Package zerodim

This article is about a function from ApCoCoA-2. If you are looking for the ApCoCoA-1 version of it, see Category:ApCoCoA-1:Package ZeroDim.

This page describes the zerodim package. The package contains various functions for computing algebraic invariants of zero-dimensional schemes and related computations. For a complete list of functions, see also Category:Package zerodim.

Algebraic Invariants

Let $K$ be a field, let $P = K [x_{1}, \dots, x_{n}]$ be the polynomial ring over $K$ in $n$ indeterminates, and let $I$ be a 0-dimensional ideal of $P$ and $R = P / I$ . Then $I$ defines a 0-dimensional scheme $X$ in the affine $n$ -space. Consider the canonical multiplication map

μ : R^{e} = R \otimes_{K} R ⟶ R, (f, g) \mapsto f g

and its kernel $J = K e r (μ)$ . Then $J / J^{2}$ is a finitely generated $R$ -module and ${A n n}_{R^{e}} (J)$ is an ideal of the enveloping algebra $R^{e}$ .

The ideal $ϑ_{N} (R / K) : = μ ({A n n}_{R^{e}} (J))$ is called the Noether different of the algebra $R / K$ .
The $R$ -module $Ω_{R / K}^{1} : = J / J^{2}$ is called the module of Kaehler differential 1-forms of the algebra $R / K$ .
The $K$ -linear map $d : R \to Ω_{R / K}^{1}, f \mapsto f \otimes 1 - 1 \otimes f + J^{2}$ , is called the universal derivation of the algebra $R / K$ .
For $m \geq 0$ , the exterior power $Ω_{R / K}^{m} : = \land_{R}^{m} Ω_{R / K}^{1}$ is called the module of Kaehler differential m-forms of the algebra $R / K$ .
For $m \geq 0$ the $m$ -th Fitting ideal $ϑ_{X}^{(m)} (R / K)$ of the module of Kaehler differential 1-forms $Ω_{R / K}^{1}$ is called the Kaehler different of the algebra $R / K$ .

More generally, for any $K$ -algebra $T / S$ , we can define the Noether different, module of Kaehler differential m-forms, Kaehler different of $T / S$ analogously. In particular, if $T / S$ is graded, then all these invariants are also graded.

Now let us embed the scheme $X$ in the projective $n$ -space via $X \subseteq D_{+} (x_{0}) \subseteq ℙ^{n}$ , where $x_{0}$ is a new indeterminate. Set $S : = P [x_{0}] = K [x_{0}, \dots, x_{n}]$ and equip $S$ with the standard grading. The homogeneous vanishing ideal of $X$ is the homogenization of $I$ with respect to $x_{0}$ and denoted by $I_{X}$ , and the homogeneous coordinate ring of $X$ is the graded 1-dimensional ring $R_{X} = S / I_{X}$ . In this case $K [x_{0}]$ is the Noetherian normalization of $R_{X}$ , and hence we can define the above invariants for the graded algebra $R_{X} / K [x_{0}]$ . Moreover, we have the following further invariants.

The graded $R_{X}$ -module $ω_{R_{X}} = {H o m}_{K [x_{0}]} (R, K [x_{0}]) (- 1)$ is called the canonical module of the algebra $R_{X} / K [x_{0}]$ .
The graded locolization $Q^{h} (R_{X})$ of $R_{X}$ at $x_{0}$ is called the homogeneous ring of quotients of $R_{X}$ .
When the scheme $X$ is reduced (more general, locally Gorenstein), there is an injection $ω_{R_{X}} ↪ Q^{h} (R_{X})$ and the inverse of $ω_{R_{X}}$ in $Q^{h} (R_{X})$ is called the Dedekind different of $R_{X} / K [x_{0}]$ .

Many interesting properties of the scheme $X$ are reflexed by the algebraic structure of the above invariants.

Package Description

The zerodim package provides functions for computing the introduced invariants of zero-dimensional schemes. In the graded case the package also provides functions for computations of the Hilbert functions of these invariants. Alias of the package is ZD.

List of main functions

MinQuotIdeal

MinQuotIdeal(P, I, J): computes a min. homog. system
          of generators of homog. ideal (I+J)/I.
     input: P=K[x[1..N]], I and J homog. ideals of P
     output: list of polys

AffineNoetherDiff

AffineNoetherDiff(P, I): computes a generating system";
          of the Noether different of algebra R/K, R=P/I";
     input: P=K[x[1..N]], I an ideal of P";
     output: list of polys";

NoetherDifferent

NoetherDifferent(P, I): computes a min.homog. gen. system
          of the Noether different of algebra R/K, R=P/I.
     input: P=K[x[1..N]], I an homog. ideal of P
     output: list of polys

NoetherDifferentRel

NoetherDifferentRel(P, Ix): computes a min.homog. gen. system
          of the Noether different of R/K[x[0]], R=P/Ix.
     input: P=K[x[0..N]], Ix vanishing ideal of a 0-dim scheme X
            in P^n_K such that intersect(X,Z(x[0])) is empty
     output: list of polys

HilbertNoetherDiff

HilbertNoetherDiff(P, I): computes the Hilbert function
          of the Noether different of R/K, R=P/I.
     input: P=K[x[1..N]], I an homog. ideal of P
     output: the Hilbert function

HilbertNoetherDiffRel

HilbertNoetherDiffRel(P, Ix): computes the Hilbert function
          of the Noether different of R/K[x[0]], R=P/Ix.
     input: P=K[x[0..N]], Ix vanishing ideal of a 0-dim scheme X
            in P^n_K such that intersect(X,Z(x[0])) is empty
     output: the Hilbert function

AffineKaehlerDiff

AffineKaehlerDiff(P,I,m): computes a generating system
          of the m-th Kaehler different of algebra R/K, R=P/I.
     input: P=K[x[1..N]], I an ideal of P, m non-neg integer
     output: list of polys

KaehlerDifferent

KaehlerDifferent(P,I,m): computes a min.homog.gen. system
          of the m-th Kaehler different of algebra R/K, R=P/I.
     input: P=K[x[1..N]], I an homog. ideal, m non-neg integer
     output: list of polys

KaehlerDifferentRel

KaehlerDifferentRel(P, Ix): computes a min. homog.gen. system
          of the Kaehler different of R/K[x[0]], R=P/Ix.
     input: P=K[x[0..N]], Ix vanishing ideal of a 0-dim scheme X
            in P^n_K such that intersect(X,Z(x[0])) is empty
     output: list of polys

HilbertKaehlerDiff

HilbertKaehlerDiff(P,I,m): computes the Hilbert function
          of the m-th Kaehler different of R/K, R=P/I.
     input: P=K[x[1..N]], I an homog. ideal, m non-neg integer
     output: the Hilbert function

HilbertKaehlerDiffRel

HilbertKaehlerDiffRel(P, Ix): computes the Hilbert function
          of the Kaehler different of R/K[x[0]], R=P/Ix.
     input: P=K[x[0..N]], Ix vanishing ideal of a 0-dim scheme X
             in P^n_K such that intersect(X,Z(x[0])) is empty
     output: the Hilbert function

AffBMAlgo

AffBMAlgo(LX,O): computes a list [GBasis,OrderIdeal,Separators]
          for a 0-dim ideal with its primary components LX.
     input: P=K[x[1..N]], LX list of 0-dim primary ideals
            q_j associated to a 0-dim ideal of P
            O list of K-bases of P/q_j
     output: [GBasis,OrderIdeal,Separators] of P/intersection(q_j)

DedekindDifferentRel

DedekindDifferentRel(P,Points): computes a min.homog.gen. system
          of the Dedekind different of R/K[x[0]], where R=P/Ix
          and Ix is the vanishing ideal of Points.
     input: P=K[x[0..N]], Points=list of points in P^n_K
            not in Z(x[0])
     output: list of polys

HilbertDedekindDiffRel

HilbertDedekindDiffRel(P,Points): computes the Hilbert function
          of the Dedekind different of R/K[x[0]], where R=P/Ix
          and Ix is the vanishing ideal of Points.
     input: P=K[x[0..N]], Points=list of points in P^n_K
            not in Z(x[0])
     output: the Hilbert function

KaehlerDiffModule

KaehlerDiffModule(P, Ix, m): computes a submodule U of P^t
          such that the module of Kaehler differential m-form
          has Omega^m(R/K)=P^t/U, R=P/Ix, t=binom{n}{m}.
     input: P=K[x[1..N]], Ix a non-zero ideal, m non-neg integer
     output: submodule with generators

HilbertKDM

HilbertKDM(P, Ix, m): computes the Hilbert function of
          the module of Kaehler differential m-form.
     input: P=K[x[1..N]], Ix a non-zero homog. ideal, 0<m<n+1
     output: HF of Omega^m(R/K)

KDMOfPoints

KDMOfPoints(P,Points,m): computes a submodule U of P^t such that
          the module of Kaehler differential m-form has
          Omega^m(R/K)=P^t/U, R=P/I_Points, t=binom{n}{m}.
     input: P=K[x[1..N]], Points=list of points, m non-neg integer
     output: submodule with generators

KDMOfProjectivePoints

KDMOfProjectivePoints(P,Points,m): computes a submodule U of P^t
          such that the module of Kaehler differential m-form has
          Omega^m(R/K)=P^t/U, R=P/I_Points, t=binom{n}{m}.
     input: P=K[x[1..N]], Points=list of projective points,
            m non-neg integer
     output: submodule with generators

KDMRel

KDMRel(P, Ix, m): computes a submodule U of P^t such that
          the module of Kaehler differential m-form of R/K[x[0]]
          has Omega^m(R/K[x[0]])=P^t/U, R=P/Ix, t=binom{n}{m}.
     input: P=K[x[0..N]], Ix a non-zero homog. ideal such that
            K[x[0]] is the Noetherian normalization of R,
            m non-neg integer
     output: submodule with generators

HilbertKDMRel

HilbertKDMRel(P, Ix, m): computes the Hilbert function of
          the module of Kaehler differential m-form of R/K[x[0]].
     input: P=K[x[0..N]], Ix a non-zero homog. ideal such that
            K[x[0]] is the Noetherian normalization of R,
            m non-neg integer";
     output: HF of Omega^m(R/K[x_0])

Examples for computations

Now let us apply the zerodim package to some concrete examples. Recall that the alias of the package is ZD, and so to call a function from this package in computation one uses ZD.functions-name.

Consider the first example, where X is the scheme defined by the homogeneous ideal Ix.

Use P ::= QQ[X[0..2]];
Ix := ideal(X[0]*X[1] -X[1]^2, X[1]^2*X[2] -X[1]*X[2]^2, 2*X[0]^2*X[2] -3*X[0]*X[2]^2 +X[2]^3);

Then we calculate the differents of X as follows:

-- Computes the Noether different of R/K:
ZD.NoetherDifferent(P,Ix); 
   []
-- Computes the Noether different of R/K[x_0]:
ZD.NoetherDifferentRel(P,Ix); 
   [X[1]^3 -2*X[1]*X[2]^2,  2*X[0]^3 -6*X[0]*X[2]^2 -2*X[1]*X[2]^2 +3*X[2]^3,  X[2]^4]
-- Computes the Hilbert function of the Noether different of R/K[x_0]:
ZD.HilbertNoetherDiffRel(P,Ix);
   H(0) = 0
   H(1) = 0
   H(2) = 0
   H(3) = 2
   H(t) = 5, for t >= 4
-- Computes the Kaehler different of R/K[x_0]:
ZD.KaehlerDifferentRel(P,Ix);
   [X[1]^3 -2*X[1]*X[2]^2,  2*X[0]^3 -6*X[0]*X[2]^2 -2*X[1]*X[2]^2 +3*X[2]^3,  X[2]^4]
-- Computes the Hilbert function of the Kaehler different of R/K[x_0]:
ZD.HilbertKaehlerDiffRel(P,Ix);
   H(0) = 0
   H(1) = 0
   H(2) = 0
   H(3) = 2
   H(t) = 5, for t >= 4

The module of Kaehler differentials 1-forms of R/K is determined by a submodule U of P^3 which is computed by:

