Package alggeozd

This page describes the alggeozd package. The package contains various functions for checking algebraic and geometric properties of zero-dimensional affine K-algebra/schemes and related computations. For a complete list of functions, see Category:Package alggeozd.

Alg-Geo Properties

Let K be a field, let I be a 0-dimensional ideal in a polynomial ring P=K[x_1,...x_n], and let R=P/I and X=Spec(P/I). In this package, we checking some algebraic and geometric properties of the scheme X or of the affine K-algebra R such as: locally Gorenstein, strict/arithmetically Gorenstein, complete intersection, Cayley-Bacharach property, and (i,j)-uniformity.

Algebraic Properties

Suppose the support of X contains s points p_1,...,p_s. For i=1,...s, the local ring of X at p_i is of the form O_i=P/q_i where q_i is a primary ideal of P. Let m_i be the maximal ideal of O_i and K_i=O_i/m_i for i=1,...,s.

• The local ring O_i is called a Gorenstein ring if the K_i-vector space (0:m_i) has dimension 1.
• The scheme X (or the ring R) is called a locally Goreinstein ring if the local ring O_i is a Gorenstein ring for every i=1,...,s.

The associated graded ring of R with respect to the standard grading is denoted by gr(R). Then gr(R) is a 0-dimenisonal local ring.

• The scheme X (or the ring R) is called a strict/arithmetically Goreinstein ring if gr(R) is a Goresntein ring.
• The local ring O_i is called a complete intersection if the ideal q_i is generated by a regular sequence of length n in P.
• The scheme X (or the ring R) is called a locally complete intersection if the local ring O_i is a complete intersection for every i=1,...,s.
• The scheme X (or the ring R) is called a complete intersection if gr(R) is a complete intersection ring.

Geometric Properties

Now we consider X as a projective subscheme of the projective n-space under the embedding X into D_+(x_0). The homogeneous coordinate ring of X is R_X and its homogeneous vanishing ideal in S=K[x_0,...,x_n] is I_X. The Hilbert function of R_X is denoted by HF_X and its regularity index is r_X. The number d_X = dim_K(R) is known as the degree of X. Suppose that X has K-rational support, that is, the maximal ideal of each O_i is generated by only linear forms. Let i,j be two positive integers. Under this assumption, we have the following notion.

• The scheme X is called (i,j)-uniform if every subscheme Y of X of degree d_Y=d_X-1 satisfies HF_Y(j)=HF_X(j).
• The scheme X is said to have the Cayley-Bacharach property if it is (1,r_X-1)-uniform.

The Cayley-Bacharach property of X has a long and rich history and it can be generalized for an arbitrary 0-dimensional scheme X (see e.g. the paper [M. Kreuzer, L.N. Long, L. Robbiano, On the Cayley-Bacharach Property, Communications in Algebra 47 (2019), 328-354]).

Package Discription

The alggeozd package provides functions for checking the introduced properties of zero-dimensional schemes. Alias of the package is AGZD.

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

Example 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: