{m_{i}+n-1}{n},$ which completes the proof. ∎

Our next corollary is an immediate consequence of Propositions 7.1 and 7.4.

Corollary 7.5**.**

Let ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a fat point scheme in $\mathbb{P}^{n}$, and let $m_{\max}=\max\{m_{1},\dots,m_{s}\}$. The following conditions are equivalent.

1. (a)

The scheme ${\mathbb{W}}$ is not reduced, i.e., $m_{\max}>1$. 2. (b)

There exists $m\in\{2,\dots,n+1\}$ such that $\operatorname{HP}_{\Omega^{m}_{R_{{\mathbb{W}}}/K}}(z)>0$. 3. (c)

$\operatorname{HP}_{\Omega^{n+1}_{R_{{\mathbb{W}}}/K}}(z)>0$.

8. The Hilbert Polynomial of

$\Omega^{n+1}_{R_{{\mathbb{W}}}/K}$ for a Fat Point Scheme ${\mathbb{W}}$

Given a fat point scheme ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ in $\mathbb{P}^{n}$, the Hilbert function of the $R_{{\mathbb{W}}}$-module $\Omega^{1}_{R_{{\mathbb{W}}}/K}$ satisfies $\operatorname{HF}_{\Omega^{1}_{R_{{\mathbb{W}}}/K}}(i)=(n+1)\operatorname{HF}_{{\mathbb{W}}}(i-1)+\operatorname{HF}_{{\mathbb{W}}}(i)-\operatorname{HF}_{\mathbb{V}}(i)$ for all $i\in\mathbb{Z}$, where $\mathbb{V}$ is the fattening of ${\mathbb{W}}$ (see [KLL, Corollary 1.9(i)]). Naturally, we still want to give a formula for the Hilbert function of the module $\Omega^{m}_{R_{{\mathbb{W}}}/K}$ for $m\geq 2$. In fact, we can calculate the Hilbert function of $\Omega^{n+1}_{R_{{\mathbb{W}}}/K}$ in the following special case.

Proposition 8.1**.**

Let ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a fat point scheme in $\mathbb{P}^{n}$ supported at a set of points $\mathbb{X}=\{P_{1},\dots,P_{s}\}$, and let $\mathbb{Y}$ be the subscheme $\mathbb{Y}=(m_{1}-1)P_{1}+\cdots+(m_{s}-1)P_{s}$ of ${\mathbb{W}}$. Suppose that $\mathbb{X}$ is contained in a hyperplane. Then we have

[TABLE]

In particular, we have $\operatorname{HF}_{\Omega^{n+1}_{R_{{\mathbb{W}}}/K}}(i)=\operatorname{HF}_{\mathbb{Y}}(i-n-1)$ for all $i\in\mathbb{Z}$.

Proof.

Assume that $\mathbb{X}\subseteq{\mathcal{Z}}^{+}(H)$, where $0\neq\!H\!=\!\sum_{i=0}^{n}a_{i}X_{i}\in S$ and $a_{0},\dots,a_{n}\!\in K$. By letting I=\big{\langle}\,\frac{\partial F}{\partial X_{i}}\mid F\in I_{{\mathbb{W}}},0\leq i\leq n\,\big{\rangle} and by Corollary 2.3, we have

[TABLE]

Write $I_{{\mathbb{W}}}=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s}^{m_{s}}$ where $\wp_{j}$ is the associated prime ideal of $P_{j}$ in $S$, and let $F\in I_{\mathbb{W}}\setminus\{0\}$. Clearly, $\frac{\partial F}{\partial X_{i}}\in\wp_{j}^{m_{j}-1}$ for $i=0,\dots,n$ and $j=1,\dots,s$, and so $\frac{\partial F}{\partial X_{i}}\in I_{\mathbb{Y}}$ for $i=0,\dots,n$. Consequently, we get $I\subseteq I_{\mathbb{Y}}$.

Now we prove $I\supseteq I_{\mathbb{Y}}$. Suppose for a contradiction that there exists a homogeneous polynomial $G$ such that $G\in I_{\mathbb{Y}}\setminus I$. Then we have $HG\in I_{{\mathbb{W}}}$. Since $H\neq 0$, we may assume that $a_{i}\neq 0$ for some $i\in\{0,\dots,n\}$. We have $\frac{\partial(HG)}{\partial X_{i}}=a_{i}G+H\frac{\partial G}{\partial X_{i}}\in I$. Since $G\notin I$, we deduce $G_{1}:=H\frac{\partial G}{\partial X_{i}}\in I_{\mathbb{Y}}\setminus I$. Futhermore, we continue to have $HG_{1}\in I_{{\mathbb{W}}}$, and so

[TABLE]

This implies that $G_{2}:=H^{2}\tfrac{\partial^{2}G}{\partial X_{i}^{2}}\in I_{\mathbb{Y}}\setminus I$. Repeating this process, we eventually get $H^{\operatorname{deg}(G)}\in I_{\mathbb{Y}}\setminus I$. On the other hand, since $G\in I_{\mathbb{Y}}$, it follows that $\operatorname{deg}(G)\geq\max\{m_{1}-1,\dots,m_{s}-1\}$. Thus we have $H^{\operatorname{deg}(G)+1}\in I_{{\mathbb{W}}}$, and consequently $H^{\operatorname{deg}(G)}=\tfrac{1}{a_{i}(\operatorname{deg}(G)+1)}\frac{\partial H^{\operatorname{deg}(G)+1}}{\partial X_{i}}\in I$, a contradiction. Therefore we get $I_{\mathbb{Y}}=I$, and hence $\Omega^{n+1}_{R_{{\mathbb{W}}}/K}\cong(S/I_{\mathbb{Y}})(-n-1)$, as desired. ∎

Let us apply the preceding proposition to a concrete case.

Example 8.2**.**

Let $K=\mathbb{Q}$, and let ${\mathbb{W}}$ and $\mathbb{Y}$ be the fat point schemes ${\mathbb{W}}=2P_{1}+3P_{2}+4P_{3}+2P_{4}+P_{5}+7P_{6}+5P_{7}$ and $\mathbb{Y}=P_{1}+2P_{2}+3P_{3}+P_{4}+6P_{6}+4P_{7}$ in $\mathbb{P}^{5}$, where $P_{1}=(1:1:1:1:1:\frac{15}{6})$, $P_{2}=(1:2:1:1:1:\frac{17}{6})$, $P_{3}=(1:1:2:1:1:\frac{18}{6})$, $P_{4}=(1:2:3:4:5:\frac{55}{6})$, $P_{5}=(1:2:2:1:1:\frac{20}{6})$, $P_{6}=(1:3:2:1:1:\frac{22}{6})$, and where $P_{7}=(1:0:0:1:1:\frac{10}{6})$. Then $\mathbb{X}=\{P_{1},\dots,P_{7}\}$ is contained in the hyperplane ${\mathcal{Z}}^{+}(X_{0}-4X_{3}+3X_{4})$. Thus Proposition 8.1 yields $\Omega^{6}_{R_{{\mathbb{W}}}/K}\cong R_{\mathbb{Y}}(-6)$. Hence the Hilbert function of $\Omega^{6}_{R_{{\mathbb{W}}}/K}$ is given by

[TABLE]

Although no formula for the Hilbert function of the module of Kähler differential $(n+1)$-forms of an equimultiple fat point scheme is known, the following theorem provides a formula for its Hilbert polynomial.

