chainComplex(CellComplex) -- compute the cellular chain complex for a cell complex

Synopsis

Function: chainComplex
Usage:

chainComplex C
Inputs:
- C, a cell complex, the cell complex for which to compute the chain complex
Optional inputs:
- Reduced => a Boolean value, default value true, that controls whether the augmented chain complexes is used
- Prune => a Boolean value, default value true, that controls whether the modules in the chain complex are pruned
Outputs:
- a chain complex, the dimension of the complex

Description

This constructs the cellular chain complex for a cell complex, taking into account the labels on the cells. The resulting cell complex will be a complex of modules over the ring associated to the cell complex. By default, the option "Reduced" is set to true, so the resulting ChainComplex has a rank 1 free module in homological degree -1.

i1 : R = QQ[x]

o1 = R

o1 : PolynomialRing

i2 : a = newSimplexCell({},x);

i3 : b1 = newCell {a,a};

i4 : b2 = newCell {a,a};

i5 : C = cellComplex(R,{b1,b2});

i6 : chainComplex C

      1      1      2
o6 = R  <-- R  <-- R
                    
     -1     0      1

o6 : ChainComplex

i7 : chainComplex(C,Reduced=>false)

      1      2
o7 = R  <-- R
             
     0      1

o7 : ChainComplex

For details see Combinatorial Commutative Algebra Section 4.1. If we restrict to the case of monomial labels, then, subject to some acyclicity conditions, the resulting complex will be (up to a shift by 1 in homological degree) a resolution of $S/I$ where $I$ is the ideal generated by the labels of the vertices. With the "Reduced" option set to false, the resulting complex will be a resolution of the ideal $I$ as a module.

i8 : R = QQ[x,y,z];

i9 : I = ideal(x,y,z);

o9 : Ideal of R

i10 : vx = newSimplexCell({},x);

i11 : vy = newSimplexCell({},y);

i12 : vz = newSimplexCell({},z);

i13 : exy = newSimplexCell({vx,vy});

i14 : exz = newSimplexCell({vx,vz});

i15 : eyz = newSimplexCell({vy,vz});

i16 : f = newSimplexCell({exy,exz,eyz});

i17 : C = cellComplex(R,{f});

i18 : betti chainComplex(C)[-1]

             0 1 2 3
o18 = total: 1 3 3 1
          0: 1 3 3 1

o18 : BettiTally

i19 : assert(betti chainComplex(C)[-1] == betti res I);

The option "Prune," also defaulted to true, controls whether the modules in the complex are pruned before being returned. With the "Prune" option set to the default of true, the resulting complex is visually nicer. However, unless the labeling ring is fine graded, some of the structure will be lost.

i20 : R = QQ[x,y]

o20 = R

o20 : PolynomialRing

i21 : a = newSimplexCell({},x);

i22 : b = newSimplexCell({},y);

i23 : e = newCell {a,b};

i24 : C = cellComplex(R,{e});

i25 : chainComplex C

       1      2      1
o25 = R  <-- R  <-- R
                     
      -1     0      1

o25 : ChainComplex

i26 : chainComplex(C,Prune=>false)

       1
o26 = R  <-- image | x 0 | <-- image | xy |
                   | 0 y |      
      -1                       1
             0

o26 : ChainComplex

Ways to use this method: