On Projective Dimension of Spline Modules

✍ Scribed by Satya Deo

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 468 KB
• Volume: 84
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.0002

✦ Synopsis

Let a region 0 of the euclidean space R d (d 1) be decomposed as a polyhedral complex g, and let S r (g) denote the set of all multivariate c r -splines on g. Then, with pointwise operations, the set S r (g) turns out to be a finitely generated torsion free module over the ring R=R[x 1 , ..., x d ] of polynomials in d variables. In this paper, the results of Billera and Rose on the freeness of this R-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 module S r (g) is not a local concept for general polyhedral complexes, as against the triangulated ones, has been constructed.