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SimplicialComplexes :: dual(SimplicialComplex)

dual(SimplicialComplex) -- the Alexander dual of a simplicial complex

Synopsis

Description

The Alexander dual of D is the simplicial complex whose faces are the complements of the nonfaces of D. The Alexander dual of a square is the disjoint union of two edges.
i1 : R = ZZ[a..d];
i2 : D = simplicialComplex {a*b,b*c,c*d,d*a}

o2 = | cd ad bc ab |

o2 : SimplicialComplex
i3 : dual D

o3 = | bd ac |

o3 : SimplicialComplex

The Alexander dual is homotopy equivalent to the complement of D in the sphere generated by all of the variables in the ring of D. In particular, it depends on the number of variables.

i4 : R = ZZ[a..e]

o4 = R

o4 : PolynomialRing
i5 : E = simplicialComplex {a*b,b*c,c*d,d*a}

o5 = | cd ad bc ab |

o5 : SimplicialComplex
i6 : dual E

o6 = | bde ace abcd |

o6 : SimplicialComplex
The projective dimension of the face ring of D equals the regularity of the face ideal of the Alexander dual of D see e.g., Corollary 5.59 of Miller-Sturmfels, Combinatorial Commutative Algebra.
i7 : R = QQ[a..f];
i8 : D = simplicialComplex monomialIdeal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f,b*c*d,b*d*e,b*e*f,c*d*f,c*e*f)

o8 = | def aef bdf bcf acf cde bce abe acd abd |

o8 : SimplicialComplex
i9 : A = dual D

o9 = | def aef bdf bcf acf cde bce abe acd abd |

o9 : SimplicialComplex
i10 : pdim (R^1/(ideal D))

o10 = 3
i11 : regularity ideal A

o11 = 3

Alexander duality interchanges extremal betti numbers of the face ideals. Following example 3.2 in Bayer-Charalambous-Popescu, Extremal betti numbers and applications to monomial ideals, we have 

i12 : R = QQ[x0,x1,x2,x3,x4,x5,x6];
i13 : D = simplicialComplex {x0*x1*x3, x1*x3*x4, x1*x2*x4, x2*x4*x5,
                     x2*x3*x5, x3*x5*x6, x3*x4*x6, x0*x4*x6,
                     x0*x4*x5, x0*x1*x5, x1*x5*x6, x1*x2*x6,
                     x0*x2*x6, x0*x2*x3}

o13 = | x3x5x6 x1x5x6 x3x4x6 x0x4x6 x1x2x6 x0x2x6 x2x4x5 x0x4x5 x2x3x5 x0x1x5 x1x3x4 x1x2x4 x0x2x3 x0x1x3 |

o13 : SimplicialComplex
i14 : I = ideal D

o14 = ideal (x0*x1*x2, x1*x2*x3, x0*x1*x4, x0*x2*x4, x0*x3*x4, x2*x3*x4,
      -----------------------------------------------------------------------
      x0*x2*x5, x1*x2*x5, x0*x3*x5, x1*x3*x5, x1*x4*x5, x3*x4*x5, x0*x1*x6,
      -----------------------------------------------------------------------
      x0*x3*x6, x1*x3*x6, x2*x3*x6, x1*x4*x6, x2*x4*x6, x0*x5*x6, x2*x5*x6,
      -----------------------------------------------------------------------
      x4*x5*x6)

o14 : Ideal of R
i15 : J = ideal dual D

o15 = ideal (x0*x1*x2*x4, x0*x2*x3*x4, x0*x1*x2*x5, x1*x2*x3*x5, x0*x3*x4*x5,
      -----------------------------------------------------------------------
      x1*x3*x4*x5, x0*x1*x3*x6, x1*x2*x3*x6, x0*x1*x4*x6, x2*x3*x4*x6,
      -----------------------------------------------------------------------
      x0*x2*x5*x6, x0*x3*x5*x6, x1*x4*x5*x6, x2*x4*x5*x6)

o15 : Ideal of R
i16 : betti res I

             0  1  2  3  4 5
o16 = total: 1 21 49 42 15 2
          0: 1  .  .  .  . .
          1: .  .  .  .  . .
          2: . 21 49 42 14 2
          3: .  .  .  .  1 .

o16 : BettiTally
i17 : betti res J

             0  1  2 3 4
o17 = total: 1 14 21 9 1
          0: 1  .  . . .
          1: .  .  . . .
          2: .  .  . . .
          3: . 14 21 7 1
          4: .  .  . 2 .

o17 : BettiTally

See also