Algorithms for Checking Zero-Dimensional Complete Intersections

TL;DR

This paper presents effective algorithms for verifying various types of complete intersection properties in 0-dimensional affine algebras, utilizing Fitting ideals and border bases, applicable over any base field.

Contribution

It introduces new algorithms based on Fitting ideals and border bases to determine complete intersection properties in 0-dimensional affine algebras, extending applicability over arbitrary fields.

Findings

Algorithms successfully identify complete intersection properties.

Methods work over arbitrary base fields.

Detection of strict complete intersections in specific ideal families.

Abstract

Given a 0-dimensional affine K-algebra R=K[x_1,...,x_n]/I, where I is an ideal in a polynomial ring K[x_1,...,x_n] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether R is a complete intersection at a maximal ideal, whether R is locally a complete intersection, and whether R is a strict complete intersection. These algorithms are based on Wiebe's characterisation of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of I form a regular sequence resp. a strict regular sequence, and they work over an arbitrary base field K. Using degree filtered border bases, we can detect strict complete intersections in certain families of 0-dimensional ideals.

Equations

$$1={\mathrm{HF}}_{\mathbb{X}}^a(0)<{\mathrm{HF}}_{\mathbb{X}}^a(1)<\cdots <{\mathrm{HF}}_{\mathbb{X}}^a(\mathrm{ri}(R_{\mathbb{X}}))=\mu$$

$${\dim }_K({\mathrm{gr}}_{\mathcal{F}}(R_{\mathbb{X}})=\sum _{i=0}^{\infty }{\dim }_K(F_i{R}_{\mathbb{X}}/F_{i-1}{R}_{\mathbb{X}})={\dim }_K(R_{\mathbb{X}})$$

$${\mathrm{HF}}_{\mathbb{X}}^a(i)=\sum _{j=0}^i{\mathrm{HF}}_{{\mathrm{gr}}_{\mathcal{F}}(R_{\mathbb{X}})}(j)\mathrm{and}\Delta {\mathrm{HF}}_{\mathbb{X}}^a(i)={\mathrm{HF}}_{{\mathrm{gr}}_{\mathcal{F}}(R_{\mathbb{X}})}(i)$$

$$R_{\mathbb{X}}^{\mathrm{hom}}/⟨x_0-1⟩\cong R_{\mathbb{X}}\mathrm{and}{R}_{\mathbb{X}}^{\mathrm{hom}}/⟨x_0⟩\cong {\mathrm{gr}}_{\mathcal{F}}(R_{\mathbb{X}})$$

$$\left(\begin{array}{ccccccc}{a}_{1j_1}& \cdots & \cdots & \cdots & \cdots & \cdots & a_{n{j}_1}\\ ⋮& & & ⋮& & & ⋮\\ a_{1j_k}& \cdots & \cdots & \cdots & \cdots & \cdots & a_{n{j}_k}\\ 0& g_{{\lambda }^{\prime }}& \cdots & -g_{{\kappa }^{\prime }}& \cdots & \cdots & 0\\ ⋮& & & ⋮& & & ⋮\\ 0& \cdots & g_{\lambda }& \cdots & -g_{\kappa }& \cdots & 0\end{array}\right)\cdot \left(\begin{array}{c}{z}_1\\ ⋮\\ z_n\end{array}\right)=\left(\begin{array}{c}{f}_{j_1}\\ ⋮\\ f_{j_k}\\ 0\\ ⋮\\ 0\end{array}\right)$$

$$\det (\mathcal{M})\cdot g_{\lambda }=\det \left(\begin{array}{ccccccc}{a}_{1j_1}& \cdots & \cdots & \cdots & f_{j_1}& \cdots & a_{n{j}_1}\\ ⋮& & ⋮& & ⋮& & ⋮\\ a_{1j_k}& \cdots & \cdots & \cdots & f_{j_k}& \cdots & a_{n{j}_k}\\ 0& g_{{\lambda }^{\prime }}& \cdots & -g_{{\kappa }^{\prime }}& 0& \cdots & 0\\ ⋮& & ⋮& & ⋮& & ⋮\\ 0& \cdots & g_{\lambda }& \cdots & 0& \cdots & 0\end{array}\right)$$

$$W=\left(\begin{array}[]{rccc}\;0&x-z&0&0\\ {-}1&1&z&z^{2}+y-1\\ \;0&0&1&z\end{array}\right)$$

$$W=\left(\begin{array}[]{ccc}\;0&x^{2}-xy&xy^{2}\\ {-}y^{2}&0&0\end{array}\right)$$

$$\begin{array}[]{l}\{y^{2}-xz,\ xy-z,\ x^{2}-y,\;\vskip 3.0pt plus 1.0pt minus 1.0pt\\ yz^{2}-{\textstyle\frac{2}{15}}z^{3}+49xz+{\textstyle\frac{98}{5}}yz-14z^{2}+{\textstyle\frac{336}{5}}x-36y-{\textstyle\frac{260}{3}}z,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ xz^{2}-{\textstyle\frac{1}{30}}z^{3}-{\textstyle\frac{91}{10}}yz-{\textstyle\frac{96}{5}}x+{\textstyle\frac{82}{3}}z,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ z^{4}-{\textstyle\frac{418}{5}}z^{3}+6699xz+{\textstyle\frac{61446}{5}}yz-\!1408z^{2}+{\textstyle\frac{210672}{5}}x-\!5292y-\!54340z\}\end{array}$$

$$W=\left(\begin{array}[]{rcccccc}-z&y&x&0&0&0\\ y&0&0&0&0&0\\ 0&0&0&yz-{\textstyle\frac{2}{15}}z^{2}&xz-{\textstyle\frac{1}{30}}z^{2}&z^{3}\end{array}\right)$$

$$G=(y^5-2y^4+y^3-xy-2y^2,x^3-2xy-x-1,x{y}^3-2x{y}^2-y,x^{2}y-y^3-y)$$

$$W=\left(\begin{array}[]{lllc}0&x^{2}&y^{3}&0\\ y^{4}&0&0&x^{2}-y^{2}\end{array}\right)$$

$$\begin{array}[]{cl}f_{1}&\!\!=(-2xy+1)\,g_{2}+(2x^{2}+2y^{2}-4y)\,g_{4}\\ f_{2}&\!\!=(-xy)\,g_{2}+(x^{2}+y^{2}-2y)\,g_{4}\\ f_{3}&\!\!=(xy^{3}-2xy^{2}+y)\,g_{2}+(-x^{2}y^{2}-y^{4}+2x^{2}y+4y^{3}-4y^{2}-x)\,g_{4}\end{array}$$

$$\mathrm{LF}(b_i)=\mathrm{DF}(f_i)+\mathrm{DF}(I_{\mathbb{X}})=f_i^{\mathrm{hom}}(0,x_1,\dots ,x_n)+\mathrm{DF}(I_{\mathbb{X}})$$

$$f_1$$

$$f_2$$

$$f_3$$

$$f_4$$

$$(-c_{41}y)x+(y)y,(x)x+(-c_{42}x)y,(y^2)x+(0)y,(xy)x+(0)y$$

$$W=\left(\begin{array}[]{cccc}-c_{41}y&x&y^{2}&xy\\ y&c_{42}x&0&0\end{array}\right)$$

$${\Omega }_{R_{\mathbb{X}}/K}^1\cong {⨁}_{i=1}^{n}Pd{x}_i/\left(I_{\mathbb{X}}\cdot {⨁}_{i=1}^{n}Pd{x}_i+⟨{\sum }_{i=1}^n\partial f/\partial x_{i}d{x}_i\mid f\in I_{\mathbb{X}}⟩\right)$$

Full text

1. Introduction

Regular sequences and complete intersections play a fundamental role in Commutative Algebra and Algebraic Geometry. Given an ideal $I$ in a polynomial ring $P=K[x_{1},\dots,x_{n}]$ over a field $K$, e.g., the vanishing ideal of an affine or projective scheme, it is therefore an important algorithmic task to check whether $I$ is a complete intersection ideal, that is, whether $I$ can be generated by ${\rm ht}(I)$ polynomials. Equivalently, we call $R=P/I$ a complete intersection ring in this case. Several approaches have been developed to tackle this problem effectively for special classes of ideals.

For instance, if the ring $R=P/I$ is local, based on a description of the structure of the $R$-algebra ${\rm Tor}^{R}(K,K)$ in [18], characterizations of the complete intersection property using the Hilbert series of ${\rm Tor}^{R}(K,K)$ were developed in [3] and [7]. However, this approach requires the calculation of the Hilbert series of a non-commutative algebra. Furthermore, in the local setting, complete intersections were characterized in [19] using the freeness of the conormal module $I/I^{2}$ and the finiteness of the projective dimension of $I$. Again, these conditions are not easy to check algorithmically. In a similar vein, if the base field $K$ has characteristic zero, one can use techniques based on the Kähler differential module $\Omega^{1}_{R/K}$, as in [4], or on Kähler differents, as in [9], but they are neither general enough for our purposes nor do they lend themselves to a generalization for families of ideals. Finally, let us mention that effective algorithms have been developed in [5] for checking the complete intersection property of toric ideals defining affine semigroup rings.

