Dualizes the chain complex cc, i.e., dc is the chain complex with modules dcp = (dual cc(-p)) and differentials dcp →dcp-1 the transpose of cc-p+1 →ccp
Other than the M2 method (dual,ChainComplex) we do not introduce an alternating sign.
i1 : R = QQ[x_1..x_4,z_1..z_4, T] o1 = R o1 : PolynomialRing |
i2 : I = ideal(z_2*z_3-z_1*z_4,x_4*z_3-x_3*z_4,x_2*z_2-x_1*z_4,x_4*z_1-x_3*z_2,x_2*z_1-x_1*z_3)
o2 = ideal (z z - z z , x z - x z , x z - x z , x z - x z , x z - x z )
2 3 1 4 4 3 3 4 2 2 1 4 4 1 3 2 2 1 1 3
o2 : Ideal of R
|
i3 : cc = res I
1 5 5 1
o3 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o3 : ChainComplex
|
i4 : betti cc
0 1 2 3
o4 = total: 1 5 5 1
0: 1 . . .
1: . 5 5 .
2: . . . 1
o4 : BettiTally
|
i5 : dc=dualComplex(cc)
1 5 5 1
o5 = 0 <-- R <-- R <-- R <-- R
-4 -3 -2 -1 0
o5 : ChainComplex
|
i6 : betti dc
-3 -2 -1 0
o6 = total: 1 5 5 1
0: . . . 1
1: . . . .
2: . . . .
3: . . 5 .
4: . . . .
5: . 5 . .
6: . . . .
7: . . . .
8: 1 . . .
o6 : BettiTally
|