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 2.2.25 Syzygies and Resolution Example
The following example, among other things, computes the resolution of ideals of sets of points.

 Example
 ``` Use R ::= QQ[x,y,z]; X1 := [[0,0,1],[1,0,1],[2,0,1],[2,1,1]]; -- 4 points in the projective -- plane X2 := [[0,0,1],[1,0,1],[0,1,1],[1,1,1]]; -- 4 more points I1 := IdealOfProjectivePoints(X1); I2 := IdealOfProjectivePoints(X2); Hilbert(R/I1); -- the Hilbert function of X1 H(0) = 1 H(1) = 3 H(x) = 4 for t >= 2 ------------------------------- Hilbert(R/I2) = Hilbert(R/I1); -- The Hilbert functions for X1 and X2 -- are the same TRUE ------------------------------- Res(R/I1); -- but the resolutions ... 0 --> R(-3)(+)R(-4) --> R^2(-2)(+)R(-3) --> R ------------------------------- Res(R/I2); -- are different. 0 --> R(-4) --> R^2(-2) --> R ------------------------------- Describe Res(R/I1); -- more information about the resolution for X1 Mat([ [xy - 2yz, y^2 - yz, x^3 - 3x^2z + 2xz^2] ]) Mat([ [y - z, x^2 - xz], [-x + 2z, 0], [0, -y] ]) ------------------------------- Syz(I1,1); -- the first syzygy module for X1 Module([y - z, -x + 2z, 0], [x^2 - xz, 0, -y]) ------------------------------- ```