In this paper we are interested in algorithms for checking the locally complete intersection property and the strict complete intersection property for a 0-dimensional ideal $I$ in $P=K[x_{1},\dots,x_{n}]$, where $K$ is an arbitrary field. In other words, we want to check these properties for a 0-dimensional affine $K$-algebra of the form $P/I$. In the language of Algebraic Geometry, the scheme ${\mathbb{X}}=\mathop{\rm Spec}\nolimits(P/I)$ is then a 0-dimensional affine scheme, and $R=P/I$ is its affine coordinate ring. Notice that every 0-dimensional projective scheme can be embedded into a basic affine open subset of $\mathbb{P}^{n}$ after possibly extending the field $K$ slightly. Then we can readily move between the languages of affine and projective geometry by using homogenization and dehomogenization. For reasons which will become clear shortly, we prefer the affine setting in this paper. In any case, 0-dimensional affine $K$-algebras $R=P/I$ have been used in many areas besides Algebraic Geometry, for instance in algebraic statistics, algebraic biology, etc.

Our main algorithms check whether a 0-dimensional affine $K$-algebra $R=P/I$ is locally a complete intersection or a strict complete intersection. They are based on a characterization of local complete intersections by H. Wiebe in [20], where it is shown that a 0-dimensional local ring is a complete intersection if and only if the 0-th Fitting ideal of its maximal ideal is non-zero. Given a 0-dimensional affine $K$-algebra $R=P/I$ whose defining ideal $I$ has a primary decomposition $I={\mathfrak{Q}}_{1}\cap\cdots\cap{\mathfrak{Q}}_{s}$, this characterization can be applied right off the bat, because the localizations are of the form $P/{\mathfrak{Q}}_{i}$, and hence again 0-dimensional local affine $K$-algebras. For checking the strict complete intersection property, we use the graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R)$ with respect to the degree filtration ${\mathcal{F}}$ instead. It has a presentation ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R)=P/\mathop{\rm DF}\nolimits(I)$, where the degree form ideal $\mathop{\rm DF}\nolimits(I)$ is homogeneous and 0-dimensional (see [13], Def. 4.2.13).

Now let us describe the contents of this paper in detail. In Section 2 we start by recalling some basic properties of a [math]-dimensional scheme ${\mathbb{X}}$ embedded in an affine space $\mathbb{A}^{n}_{K}$, where $K$ is an arbitrary field. In particular, besides recalling the degree filtration and the affine Hilbert function of the affine coordinate ring $R_{\mathbb{X}}=P/I_{\mathbb{X}}$ of ${\mathbb{X}}$, where $I_{\mathbb{X}}$ is the vanishing ideal of ${\mathbb{X}}$ in $P=K[x_{1},\dots,x_{n}]$ and $K$ is a field, we recall the degree form ideal $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$, the associated graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$, and mention their connection to the Rees algebra $\mathcal{R}_{\mathcal{F}}(R_{\mathbb{X}})$ and to Macaulay bases of $I_{\mathbb{X}}$.

Then the main part of the paper starts in Section 3. After defining what we mean by a complete intersection at a maximal ideal and by the property of being locallly a complete intersection, we recall Wiebe’s characterization mentioned above (see [20], Satz 3). The central tool for computing the 0-th Fitting ideal needed in this characterization is given in Proposition 3.3. For a maximal ideal $\mathfrak{M}$ of $P$ and an $\mathfrak{M}$-primary ideal ${\mathfrak{Q}}$, it suffices to write the generators of ${\mathfrak{Q}}$ in terms of a regular sequence generating $\mathfrak{M}$ and to compute the maximal minors of the coefficient matrix. As an immediate consequence, we obtain Algorithm 3.4 for checking if a 0-dimensional affine $K$-algebra $R_{\mathbb{X}}=P/I_{\mathbb{X}}$ is a complete intersection at a maximal ideal, and by combining this with the computation of a primary decomposition of $I_{\mathbb{X}}$, we get an algorithm for checking whether $R_{\mathbb{X}}$ is locally a complete intersection. Moreover, the non-zero minors provide us with regular sequences generating ${\mathfrak{Q}}\,P_{\mathfrak{M}}$, as Proposition 3.7 shows. Suitable examples at the end of the section illustrate the merits and some hidden features of these algorithms.

In Section 4 further notions are recalled. More precisely, strict regular sequences, strict Gorenstein rings, and strict complete intersections enter the game. In particular, the ideal $I_{\mathbb{X}}$ is called a strict complete intersection if its degree form ideal $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ is generated by a homogeneous regular sequence. Since the graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})=P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ is 0-dimensional and local, we can use Proposition 3.3 and Wiebe’s result again to construct Algorithm 4.4 for checking the strict complete intersection property. Moreover, we also get a description of which generators of $I_{\mathbb{X}}$ form a strict regular sequence (see Corollary 4.5) and illustrate everything via some explicit examples.

In the last section of the paper, we present a second algorithm for checking the strict complete intersection property via border bases which generalizes to certain families of 0-dimensional ideals. Based on the notion of a degree filtered $K$-basis of $R_{\mathbb{X}}$, we define degree filtered ${\mathcal{O}}$-border bases. They have several nice characterizations, the most useful one here being the property that the degree forms of the border basis polynomials form an ${\mathcal{O}}$-border basis of the degree form ideal (see Proposition 5.3). Then we obtain Algorithm 5.4 which checks for the strict complete intersection property using a degree filtered border basis. This version allows us to detect all strict complete intersections within certain families of 0-dimensional ideals, as illustrated by Example 5.5 and applied further in [11]. Finally, in Remark 5.6, we compare the algorithms of this paper with the methods based on Jacobian matrices, Kähler differentials, and Kähler differents mentioned above.

All examples in this paper were computed using the computer algebra system CoCoA (see [2]). Unless explicitly stated otherwise, we adhere to the notation and definitions provided in [12], [13], and [14].

2. Zero-Dimensional Affine Schemes

In the following we always work over an arbitrary field $K$ and let $\mathbb{A}^{n}_{K}$ be the affine $n$-space over $K$. We fix a coordinate system, so that the affine coordinate ring of $\mathbb{A}^{n}_{K}$ is given by the polynomial ring $P=K[x_{1},\dots,x_{n}]$. Thus a 0-dimensional subscheme ${\mathbb{X}}$ of $\mathbb{A}^{n}_{K}$ is defined by a 0-dimensional ideal $I_{\mathbb{X}}$ in $P\!$, and its affine coordinate ring is $R_{\mathbb{X}}=P/I_{\mathbb{X}}$. Consequently, the vector space dimension $\mu=\dim_{K}(R_{\mathbb{X}})$ is finite and equal to the length of the scheme ${\mathbb{X}}$.

Since we are keeping the coordinate system fixed at all times, we have further invariants of ${\mathbb{X}}$. Recall that the degree filtration $\widetilde{\mathcal{F}}=(F_{i}P)_{i\in\mathbb{Z}}$ on $P$ is given by $F_{i}P=\{f\in P\setminus\{0\}\mid\deg(f)\leq i\}\cup\{0\}$ for all $i\in\mathbb{Z}$. Then the induced filtration ${\mathcal{F}}=(F_{i}R_{\mathbb{X}})_{i\in\mathbb{Z}}$, where $F_{i}R_{\mathbb{X}}=F_{i}P/(F_{i}P\cap I_{\mathbb{X}})$, is called the degree filtration on $R_{\mathbb{X}}$. It is easy to see that the degree filtration on $R_{\mathbb{X}}$ is increasing, exhaustive and orderly in the sense that every element $f\in R_{\mathbb{X}}\setminus\{0\}$ has an order $\mathop{\rm ord}\nolimits_{\mathcal{F}}(f)=\min\{i\in\mathbb{Z}\mid f\in F_{i}R_{\mathbb{X}}\setminus F_{i-1}R_{\mathbb{X}}\}$.

Definition 2.1**.**

Let ${\mathbb{X}}$ be a 0-dimensional subscheme of $\mathbb{A}^{n}_{K}$ as above.

1. (a)

The map $\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}:\mathbb{Z}\longrightarrow\mathbb{Z}$ given by $i\mapsto\dim_{K}(F_{i}R_{\mathbb{X}})$ is called the affine Hilbert function of ${\mathbb{X}}$. 2. (b)

The number $\mathop{\rm ri}\nolimits(R_{\mathbb{X}})=\min\{i\in\mathbb{Z}\mid\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(j)=\mu$ for all $j\geq i\}$ is called the regularity index of ${\mathbb{X}}$. 3. (c)

The first difference function $\Delta\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(i)=\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(i)-\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(i-1)$ of $\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}$ is called the Castelnuovo function of ${\mathbb{X}}$, and the number $\Delta_{\mathbb{X}}=\Delta\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(\mathop{\rm ri}\nolimits(R_{\mathbb{X}}))$ is the last difference of ${\mathbb{X}}$.

It is well-known that $\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}$ satisfies $\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(i)=0$ for $i<0$ and

[TABLE]

as well as $\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(i)=\mu$ for $i\geq\mathop{\rm ri}\nolimits(R_{\mathbb{X}})$. The affine Hilbert function of ${\mathbb{X}}$ is related to the following objects.