U := ZD.KaehlerDiffModule(P,Ix,1); indent(U);
   SubmoduleRows(F, matrix([
     [X[1], X[0] -2*X[1], 0],
     [0, 2*X[1]*X[2] -X[2]^2, X[1]^2 -2*X[1]*X[2]],
     [4*X[0]*X[2] -3*X[2]^2, 0, 2*X[0]^2 -6*X[0]*X[2] +3*X[2]^2],
     [X[0]*X[1] -X[1]^2, 0, 0],
     [2*X[0]^2*X[2] -3*X[0]*X[2]^2 +X[2]^3, 0, 0],
     [0, X[0]*X[1] -X[1]^2, 0],
     [0, 2*X[0]^2*X[2] -3*X[0]*X[2]^2 +X[2]^3, 0],
     [0, 0, X[0]*X[1] -X[1]^2],
     [0, 0, X[1]^2*X[2] -X[1]*X[2]^2],
     [0, 0, 2*X[0]^2*X[2] -3*X[0]*X[2]^2 +X[2]^3]
   ]))

Next, let us consider the example, where X is given by a set of 10 points:

Use P::=QQ[X[0..2]];
Points := [[1,1,0], [1,3,0], [1,1,1], [1,2,1], [1,3,1], [1,0,2], [1,1,2], [1,2,2], [1,3,2], [1,3,3]];

We can compute the Dedekind different of X and its Hilbert function by:

ZD.DedekindDifferentRel(P,Points);
   [X[2]^6,  X[1]*X[2]^5,  X[0]*X[2]^5,  X[1]^2*X[2]^4,  X[0]*X[1]*X[2]^4,  
    X[0]^2*X[2]^4,  X[1]^3*X[2]^3,  X[0]*X[1]^2*X[2]^3,  X[1]^6,  X[0]*X[1]^5]
ZD.HilbertDedekindDiffRel(P,Points);
   H(0) = 0
   H(1) = 0
   H(2) = 0
   H(3) = 0
   H(4) = 0
   H(5) = 0
   H(t) = 10, for t >= 6

The module of Kaehler differential 3-forms of R/K can be computed by

ZD.KDMOfProjectivePoints(P,Points,3);
   submodule(FreeModule(RingWithID(144, "QQ[X[0],X[1],X[2]]"), 1), 
     [[(-1/2)*X[0]*X[1]^2 +(-19/27)*X[1]^3 +(-85/18)*X[0]^2*X[2] +(563/27)*X[0]*X[1]*X[2] 
       +(85/27)*X[1]^2*X[2] +(-61/3)*X[0]*X[2]^2 +(-301/18)*X[1]*X[2]^2 +(47/3)*X[2]^3], 
      [-X[0]^2*X[2] +(3/2)*X[0]*X[2]^2 +(-1/2)*X[2]^3], [-2*X[0]^2*X[1] +4*X[0]*X[1]^2 
       +(-4/3)*X[1]^3 +X[1]^2*X[2] +(-23/6)*X[0]*X[2]^2 +(-4/3)*X[1]*X[2]^2 +(13/6)*X[2]^3], 
      [-X[0]^3 +(13/3)*X[0]*X[1]^2 +(-16/9)*X[1]^3 +(11/6)*X[1]^2*X[2] +(-253/36)*X[0]*X[2]^2 
       +(-22/9)*X[1]*X[2]^2 +(143/36)*X[2]^3], 
      [(40/9)*X[0]*X[1]^2 +(-52/27)*X[1]^3 +X[0]*X[1]*X[2] +(19/9)*X[1]^2*X[2] 
       +(-563/54)*X[0]*X[2]^2 +(-85/27)*X[1]*X[2]^2 +(301/54)*X[2]^3], 
      [2*X[0]*X[1]*X[2] -2*X[0]*X[2]^2 +(-3/2)*X[1]*X[2]^2 +(3/2)*X[2]^3], 
      [3*X[0]^2*X[2] +(-11/3)*X[0]*X[2]^2 +X[2]^3], 
      [2*X[0]*X[1]^2 +(-4/3)*X[1]^3 -2*X[0]*X[2]^2 +(23/6)*X[1]*X[2]^2 +(-5/2)*X[2]^3], 
      [3*X[0]^2*X[1] +(-13/9)*X[1]^3 +(-11/3)*X[0]*X[2]^2 +(253/36)*X[1]*X[2]^2 +(-55/12)*X[2]^3], 
      [4*X[0]^3 +(-40/27)*X[1]^3 +(-1/2)*X[1]^2*X[2] +(-85/18)*X[0]*X[2]^2 +(563/54)*X[1]*X[2]^2 +(-61/9)*X[2]^3]
     ])

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