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AlgebraicSplines :: idealsComplex

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 Δ.

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 , t ])
              0   1   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 , t ])
              0   1   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 HH1, 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 :