Cyclic Polytopes -- Constructing minimal resolutions for Stanley-Reisner rings of boundary complexes of cyclic polytopes

In the following example we construct the minimal resolution of the Stanley-Reisner ring of the codimension 4 cyclic polytope Δ(4,8) from those of the cyclic polytopes Δ(2,6) and Δ(4,7) (the last one being Pfaffian).

Of course this process can be iterated, for details see

J. Boehm, S. Papadakis: On the structure of Stanley-Reisner rings associated to cyclic polytopes, http://arxiv.org/abs/0912.2152

 `i1 : K=QQ;` ```i2 : C26=delta(2,K[z,x_2..x_6]) o2 = {z x , x x , x x , x x , z x , x x } 2 2 3 3 4 4 5 6 5 6 o2 : complex with 6 facets on the vertices z x x x x x 2 3 4 5 6``` ```i3 : R=K[z,x_1..x_7] o3 = R o3 : PolynomialRing``` ```i4 : J=sub(complexToIdeal C26,R) o4 = ideal (x x , x x , x x , x x , x x , z*x , x x , z*x , z*x ) 4 6 3 6 2 6 3 5 2 5 5 2 4 4 3 o4 : Ideal of R``` `i5 : c26=res J;` ```i6 : betti c26 0 1 2 3 4 o6 = total: 1 9 16 9 1 0: 1 . . . . 1: . 9 16 9 . 2: . . . . 1 o6 : BettiTally``` ```i7 : C47=delta(4,K[x_1..x_7]) o7 = {x x x x , x x x x , x x x x , x x x x , x x x x , 1 2 3 4 1 2 4 5 2 3 4 5 1 2 5 6 2 3 5 6 ------------------------------------------------------------------------ x x x x , x x x x , x x x x , x x x x , x x x x , x 3 4 5 6 1 2 3 7 1 3 4 7 1 4 5 7 1 2 6 7 2 ------------------------------------------------------------------------ x x x , x x x x , x x x x , x x x x } 3 6 7 3 4 6 7 1 5 6 7 4 5 6 7 o7 : complex with 14 facets on the vertices x x x x x x x 1 2 3 4 5 6 7``` ```i8 : I=sub(complexToIdeal C47,R) o8 = ideal (x x x , x x x , x x x , x x x , x x x , x x x , x x x ) 3 5 7 2 5 7 2 4 7 2 4 6 1 4 6 1 3 6 1 3 5 o8 : Ideal of R``` `i9 : c47=res I;` ```i10 : betti c47 0 1 2 3 o10 = total: 1 7 7 1 0: 1 . . . 1: . . . . 2: . 7 7 . 3: . . . . 4: . . . 1 o10 : BettiTally``` `i11 : cc=kustinMillerComplex(c47,c26,K[x_8]);` ```i12 : betti cc 0 1 2 3 4 o12 = total: 1 16 30 16 1 0: 1 . . . . 1: . . . . . 2: . 16 30 16 . 3: . . . . . 4: . . . . 1 o12 : BettiTally```

We compare with the combinatorics, i.e., check that the Kustin-Miller complex at the special fiber z=0 indeed resolves the Stanley-Reisner ring of Δ(4,8).

 `i13 : R'=K[x_1..x_8];` ```i14 : C48=delta(4,R') o14 = {x x x x , x x x x , x x x x , x x x x , x x x x , 1 2 3 4 1 2 4 5 2 3 4 5 1 2 5 6 2 3 5 6 ----------------------------------------------------------------------- x x x x , x x x x , x x x x , x x x x , x x x x , 3 4 5 6 1 2 6 7 2 3 6 7 3 4 6 7 4 5 6 7 ----------------------------------------------------------------------- x x x x , x x x x , x x x x , x x x x , x x x x , 1 2 3 8 1 3 4 8 1 4 5 8 1 5 6 8 1 2 7 8 ----------------------------------------------------------------------- x x x x , x x x x , x x x x , x x x x , x x x x } 2 3 7 8 3 4 7 8 4 5 7 8 1 6 7 8 5 6 7 8 o14 : complex with 20 facets on the vertices x x x x x x x x 1 2 3 4 5 6 7 8``` ```i15 : I48=complexToIdeal C48 o15 = ideal (x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , 4 6 8 3 6 8 2 6 8 3 5 8 2 5 8 2 4 8 3 5 7 2 5 7 ----------------------------------------------------------------------- x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x ) 1 5 7 2 4 7 1 4 7 1 3 7 2 4 6 1 4 6 1 3 6 1 3 5 o15 : Ideal of R'``` ```i16 : betti res I48 0 1 2 3 4 o16 = total: 1 16 30 16 1 0: 1 . . . . 1: . . . . . 2: . 16 30 16 . 3: . . . . . 4: . . . . 1 o16 : BettiTally``` ```i17 : I48==sub(ideal cc.dd_1,R') o17 = true```

We finish the example by printing the differentials of the Kustin-Miller complex:

 ```i18 : print cc.dd_1 | x_1x_3x_5 x_1x_3x_6 x_1x_4x_6 x_2x_4x_6 x_2x_4x_7 x_2x_5x_7 x_3x_5x_7 x_8zx_3-x_1x_3x_7 x_8zx_4-x_1x_4x_7 x_8x_2x_4 x_8zx_5-x_1x_5x_7 x_8x_2x_5 x_8x_3x_5 x_8x_2x_6 x_8x_3x_6 x_8x_4x_6 |``` ```i19 : print cc.dd_2 {3} | 0 0 -x_6 -x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 | {3} | 0 -x_4 x_5 0 0 0 0 0 0 0 0 0 0 0 0 x_7 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 | {3} | -x_2 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_7 0 0 0 0 0 0 0 x_8 0 0 0 0 | {3} | x_1 0 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 | {3} | 0 0 0 0 0 -x_5 x_6 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 | {3} | 0 0 0 0 -x_3 x_4 0 0 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 | {3} | 0 0 0 x_1 x_2 0 0 0 0 0 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 | {3} | 0 0 0 0 0 0 0 0 x_4 0 x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 x_2 -x_3 0 0 0 0 x_5 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 -z 0 0 0 0 0 0 x_5 0 0 0 x_6 0 0 0 0 0 0 0 -x_6 -x_7 0 0 | {3} | 0 0 0 0 0 0 0 0 0 x_2 -x_3 x_3 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 -z 0 0 x_3 0 -x_4 0 0 0 0 0 0 x_6 0 0 0 0 0 0 -x_7 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 -z -x_2 0 0 0 0 0 0 0 0 0 x_6 -x_1 0 0 0 0 0 -x_7 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 x_4 0 -x_5 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -z -x_2 0 0 0 x_4 0 -x_5 0 -x_1 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -z 0 -x_2 -x_3 0 0 0 0 -x_1 0 0 0 0 |``` ```i20 : print cc.dd_3 {4} | 0 0 0 x_7 0 0 0 0 0 -x_8 0 0 0 0 0 0 | {4} | 0 0 0 0 -x_7 0 0 0 0 0 -x_8 0 0 0 0 0 | {4} | 0 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 0 | {4} | 0 0 0 0 0 0 x_6 0 0 0 0 0 -x_8 0 0 0 | {4} | -x_1 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 | {4} | 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 | {4} | 0 0 0 x_1 0 0 0 0 0 0 0 0 0 0 0 -x_8 | {4} | 0 -x_5 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 -x_5 0 x_6 0 0 0 0 0 0 0 0 0 0 0 | {4} | -x_3 x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 x_4 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 | {4} | x_2 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 | {4} | -z 0 0 0 0 0 0 -x_6 0 0 0 0 0 -x_7 0 0 | {4} | 0 x_2 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 -z 0 0 0 0 0 0 -x_6 0 0 0 0 0 -x_7 0 | {4} | 0 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 | {4} | 0 0 0 x_2 x_3 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 -z 0 0 0 0 x_5 x_1 0 0 0 0 0 0 | {4} | 0 0 0 -z 0 x_3 0 0 0 x_1 0 0 0 0 0 0 | {4} | 0 0 0 0 -z -x_2 0 0 0 0 -x_1 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 x_3 -x_4 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 -z -x_2 0 0 0 -x_1 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 -x_6 -x_7 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 -x_4 x_5 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 -x_2 x_3 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 x_1 0 0 0 0 0 -x_7 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 x_6 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_3 x_4 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 x_1 x_2 0 0 |```