The Kustin-Miller complex construction for the Jerry example which can be found in Section 5.7 of
Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geometry 13 (2004) 249-268, http://arxiv.org/abs/math/0111195
Here we pass from a Pfaffian to a codimension 4 variety.
i1 : R = QQ [x_1..x_3, z_1..z_4]
o1 = R
o1 : PolynomialRing
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i2 : I = ideal(-z_2*z_3+z_1*x_1,-z_2*z_4+z_1*x_2,-z_3*z_4+z_1*x_3,-z_3*x_2+z_2*x_3,z_4*x_1-z_3*x_2)
o2 = ideal (x z - z z , x z - z z , x z - z z , x z - x z , - x z +
1 1 2 3 2 1 2 4 3 1 3 4 3 2 2 3 2 3
------------------------------------------------------------------------
x z )
1 4
o2 : Ideal of R
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i3 : cI=res I
1 5 5 1
o3 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o3 : ChainComplex
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i4 : betti cI
0 1 2 3
o4 = total: 1 5 5 1
0: 1 . . .
1: . 5 5 .
2: . . . 1
o4 : BettiTally
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i5 : J = ideal (z_1..z_4)
o5 = ideal (z , z , z , z )
1 2 3 4
o5 : Ideal of R
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i6 : cJ=res J
1 4 6 4 1
o6 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o6 : ChainComplex
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i7 : betti cJ
0 1 2 3 4
o7 = total: 1 4 6 4 1
0: 1 4 6 4 1
o7 : BettiTally
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i8 : cc=kustinMillerComplex(cI,cJ,QQ[T]);
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i9 : S=ring cc
o9 = S
o9 : PolynomialRing
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i10 : cc
1 9 16 9 1
o10 = S <-- S <-- S <-- S <-- S
0 1 2 3 4
o10 : ChainComplex
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i11 : betti cc
0 1 2 3 4
o11 = total: 1 9 16 9 1
0: 1 . . . .
1: . 9 16 9 .
2: . . . . 1
o11 : BettiTally
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i12 : isExactRes cc
o12 = true
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i13 : print cc.dd_1
| x_1z_1-z_2z_3 x_2z_1-z_2z_4 x_3z_1-z_3z_4 x_3z_2-x_2z_3 x_2z_3-x_1z_4 Tz_1-x_1z_4 -x_1x_2+Tz_2 -x_1x_3+Tz_3 -x_2x_3+Tz_4 |
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i14 : print cc.dd_2
{2} | -x_2 -x_3 0 0 z_4 0 0 0 0 0 0 T 0 0 0 0 |
{2} | x_1 0 -x_3 z_3 -z_3 -x_1 0 0 0 0 0 0 T 0 0 0 |
{2} | 0 x_1 x_2 -z_2 0 0 -x_1 0 -x_2 0 0 0 0 T 0 0 |
{2} | 0 -z_3 -z_4 z_1 0 0 0 -x_1 0 -x_2 0 0 0 0 T 0 |
{2} | -z_2 -z_3 0 0 z_1 0 0 0 -z_4 -x_2 -x_3 x_1 0 0 0 T |
{2} | 0 0 0 0 0 z_2 z_3 0 z_4 0 0 -x_1 -x_2 -x_3 0 0 |
{2} | 0 0 0 0 0 -z_1 0 z_3 0 z_4 0 z_3 z_4 0 -x_3 0 |
{2} | 0 0 0 0 0 0 -z_1 -z_2 0 0 z_4 0 0 z_4 x_2 -x_2 |
{2} | 0 0 0 0 0 0 0 0 -z_1 -z_2 -z_3 0 0 0 0 x_1 |
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i15 : print cc.dd_3
{3} | 0 -z_4 0 x_3 -T 0 0 0 0 |
{3} | 0 0 -z_4 -x_2 0 -T 0 0 0 |
{3} | 0 0 z_3 x_1 0 0 -T x_1 0 |
{3} | -x_1 -x_2 0 0 0 0 0 -T 0 |
{3} | 0 -x_2 -x_3 0 0 0 0 0 -T |
{3} | -z_3 -z_4 0 0 0 0 0 -x_3 0 |
{3} | z_2 0 -z_4 0 0 0 0 x_2 -x_2 |
{3} | -z_1 0 0 -z_4 x_2 0 0 -z_4 0 |
{3} | 0 z_2 z_3 0 0 0 0 0 x_1 |
{3} | 0 -z_1 0 z_3 -x_1 0 0 0 0 |
{3} | 0 0 -z_1 -z_2 0 -x_1 0 0 0 |
{3} | 0 0 0 0 -x_2 -x_3 0 0 z_4 |
{3} | 0 0 0 0 x_1 0 -x_3 z_3 -z_3 |
{3} | 0 0 0 0 0 x_1 x_2 -z_2 0 |
{3} | 0 0 0 0 0 -z_3 -z_4 z_1 0 |
{3} | 0 0 0 0 -z_2 -z_3 0 0 z_1 |
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i16 : print cc.dd_4
{4} | -x_2x_3+Tz_4 |
{4} | x_1x_3-Tz_3 |
{4} | -x_1x_2+Tz_2 |
{4} | -Tz_1+x_1z_4 |
{4} | -x_3z_1+z_3z_4 |
{4} | x_2z_1-z_2z_4 |
{4} | -x_1z_1+z_2z_3 |
{4} | x_2z_3-x_1z_4 |
{4} | -x_3z_2+x_2z_3 |
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