Theorem 8.3**.**

Let $\mathbb{X}=\{P_{1},\dots,P_{s}\}\subseteq\mathbb{P}^{n}$ be a set of $s$ distinct $K$-rational points, and let $\nu\geq 1$. Then we have $\operatorname{HP}_{\Omega^{n+1}_{R_{(\nu+1)\mathbb{X}}/K}}(z)=\operatorname{HP}_{\nu\mathbb{X}}(z)$.

Proof.

Let $I=\langle\,\frac{\partial F}{\partial X_{i}}\,\mid\,F\in I_{\mathbb{X}},0\leq i\leq n\,\rangle$. By Corollary 2.3, we have $\Omega^{n+1}_{R_{\mathbb{X}}/K}\cong(S/I)(-n-1)$. So, Proposition 7.1 implies $\operatorname{HF}_{\Omega^{n+1}_{R_{\mathbb{X}}/K}}(i)=\operatorname{HF}_{S/I}(i-n-1)=0$ for $i\gg 0$. Let $\mathfrak{M}$ denote the homogeneous maximal ideal of $S$. There exists a number $t_{1}\in\mathbb{N}$ such that $I_{t_{1}+i}=\mathfrak{M}_{t_{1}+i}$ for all $i\in\mathbb{N}$. Moreover, by [HC, 4.2], there is $t_{2}\in\mathbb{N}$ such that $(I_{\nu\mathbb{X}})_{t_{2}+i}=(I_{\mathbb{X}}^{\nu})_{t_{2}+i}$ for all $i\in\mathbb{N}$. Let $t=\max\{t_{1},t_{2},r_{\nu\mathbb{X}}+1\}$, let $r=\binom{n+t}{n}-s$, let $\{F_{1},\dots,F_{r}\}$ be a $K$-basis of the $K$-vector space $(I_{\mathbb{X}})_{t}$, and let

[TABLE]

Clearly, we have $I_{(\nu+1)\mathbb{X}}\subseteq J\subseteq I_{\nu\mathbb{X}}$. Since $\Omega^{n+1}_{R_{(\nu+1)\mathbb{X}}/K}\cong(S/J)(-n-1)$, and since $I_{\nu\mathbb{X}}$ is generated by elements of degrees $\leq t$ (cf. [GM, Proposition 1.1]), it suffices to show that $J_{i}=(I_{\nu\mathbb{X}})_{i}$ for some $i\geq t$.

We observe that

[TABLE]

Here the last two equalities follow from Euler’s relation and the fact that $(I_{\mathbb{X}})_{t+1}=(I_{\mathbb{X}})_{t}\mathfrak{M}_{1}=\langle F_{j}\mid 1\leq j\leq r\rangle_{K}\mathfrak{M}_{1}$. (Note that $I_{\mathbb{X}}$ can be generated in degrees $\leq t$.) Therefore we get equalities

[TABLE]

Thus we only need to prove the inclusion

[TABLE]

To this end, we first prove that $F_{j_{1}}^{\nu-k}F_{j_{2}}\cdots F_{j_{k+1}}\tfrac{\partial F_{j_{1}}}{\partial X_{i_{1}}}\cdots\tfrac{\partial F_{j_{1}}}{\partial X_{i_{k+1}}}\in J$ for all $i_{1},\dots,i_{k+1}\in\{0,\dots,n\}$ and $j_{1},\dots,j_{k+1}\in\{1,\dots,r\}$ and $0\leq k\leq\nu$. We proceed by induction on $k$. If $k=0$, for $0\leq i_{1}\leq n$ and $1\leq j_{1}\leq r$ we have

[TABLE]

If $k=1$, for $i_{1},i_{2}\in\{0,\dots,n\}$ and $j_{1},j_{2}\in\{1,\dots,r\}$, we get

[TABLE]

Now we assume that $2\leq k\leq\nu$ and $F_{j^{\prime}_{1}}^{\nu-(k-1)}F_{j^{\prime}_{2}}\cdots F_{j^{\prime}_{k}}\tfrac{\partial F_{j^{\prime}_{1}}}{\partial X_{i^{\prime}_{1}}}\cdots\tfrac{\partial F_{j^{\prime}_{1}}}{\partial X_{i^{\prime}_{k}}}\in J$ for all $i^{\prime}_{1},\dots,i^{\prime}_{k}\in\{0,\dots,n\}$ and $j^{\prime}_{1},\dots,j^{\prime}_{k}\in\{1,\dots,r\}$. Let $i_{1},\dots,i_{k+1}\in\{0,\dots,n\}$ and $j_{1},\dots,j_{k+1}\in\{1,\dots,r\}$. We have

[TABLE]

By the inductive hypothesis, we have $F_{j_{1}}^{\nu-(k-1)}\tfrac{F_{j_{2}}\cdots F_{j_{k+1}}}{F_{j_{l}}}\tfrac{\partial F_{j_{1}}}{\partial X_{i_{2}}}\cdots\tfrac{\partial F_{j_{1}}}{\partial X_{i_{k+1}}}\in J$ for all $l=2,\dots,k+1$. Hence we get $F_{j_{1}}^{\nu-k}F_{j_{2}}\cdots F_{j_{k+1}}\tfrac{\partial F_{j_{1}}}{\partial X_{i_{1}}}\cdots\tfrac{\partial F_{j_{1}}}{\partial X_{i_{k+1}}}\in J$.

