CoCoA:Syz
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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
<type>groebner</type> <type>groebner-basic</type> <type>ideal</type> <type>list</type> <type>module</type> <type>cocoaserver</type>