# Tom -- The Kustin-Miller complex for Tom

The Kustin-Miller complex construction for the Tom example which can be found in Section 5.5 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 codimension 4.

 ```i1 : R=QQ[x_1..x_4,z_1..z_4] o1 = R o1 : PolynomialRing``` ```i2 : b2=matrix {{ 0, x_1, x_2, x_3, x_4}, {-x_1, 0, 0, z_1, z_2}, {-x_2, 0, 0, z_3, z_4}, {-x_3, -z_1, -z_3, 0, 0}, {-x_4, -z_2, -z_4, 0, 0}}; 5 5 o2 : Matrix R <--- R``` `i3 : cI=resBE b2;` ```i4 : betti cI 0 1 2 3 o4 = total: 1 5 5 1 0: 1 . . . 1: . 5 5 . 2: . . . 1 o4 : BettiTally``` ```i5 : J = ideal (z_1..z_4) o5 = ideal (z , z , z , z ) 1 2 3 4 o5 : Ideal of R``` ```i6 : cJ=res J 1 4 6 4 1 o6 = R <-- R <-- R <-- R <-- R <-- 0 0 1 2 3 4 5 o6 : ChainComplex``` ```i7 : betti cJ 0 1 2 3 4 o7 = total: 1 4 6 4 1 0: 1 4 6 4 1 o7 : BettiTally``` `i8 : cc=kustinMillerComplex(cI,cJ,QQ[T]);` ```i9 : S=ring cc o9 = S o9 : PolynomialRing``` ```i10 : cc 1 9 16 9 1 o10 = S <-- S <-- S <-- S <-- S 0 1 2 3 4 o10 : ChainComplex``` ```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``` ```i12 : isExactRes cc o12 = true``` ```i13 : print cc.dd_1 | z_2z_3-z_1z_4 -x_4z_3+x_3z_4 x_4z_1-x_3z_2 x_2z_2-x_1z_4 -x_2z_1+x_1z_3 -x_1x_3+Tz_1 -x_1x_4+Tz_2 -x_2x_3+Tz_3 -x_2x_4+Tz_4 |``` ```i14 : print cc.dd_2 {2} | 0 x_1 x_2 x_3 x_4 0 0 0 0 0 0 T 0 0 0 0 | {2} | -x_1 0 0 z_1 z_2 0 0 -x_1 0 0 x_2 0 T 0 0 0 | {2} | -x_2 0 0 z_3 z_4 -x_1 0 0 -x_2 0 0 0 0 T 0 0 | {2} | -x_3 -z_1 -z_3 0 0 0 0 -x_3 -x_3 -x_4 0 -x_3 0 0 T 0 | {2} | -x_4 -z_2 -z_4 0 0 0 x_3 0 0 0 0 0 0 0 0 T | {2} | 0 0 0 0 0 z_2 z_3 0 z_4 0 0 z_4 0 -x_4 0 x_2 | {2} | 0 0 0 0 0 -z_1 0 z_3 0 z_4 0 0 0 x_3 -x_2 0 | {2} | 0 0 0 0 0 0 -z_1 -z_2 0 0 z_4 -z_2 x_4 0 0 -x_1 | {2} | 0 0 0 0 0 0 0 0 -z_1 -z_2 -z_3 0 -x_3 0 x_1 0 |``` ```i15 : print cc.dd_3 {3} | 0 -z_2 0 z_4 -T 0 0 x_3 0 | {3} | x_3 x_4 0 0 0 -T 0 0 0 | {3} | 0 0 -x_3 -x_4 0 0 -T 0 0 | {3} | -x_1 0 x_2 0 0 0 0 -T 0 | {3} | 0 -x_1 0 x_2 0 0 0 0 -T | {3} | -z_3 -z_4 0 0 0 x_2 0 0 0 | {3} | z_2 0 -z_4 0 0 0 0 x_4 0 | {3} | -z_1 0 0 -z_4 0 0 -x_2 -x_3 0 | {3} | 0 z_2 z_3 0 0 -x_1 0 -x_3 0 | {3} | 0 -z_1 0 z_3 0 0 0 0 -x_3 | {3} | 0 0 -z_1 -z_2 0 0 -x_1 0 0 | {3} | 0 0 0 0 0 x_1 x_2 x_3 x_4 | {3} | 0 0 0 0 -x_1 0 0 z_1 z_2 | {3} | 0 0 0 0 -x_2 0 0 z_3 z_4 | {3} | 0 0 0 0 -x_3 -z_1 -z_3 0 0 | {3} | 0 0 0 0 -x_4 -z_2 -z_4 0 0 |``` ```i16 : print cc.dd_4 {4} | -x_2x_4+Tz_4 | {4} | x_2x_3-Tz_3 | {4} | -x_1x_4+Tz_2 | {4} | x_1x_3-Tz_1 | {4} | z_2z_3-z_1z_4 | {4} | -x_4z_3+x_3z_4 | {4} | x_4z_1-x_3z_2 | {4} | x_2z_2-x_1z_4 | {4} | -x_2z_1+x_1z_3 |```