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 Syz

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

 Description
In the first two forms this function computes the syzygy module of a list of polynomials or vectors. SyzOfGens(I) is the same as Syz(Gens(I)).

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.

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

 Example
 ``` Use R ::= QQ[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, x, y); Syz(Gens(I)); SyzOfGens(I); Syz(I, 1); Module([[1, -1, 0], [0, y, -x]]) ------------------------------- Module([[1, -1, 0], [0, y, -x]]) ------------------------------- Module([[x, -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) ------------------------------- ```