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

## Synopsis

• Usage:
C = idealsComplex(V,F,r)
• Inputs:
• V, a list, list of vertex coordinates of $\Delta$
• F, a list, list of facets of $\Delta$ (each facet is recorded as a list of indices of vertices taken from V)
• r, an integer, integer, desired degree of smoothness
• Optional inputs:
• BaseRing (missing documentation) => a ring, default value null,
• Homogenize (missing documentation) => , default value true,
• CoefficientRing (missing documentation) => a ring, default value QQ,
• VariableName (missing documentation) => , default value t,
• Outputs:
• C, , Billera-Schenck-Stillman chain complex of ideals

## 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.

• cellularComplex -- create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary