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, Le N. Long, L. Robbiano, On the Cayley-Bacharach Property, Communications in Algebra 47 (2019), 328-354]).

Package Discription

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

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: