CoCoA:Syz

Syz

syzygy modules

Description

In the first two forms this function computes the syzygy module of a list

of polynomials or vectors. In the last form this function returns the

specified syzygy module of the minimal free resolution of M which must be homogeneous. As a side effect, it computes the Groebner basis of M.

The coefficient ring must be a field.

Example

  Use R ::= Q[x,y,z];
  Syz([x^2-y,xy-z,xy]);
  Module([0, xy, -xy + z], [z, x^2 - y, -x^2 + y], [yz, -y^2, y^2 - xz],
[xy, 0, -x^2 + y])
-------------------------------
  I := Ideal(x^2-yz, xy-z^2, xyz);
  Syz(I,0);
Module([x^2 - yz], [xy - z^2], [xyz])
-------------------------------
  Syz(I,1);
  Module([-x^2 + yz, xy - z^2, 0], [xz^2, -yz^2, -y^2 + xz], [z^3, 0,
-xy + z^2], [0, z^3, -x^2 + yz])
-------------------------------
  Syz(I,2);
Module([0, z, -x, y], [-z^2, -x, y, -z])
-------------------------------
  Syz(I,3);
Module([[0]])
-------------------------------
  Res(I);
0 --> R^2(-6) --> R(-4)(+)R^3(-5) --> R^2(-2)(+)R(-3)
-------------------------------

For fine control and monitoring of Groebner basis calculations, see
<ref>The Interactive Groebner Framework</ref> and <ref>Introduction to Panels</ref>.

Syntax

Syz(L:LIST of POLY):MODULE
Syz(L:LIST of VECTOR):MODULE
Syz(M:IDEAL or MODULE, Index:INT):MODULE

Introduction to Groebner Bases in CoCoA

GBasis5, and more

   <type>groebner</type>
   <type>groebner-basic</type>
   <type>ideal</type>
   <type>list</type>
   <type>module</type>
   <type>cocoaserver</type>