CoCoA:Syz

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 <command>
   <title>Syz</title>
   <short_description>syzygy modules</short_description>
   <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. <par/> 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>. </example>

   </description>
   <syntax>

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

   <see>Introduction to Groebner Bases in CoCoA</see>
   <see>GBasis5, and more</see>
   <type>groebner</type>
   <type>groebner-basic</type>
   <type>ideal</type>
   <type>list</type>
   <type>module</type>
   <type>cocoaserver</type>
   <key>syz</key>
   <key>syzygies</key>
   <key>syzygy</key>
 </command>