Modules of splines on polyhedral complexes. (English) Zbl 0848.52003
For a finite polyhedral complex \(\Delta\) in real \(d\)-space let \(S^r (\Delta)\) denote the set of all \(r\) times differential splines (piecewise polynomial functions) on \(\Delta\). This is a module over the ring \(R\) of polynomial functions on \(d\)-space.
The main result of the paper is a characterization of the projective dimension \(\text{pd}_R (S^r (\Delta))\) of \(S^r (\Delta)\) in terms of \(\Delta\). For simplicial complexes \(\Delta\) the projective dimension of \(S^0 (\Delta)\) is determined by the combinatorics of \(\Delta\). However, this is not true for arbitrary \(\Delta\). There exist combinatorially equivalent complexes \(\Delta\) in 3-space such that the corresponding modules \(S^0 (\Delta)\) have different projective dimensions.
The main result of the paper is a characterization of the projective dimension \(\text{pd}_R (S^r (\Delta))\) of \(S^r (\Delta)\) in terms of \(\Delta\). For simplicial complexes \(\Delta\) the projective dimension of \(S^0 (\Delta)\) is determined by the combinatorics of \(\Delta\). However, this is not true for arbitrary \(\Delta\). There exist combinatorially equivalent complexes \(\Delta\) in 3-space such that the corresponding modules \(S^0 (\Delta)\) have different projective dimensions.
Reviewer: E.Schulte (MR 93h:52015)
MSC:
52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
13D05 | Homological dimension and commutative rings |
41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
52B99 | Polytopes and polyhedra |
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