Definition 2.2**.**

Let ${\mathbb{X}}$ be a 0-dimensional subscheme of $\mathbb{A}^{n}_{K}$ as above.

1. (a)

For every polynomial $f\in P\setminus\{0\}$, its homogeneous component of highest degree is called the degree form of $f$ and is denoted by $\mathop{\rm DF}\nolimits(f)$. 2. (b)

The ideal $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})=\langle\mathop{\rm DF}\nolimits(f)\mid f\in I_{\mathbb{X}}\setminus\{0\}\rangle$ is called the degree form ideal of $I_{\mathbb{X}}$. 3. (c)

The ring $\mathop{\rm gr}\nolimits_{\widetilde{{\mathcal{F}}}}(P)=\bigoplus_{i\in\mathbb{Z}}F_{i}P/F_{i-1}P$ is called the associated graded ring of $P$ with respect to $\widetilde{{\mathcal{F}}}$. 4. (d)

The ring $\mathop{\rm gr}\nolimits_{\mathcal{F}}(R_{\mathbb{X}})=\bigoplus_{i\in\mathbb{Z}}F_{i}R_{\mathbb{X}}/F_{i-1}R_{\mathbb{X}}$ is called the associated graded ring of $R_{\mathbb{X}}$ with respect to ${\mathcal{F}}$. 5. (e)

For an element $f\in R_{\mathbb{X}}\setminus\{0\}$ of order $d=\mathop{\rm ord}\nolimits_{\mathcal{F}}(f)$, the residue class $\mathop{\rm LF}\nolimits(f)=f+F_{d-1}R_{\mathbb{X}}$ in ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$ is called the leading form of $f$ with respect to ${\mathcal{F}}$.

We observe that in our setting the associated graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$ is a 0-dimensional local ring whose maximal ideal is generated by the residue classes of the indeterminates. Its $K$-vector space dimension is given by

[TABLE]

For actual computations involving the associated graded ring and leading forms, we can use the following observations.

Remark 2.3**.**

Notice that there is a canonical isomorphism of graded $K$-algebras $\mathop{\rm gr}\nolimits_{\widetilde{{\mathcal{F}}}}(P)\cong P$ which allows us to identify ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})\cong P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ (see [13], Props. 6.5.8 and 6.5.9).

In order to represent the leading form of a non-zero element $f\in R_{\mathbb{X}}$ as a residue class in $P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$, we first have to represent $f$ by a polynomial $F\in P$ with $\deg(F)=\mathop{\rm ord}\nolimits_{\mathcal{F}}(f)$. This can, for instance, be achieved by taking any representative $\widetilde{F}\in P$ of $f$ and computing the normal form $F=\mathop{\rm NF}\nolimits_{\sigma,I_{\mathbb{X}}}(\widetilde{F})$ with respect to a degree compatible term ordering $\sigma$. Then the degree form $\mathop{\rm DF}\nolimits(F)$ represents the leading form $\mathop{\rm LF}\nolimits_{\mathcal{F}}(f)$ with respect to the isomorphism ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})\cong P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$.

The affine Hilbert function of ${\mathbb{X}}$ and the (usual) Hilbert function of ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$ are connected by

[TABLE]

for all $i\geq 0$. Since the Hilbert function of ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$ can be calculated using a suitable Gröbner basis of $I_{\mathbb{X}}$ (see [13], Section 4.3), this formula allows us to compute the affine Hilbert function of ${\mathbb{X}}$.

The ring $R_{\mathbb{X}}$ and its graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$ are connected by the following flat family. (For further details and proofs, see [13], Section 4.3.B.)

Remark 2.4**.**

Let $x_{0}$ be a further indeterminate.

1. (a)

The ring ${\mathcal{R}_{\widetilde{{\mathcal{F}}}}(P)=\bigoplus_{i\in\mathbb{Z}}F_{i}P\cdot x_{0}^{i}}_{\mathstrut}$ is called the Rees algebra of $P$ with respect to ${\widetilde{{\mathcal{F}}}}$. Every non-zero homogeneous element of the Rees algebra is of the form $f\cdot x_{0}^{d+j}$ with $d=\deg(f)$ and $j\geq 0$. By identifying it with the polynomial $f^{\rm hom}\cdot x_{0}^{j}$, we obtain an isomorphism of graded $K[x_{0}]$-algebras $\mathcal{R}_{\widetilde{{\mathcal{F}}}}(P)\cong K[x_{0},x_{1},\dots,x_{n}]=:\overline{P}$. Here $f^{\rm hom}$ is the usual homogenization $f^{\rm hom}=x_{0}^{\deg(f)}\cdot f({\textstyle\frac{x_{1}}{x_{0}}},\dots,{\textstyle\frac{x_{n}}{x_{0}}})$. 2. (b)

Similarly, using the degree filtration ${\mathcal{F}}$ on $R_{\mathbb{X}}$, we have the Rees algebra $\mathcal{R}_{\mathcal{F}}(R_{\mathbb{X}})=\bigoplus_{i\in\mathbb{Z}}F_{i}R_{\mathbb{X}}\cdot x_{0}^{i}$. Now we let $I_{\mathbb{X}}^{\rm hom}=\langle f^{\rm hom}\mid f\in I_{\mathbb{X}}\setminus\{0\}\rangle$ be the homogenization of $I_{\mathbb{X}}$. Then the above isomorphism induces an isomorphism of graded $K[x_{0}]$-algebras $\mathcal{R}_{\mathcal{F}}(R_{\mathbb{X}})\cong\overline{P}/I_{\mathbb{X}}^{\rm hom}=:R_{\mathbb{X}}^{\rm hom}$.

Geometrically speaking, the ring $R_{\mathbb{X}}^{\rm hom}$ is the homogeneous coordinate ring of the 0-dimensional scheme obtained by embedding ${\mathbb{X}}=\mathop{\rm Spec}\nolimits(P/I_{\mathbb{X}})\subset\mathbb{A}^{n}_{K}$ into the projective $n$-space via $\mathbb{A}^{n}_{K}\cong D_{+}(x_{0})\subset\mathbb{P}^{n}$. 3. (c)

The ring $R_{\mathbb{X}}^{\rm hom}$ is a free (and hence flat) $K[x_{0}]$-module. The elements $x_{0}$ and $x_{0}-1$ are non-zerodivisors for $R_{\mathbb{X}}^{\rm hom}$, and we have canonical isomorphisms

[TABLE]

In other words, the algebra homomorphism $K[x_{0}]\longrightarrow R_{\mathbb{X}}^{\rm hom}$ defines a flat family whose special fiber is ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$ and whose general fiber is $R_{\mathbb{X}}$. 4. (d)

A set $\{f_{1},\dots,f_{s}\}\subseteq I_{\mathbb{X}}$ with $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})=\langle\mathop{\rm DF}\nolimits(f_{1}),\dots,\mathop{\rm DF}\nolimits(f_{s})\rangle$ is called a Macaulay basis of $I_{\mathbb{X}}$. In this case the polynomials generate $I_{\mathbb{X}}$ and we have $I_{\mathbb{X}}^{\rm hom}=\langle f_{1}^{\rm hom},\dots,f_{s}^{\rm hom}\rangle$ (see [13], Thm. 4.3.19).

3. Locally Complete Intersections

In this section we analyse a property of a 0-dimensional subscheme ${\mathbb{X}}$ of $\mathbb{A}^{n}_{K}$, namely the property of being locally a complete intersection. For this purpose, we continue to use the definitions and notation introduced in the preceding section. Recall that a 0-dimensional local ring of the form $P_{\mathfrak{M}}/I$ with a field $K$, a maximal ideal $\mathfrak{M}$, and a 0-dimensional ideal $I$, is called a complete intersection if $I$ can be generated by a regular sequence of length $n$ in $P_{\mathfrak{M}}$. In our setting, the following versions of this notion will be considered.

Definition 3.1**.**

Let ${\mathbb{X}}$ be a 0-dimensional subscheme of $\mathbb{A}^{n}_{K}$ as above, let $I_{\mathbb{X}}={\mathfrak{Q}}_{1}\cap\cdots\cap{\mathfrak{Q}}_{s}$ be the primary decomposition of $I_{\mathbb{X}}$, and let $\mathfrak{m}_{1},\dots,\mathfrak{m}_{s}$ be the maximal ideals of $R_{\mathbb{X}}$.

1. (a)

The scheme ${\mathbb{X}}$ is called a complete intersection at $\mathfrak{m}_{i}$ if the local ring $(R_{\mathbb{X}})_{\mathfrak{m}_{i}}\cong P/{\mathfrak{Q}}_{i}$ is a complete intersection. 2. (b)

We say that ${\mathbb{X}}$ is a locally complete intersection scheme if it is a complete intersection at $\mathfrak{m}_{i}$ for each $i\in\{1,\dots,s\}$.

In the following we are looking for characterizations of 0-dimensional locally complete intersection schemes which can be checked algorithmically. Our approach is inspired by the following result of H. Wiebe (cf. [20], Satz 3). Here the $i$-th Fitting ideal of a module $M$ is denoted by $\mathop{\rm Fitt}\nolimits_{i}(M)$. (For a definition and basic properties of Fitting ideals, see [16], Appendix D.)

