# coComplex -- Make a co-complex.

## Synopsis

• Usage:
coComplex(R,facelist)
coComplex(R,facelist,facetlist)
coComplex(R,facelist,Rdual)
coComplex(R,facelist,facetlist,Rdual)
• Inputs:
• R, ,
• facelist, a list, of the faces, sorted by dimension
• facetlist, a list, of the facets, sorted by dimension
• Rdual, ,
• Outputs:

## Description

Make a co-complex from a list of faces and/or facets.

This is mostly used internally but may be occasionally useful for the end user.

 i1 : R=QQ[x_0..x_5] o1 = R o1 : PolynomialRing i2 : C=boundaryCyclicPolytope(3,R) o2 = 2: x x x x x x x x x x x x x x x x x x x x x x x x 0 1 2 0 2 3 0 3 4 0 1 5 1 2 5 2 3 5 0 4 5 3 4 5 o2 : complex of dim 2 embedded in dim 5 (printing facets) equidimensional, simplicial, F-vector {1, 6, 12, 8, 0, 0, 0}, Euler = 1 i3 : grading R o3 = | -1 -1 -1 -1 -1 | | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 1 0 0 | | 0 0 0 1 0 | | 0 0 0 0 1 | 6 5 o3 : Matrix ZZ <--- ZZ i4 : dC=dualize C o4 = 2: v v v v v v v v v v v v v v v v v v v v v v v v 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 o4 : co-complex of dim 2 embedded in dim 5 (printing facets) equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1 i5 : fdC=fc dC o5 = {{}, {}, {}, {v v v , v v v , v v v , v v v , v v v , v v v , v v v , 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 ------------------------------------------------------------------------ v v v }, {v v v v , v v v v , v v v v , v v v v , v v v v , v v v v , 3 4 5 0 1 2 3 0 1 2 4 0 1 2 5 0 1 3 4 0 1 4 5 0 2 3 4 ------------------------------------------------------------------------ v v v v , v v v v , v v v v , v v v v , v v v v , v v v v }, 0 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 ------------------------------------------------------------------------ {v v v v v , v v v v v , v v v v v , v v v v v , v v v v v , 0 1 2 3 4 0 1 2 3 5 0 1 2 4 5 0 1 3 4 5 0 2 3 4 5 ------------------------------------------------------------------------ v v v v v }, {v v v v v v }} 1 2 3 4 5 0 1 2 3 4 5 o5 : List i6 : Rdual=simplexRing dC o6 = Rdual o6 : PolynomialRing i7 : grading Rdual o7 = | -1 -1 -1 -1 5 | | -1 -1 -1 5 -1 | | -1 -1 5 -1 -1 | | -1 5 -1 -1 -1 | | 5 -1 -1 -1 -1 | | -1 -1 -1 -1 -1 | 6 5 o7 : Matrix QQ <--- QQ i8 : dC1=coComplex(Rdual,fdC) o8 = 2: v v v v v v v v v v v v v v v v v v v v v v v v 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 o8 : co-complex of dim 2 embedded in dim 5 (printing facets) equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1 i9 : dC==dC1 o9 = true

## Caveat

If both the list of faces and facets is specified there is no consistency check.