A SPECTRAL SEQUENCE FOR SPLINES

We define a complex R/J of graded modules on a d-dimensional simplicia l complex Delta, so that the top homology module of this complex consi sts of piecewise polynomial functions (splines) of smoothness r on the cone of Delta. In a series of papers, Billera and Rose [Trans. Amer. Math. Sec. 310 (1988), 325-340; Comput. Geom. 6 (1991), 107-128; Math, Z. 209 (1992), 485-497] used a similar approach to study the dimensio n of the spaces of splines on Delta, but with a complex substantially different from R/J. We obtain bounds on the dimension of the homology modules H-i(R/J) for all i < d and find a spectral sequence which rela tes these modules to the spline module. We use this to give simple con ditions governing the projective dimension of the spline module. We al so prove that if the spline module is free, then it is determined enti rely by local data; that is, by the arrangements of hyperplanes incide nt to the various dimensional faces of Delta. (C) 1997 Academic Press.