Proposition 3.2**.**

Let $R$ be a local ring with maximal ideal $\mathfrak{m}$. Then the following conditions are equivalent.

1. (a)

The ring $R$ is a 0-dimensional complete intersection. 2. (b)

We have $\mathop{\rm Fitt}\nolimits_{0}(\mathfrak{m})\neq\langle 0\rangle$.

Now we turn this characterization into an algorithm by specifying a way to compute the 0-th Fitting ideal of the maximal ideal in the setting of this paper. The next proposition provides the central tool for this algorithm. Recall that a maximal ideal in the polynomial ring can always be generated by a regular sequence of length $n$ (see [14], Cor. 5.3.14).

Proposition 3.3**.**

Let $P=K[x_{1},\dots,x_{n}]$, let $\mathfrak{M}$ be a maximal ideal of $P$, and let $\{g_{1},\dots,g_{n}\}$ be a regular sequence which generates $\mathfrak{M}$. Then let ${\mathfrak{Q}}\subset P$ be an $\mathfrak{M}$-primary ideal, let $\{f_{1},\dots,f_{r}\}$ be a system of generators of ${\mathfrak{Q}}$, let $R=P/{\mathfrak{Q}}$, and let $\mathfrak{m}=\mathfrak{M}/{\mathfrak{Q}}$. For $j=1,\dots,r$, write $f_{j}=\sum_{i=1}^{n}a_{ij}g_{i}$, and form the matrix $W\in\mathop{\rm Mat}\nolimits_{n,r}(P)$ of size $n\times r$ whose columns are given by $\sum_{i=1}^{n}a_{ij}e_{i}$ for $j=1,\dots,r$. Then the 0-th Fitting ideal $\mathop{\rm Fitt}\nolimits_{0}(\mathfrak{m})$ is generated by the residue classes in $R$ of the minors of order $n$ of the matrix $W$.

Proof.

Notice that we have $r\geq n$, since the height of ${\mathfrak{Q}}$ is $n$. A presentation of the $P$-module $\mathfrak{m}$ is given by $\varepsilon:\;P^{n}\longrightarrow\mathfrak{m}$ with $\varepsilon(e_{i})=\bar{g}_{i}$ for $i=1,\dots,n$. Let us first show that the kernel of $\varepsilon$ is generated by the elements in the set

[TABLE]

It is clear that all these elements are contained in $\mathop{\rm Ker}\nolimits(\varepsilon)$. Conversely, let $\sum_{i=1}^{n}b_{i}e_{i}\in\mathop{\rm Ker}\nolimits(\varepsilon)$. Then $\sum_{i=1}^{n}b_{i}g_{i}\in{\mathfrak{Q}}$ implies that there exist polynomials $c_{k}\in P$ such that $\sum_{i=1}^{n}b_{i}g_{i}=\sum_{k=1}^{r}c_{k}f_{k}$. Then we have the equality $\sum_{i=1}^{n}(b_{i}-\sum_{k=1}^{r}c_{k}a_{ik})\,g_{i}=0$, and the fact that the syzygy module of a regular sequence is generated by the trivial syzygies implies that the element $\sum_{i=1}^{n}(b_{i}-\sum_{k=1}^{r}c_{k}a_{ik})\,e_{i}$ is contained in the module generated by the first set in $(\ast)$. Therefore $\sum_{i=1}^{n}b_{i}e_{i}$ is in the module generated by the two sets in $(\ast)$.

It follows that the ideal $\mathop{\rm Fitt}\nolimits_{0}(\mathfrak{m})$ is generated by the residue classes in $R=P/{\mathfrak{Q}}$ of the minors of order $n$ of the matrix $V$ whose columns are given by the tuples in $(\ast)$. It remains to show that all minors of $V$ which involve a column corresponding to a trivial syzygy belong to ${\mathfrak{Q}}$.

For this, we may assume that there exist a number ${k\in\{1,\dots,n-1\}}$ and indices $1\leq j_{1}<\cdots<j_{k}\leq n$ such that the columns of type $(a_{1j},\dots,a_{nj})^{\,\rm tr}$ involved in the minor are exactly the columns $j_{1},\dots,j_{k}$, and the remaining columns correspond to trivial syzygies. W.l.o.g. let the last column correspond to the trivial syzygy $g_{\lambda}e_{\kappa}-g_{\kappa}e_{\lambda}$ with $1\leq\kappa<\lambda\leq n$. Then the tuple $(g_{1},\dots,g_{n})$ is a solution of the linear system

[TABLE]

where the rows of the left-hand side matrix are the columns of $V$ corresponding to the chosen minor. If the determinant of the matrix $\mathcal{M}$ on the left-hand side is zero, there is nothing to show. Hence we may assume that it is non-zero, and since $P$ is an integral domain, the system has a unique solution $(g_{1},\dots,g_{n})$ given by Cramer’s Rule. In particular, we have

[TABLE]

Developing the determinant on the right-hand side first via the last row and then via the $\lambda$-th column yields $\det(\mathcal{M})\cdot g_{\lambda}\in g_{\lambda}\cdot\langle f_{j_{1}},\dots,f_{j_{k}}\rangle$. Since $g_{\lambda}$ is a non-zerodivisor in $P$, we can cancel it. Then we obtain $\det(\mathcal{M})\in\langle f_{j_{1}},\dots,f_{j_{k}}\rangle\subseteq{\mathfrak{Q}}$, and this finishes the proof. ∎

By applying this proposition to a local ring $P/{\mathfrak{Q}}$, we can formulate an algorithm to check whether $P/{\mathfrak{Q}}$ is a complete intersection.

Algorithm 3.4**.**

**(Checking a Complete Intersection at $\mathfrak{m}$)

**Let $P=K[x_{1},\dots,x_{n}]$, let $\mathfrak{M}$ be a maximal ideal of $P$, and let $\{g_{1},\dots,g_{n}\}$ be a regular sequence which generates $\mathfrak{M}$. Then let ${\mathfrak{Q}}\subset P$ be an $\mathfrak{M}$-primary ideal, and let $\{f_{1},\dots,f_{r}\}$ be a system of generators of ${\mathfrak{Q}}$. The following instructions define an algorithm which checks whether ${\mathfrak{Q}}P_{\mathfrak{M}}$ is generated by a regular sequence and returns the corresponding Boolean value.

1. (1)

For $j=1,\dots,r$, write $f_{j}=\sum_{k=1}^{n}a_{kj}g_{k}$. 2. (2)

Form the matrix $W$ of size $n\times r$ whose columns are given by $\sum_{k=1}^{n}a_{kj}e_{k}$. 3. (3)

Calculate the tuple of residue classes in $P/{\mathfrak{Q}}$ of the minors of order $n$ of $W.$ If the result is different from $(0,\dots,0)$, return TRUE. Otherwise, return FALSE and stop.

Proof.

In view of Proposition 3.2, it suffices to show that Steps (2) and (3) calculate the ideal $\mathop{\rm Fitt}\nolimits_{0}(\mathfrak{M}/{\mathfrak{Q}})$. This follows from Proposition 3.3. ∎

Remark 3.5**.**

For checking if an ideal in $P$ is maximal, and if this is the case, to compute a regular sequence which generates it, we refer the reader to the algorithms in [14], Ch. 5 and [1].

A solution for the task of checking whether a [math]-dimensional scheme is locally a complete intersection scheme follows easily.

Algorithm 3.6**.**

**(Checking Locally Complete Intersections)

**Let ${\mathbb{X}}$ be a 0-dimensional scheme in $\mathbb{A}^{n}_{K}$, and let $P/I_{\mathbb{X}}$ be the affine coordinate ring of ${\mathbb{X}}$. The following instructions define an algorithm which checks whether ${\mathbb{X}}$ is locally a complete intersection and returns the corresponding Boolean value.

Compute the primary decomposition $I_{\mathbb{X}}={\mathfrak{Q}}_{1}\cap\cdots\cap{\mathfrak{Q}}_{s}$ of the ideal $I_{\mathbb{X}}$.

For $i=1,\dots,s$, check whether $P/{\mathfrak{Q}}_{i}$ is a complete intersection ring using Algorithm 3.4. If the answer is TRUE for every $i\in\{1,\dots s\}$, return TRUE. Otherwise, return FALSE.

If the result of Algorithm 3.4 is TRUE, then a more careful examination of Step (4) produces the following extra information.

Proposition 3.7**.**

In the setting of Algorithm 3.4, the following conditions are equivalent.

1. (a)

The residue class in $P/{\mathfrak{Q}}$ of the minor involving columns $(j_{1},\dots,j_{n})$ of the matrix $W$ is non-zero. 2. (b)

The polynomials $f_{j_{1}},\dots,f_{j_{n}}$ form a regular sequence in $P_{\mathfrak{M}}$ which generates ${\mathfrak{Q}}P_{\mathfrak{M}}$.

Moreover, if $f_{j_{1}},\dots,f_{j_{n}}$ satisfy the equivalent conditions above, and if we have $\mathop{\rm Rad}\nolimits(\langle f_{j_{1}},\dots,f_{j_{n}}\rangle)=\mathfrak{M}$, then $\langle f_{j_{1}},\dots,f_{j_{n}}\rangle={\mathfrak{Q}}$.

