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

## Synopsis

• Function: dual
• Usage:
dual D
• Inputs:
• Outputs:
• , the Alexander dual of D

## 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