Consequently, if $k=\nu$, then we have $F_{j_{1}}F_{j_{2}}\cdots F_{j_{\nu}}\tfrac{\partial F_{j}}{\partial X_{i_{1}}}\cdots\tfrac{\partial F_{j}}{\partial X_{i_{\nu+1}}}\in J$ for all $i_{1},\dots,i_{\nu+1}\in\{0,\dots,n\}$ and $j,j_{1},\dots,j_{\nu}\in\{1,\dots,r\}$. On the other hand, the $K$-vector space $\langle\frac{\partial F_{j}}{\partial X_{i}}\mid 0\leq i\leq n,1\leq j\leq r\rangle_{K}^{\nu r+1}$ has a system of generators consisting of elements of the form

[TABLE]

where $\alpha_{ij}\in\mathbb{N}$ satisfy $\sum_{j=1}^{r}\sum_{i=0}^{n}\alpha_{ij}=\nu r+1$. Set $\alpha_{j}:=\sum_{i=0}^{n}\alpha_{ij}$ for $j=1,\dots,r$. Since $\sum_{j=1}^{r}\alpha_{j}=\nu r+1$, there is an index $j\in\{1,\dots,r\}$ such that $\alpha_{j}\geq\nu+1$. Also, we have $\nu rn+1\geq\nu r+1$. It follows that any element of the $K$-vector space

[TABLE]

is a sum of elements of the form $F_{j_{1}}\cdots F_{j_{\nu}}\tfrac{\partial F_{j}}{\partial X_{i_{1}}}\cdots\frac{\partial F_{j}}{\partial X_{i_{\nu+1}}}G$, where $i_{1},\dots,i_{\nu+1}\in\{0,\dots,n\}$, where $j,j_{1},\dots,j_{\nu}\in\{1,\dots,r\}$, and where $G\in S$ is a homogeneous polynomial of degree $\operatorname{deg}(G)=\nu(rn-1)(t-1)$. Therefore we get

[TABLE]

and this completes the proof. ∎

The following corollary follows immediately from Theorem 8.3.

Corollary 8.4**.**

Let $\mathbb{X}=\{P_{1},\dots,P_{s}\}\subseteq\mathbb{P}^{n}$ be a set of $s$ distinct $K$-rational points, and let $\nu\geq 1$. Then we have $\operatorname{HP}_{\Omega^{n+1}_{R_{(\nu+1)\mathbb{X}}/K}}(z)=s\binom{\nu+n-1}{n}$.

In the last part of this section we study the module of Kähler differential 2-forms of fat point schemes ${\mathbb{W}}$ in $\mathbb{P}^{n}$. Let us begin with the following sequence of graded $R_{{\mathbb{W}}}$-modules.

Proposition 8.5**.**

Let ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a fat point scheme in $\mathbb{P}^{n}$, and let ${\mathbb{W}}^{(i)}$ be the $i$-th fattening of ${\mathbb{W}}$ for $i\geq 1$. Then the sequence of graded $R_{{\mathbb{W}}}$-modules

[TABLE]

is a complex, where $\alpha(F+I_{{\mathbb{W}}^{(2)}})=dF+I_{{\mathbb{W}}^{(1)}}\Omega^{1}_{S/K}$, where $\beta(GdX_{i}+I_{{\mathbb{W}}^{(1)}}\Omega^{1}_{S/K})=d(GdX_{i})+I_{{\mathbb{W}}}\Omega^{2}_{S/K}$, and where $\gamma(H+I_{{\mathbb{W}}}\Omega^{2}_{S/K})=H+(I_{{\mathbb{W}}}\Omega^{2}_{S/K}+dI_{{\mathbb{W}}}\Omega^{1}_{S/K})$. Moreover, the following statements hold true.

1. (a)

The map $\alpha$ is injective. 2. (b)

The map $\gamma$ is surjective. 3. (c)

We have $\operatorname{Im}(\beta)=\operatorname{Ker}(\gamma)$. 4. (d)

For all $i\geq 0$, we have

[TABLE]

Proof.

To prove (a), we note that the arguments in the proof of [KLL, Theorem 1.7] shown that the map $\alpha$ is a homogeneous injection. Claims (b) and (c) are implied by the presentation

[TABLE]

Since $d\circ d=0$, it follows that $\beta\circ\alpha=0$. Therefore the sequence ($\mathcal{C}$) is a complex. In addition, claim (d) follows from the fact that ($\mathcal{C}$) is a complex and (c). ∎

Now we look at the special case of an equimultiple fat point scheme ${\mathbb{W}}=\nu\mathbb{X}$ in $\mathbb{P}^{2}$. In this case, by taking the homogeneous components at a large degree $i$ of the complex $(\mathcal{C})$, we have the following exact sequence of $K$-vector spaces.

Proposition 8.6**.**

Let $\mathbb{X}=\{P_{1},\dots,P_{s}\}\subseteq\mathbb{P}^{2}$ be a set of $s$ distinct $K$-rational points, and let $\nu\geq 1$. For $i\gg 0$, the sequence of $K$-vector spaces

[TABLE]

is exact. Here the maps $\alpha$, $\beta$ and $\gamma$ are defined as in Proposition 8.5.

Proof.

By Proposition 8.5, we only need to prove $\operatorname{Im}(\alpha)=\operatorname{Ker}(\beta)$. Hence it suffices to show that the Hilbert polynomial of $\Omega^{2}_{R_{\nu\mathbb{X}}/K}$ satisfies

[TABLE]

In $\mathbb{P}^{2}$, Proposition 2.4 yields the exact sequence of graded $R_{\nu\mathbb{X}}$-modules

[TABLE]

Moreover, we have $\operatorname{HP}_{\Omega^{3}_{R_{\nu\mathbb{X}}/K}}(z)=\operatorname{HP}_{(\nu-1)\mathbb{X}}(z)$ by Theorem 8.3. Hence, by applying [KLL, Corollary 1.9], we get

[TABLE]

and the claim follows. ∎

The following formula for the Hilbert polynomial of $\Omega^{2}_{R_{\nu\mathbb{X}}/K}$ can be extracted from the proof of this proposition.

Corollary 8.7**.**

Let $\mathbb{X}=\{P_{1},\dots,P_{s}\}\subseteq\mathbb{P}^{2}$ be a set of $s$ distinct $K$-rational points, and let $\nu\geq 1$. Then we have $\operatorname{HP}_{\Omega^{2}_{R_{\nu\mathbb{X}}/K}}(z)=\frac{1}{2}(3\nu^{2}-\nu-2)s$.

In general, we do not have an explicit formula for the Hilbert polynomial of $\Omega^{n+1}_{R_{{\mathbb{W}}}/K}$ for a non-reduced fat point scheme ${\mathbb{W}}$ in $\mathbb{P}^{n}$. However, we propose the following conjecture.

Conjecture 8.8**.**

Let ${\mathbb{W}}$ be the fat point scheme ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ in $\mathbb{P}^{n}$, and let $\mathbb{Y}=(m_{1}-1)P_{1}+\cdots+(m_{s}-1)P_{s}$. Then we have $\operatorname{HP}_{\Omega^{n+1}_{R_{{\mathbb{W}}}/K}}(z)=\operatorname{HP}_{\mathbb{Y}}(z)$.

The above conjecture holds true for $n=1$ by Proposition 3.1, and for $n\geq 2$ such that ${\mathbb{W}}$ is an equimultiple fat point scheme in $\mathbb{P}^{n}$ by Theorem 8.3. The following example provides a further instance in which the conjecture holds.