Proof.

Since (b)$\Rightarrow$(a) follows immediately from Algorithm 3.4, let us prove that (a)$\Rightarrow$(b). First, we show that ${\mathfrak{Q}}P_{\mathfrak{M}}$ is generated by $\{f_{j_{1}},\dots,f_{j_{n}}\}$. By Algorithm 3.4, we know that there exist polynomials $h_{1},\dots,h_{n}\in P$ such that $(h_{1},\dots,h_{n})$ is a regular sequence in $P_{\mathfrak{M}}$ which generates ${\mathfrak{Q}}P_{\mathfrak{M}}$. For $k=1,\dots,n$, we write $h_{k}=\sum_{\ell=1}^{n}b_{\ell k}g_{\ell}$ and $f_{j_{k}}=\sum_{\ell=1}^{n}{\textstyle\frac{c_{\ell k}}{a_{k}}}h_{\ell}$ with some polynomials $a_{k},b_{\ell k},c_{\ell k}\in P$ such that $a_{k}\notin\mathfrak{M}$. Then $\det(\bar{b}_{\ell k})$ is a non-zero element of the ring $P/{\mathfrak{Q}}$ which generates the socle of this ring (cf. [20], Satz 3). Moreover, we have the equality $f_{j_{k}}=\sum_{m=1}^{n}(\sum_{\ell=1}^{n}b_{m\ell}{\textstyle\frac{c_{\ell k}}{a_{k}}})\,g_{m}$. By the hypothesis, also the residue class of the element $\det((\sum_{\ell=1}^{n}b_{m\ell}{\textstyle\frac{c_{\ell k}}{a_{k}}}^{\mathstrut})_{m,k})=\det(b_{\ell k})\cdot\det({\textstyle\frac{c_{\ell k}}{a_{k}}})$ is non-zero in $P/{\mathfrak{Q}}$. Since we know that $\det(\bar{b}_{\ell k})$ generates the socle of $P/{\mathfrak{Q}}$, the element $\det({\textstyle\frac{\bar{c}_{\ell k}}{\bar{a}_{k}}}^{\mathstrut})$ is invertible in this ring. Hence we have $\det({\textstyle\frac{c_{\ell k}}{a_{k}}})\notin\mathfrak{M}P_{\mathfrak{M}}$, and thus the matrix $({\textstyle\frac{c_{\ell k}}{a_{k}}})_{\ell,k}$ is invertible over $P_{\mathfrak{M}}$. Consequently, also $\{f_{j_{1}},\dots,f_{j_{n}}\}$ generates ${\mathfrak{Q}}P_{\mathfrak{M}}$, and the fact that $(f_{j_{1}},\dots,f_{j_{n}})$ is a regular sequence in $P_{\mathfrak{M}}$ follows from [6], Theorem 2.1.2.

To prove the last claim, we let ${\mathfrak{Q}}^{\prime}=\langle f_{j_{1}},\dots,f_{j_{n}}\rangle$. If the two equivalent conditions are satisfied, we have ${\mathfrak{Q}}^{\prime}P_{\mathfrak{M}}={\mathfrak{Q}}P_{\mathfrak{M}}$. Then $\mathop{\rm Rad}\nolimits(\langle f_{j_{1}},\dots,f_{j_{n}}\rangle){=\mathfrak{M}}$ means that ${\mathfrak{Q}}^{\prime}$ is an $\mathfrak{M}$-primary ideal, and we conclude that ${\mathfrak{Q}}^{\prime}={\mathfrak{Q}}$. ∎

Let us see some examples which illustrate Algorithm 3.4 and Proposition 3.7.

Example 3.8**.**

Let $P=\mathbb{Q}[x]$, $\mathfrak{M}=\langle x\rangle$, and ${\mathfrak{Q}}=\langle x(x-1),\;x(x-2)\rangle$. Let us apply Algorithm 3.4 to check whether the ring $R=P/{\mathfrak{Q}}$ is a local complete intersection at $\mathfrak{M}$.

The residue classes modulo ${\mathfrak{Q}}$ of the $1$-minors of $W$ are $-1$ and $-2$. Hence we deduce that ${\mathfrak{Q}}P_{\mathfrak{M}}$ is generated by a regular sequence. Since ${\mathfrak{Q}}=\mathfrak{M}$, it is clear that both $x(x-1)$ and $x(x-2)$ generate ${\mathfrak{Q}}P_{\mathfrak{M}}$. However, neither of these polynomials generates ${\mathfrak{Q}}$. Moreover, observe that $\langle x(x-1)\rangle={\mathfrak{Q}}\cap\langle x-1\rangle$ and $\langle x(x-2)\rangle={\mathfrak{Q}}\cap\langle x-2\rangle$.

Example 3.9**.**

Let $K=\mathbb{Q}$, let $P=K[x,y,z]$, and let ${\mathbb{X}}$ be the 0-dimensional subscheme of $\mathbb{A}^{2}_{K}$ defined by the ideal $I_{\mathbb{X}}=\langle f_{1},\dots,f_{4}\rangle$, where $f_{1}=z^{2}-y$, $f_{2}=x^{2}-2xz+y$, $f_{3}=yz-z-1$, and $f_{4}=y^{2}-y-z$.

The calculation of the primary decomposition of $I_{\mathbb{X}}$ yields that $I_{\mathbb{X}}$ is a primary ideal and its radical is the maximal ideal $\mathfrak{M}=\langle x-z,\,y-z^{2},\allowbreak z^{3}-z-1\rangle$. Here the polynomials $g_{1}=x-z$, $g_{2}=y-z^{2}$, and $g_{3}=z^{3}-z-1$ form a regular sequence which generates $\mathfrak{M}$. Let us use Algorithm 3.4 to check whether ${\mathbb{X}}$ is a complete intersection scheme.

Thus we represent $f_{1},\dots,f_{4}$ as required by Step (2) of the algorithm and get the matrix

[TABLE]

The tuple of residue classes in the ring $P/I_{\mathbb{X}}$ of the minors of order 3 of $W$ is $(\bar{x}-\bar{z},\;\bar{x}\bar{z}-\bar{y},\;0,\;-\bar{x}\bar{y}+\bar{x}+1)$. Therefore we deduce from Algorithm 3.4 that the scheme ${\mathbb{X}}$ is locally a complete intersection.

By Proposition 3.7, there are three triples of polynomials $\{f_{1},f_{2},f_{3}\}$, $\{f_{1},f_{2},f_{4}\}$, $\{f_{2},f_{3},f_{4}\}$ with the property that each triple forms a regular sequence in $P_{\mathfrak{M}}$ which generates $I_{\mathbb{X}}P_{\mathfrak{M}}$, since the residue classes of the corresponding minors are non-zero in $P/I_{\mathbb{X}}$.

It is interesting to observe that the two triples $\{f_{1},f_{2},f_{3}\}$ and $\{f_{2},f_{3},f_{4}\}$ generate ideals whose radical is $\mathfrak{M}$, hence they generate $I_{\mathbb{X}}$ according to the last claim of Proposition 3.7. On the other hand, the triple $\{f_{1},f_{2},f_{4}\}$ does not generate $I_{\mathbb{X}}$, since we have the equality $\langle f_{1},f_{2},f_{4}\rangle=I_{\mathbb{X}}\cap\langle z,y,x^{2}\rangle$.

Example 3.10**.**

Let $K=\mathbb{Q}$, let $P=K[x,y]$, and let ${\mathfrak{Q}}=\langle f_{1},f_{2},f_{3}\rangle$, where $f_{1}=y^{3}-x^{2}$, $f_{2}=x^{3}-x^{2}y$, and $f_{3}=x^{2}y^{2}$. It turns out that the ideal ${\mathfrak{Q}}$ is a [math]-dimensional $\mathfrak{M}$-primary ideal for the maximal ideal $\mathfrak{M}=\langle x,\,y\rangle$. We represent $f_{1},\ f_{2},\ f_{3}$ as required by Step (2) of Algorithm 3.4 and get the matrix

[TABLE]

The tuple of residue classes in the ring $P/{\mathfrak{Q}}$ of the minors of order 2 of $W$ is $(-\bar{x}^{2}\bar{y},\,0,\,0)$. Therefore $(f_{1},\,f_{2})$ is a regular sequence in $P_{\mathfrak{M}}$ which generates ${\mathfrak{Q}}P_{\mathfrak{M}}$. Notice that $\{f_{1},\,f_{2}\}$ does not generate ${\mathfrak{Q}}$, since we have the equality $\langle f_{1},\,f_{2}\rangle={\mathfrak{Q}}\cap\langle x-1,\,y-1\rangle$.

4. Strict Complete Intersections via Macaulay Bases

As in the preceding sections, we work over an arbitrary field $K$, we let ${\mathbb{X}}$ be a 0-dimensional subscheme of $\mathbb{A}^{n}_{K}$, and we denote the vanishing ideal of ${\mathbb{X}}$ in $P$ by $I_{\mathbb{X}}$. Recall that a 0-dimensional local ring $S$ is said to be Gorenstein if its socle, i.e., the annihilator of its maximal ideal $\mathfrak{n}$, is a 1-dimensional $S/\mathfrak{n}$-vector space. In this setting, we consider the following properties of ${\mathbb{X}}$.

Definition 4.1**.**

Let ${\mathbb{X}}$ be a [math]-dimensional subscheme of $\mathbb{A}^{n}_{K}$.

1. (a)

The scheme ${\mathbb{X}}$ is called a strict Gorenstein scheme if the associated graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})\cong P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ is Gorenstein. 2. (b)

The scheme ${\mathbb{X}}$ is called a strict complete intersection scheme if the degree form ideal $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ is generated by a homogeneous regular sequence.

Notice that in our setting the associated graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$ is a 0-dimensional local ring with maximal ideal $\langle\bar{x}_{1},\dots,\bar{x}_{n}\rangle$. The following definition is useful to describe the vanishing ideal of a strict complete intersection.

Definition 4.2**.**

Let $f_{1},\dots,f_{r}\in P\setminus K$. Then $(f_{1},\dots,f_{r})$ is said to be a strict regular sequence if $(\mathop{\rm DF}\nolimits(f_{1}),\dots,\mathop{\rm DF}\nolimits(f_{r}))$ is a homogeneous regular sequence.

Let us collect some basic properties of [math]-dimensional strict complete intersection schemes.

Proposition 4.3**.**

Let ${\mathbb{X}}$ be a 0-dimensional subscheme of $\mathbb{A}^{n}_{K}$.

1. (a)

If ${\mathbb{X}}$ is a strict complete intersection scheme, then there exist $f_{1},\dots,f_{n}\in I_{\mathbb{X}}$ such that $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})=\langle\mathop{\rm DF}\nolimits(f_{1}),\dots,\mathop{\rm DF}\nolimits(f_{n})\rangle$. 2. (b)

Let $f_{1},\dots,f_{n}\in I_{\mathbb{X}}$ be a strict regular sequence such that we have $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})=\langle\mathop{\rm DF}\nolimits(f_{1}),\dots,\mathop{\rm DF}\nolimits(f_{n})\rangle$. Then $(f_{1},\dots,f_{n})$ is a regular sequence which generates $I_{\mathbb{X}}$. In particular, every strict complete intersection scheme ${\mathbb{X}}$ is locally a complete intersection scheme. 3. (c)

Every strict complete intersection scheme ${\mathbb{X}}$ is a strict Gorenstein scheme, and every locally complete intersection scheme ${\mathbb{X}}$ is a locally Gorenstein scheme.

Proof.

First we show (a). Since every non-zero homogeneous element of $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ is a degree form, it follows that $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ can be generated by a regular sequence of the form $(\mathop{\rm DF}\nolimits(f_{1}),\dots,\mathop{\rm DF}\nolimits(f_{n}))$.

The first claim in (b) is, for instance, shown in [8], Lemma 1.8 and Prop. 1.9. The second claim is then a consequence of the observation that $R_{\mathbb{X}}$ is semi-local and its local rings satisfy $P/{\mathfrak{Q}}_{i}\;\cong\;(R_{\mathbb{X}})_{\mathfrak{m}_{i}}\;\cong\;P_{\mathfrak{M}_{i}}/\langle f_{1},\dots,f_{n}\rangle P_{\mathfrak{M}_{i}}$.

Finally, to prove (c), we note that every 0-dimensional local complete intersection ring is Gorenstein (see for instance [6], Prop. 3.1.20). ∎

Recall that the Castelnuovo function of ${\mathbb{X}}$ is said to be symmetric if we have $\Delta\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(\mathop{\rm ri}\nolimits(R_{\mathbb{X}})-i)=\Delta\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(i)$ for all $i\in\mathbb{Z}$. Now we are ready to formulate the following algorithm.

Algorithm 4.4**.**

**(Checking Strict Complete Intersections, I)

**Let ${\mathbb{X}}$ be a 0-dimensional scheme in $\mathbb{A}^{n}_{K}$, and let $R_{\mathbb{X}}=P/I_{\mathbb{X}}$ be the affine coordinate ring of ${\mathbb{X}}$. The following instructions define an algorithm which checks whether ${\mathbb{X}}$ is a strict complete intersection scheme and returns the corresponding Boolean value.

1. (1)

Compute a Macaulay basis of $I_{\mathbb{X}}$, i.e., compute polynomials $\{f_{1},\dots,f_{r}\}$ in $I_{\mathbb{X}}$ whose degree forms generate the degree form ideal $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$. 2. (2)

Check if the Castelnuovo function of ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})\cong P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ is symmetric. If this is not the case, return FALSE and stop. 3. (3)

For $j=1,\dots,r$, write $\mathop{\rm DF}\nolimits(f_{j})=\sum_{i=1}^{n}a_{ij}x_{i}$. Form the matrix $W$ of size $n\times r$ whose columns are given by $\sum_{i=1}^{n}a_{ij}e_{i}$ for $j=1,\dots,r$. 4. (4)

Calculate the tuple of the residue classes in $P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ of the minors of order $n$ of $W$. If one of these residue classes is non-zero, return TRUE. Otherwise, return FALSE.

Proof.

As every strict complete intersection is a strict Gorenstein scheme, Step (2) produces the correct answer if the affine Hilbert function of ${\mathbb{X}}$ is not symmetric. By the preceding proposition, Steps (3) and (4) compute a tuple of elements of $P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ whose entries generate the 0-th Fitting ideal of the maximal ideal $\langle\bar{x}_{1},\dots,\bar{x}_{n}\rangle$ of that ring. Thus the correctness of the answer in Step (4) follows from Proposition 3.3. ∎

In Proposition 3.7 we described a condition under which we can obtain a regular sequence which generates ${\mathfrak{Q}}P_{\mathfrak{M}}$. Here we can do better.

Corollary 4.5**.**

In the setting of Algorithm 4.4, the following conditions are equivalent.

1. (a)

The residue class of the minor involving columns $(j_{1},\dots,j_{n})$ of the matrix $W$ is non-zero. 2. (b)

The polynomials $(f_{j_{1}},\dots,f_{j_{n}})$ form a strict regular sequence which generates the ideal $I_{\mathbb{X}}$.

Proof.

The proof is analogous to the proof of Proposition 3.7. The main difference is that in the proof of (a)$\Rightarrow$(b) of that proposition we show that $\det(c^{\mathstrut}_{\ell k})$ is not contained in $\mathfrak{M}P_{\mathfrak{M}}$. On the other hand, in the current setting the determinant $\det(c^{\mathstrut}_{\ell k})$ is a homogeneous polynomial, and $\mathfrak{M}$ is the ideal generated by the indeterminates. Consequently, $\det(c^{\mathstrut}_{\ell k})$ is a non-zero constant, and hence invertible in $P$. ∎

The next examples show Algorithm 4.4 and Corollary 4.5 at work.

Example 4.6**.**

Let $K=\mathbb{Q}$, let $P=K[x,y,z]$, and let ${\mathbb{X}}$ be the reduced subscheme of $\mathbb{A}^{3}_{K}$ consisting of the eight points $p_{1}=(0,0,0)$, $p_{2}=(1,1,1)$, $p_{3}=(-1,1,-1)$, $p_{4}=(2,4,8)$, $p_{5}=(-2,4,-8)$, $p_{6}=(3,9,27)$, $p_{7}=(-3,9,-27)$, and $p_{8}=(4,16,64)$ on the twisted cubic curve $T=\{(t,t^{2},t^{3})\in\mathbb{A}^{3}_{K}\mid t\in K\}$. The reduced $\tt DegRevLex$-Gröbner basis of the vanishing ideal $I_{\mathbb{X}}$ is

[TABLE]

and hence $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})=\langle y^{2}-xz,\ xy,\ x^{2},\ yz^{2}-{\textstyle\frac{2}{15}}z^{3},\ xz^{2}-{\textstyle\frac{1}{30}}z^{3},\ z^{4}\rangle$.

In Step (2) we calculate the Castelnuovo function $(1,3,3,1)$ and notice that it is symmetric.

Next we write the generators of $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ in the form $y^{2}-xz=(-z)x+(y)y+(0)z$, etc., and get the matrix

[TABLE]

Since the residue classes of all maximal minors of $W$ are zero in $P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$, we conclude that ${\mathbb{X}}$ is not a strict complete intersection scheme. Notice, however, that we can use [10], Algorithm 4.6 and Theorem 6.8 to check that ${\mathbb{X}}$ is a strict Gorenstein scheme.

A positive answer is obtained in the following case.

Example 4.7**.**

Let $K=\mathbb{Q}$, let $P=K[x,y]$, let $f_{1}=x^{3}-x-2y^{5}+4y^{4}-2y^{3}+4y^{2}-1$, $f_{2}=xy-y^{5}+2y^{4}-y^{3}+2y^{2}$, and $f_{3}=y^{7}-4y^{6}+5y^{5}-4y^{4}+4y^{3}-y$, and let ${\mathbb{X}}$ be the 0-dimensional subscheme of $\mathbb{A}^{2}_{K}$ defined by $I_{\mathbb{X}}=\langle f_{1},\ f_{2},\ f_{3}\rangle$. Then the reduced ${\tt DegRevLex}$-Gröbner basis of $I_{\mathbb{X}}$ turns out to be

[TABLE]

and hence we have $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})=\langle y^{5},\ x^{3},\ xy^{3},\ x^{2}y-y^{3}\rangle$.

