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 syz

syzygy modules

 Syntax
 Syz(L: LIST of RINGELEM): MODULE Syz(M: IDEAL|MODULE, Index: INT): MODULE

 Description
In the first two forms this function computes the syzygy module of a list of polynomials or module elements. 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. (***** NOT YET IMPLEMENTED *****)

The coefficient ring must be a field.

 Example
 /**/ use R ::= QQ[x,y,z]; /**/ indent(Syz([x^2-y-1, y^3-z, x^2-y, y^3-z])); SubmoduleRows(F, matrix( [y^3 -z, 0, 0, -x^2 +y +1], [0, 1, 0, -1], [x^2 -y, 0, -x^2 +y +1, 0], [0, 0, y^3 -z, -x^2 +y] )) ------------------------------- /**/ I := ideal(x, x, y); /**/ syz(gens(I)); submodule(FreeModule(..), [[1, -1, 0], [0, y, -x]]) /**/ SyzOfGens(I); submodule(FreeModule(..), [[1, -1, 0], [0, y, -x]]) Syz(I, 1); -- NOT YET IMPLEMENTED Module([[x, -y]]) ------------------------------- I := ideal(x^2-yz, xy-z^2, xyz); -- NOT YET IMPLEMENTED Syz(I,0); Module([x^2 - yz], [xy - z^2], [xyz]) ------------------------------- Syz(I,1); -- NOT YET IMPLEMENTED 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(-6)^2 --> R(-4)(+)R(-5)^3 --> R(-2)^2(+)R(-3) -------------------------------