A Spectral Sequence for Splines

Abstract

We define a complex R RrJ J of graded modules on a d-dimensional simplicial complex ⌬, so that the top homology module of this complex consists of piecewise Ž . polynomial functions splines of smoothness r on the cone of ⌬. In a series of w Ž . papers, Billera and Rose Trans. Amer. Math. Soc. 310 1988 , 325᎐340; Comput. Ž . Ž . x Geom. 6 1991 , 107᎐128; Math. Z. 209 1992 , 485᎐497 used a similar approach to study the dimension of the spaces of splines on ⌬, but with a complex substantially different from R RrJ J. We obtain bounds on the dimension of the homology modules Ž . H R R rJ J for all i -d and find a spectral sequence which relates these modules to i 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 ⌬. ᮊ 1997 Academic Press

Subdivision and Spline Spaces

“…By Theorem 2.8, H k (Q) ≃ S r ( ∆ ′′ ), modulo constant splines. The result follows from the long exact sequence in homology associated with the short exact sequence of complexes in Proposition 2.6 coupled with Theorem 4.10 of [7].…”

“…As a consequence, we obtain dimension formulas and explicit bases for several commonly used subdivisions, their multivariate generalizations, as well as on various intermediate subdivisions. For these subdivisions, S r (∆ ′ ) is free, and a generalization [7] of Schumaker's lower bound for the planar case [10] gives the correct dimension.…”

Equivariant cohomology, syzygies and orbit structure

“…Moreover, it is implicit in papers of De Concini-Procesi-Vergne on transversally elliptic operators, splines and the infinitesimal index [13], [14]. It is also related to Schenck's work on splines and equivariant Chow groups of toric varieties [28], [29] and to the generalization of intersection cohomology for toric varieties studied by Barthel-Brasselet-Fieseler-Kaup [4].…”

Generalized splines and graphic arrangements

“…With our grading conventions, the total com-plexP •• has graded pieces i−j=kP ij labelled by the difference of i and j. The following result is essentially a translation of [16,Lemma 4.11] into our context.…”