Consider a cyclic polytope C of dimension d and vertices the variables of R, and the Stanley-Reisner ideal I of the boundary complex of C.
This function computes the minimal resolution of the unprojection deformation of R/I with base z (for d even).
This is obtained by doing a single unprojection step
Δ(d,m) and Δ(d-2,m-1) →Δ(d,m+1)
So for z = 0 (d even) we obtain the minimal resolution of I.
The unprojection variable (called T in the paper) corresponds to the last variable of R.
i1 : R=QQ[x_1..x_8]; |
i2 : Z=QQ[z]; |
i3 : cc=cycResDef(4,R,Z); |
i4 : isExact cc o4 = true |
i5 : S=ring cc o5 = S o5 : PolynomialRing |
i6 : betti cc
0 1 2 3 4
o6 = total: 1 16 30 16 1
0: 1 . . . .
1: . . . . .
2: . 16 30 16 .
3: . . . . .
4: . . . . 1
o6 : BettiTally
|
i7 : 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 | |
i8 : 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 |
|
i9 : 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 |
|
i10 : print cc.dd_4
{5} | x_8x_4x_6 |
{5} | x_8x_3x_6 |
{5} | x_8x_2x_6 |
{5} | -x_8x_3x_5 |
{5} | x_8x_2x_5 |
{5} | -x_8zx_5+x_1x_5x_7 |
{5} | x_8x_2x_4 |
{5} | -x_8zx_4+x_1x_4x_7 |
{5} | -x_8zx_3+x_1x_3x_7 |
{5} | -x_3x_5x_7 |
{5} | -x_2x_5x_7 |
{5} | -x_2x_4x_7 |
{5} | x_2x_4x_6 |
{5} | -x_1x_4x_6 |
{5} | -x_1x_3x_6 |
{5} | -x_1x_3x_5 |
|
i11 : C=delta(4,R)
o11 = {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
o11 : complex with 20 facets on the vertices x x x x x x x x
1 2 3 4 5 6 7 8
|
i12 : I=complexToIdeal C
o12 = 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
o12 : Ideal of R
|
i13 : betti res I
0 1 2 3 4
o13 = total: 1 16 30 16 1
0: 1 . . . .
1: . . . . .
2: . 16 30 16 .
3: . . . . .
4: . . . . 1
o13 : BettiTally
|
i14 : cs=substitute(cc,R)
1 16 30 16 1
o14 = R <-- R <-- R <-- R <-- R
0 1 2 3 4
o14 : ChainComplex
|
i15 : isExact cs o15 = true |
i16 : I==ideal(cs.dd_1) o16 = true |