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}};
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i2 : F = {{0,1,2},{0,2,3},{0,1,3}};
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i3 : C = idealsComplex(V,F,1); |
i4 : prune HH C
o4 = 0 : 0
2
1 : (QQ[t ..t ])
0 2
2 : 0
o4 : GradedModule
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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}};
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i6 : F = {{0,1,2},{0,1,3,4},{1,2,4,5},{0,2,3,5}};
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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
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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.
The object idealsComplex is a method function with options.