A spectral sequence for splines
نویسنده
چکیده
We de ne a complex R=J of graded modules on a d-dimensional simplicial complex , so that the top homology module of this complex consists of piecewise polynomial functions (splines) of smoothness r on the cone of . In a series of papers ([4], [5], [6]), Billera and Rose used a similar approach to study the dimension of the spaces of splines on , but with a complex substantially di erent fromR=J . We obtain bounds on the dimension of the homology modules Hi(R=J ), for all i < d, and nd a spectral sequence which relates these modules to the spline module. We use this to give simple conditions governing the projective dimension of the spline module. We also prove that if the spline module is free, then it is determined entirely by local data; that is, by the arrangements of hyperplanes incident to the various dimensional faces of .
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