On Projective Dimension of Spline Modules

AbstractLet a regionΩof the euclidean spaceRd(d⩾1) be decomposed as a polyhedral complex □, and letSr(□) denote the set of all multivariatecr-splines on □. Then, with pointwise operations, the setSr(□) turns out to be a finitely generated torsion free module over the ringR=R[x1, …,xd] of polynomials indvariables. In this paper, the results of Billera and Rose on the freeness of thisR-module on triangulated regions are extended to the projective dimension of this module and on arbitrary polygonal subdivisions. Possible relationships between the projective dimensions of the spline modules on subcomplexes have been established. Examples illustrating the theorems and counterexamples limiting the possibilities have been presented. In particular, an example showing that freeness of the spline moduleSr(□) is not a local concept for general polyhedral complexes, as against the triangulated ones, has been constructed