Example 8.9**.**

Let $\mathbb{X}=\{P_{1},...,P_{10}\}$ be the set of 10 points on the twisted cubic curve in $\mathbb{P}^{3}$ given by $P_{1}=(1:1:1:1)$, $P_{2}=(1:-1:1:-1)$, $P_{3}=(1:2:4:8)$, $P_{4}=(1:-2:4:-8)$, $P_{5}=(1:3:9:27)$, $P_{6}=(1:-3:9:-27)$, $P_{7}=(1:4:16:64)$, $P_{8}=(1:-4:16:-64)$, $P_{9}=(1:-5:25:-125)$, and $P_{10}=(1:6:36:216)$. Let ${\mathbb{W}}=P_{1}+2P_{2}+4P_{3}+3P_{4}+4P_{5}+2P_{6}+3P_{7}+7P_{8}+5P_{9}+6P_{10}$, and let $\mathbb{Y}=P_{2}+3P_{3}+2P_{4}+3P_{5}+P_{6}+2P_{7}+6P_{8}+4P_{9}+5P_{10}$. A calculation yields that the Hilbert polynomials of $\Omega^{n+1}_{R_{{\mathbb{W}}}/K}$ and of $R_{\mathbb{Y}}$ are the same and equal to 141.

9. Fat Point Schemes Supported on Non-Singular Conics

In Proposition 3.1, we described concretely the Hilbert function of the module of Kähler differential $m$-forms when ${\mathbb{W}}$ is a fat point scheme in $\mathbb{P}^{1}$. This result leads us to the following question:

Question 9.1*.*

Can one compute explicitly the Hilbert function of the bi-graded $R_{{\mathbb{W}}}$-algebra $\Omega_{R_{{\mathbb{W}}}/K}$ for a fat point scheme ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ in $\mathbb{P}^{2}$?

In this section we answer some parts of this question. In particular, we give concrete formulas for the Hilbert function of the bi-graded algebra $\Omega_{R_{{\mathbb{W}}}/K}$ if ${\mathbb{W}}$ is an equimultiple fat point scheme in $\mathbb{P}^{2}$ whose support $\mathbb{X}$ lies on a non-singular conic.

In what follows, we let $\mathcal{C}=\mathcal{Z}^{+}(C)$ be a non-singular conic defined by a quadratic polynomial $C\in S=K[X_{0},X_{1},X_{2}]$, we let $\mathbb{X}=\{P_{1},\dots,P_{s}\}$ be a set of $s$ distinct $K$-rational points on $\mathcal{C}$, and we let ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a fat point scheme in $\mathbb{P}^{2}$ supported at $\mathbb{X}$. Suppose that $0\leq m_{1}\leq\cdots\leq m_{s}$ and $s\geq 4$. Then it was shown in [Cat, Proposition 2.2] that the regularity index of ${\mathbb{W}}$ is

[TABLE]

Moreover, the Hilbert function of ${\mathbb{W}}$ can be effectively computed from the Hilbert function of a certain subscheme $\mathbb{Y}$ of ${\mathbb{W}}$ (see [Cat, Theorem 3.1]).

The Hilbert function of the module of Kähler differential $1$-forms $\Omega^{1}_{R_{{\mathbb{W}}}/K}$ satisfies the following conditions.

Theorem 9.2**.**

In the setting above, let $\mu=\sum_{j=1}^{s}m_{j}+s$ and $\varrho=m_{s}+m_{s-1}$.

1. (a)

If $\mu\geq 2\varrho+4$ then

[TABLE] 2. (b)

If $\mu\leq 2\varrho+3$ and $\mathbb{Y}:=(m_{1}+1)P_{1}+\cdots+(m_{s-2}+1)P_{s-2}+m_{s-1}P_{s-1}+m_{s}P_{s}$, then

[TABLE]

Proof.

Let $\mathbb{V}$ be the fat point scheme $\mathbb{V}=(m_{1}+1)P_{1}+\cdots+(m_{s}+1)P_{s}$ containing ${\mathbb{W}}$. By [KLL, Corollary 1.9(i)], we have

[TABLE]

for all $i\in\mathbb{Z}$. Also, we have $r_{\mathbb{V}}=\max\{\varrho+1,\lfloor\frac{\mu}{2}\rfloor\}$.

(a) Consider the case $\mu\geq 2\varrho+4$. In this case, we have $r_{\mathbb{V}}=\lfloor\frac{\mu}{2}\rfloor$. Since $s\geq 4$, we get the inequality

[TABLE]

So, [KLL, Corollary 1.9(iii)] yields $\operatorname{ri}(\Omega^{1}_{R_{{\mathbb{W}}}/K})\leq r_{\mathbb{V}}=\lfloor\frac{\mu}{2}\rfloor$. Moreover, we see that

[TABLE]

for all $i\geq 0$, and hence $\operatorname{ri}(\Omega^{1}_{R_{{\mathbb{W}}}/K})=r_{\mathbb{V}}$. Consequently, we can apply [Cat, Theorem 3.1] to work out the Hilbert function of $\Omega^{1}_{R_{{\mathbb{W}}}/K}$ with respect to the value of the degree $i$ as follows:

1. (i)

For $i\geq r_{\mathbb{V}}=\lfloor\frac{\mu}{2}\rfloor$, the Hilbert function of $\Omega^{1}_{R_{{\mathbb{W}}}/K}$ satisfies

[TABLE] 2. (ii)

Let $r_{{\mathbb{W}}}+2\leq i<\lfloor\frac{\mu}{2}\rfloor$. Then we have $\operatorname{HF}_{{\mathbb{W}}}(i-2)=\operatorname{HF}_{{\mathbb{W}}}(i-1)=\operatorname{HF}_{{\mathbb{W}}}(i)=\operatorname{deg}({\mathbb{W}})=\sum_{j=1}^{s}\binom{m_{j}+1}{2}$ and $\operatorname{HF}_{\mathbb{V}}(i)=2i+1+\operatorname{HF}_{{\mathbb{W}}}(i-2)$. It follows that

[TABLE] 3. (iii)

In the case $i=r_{{\mathbb{W}}}+1$, we have $\operatorname{HF}_{{\mathbb{W}}}(i-1)=\operatorname{HF}_{{\mathbb{W}}}(i)=\sum_{j=1}^{s}\binom{m_{j}+1}{2}$ and $\operatorname{HF}_{\mathbb{V}}(i)=2i+1+\operatorname{HF}_{{\mathbb{W}}}(i-2)$. Thus

[TABLE] 4. (iv)

If $0\leq i\leq r_{{\mathbb{W}}}$, we have $\operatorname{HF}_{\mathbb{V}}(i)=2i+1+\operatorname{HF}_{{\mathbb{W}}}(i-2)$ and

[TABLE]

Altogether, we have proved the formula for the Hilbert function of $\Omega^{1}_{R_{{\mathbb{W}}}/K}$ in the case $\mu\geq 2\varrho+4$.

