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idealsComplex -- creates the Billera-Schenck-Stillman chain complex of ideals

Synopsis

Description

This method returns the Billera-Schenck-Stillman chain complex of ideals whose top homology is the module of non-trivial splines on $\Delta$.

i1 : V = {{0,0},{1,0},{0,1},{-1,-1}};
 
i2 : F = {{0,1,2},{0,2,3},{0,1,3}};
 
i3 : C = idealsComplex(V,F,1);
 
i4 : prune HH C

o4 = 0 : 0            

                     2
     1 : (QQ[t ..t ])
              0   2

     2 : 0            

o4 : GradedModule

The output from the above example shows that there is only one nonvanishing homology, and it is free as a module over the polynomial ring in three variables.

i5 : V = {{-1,-1},{1,-1},{0,1},{-2,-2},{2,-2},{0,2}};
 
i6 : F = {{0,1,2},{0,1,3,4},{1,2,4,5},{0,2,3,5}};
 
i7 : C = idealsComplex(V,F,1);
 
i8 : prune HH C

o8 = 0 : cokernel {2} | 8t_0 0        8t_1-2t_2 -2t_2    -2t_2    0        |
                  {2} | t_2  8t_0+t_2 t_2       8t_1+t_2 t_2      t_2      |
                  {2} | -t_2 -t_2     t_2       t_2      8t_1+t_2 8t_0-t_2 |

                     3
     1 : (QQ[t ..t ])
              0   2                                                         

     2 : 0                                                                  

o8 : GradedModule

The output from the above example shows that there are two nonvanishing homologies, but the spline module, which is (almost) the homology HH_1, is still free. This shows that freeness of the spline module does not depend on vanishing of lower homologies if the underlying complex is polyhedral.

See also

Ways to use idealsComplex :

For the programmer

The object idealsComplex is a method function with options.