Macaulay2 » Documentation
Packages » KustinMiller :: Cyclic Polytopes
next | previous | forward | backward | up | index | toc

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 cyclic polytope $\Delta(4,8)$ of embedding codimension 4 (as a subcomplex of the simplex on 8 vertices) from those of the cyclic polytopes $\Delta(2,6)$ and $\Delta(4,7)$ (the last one being Pfaffian).

This process can be iterated to give a recursive construction of the resolutions of all cyclic polytopes, for details see

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

i1 : K=QQ;
 
i2 : C26=delta(2,K[z,x_2..x_6])

o2 = simplicialComplex | x_5x_6 zx_6 x_4x_5 x_3x_4 x_2x_3 zx_2 |

o2 : SimplicialComplex
 
i3 : R=K[z,x_1..x_7]

o3 = R

o3 : PolynomialRing
 
i4 : J=sub(ideal C26,R)

o4 = ideal (z*x , z*x , x x , z*x , x x , x x , x x , x x , x x )
               3     4   2 4     5   2 5   3 5   2 6   3 6   4 6

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 = simplicialComplex | x_4x_5x_6x_7 x_1x_5x_6x_7 x_3x_4x_6x_7 x_2x_3x_6x_7 x_1x_2x_6x_7 x_1x_4x_5x_7 x_1x_3x_4x_7 x_1x_2x_3x_7 x_3x_4x_5x_6 x_2x_3x_5x_6 x_1x_2x_5x_6 x_2x_3x_4x_5 x_1x_2x_4x_5 x_1x_2x_3x_4 |

o7 : SimplicialComplex
 
i8 : I=sub(ideal 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 )
             1 3 5   1 3 6   1 4 6   2 4 6   2 4 7   2 5 7   3 5 7

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, that is, check that the Kustin-Miller complex at the special fiber z=0 indeed resolves the Stanley-Reisner ring of $\Delta(4,8)$.

i13 : R'=K[x_1..x_8];
 
i14 : C48=delta(4,R')

o14 = simplicialComplex | x_5x_6x_7x_8 x_1x_6x_7x_8 x_4x_5x_7x_8 x_3x_4x_7x_8 x_2x_3x_7x_8 x_1x_2x_7x_8 x_1x_5x_6x_8 x_1x_4x_5x_8 x_1x_3x_4x_8 x_1x_2x_3x_8 x_4x_5x_6x_7 x_3x_4x_6x_7 x_2x_3x_6x_7 x_1x_2x_6x_7 x_3x_4x_5x_6 x_2x_3x_5x_6 x_1x_2x_5x_6 x_2x_3x_4x_5 x_1x_2x_4x_5 x_1x_2x_3x_4 |

o14 : SimplicialComplex
 
i15 : I48=ideal 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 ,
              1 3 5   1 3 6   1 4 6   2 4 6   1 3 7   1 4 7   2 4 7   1 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 )
       2 5 7   3 5 7   2 4 8   2 5 8   3 5 8   2 6 8   3 6 8   4 6 8

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    |

See also