(b) Next we consider the case $\mu\leq 2\varrho+3$. In this case, the relation between Hilbert functions of $\mathbb{V}$ and of $\mathbb{Y}$ follows from [Cat, Theorem 3.1]. Also, we have $r_{\mathbb{V}}=\varrho+1$ and $r_{{\mathbb{W}}}=\varrho-1<r_{\mathbb{V}}$, and hence $\operatorname{ri}(\Omega^{1}_{R_{{\mathbb{W}}}/K})=r_{\mathbb{V}}$. Therefore a similar argument as in the first case yields the desired formula for the Hilbert function of $\Omega^{1}_{R_{{\mathbb{W}}}/K}$. ∎

It is worth noting that [Cat, Theorem 3.1] and Theorem 9.2 give us a procedure for computing the Hilbert function of the module of Kähler differential $1$-forms of $R_{{\mathbb{W}}}/K$ from some suitable fat point schemes. Moreover, $\operatorname{HF}_{\Omega^{1}_{R_{{\mathbb{W}}}/K}}$ is completely determined by $s$ and the multiplicities $m_{1},\dots,m_{s}$.

Remark 9.3**.**

On a non-singular conic $\mathcal{C}$, let ${\mathbb{W}}$ be a complete intersection of type $(2,d)$. Let $P\in{\mathbb{W}}$, and let $\mathbb{Y}={\mathbb{W}}\setminus\{P\}$. The regularity index of the scheme $\mathbb{Y}$ is $d-1$. Using Theorem 9.2 we see that the Hilbert function of $\Omega^{1}_{R_{\mathbb{Y}}/K}$ is independent of the choice of the point $P$.

The following proposition can be used to find out a connection between the Hilbert functions of $\Omega^{3}_{R_{{\mathbb{W}}}/K}$ (as well as of $\Omega^{2}_{R_{{\mathbb{W}}}/K}$) and of a suitable subscheme of ${\mathbb{W}}$ if ${\mathbb{W}}$ is an equimultiple fat point scheme, i.e., if $m_{1}=\cdots=m_{s}=\nu$. This proposition follows from [Cat, Proposition 4.3].

Proposition 9.4**.**

In the setting of Theorem 9.2, we define

[TABLE]

if $\mu-s\geq 2\varrho$ and

[TABLE]

otherwise. Further, let $q=\max\{\varrho,\lfloor\tfrac{\mu-s+1}{2}\rfloor\}$, let $\{G_{1},\dots,G_{r}\}$ be a minimal homogenous system of generators of $I_{\mathbb{Y}}$, and let $L$ be the linear form passing through $P_{s-1}$ and $P_{s}$.

1. (a)

If $\mu-s\geq 2\varrho$ and $\mu-s$ is odd, there exist $F_{1},F_{2}\in(I_{{\mathbb{W}}})_{q}$ such that the set $\{CG_{1},\dots,CG_{r},F_{1},F_{2}\}$ is a minimal homogeneous system of generators of $I_{{\mathbb{W}}}$. 2. (b)

If $\mu-s\geq 2\varrho$ and $\mu-s$ is even, there exists $F\in(I_{{\mathbb{W}}})_{q}$ such that the set $\{CG_{1},\dots,CG_{r},F\}$ is a minimal homogeneous system of generators of $I_{{\mathbb{W}}}$. 3. (c)

If $\mu-s<2\varrho$, there exists $G\in(I_{{\mathbb{W}}})_{q}$ such that the set $\{LG_{1},\dots,LG_{r},G\}$ is a minimal homogeneous system of generators of $I_{{\mathbb{W}}}$.

In particular, if $\mathbb{X}=\{P_{1},\dots,P_{s}\}$ is a set of $s$ distinct $K$-rational points on a non-singular conic $\mathcal{C}$, then, for every $k\geq 1$, there is a minimal homogeneous system of generators of $I_{k\mathbb{X}}$ of the following form.

Corollary 9.5**.**

Let $s\geq 4$, let $\mathbb{X}=\{P_{1},\dots,P_{s}\}$ be a set of $s$ distinct $K$-rational points on a non-singular conic $\mathcal{C}=\mathcal{Z}^{+}(C)$, let $\{C,G_{1},\dots,G_{t}\}$ be a minimal homogeneous system of generators of $I_{\mathbb{X}}$, and let $k\geq 1$.

1. (a)

If $s=2v$ for some $v\in\mathbb{N}$, then there exists a minimal homogeneous system of generators of $I_{k\mathbb{X}}$ of the form

[TABLE]

where $F_{j1}\in(I_{j\mathbb{X}})_{jv}$ for all $j=2,\dots,k$. 2. (b)

If $s=2v+1$ for some $v\in\mathbb{N}$ and $k$ is even, then there exists a minimal homogeneous system of generators of $I_{k\mathbb{X}}$ of the form

[TABLE]

where $F_{jl}\in(I_{j\mathbb{X}})_{q_{j}}$ with $q_{j}=\lfloor\frac{j(2v+1)+1}{2}\rfloor$ for every $j=2,\dots,k$ and $l=1,2$. 3. (c)

If $s=2v+1$ for some $v\in\mathbb{N}$ and $k$ is odd, then there exists a minimal homogeneous system of generators of $I_{k\mathbb{X}}$ of the form

[TABLE]

where $F_{jl}\in(I_{j\mathbb{X}})_{q_{j}}$ with $q_{j}=\lfloor\frac{j(2v+1)+1}{2}\rfloor$ for every $j=2,\dots,k$ and $l=1,2$.

Proof.

Since $s\geq 4$, we have $q_{k}=\max\{2k,\lfloor\frac{sk+1}{2}\rfloor\}=\lfloor\frac{sk+1}{2}\rfloor$ for every $k\geq 1$. By Proposition 9.4 and by induction on $k$, we get the claimed minimal homogeneous system of generators of the ideal $I_{k\mathbb{X}}$. ∎

Now we present a relation between the Hilbert functions of the module of Kähler differential $3$-forms $\Omega^{3}_{R_{\nu\mathbb{X}}/K}$ and of $S/\mathfrak{M}I_{(\nu-1)\mathbb{X}}$, where $\mathfrak{M}=\langle X_{0},\dots,X_{n}\rangle$ is the homogeneous maximal ideal of $S$. Here we make the convention that $I_{(\nu-1)\mathbb{X}}:=\langle 1\rangle$ if $\nu=1$.

Theorem 9.6**.**

Let $s\geq 4$ and $\nu\geq 1$, and let $\mathbb{X}=\{P_{1},\dots,P_{s}\}\subseteq\mathbb{P}^{2}$ be a set of $s$ distinct $K$-rational points which lie on a non-singular conic $\mathcal{C}=\mathcal{Z}^{+}(C)$. Then we have $\Omega^{3}_{R_{\nu\mathbb{X}}/K}\cong(S/\mathfrak{M}I_{(\nu-1)\mathbb{X}})(-3)$. In particular, for all $i\in\mathbb{N}$, we have

[TABLE]

Proof.