In Step (2) we get the Castelnuovo function $(1,\,2,\,3,\,2,\,1)$ which is symmetric. Next we write $y^{5}=(0)x+(y^{4})y$, etc., and get the matrix

[TABLE]

Here the tuple of residue classes in $P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ of the minors of order 2 of $W$ is given by $(0,\;0,\;0,\;0,\;-\bar{y}^{4},\;0)$, and therefore ${\mathbb{X}}$ is a strict complete intersection scheme.

Notice that the non-zero minor of order 2 is obtained by selecting the second and the fourth column of $W$. The corresponding polynomials satisfy $\mathop{\rm DF}\nolimits(g_{2})=x^{3}$ and $\mathop{\rm DF}\nolimits(g_{4})=x^{2}y-y^{3}$. They form a regular sequence which generates $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$. In particular, note that we have $y^{5}=(xy)x^{3}+(-x^{2}-y^{2})(x^{2}y-y^{3})$ and $xy^{3}=(y)x^{3}+(-x)(x^{2}y-y^{3})$.

Hence the polynomials $g_{2}=x^{3}-2xy-x-1$ and $g_{4}=x^{2}y-y^{3}-y$ form a strict regular sequence. It turns out that

[TABLE]

5. Strict Complete Intersections via Border Bases

In this section we continue to work in the setting defined at the beginning of Section 2. Our goal is to construct another algorithm for checking the strict complete intersection property which is based on the computation of a degree filtered border basis of $I_{\mathbb{X}}$. This algorithm has the advantage of allowing us to check which ideals in certain families of ideals are strict complete intersections (see Example 5.5 and [11]). First of all, we define the following useful kind of vector space bases of $R_{\mathbb{X}}$.

Definition 5.1**.**

A tuple $B=(b_{1},\dots,b_{\mu})\in R_{\mathbb{X}}^{\mu}$ is said to be a degree filtered $K$-basis of $R_{\mathbb{X}}$ if the set $F_{i}B=B\cap F_{i}R_{\mathbb{X}}$ is a $K$-basis of $F_{i}R_{\mathbb{X}}$ for every $i\in\mathbb{Z}$ and if $\mathop{\rm ord}\nolimits_{\mathcal{F}}(b_{1})\leq\cdots\leq\mathop{\rm ord}\nolimits_{\mathcal{F}}(b_{\mu})$.

In addition to these properties, we may (and shall) assume that $b_{1}=1$ in each degree filtered $K$-basis of $R_{\mathbb{X}}$. For a discussion of this notion, we refer to [10], Remark 2.10 and Example 2.11. Degree filtered bases of $R_{\mathbb{X}}$ can be characterized as follows.

Proposition 5.2**.**

Let ${\mathbb{X}}$ be a 0-dimensional subscheme of $\mathbb{A}^{n}_{K}$ as above, and let $B=(b_{1},\dots,b_{\mu})\in R_{\mathbb{X}}^{\mu}$ with $\mathop{\rm ord}\nolimits_{\mathcal{F}}(b_{1})\leq\cdots\leq\mathop{\rm ord}\nolimits_{\mathcal{F}}(b_{\mu})$. Then the following conditions are equivalent.

1. (a)

The tuple $B$ is a degree filtered $K$-basis of $R_{\mathbb{X}}$. 2. (b)

The leading form tuple $\mathop{\rm LF}\nolimits(b_{1}),\dots,\mathop{\rm LF}\nolimits(b_{\mu})$ is a $K$-basis of the graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})\cong P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$. 3. (c)

There exist polynomials $f_{1},\dots,f_{\mu}\in P$ such that $b_{i}=f_{i}+I_{\mathbb{X}}$ for $i=1,\dots,\mu$ and $B^{\rm hom}=(f_{1}^{\rm hom}+I_{\mathbb{X}}^{\rm hom},\dots,f_{\mu}^{\rm hom}+I_{\mathbb{X}}^{\rm hom})$ is a $K[x_{0}]$-module basis of $R_{\mathbb{X}}^{\rm hom}$.

Proof.

To show that (a) implies (b), we note that $\dim_{K}({\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}}))=\dim_{K}(R_{\mathbb{X}})$ implies that it suffices to prove linear independence. Assume that there exists a tuple $(a_{1},\dots,a_{\mu})\in K^{\mu}\setminus\{0\}$ such that we have $a_{1}\mathop{\rm LF}\nolimits(b_{1})+\cdots+a_{\mu}\mathop{\rm LF}\nolimits(b_{\mu}){=0}$. Since the elements $\mathop{\rm LF}\nolimits(b_{i})$ are homogeneous in the graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$, we may assume that there is a degree $d$ such that $a_{i}=0$ for $\mathop{\rm ord}\nolimits_{\mathcal{F}}(b_{i})\neq d$. Let $k,k+1,\dots,\ell\in\{1,\dots,\mu\}$ be the indices for which $\mathop{\rm ord}\nolimits_{\mathcal{F}}(b_{i})=d$. Then $a_{k}\mathop{\rm LF}\nolimits(b_{k})+\cdots+a_{\ell}\mathop{\rm LF}\nolimits(b_{\ell})=0$ implies $a_{k}b_{k}+\cdots+a_{\ell}b_{\ell}\in F_{d-1}R_{\mathbb{X}}=\langle b_{1},\dots,b_{k-1}\rangle_{K}$, in contradiction to the linear independence of $F_{d}B=(b_{1},\dots,b_{\ell})$.

To prove that (b) implies (c), we choose $f_{i}\in P$ such that $b_{i}=f_{i}+I_{\mathbb{X}}$ and $\deg(f_{i})=\mathop{\rm ord}\nolimits_{\mathcal{F}}(b_{i})$ for $i=1,\dots,\mu$. (If $f_{i}$ does not satisfy the second condition, replace it by $\mathop{\rm NF}\nolimits_{\sigma,I_{\mathbb{X}}}(f_{i})$ for some degree compatible term ordering $\sigma$.) Then we have $\mathop{\rm LF}\nolimits(b_{i})=\mathop{\rm DF}\nolimits(f_{i})+\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ for $i=1,\dots,\mu$. Now the claim is a consequence of Remark 2.4.c and $\mathop{\rm DF}\nolimits(f_{i})=f_{i}^{\rm hom}(0,x_{1},\dots,x_{n})$ for $i=1,\dots,\mu$.

Finally we show that (c) implies (a). In view of the isomorphisms in Remark 2.4.c, the elements $b_{i}=f_{i}+I_{\mathbb{X}}=f_{i}^{\rm hom}(1,x_{1},\dots,x_{n})+I_{\mathbb{X}}$ with $i\in\{1,\dots,\mu\}$ form a $K$-basis of $R_{\mathbb{X}}$, and the elements

[TABLE]

with $i\in\{1,\dots,\mu\}$ form a $K$-basis of ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})$. From the latter property and the formula $\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(i)=\sum_{j=0}^{i}\mathop{\rm HF}\nolimits_{{\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})}(j)$ we deduce that the number of elements in $F_{i}B$ equals $\mathop{\rm HF}\nolimits^{a}_{\mathbb{X}}(i)$ for all $i\geq 0$. Hence $F_{i}B$ is a $K$-basis of $F_{i}R_{\mathbb{X}}$ for all $i\geq 0$, and the proof is complete. ∎

An even more useful type of degree-filtered $K$-bases is described in the following proposition. We recall that $\mathbb{T}^{n}=\{x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}\mid\alpha_{i}\geq 0\}$ is the monoid of terms in $P$.

Proposition 5.3**.**

Let ${\mathcal{O}}=\{t_{1},\dots,t_{\mu}\}$ be an order ideal of terms in $\mathbb{T}^{n}$ such that $\deg(t_{1})\leq\cdots\leq\deg(t_{\mu})$. Let $\partial{\mathcal{O}}=\{b_{1},\dots,b_{\nu}\}$ be the border of ${\mathcal{O}}$, and let $G=\{g_{1},\dots,g_{\nu}\}$ be an ${\mathcal{O}}$-border prebasis of the ideal $I_{\mathbb{X}}$, where we have $g_{j}=b_{j}-\sum_{i=1}^{\mu}\gamma_{ij}t_{i}$ with $\gamma_{1j},\dots,\gamma_{\mu j}\in K$ for $j=1,\dots,\nu$. Then the following conditions are equivalent.

1. (a)

The terms in ${\mathcal{O}}$ form a degree filtered $K$-basis of $R_{\mathbb{X}}$. 2. (b)

The terms in ${\mathcal{O}}$ form a $K$-basis of ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})\cong P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$. 3. (c)

The ideal $I_{\mathbb{X}}$ has an ${\mathcal{O}}$-border basis and for all $i\geq 0$ we have $\mathop{\rm HF}\nolimits^{a}_{{\mathbb{X}}}(i)=\#\{j\in\{1,\dots,\mu\}\mid\deg(t_{j})=i\}$. 4. (d)

The ideal $I_{\mathbb{X}}$ has an ${\mathcal{O}}$-border basis and we have $b_{j}\in\mathop{\rm DF}\nolimits(g_{j})$ for $j=1,\dots,\nu$. 5. (e)

The polynomials in $\mathop{\rm DF}\nolimits(G)=\{\mathop{\rm DF}\nolimits(g_{1}),\dots,\mathop{\rm DF}\nolimits(g_{\nu})\}$ form an ${\mathcal{O}}$-border basis of $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$, hence they generate $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$.