Let $\mathcal{B}_{1}=\{C,G_{1},\dots,G_{t}\}$ be a minimal homogeneous system of generators of $I_{\mathbb{X}}$, and let $J^{(\nu)}=\langle\frac{\partial F}{\partial X_{i}}\mid F\in I_{\nu\mathbb{X}},0\leq i\leq n\rangle$. Note that $r_{\mathbb{X}}=\lfloor\frac{s}{2}\rfloor$. According to [GM, Proposition 1.1], we may assume that $2\leq\operatorname{deg}(G_{j})\leq\lfloor\frac{s}{2}\rfloor+1$ for $j=1,\dots,t$. By Corollary 2.3, we have $\Omega^{3}_{R_{\nu\mathbb{X}}/K}\cong(S/J^{(\nu-1)})(-3)$. Moreover, since $\mathcal{C}$ is a non-singular conic, we have $\langle\tfrac{\partial C}{\partial X_{i}}\mid 0\leq i\leq 2\rangle=\mathfrak{M}$, and hence $J^{(1)}=\mathfrak{M}$. Thus it suffices to prove the equality $J^{(\nu+1)}=\mathfrak{M}I_{\nu\mathbb{X}}$ for all $\nu\geq 1$. For $k\geq 2$, let $\mathcal{B}_{k}$ be the minimal homogeneous system of generators of $I_{k\mathbb{X}}$ constructed in Corollary 9.5. We see that $\operatorname{deg}(F_{(k+1)1})\geq\operatorname{deg}(F_{k1})+2$ for all $k\geq 2$ and that $\operatorname{deg}(F_{21})=s$.

If $s=4$ then $\mathbb{X}$ is a complete intersection of type $(2,2)$, and so we may assume $\mathcal{B}_{1}=\{C,G_{1}\}$ with $\operatorname{deg}(G_{1})=2$. This implies $\operatorname{deg}(F_{21})=\operatorname{deg}(G_{1})+2$. In the case $s\geq 5$ we have $\operatorname{deg}(F_{21})=s\geq\lfloor\frac{s}{2}\rfloor+3\geq\max\{\operatorname{deg}(G_{j})\mid 1\leq j\leq t\}+2$. Hence the inclusion $J^{(\nu+1)}\subseteq\mathfrak{M}I_{\nu\mathbb{X}}$ holds true for all $\nu\geq 1$.

Now we proceed by induction on $\nu$ to prove that $C^{k-1}\mathfrak{M}I_{\nu\mathbb{X}}\subseteq J^{(\nu+k)}$ for all $k\geq 1$. In the following, let $k\geq 1$, $0\leq i\leq 2$ and $1\leq j\leq t$. If $\nu=1$, we have

[TABLE]

Since $\tfrac{\partial G_{j}}{\partial X_{i}}\in\mathfrak{M}$, we also have

[TABLE]

Hence we get $C^{k-1}\mathfrak{M}I_{\mathbb{X}}\subseteq J^{(1+k)}$ for all $k\geq 1$, and the claim holds true for $\nu=1$. Next we assume that $\nu\geq 2$ and that $C^{k-1}\mathfrak{M}I_{l\mathbb{X}}\subseteq J^{(l+k)}$ for $1\leq l\leq\nu-1$ and all $k\geq 1$. We distinguish the following two cases.

Case (a):  Suppose that $s$ is even. Using Corollary 9.5(a) we write

[TABLE]

where $F_{l1}\in(I_{l\mathbb{X}})_{sl/2}$. Note that $\tfrac{\partial F_{l1}}{\partial X_{i}}\in\mathfrak{M}I_{(l-1)\mathbb{X}}$ for $2\leq l\leq\nu$. It follows from the inductive hypothesis that $C^{k+\nu-l}\tfrac{\partial F_{l1}}{\partial X_{i}}\in C^{k+\nu-l}\mathfrak{M}I_{(l-1)\mathbb{X}}\subseteq J^{(\nu+k)}$. Thus, for $2\leq l\leq\nu$, we have

[TABLE]

As above we have $C^{k-1+\nu}\mathfrak{M}\subseteq J^{(\nu+k)}$ and $C^{k+\nu-2}G_{1}\mathfrak{M}\subseteq J^{(\nu+k)}$. Therefore we obtain $C^{k-1}\mathfrak{M}I_{\nu\mathbb{X}}\subseteq J^{(\nu+k)}$ for all $k\geq 1$, as desired.

Case (b):  Suppose that $s$ is odd. In this case the minimal homogeneous system of generators $\mathcal{B}_{\nu}$ of $I_{\nu\mathbb{X}}$ is given by

[TABLE]

if $\nu=2l$ and

[TABLE]

if $\nu=2l+1$, where $F_{ju}\in(I_{j\mathbb{X}})_{\lfloor\frac{js+1}{2}\rfloor}$ (see Corollary 9.5(b),(c)). Thus we can use the same argument as in case (a) and get $C^{k-1}\mathfrak{M}I_{\nu\mathbb{X}}\subseteq J^{(\nu+k)}$ for all $k\geq 1$.

Altogether, we have shown that $C^{k-1}\mathfrak{M}I_{\nu\mathbb{X}}\subseteq J^{(\nu+k)}$ for all $\nu,k\geq 1$. In particular, if $k=1$, then we have $\mathfrak{M}I_{\nu\mathbb{X}}\subseteq J^{(\nu+1)}$, and the proof is complete. ∎

If $\nu=1$, we have $\operatorname{HF}_{\Omega^{3}_{R_{\mathbb{X}}/K}}(i)=0$ for $i\neq 3$ and $\operatorname{HF}_{\Omega^{3}_{R_{\mathbb{X}}/K}}(3)=1$. If $\nu\geq 2$, this Hilbert function can be described explicitly as follows.

Proposition 9.7**.**

In the setting of Theorem 9.6, let $\mathcal{B}_{1}=\{C,G_{1},\dots,G_{t}\}$ be a minimal homogeneous system of generators of $I_{\mathbb{X}}$ as constructed in Corollary 9.5, let $d_{j}=\operatorname{deg}(G_{j})$ for $j=1,\dots,t$, and let $\nu\geq 2$. Suppose that $d_{1}\leq\dots\leq d_{t}$. Then we have

[TABLE]

Here we let $h_{i}=\#\{G\in\mathcal{B}_{1}\mid\operatorname{deg}(G)=i+1-2\nu\}$, and $\delta_{i}$ is defined as follows.

1. (a)

If $s=4$ then $\delta_{i}=\nu-2$ if $i=2\nu+1$ and $\delta_{i}=0$ otherwise. 2. (b)

If $s=5$ then

[TABLE] 3. (c)

If $s\geq 6$ then

[TABLE]

Proof.