If these conditions are satisfied, we say that $G$ is a degree filtered ${\mathcal{O}}$-border basis of $I_{\mathbb{X}}$.

Proof.

The equivalence of (a) and (b) was shown in Proposition 5.2. The equivalence of (a) and (c) follows immediately from the definitions. The equivalence of (b) and (d) follows from [15], Thm. 2.4. The implication (b)$\Rightarrow$(e) was also shown in [15], Thm. 2.4, and (e)$\Rightarrow$(b) holds by definition. ∎

Using a degree filtered border basis as described in the above proposition, we can formulate another version of Algorithm 4.4 for checking the strict complete intersection property.

Algorithm 5.4**.**

**(Checking Strict Complete Intersections, II)

**Let ${\mathbb{X}}$ be a 0-dimensional scheme in $\mathbb{A}^{n}_{K}$, let $R_{\mathbb{X}}=P/I_{\mathbb{X}}$ be the affine coordinate ring of ${\mathbb{X}}$, and let $\mu=\dim_{K}(R_{\mathbb{X}})$. The following instructions define an algorithm which checks whether ${\mathbb{X}}$ is a strict complete intersection and returns the corresponding Boolean value.

1. (1)

Compute an order ideal ${\mathcal{O}}=(t_{1},\dots,t_{\mu})$ in $\mathbb{T}^{n}$ such that $I_{\mathbb{X}}$ has a degree filtered ${\mathcal{O}}$-border basis. Let ${\mathop{\Varrho}\,}=\deg(t_{\mu})$. 2. (2)

For $i=0,\dots,{\mathop{\Varrho}\,}$, find $h_{i}=\#\{j\in\{1,\dots,\mu\}\mid\deg(t_{j})=i\}$. 3. (3)

For $i=0,\dots,\lfloor{\mathop{\Varrho}\,}/2\rfloor$, check whether $h_{{\mathop{\Varrho}\,}-i}=h_{i}$. If this is not the case, return FALSE and stop. 4. (4)

Let $G=\{g_{1},\dots,g_{\nu}\}$ be the ${\mathcal{O}}$-border basis of $I_{\mathbb{X}}$. For $j=1,\dots,\nu$, write $\mathop{\rm DF}\nolimits(g_{j})=\sum_{i=1}^{n}h_{ij}x_{i}$ with homogeneous polynomials $h_{ij}\in P$. 5. (5)

Form the matrix $W$ of size $n\times\nu$ whose columns are given by $\sum_{i=1}^{n}h_{ij}e_{i}$ for $j=1,\dots,\nu$. 6. (6)

Calculate the tuple of residue classes in $P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ of the minors of order $n$ of $W$. 7. (7)

If the result is different from $(0,\dots,0)$, return TRUE. Otherwise return FALSE.

Proof.

By Step (1), the residue classes of the elements of ${\mathcal{O}}$ form a $K$-basis of the ring $P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$. Since every strict complete intersection is a strict Gorenstein ring, we can stop in Step (3) if the Castelnuovo function $(h_{0},\dots,h_{\mathop{\Varrho}\,})$ of this ring is not symmetric.

Now we use Algorithm 4.4 for the ring $P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$. Notice that the degree forms $\mathop{\rm DF}\nolimits(g_{1}),\dots,\mathop{\rm DF}\nolimits(g_{\nu})$ generate $\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ by Proposition 5.3.e. In Step (4) we write them as linear combinations of $x_{1},\dots,x_{n}$. Hence the correctness follows from Algorithm 4.4. ∎

Next example shows how the methods described in this paper allow us to detect strict complete intersections within certain families of [math]-dimensional schemes. For a more detailed discussion of this topic see [11].

Example 5.5**.**

Let $K$ be a field, let $P=K[x,y]$, and let ${\mathcal{O}}$ be the order ideal ${\mathcal{O}}=\{1,y,x,xy\}$. According to [15], Example 3.8, the ideal $I=\langle f_{1},f_{2},f_{3},f_{4}\rangle$ where

[TABLE]

is the defining ideal of the universal family of all subschemes of length four of the affine plane which have the property that their coordinate ring admits $\overline{\mathcal{O}}$ as a vector space basis. The parameters $c_{21}$, $c_{23}$, $c_{32}$, $c_{34}$, $c_{41}$, $c_{42}$, $c_{43}$, and $c_{44}$ are free. In other words, the family is parametrized by an $8$-dimensional affine space. Since the degree form ideal $\mathop{\rm DF}\nolimits(I)$ is generated by the degree forms of $f_{1},f_{2},f_{3},f_{4}$, we have $\mathop{\rm DF}\nolimits(I)=\langle y^{2}-c_{41}xy,\ x^{2}-c_{42}xy,\ xy^{2},\ x^{2}y\rangle$. To compute the locus of strict complete intersections in $\mathbb{A}^{8}_{K}$, we write the generators of $\mathop{\rm DF}\nolimits(I)$ in the form

[TABLE]

We get the matrix

[TABLE]

Then the only non-zero maximal minor of $W$ modulo $\mathop{\rm DF}\nolimits(I)$ is given by $(1-c_{41}c_{42})xy$. In conclusion, outside the hypersurface in $\mathbb{A}^{8}_{K}$ defined by $1-c_{41}c_{42}=0$, all ideals define a strict complete intersection scheme.

Finally, we note that one can also use the Kähler different of $R_{\mathbb{X}}$ to check whether this ring is a strict complete intersection or locally a complete intersection. However, this approach introduces constraints on the characteristic of the base field. Let us formulate the characterizations underlying the algorithms using Kähler differents and leave the details to the interested reader.

Remark 5.6**.**

Let ${\mathbb{X}}$ be a 0-dimensional subscheme of $\mathbb{A}^{n}_{K}$, and let $R_{\mathbb{X}}=P/I_{\mathbb{X}}$ be the affine coordinate ring of ${\mathbb{X}}$.

1. (a)

The module of Kähler differentials $\Omega^{1}_{R_{\mathbb{X}}/K}$ is given by the presentation

[TABLE] 2. (b)

The Kähler different $\vartheta_{R_{\mathbb{X}}}$ of the $K$-algebra $R_{\mathbb{X}}$ is the ideal in $R_{\mathbb{X}}$ generated by residue classes of the maximal minors of the Jacobian matrix $\mathop{\rm Jac}\nolimits(f_{1},\dots,f_{r})=(\partial f_{i}/\partial x_{j})_{i,j}$, where $\{f_{1},\dots,f_{r}\}$ is a system of generators of $I_{\mathbb{X}}$. For further details about Kähler differents, see [16], § 10. 3. (c)

The Kähler different $\vartheta_{{\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})}$ of the associated graded ring ${\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})\cong P/\mathop{\rm DF}\nolimits(I_{\mathbb{X}})$ is defined similarly. 4. (d)

Suppose that $\mathop{\rm char}\nolimits(K)$ does not divide $\mu$. Then ${\mathbb{X}}$ is a strict complete intersection if and only if $\vartheta_{{\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})}$ is non-zero. 5. (e)

Again, suppose that $\mathop{\rm char}\nolimits(K)$ does not divide $\mu$. Then ${\mathbb{X}}$ is locally a complete intersection if and only if the image of $\vartheta_{R_{\mathbb{X}}}$ in every local ring of $R_{\mathbb{X}}$ is non-zero. (For instance, if we know the principal idempotents $f_{1},\dots,f_{s}$ of $R_{\mathbb{X}}$, it suffices to check that $f_{i}\cdot\vartheta_{R_{\mathbb{X}}}\neq\langle 0\rangle$ for $i=1,\dots,s$.)

The following easy example shows that the assumption on the characteristic of $K$ is necessary for the approach via Kähler differents to work, while the approach based on Wiebe’s result (see Proposition 3.2) works in general.

Example 5.7**.**

Let $p$ be a prime number, let $K=\mathbb{F}_{p}$, let $P=K[x]$, and let ${\mathbb{X}}$ be the 0-dimensional subscheme of $\mathbb{A}^{1}_{K}$ defined by $I_{\mathbb{X}}=\langle x^{p}\rangle$.

When we use Algorithm 4.4 to check whether ${\mathbb{X}}$ is a strict complete intersection, we find the matrix $W=(x^{p-1})$ whose determinant yields the relation $\bar{x}^{p-1}\in{\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})\setminus\{0\}$. Hence we conclude that ${\mathbb{X}}$ is a strict complete intersection.

However, the Jacobian matrix is $\mathop{\rm Jac}\nolimits(x^{p})=(0)$ and therefore we have $\vartheta_{{\mathop{\rm gr}\nolimits_{\mathcal{F}}}(R_{\mathbb{X}})}=\langle 0\rangle$. Thus the Kähler different does not yield the correct answer.

Acknowledgements. The first and second authors were partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant number 101.04-2019.07. The third author thanks the University of Passau for its hospitality and support during part of the preparation of this paper.