By Theorem 9.6, we have

[TABLE]

for all $i\in\mathbb{Z}$. Note that we have $r_{(\nu-1)\mathbb{X}}=\lfloor\frac{s(\nu-1)}{2}\rfloor$. If $i\geq r_{(\nu-1)\mathbb{X}}+3$, we obtain $\operatorname{HF}_{(\nu-1)\mathbb{X}}(i-3)=s\binom{\nu}{2}$. Otherwise, we have $\operatorname{HF}_{(\nu-1)\mathbb{X}}(i-3)=\operatorname{HF}_{\nu\mathbb{X}}(i-1)-2i+1$ by [Cat, Theorem 3.1]. For every $i\geq 0$, we set

[TABLE]

Hence the claimed shape of the Hilbert function of $\Omega^{3}_{R_{\nu\mathbb{X}}/K}$ follows immediately. Now we apply Corollary 9.5 to compute the values of $\delta_{i}$. We look at degrees of the elements in the tuple

[TABLE]

and get the tuple of degrees

[TABLE]

Consider the following cases.

(a) Suppose that $s=4$. Then $\mathcal{B}_{1}=\{C,G_{1}\}$ with $\operatorname{deg}(G_{1})=2$ and every element in $\mathcal{A}^{*}$ equals $2\nu-2$. Also, $\#(\mathcal{B}_{\nu-1})_{i-3}>0$ only if $i-3=2\nu-2$. In the case $i=2\nu+1$, we have $\#(\mathcal{B}_{\nu-1})_{i-3}=\nu$ and $h_{i}=2$, and so $\delta_{i}=\nu-2$. Clearly, $\delta_{i}=0$ if $i\neq 2\nu+1$.

(b) Suppose that $s=5$. We write $\operatorname{deg}(C^{\nu-(k+1)}F_{k1})=2\nu-2(k+1)+\lfloor\tfrac{5k+1}{2}\rfloor$ for $k=2,\dots,\nu-1$. It is easy to see that $\operatorname{deg}(C^{\nu-(k+1)}F_{k1})=\operatorname{deg}(C^{\nu-(k+2)}F_{k+1\>\!1})$ if $k$ is odd and $\operatorname{deg}(C^{\nu-(k+1)}F_{k1})+1=\operatorname{deg}(C^{\nu-(k+2)}F_{k+1\>\!1})$ otherwise.

• (i)

If $\nu$ is even then we have

[TABLE]

• (ii)

If $\nu$ is odd then we have

[TABLE]

(c) Suppose that $s\geq 6$. In this case the sequence of elements in $\mathcal{A}^{*}$ is strictly increasing.

• (i)

If $s$ is even then

[TABLE]

• (ii)

If $s$ is odd then

[TABLE]

Altogether, the claims follow. ∎

Example 9.8**.**

Let $K=\mathbb{Q}$, and let $\mathbb{X}=\{P_{1},\dots,P_{8}\}\subseteq\mathbb{P}^{2}$ be the set of 8 points given by $P_{1}=(1:1:0)$, $P_{2}=(1:3:0)$, $P_{3}=(1:0:1)$, $P_{4}=(1:4:1)$, $P_{5}=(1:0:3)$, $P_{6}=(1:1:4)$, $P_{7}=(1:4:3)$, and $P_{8}=(1:3:4)$. Then $\mathbb{X}$ is contained in the non-singular conic defined by $C=3X_{0}^{2}-4X_{0}X_{1}+X_{1}^{2}-4X_{0}X_{2}+X_{2}^{2}$. In particular, $\mathbb{X}$ is a complete intersection of type $(2,4)$, and hence the set $\mathcal{B}_{1}$ constructed in Corollary 9.5 is of the form $\mathcal{B}_{1}=\{C,G_{1}\}$ with $\operatorname{deg}(G_{1})=4$. The Hilbert functions of $\nu\mathbb{X}$, $1\leq\nu\leq 3$, are given by

[TABLE]

Moreover, we have $\operatorname{HF}_{\Omega^{3}_{R_{\mathbb{X}}/K}}:\ 0\ 0\ 0\ 1\ 0\ 0\cdots$. Now we apply Proposition 9.7 to compute the Hilbert functions of $\Omega^{3}_{R_{\nu\mathbb{X}}/K}$, where $\nu=2,3$. Since $s=8$ is even, we have

[TABLE]

If $\nu=2$, then we have $\delta_{i}=0$ for all $i\geq 1$ and

[TABLE]

An application of Proposition 9.7 yields

[TABLE]

Hence we get $\operatorname{HF}_{\Omega^{3}_{R_{2\mathbb{X}}/K}}:\ 0\ 0\ 0\ 1\ 3\ 6\ 7\ 9\ 8\ 8\cdots$.

Similarly, for $\nu=3$ we have

[TABLE]

Therefore we get $\operatorname{HF}_{\Omega^{3}_{R_{3\mathbb{X}}/K}}:\ 0\ 0\ 0\ 1\ 3\ 6\ 10\ 15\ 18\ 23\ 25\ 24\ 24\cdots$.

Proposition 9.9**.**

Let $s\geq 4$ and $\nu\geq 1$, and let $\mathbb{X}=\{P_{1},\dots,P_{s}\}\subseteq\mathbb{P}^{2}$ be a set of $s$ distinct $K$-rational points which lie on a non-singular conic $\mathcal{C}=\mathcal{Z}^{+}(C)$. For $i\geq 0$, let $h_{i},\delta_{i}$ be defined as in Proposition 9.7.

1. (a)

If $\nu=1$, we have

[TABLE] 2. (b)

For every $\nu\geq 2$, let $\mu=\lfloor\frac{s\nu}{2}\rfloor+2$ and $t=\lfloor\frac{s(\nu-1)}{2}\rfloor+3$. Then we have

[TABLE]

Proof.

Clearly, $\operatorname{HF}_{\Omega^{2}_{R_{\nu\mathbb{X}}/K}}(i)=0$ for $i\leq 1$. By Proposition 2.4, we have an exact sequence of graded $R_{\nu\mathbb{X}}$-modules

[TABLE]

For $i\geq 2$, we get

[TABLE]

where $(*)$ follows from [KLL, Corollary 1.9(i)].

We consider the case $\nu=1$. We see that $\operatorname{HF}_{2\mathbb{X}}(3)=10$, so $\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}(3)=3\operatorname{HF}_{\mathbb{X}}(2)-10+1=3\operatorname{HF}_{\mathbb{X}}(2)-9$. For $i<s$ and $i\neq 3$, we have $\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}(i)=3\operatorname{HF}_{\mathbb{X}}(i-1)-(2i+1+\operatorname{HF}_{\mathbb{X}}(i-2)).$ For $i\geq s$ we get $\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K}}(i)=3\operatorname{HF}_{\mathbb{X}}(i-1)-\operatorname{HF}_{2\mathbb{X}}(i)=3s-3s=0$. Thus claim (a) follows.

Now we suppose $\nu\geq 2$. By Proposition 9.7 we have

[TABLE]

for $i\geq\lfloor\frac{s(\nu+1)}{2}\rfloor$. If $i<\lfloor\frac{s(\nu+1)}{2}\rfloor$, then $\operatorname{HF}_{(\nu+1)\mathbb{X}}(i)=2i+1+\operatorname{HF}_{\nu\mathbb{X}}(i-2)$ by [Cat, Theorem 3.1]. So, for $\mu\leq i<\lfloor\frac{s(\nu+1)}{2}\rfloor$, we have

[TABLE]

For $t\leq i<\mu$, we get

[TABLE]

For $i<t$, it follows from Proposition 9.7 again that

[TABLE]

Therefore claim (b) is completely proved. ∎

To end this section, we apply the preceding proposition to compute the Hilbert function of $\Omega^{2}_{R_{\nu\mathbb{X}}/K}$ in a concrete case.

Example 9.10**.**

Let $\mathbb{X}$ be the complete intersection given in Example 9.8. For $1\leq\nu\leq 3$, an application of Proposition 9.9 yields

[TABLE]

10. The Kähler Differential Algebra of $R_{\mathbb{X}}/K[x_{0}]$

Given a [math]-dimensional scheme $\mathbb{X}\subseteq\mathbb{P}^{n}$ such that no point in its support is contained in the hyperplane $\mathcal{Z}^{+}(X_{0})$, the element $x_{0}=X_{0}+I_{\mathbb{X}}$ is a non-zerodivisor of $R_{\mathbb{X}}$. Hence there is a short exact sequence of graded $R_{\mathbb{X}}$-modules

[TABLE]

(see [Kun, Proposition 3.24]). As noted in Section 2, the homogeneous $R_{\mathbb{X}}$-linear map $\gamma:\Omega^{1}_{R_{\mathbb{X}}/K}\rightarrow R_{\mathbb{X}}$ satisfies $\gamma(dx_{i})=x_{i}$ for all $i=0,\dots,n$. If $fdx_{0}=0$ for some homogeneous element $f\in R_{\mathbb{X}}$, then $\gamma(fdx_{0})=fx_{0}=0$, and so $f=0$, since $x_{0}$ is a non-zerodivisor of $R_{\mathbb{X}}$. Hence we have ${\rm Ann}_{R_{\mathbb{X}}}(dx_{0})=\langle 0\rangle$. Consequently, we obtain

[TABLE]

for all $i\in\mathbb{Z}$ and $\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K[x_{0}]})\leq\max\{r_{\mathbb{X}}+1,\operatorname{ri}(\Omega^{1}_{R_{\mathbb{X}}/K})\}$.

Thus the Hilbert functions of $\Omega^{1}_{R_{\mathbb{X}}/K}$ and $\Omega^{1}_{R_{\mathbb{X}}/K[x_{0}]}$ are strongly related, and it is straightforward to transfer our earlier results to results about $\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}$. For instance, for a [math]-dimensional subscheme of $\mathbb{P}^{1}$, an application of Proposition 3.1 yields the following property.

Proposition 10.1**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{1}$ be a [math]-dimensional scheme, and let $I_{\mathbb{X}}=\langle F\rangle$, where $F=\prod_{i=1}^{s}(X_{1}-a_{i}X_{0})^{m_{i}}$ for some $s$, $m_{1},\dots,m_{s}\geq 1$, and $a_{i}\in K$ with $a_{i}\neq a_{j}$ for $i\neq j$, and let $\mu=\sum_{i=1}^{s}m_{i}$. Then the Hilbert functions of the modules of Kähler differential $m$-forms of $R_{\mathbb{X}}/K[x_{0}]$ are given by

[TABLE]

and $\operatorname{HF}_{\Omega^{2}_{R_{\mathbb{X}}/K[x_{0}]}}(i)=0$ for all $i\in\mathbb{Z}$.

In general, for $1\!\leq m\leq\!n+1$, the module of Kähler differential $m$-forms $\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}$ has a presentation $\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}\!\cong\Omega^{m}_{S/K[x_{0}]}/\!(I_{\mathbb{X}}\Omega^{m}_{S/K[x_{0}]}+d_{S/K[x_{0}]}I_{\mathbb{X}}\Omega^{m-1}_{S/K[x_{0}]})$ (cf. ​[Kun, Proposition 4.12]). Moreover, according to [SS, X.83], the above exact sequence induces an exact sequence of graded $R_{\mathbb{X}}$-modules

[TABLE]

By using these results and by applying the methods given in Propositions 4.2, 5.3, 6.2 and 7.4, we get the following properties for the Hilbert function and regularity index of $\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}$. We leave the detailed proofs of these properties to the interested reader.

Proposition 10.2**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional scheme, and let $1\leq m\leq n+1$.

1. (a)

For $i<m$, we have $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}}(i)=0$. 2. (b)

For $m\leq i<\alpha_{\mathbb{X}}+m-1$, we have $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}}(i)=\binom{n}{m}\cdot\binom{n+i-m}{n}$. 3. (c)

The Hilbert polynomial of $\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}$ is constant. 4. (d)

We have $\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}}(r_{\mathbb{X}}+m)\geq\operatorname{HF}_{\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}}(r_{\mathbb{X}}+m+1)\geq\cdots$, and if $\operatorname{ri}(\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]})\geq r_{\mathbb{X}}+m$ then

[TABLE] 5. (e)

The regularity indices of $\Omega^{m}_{R_{\mathbb{X}}/K[x_{0}]}$ satisfies

[TABLE]

Finally, for non-reduced fat point schemes in $\mathbb{P}^{n}$, the Hilbert function and regularity index of $\Omega^{m}_{R_{{\mathbb{W}}}/K[x_{0}]}$ have the following properties.

Proposition 10.3**.**

Let ${\mathbb{W}}=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a fat point scheme in $\mathbb{P}^{n}$, and let $\mathbb{V}$ be the fattening of ${\mathbb{W}}$. Suppose that $m_{i}\geq 2$ for some $i\in\{1,\dots,s\}$, and let $1\leq m\leq n+1$.

1. (a)

The Hilbert polynomial of $\Omega^{m}_{R_{{\mathbb{W}}}/K[x_{0}]}$ is bounded by

[TABLE] 2. (b)

We have

[TABLE] 3. (c)

If $m_{1}\!\leq\!\cdots\leq m_{s}$ and $\operatorname{Supp}({\mathbb{W}})\!=\{P_{1},\!\dots,P_{s}\}$ is in general position, then

[TABLE]

Acknowledgment

This paper is partially based on the second author’s dissertation [Lin]. The authors are grateful to Lorenzo Robbiano for his encouragement to elaborate some of the results presented here. The first author thanks Hue University’s College of Education (Vietnam) for the hospitality he experienced during part of the preparation of this paper. The second author would also like to acknowledge her financial support from OeAD.