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 |
References:
| [1] | [B] Billera, L.: The algebra of continuous piecewise polynomials. Adv. Math.76, 170–183 (1989) · Zbl 0703.13015 · doi:10.1016/0001-8708(89)90047-9 |
| [2] | [BR1] Billera, L., Rose, L.: Modules of piecewise polynomials and their freeness. Math. Z.209, 485–497 (1992) · Zbl 0891.13004 · doi:10.1007/BF02570848 |
| [3] | [BR2] Billera, L., Rose, L.: A dimension series for multivariant splines. Discrete Comput. Geom. 6 (1991) |
| [4] | [G] Godement, R.: Topologie algébrique et théorie des faisccaux. Paris: Hermann 1958 |
| [5] | [K] Kunz, E.: Introduction to commutative algebra and algebraic geometry. Boston: Birkhäuser 1985 · Zbl 0563.13001 |
| [6] | [M] Matsumura, H.: Commutative algebra. New York: Benjamin 1970 · Zbl 0211.06501 |
| [7] | [Mu] Munkres, J.: Topological results in combinatorics. Mich. Math. J.31, 113–127 (1984) · Zbl 0585.57014 · doi:10.1307/mmj/1029002969 |
| [8] | [N] Northcott, D.: Finite free resolutions. London: Cambridge University Press 1976 · Zbl 0328.13010 |
| [9] | [R] Rose, L.: The structure of modules of splines over polynomial rings. Ph.D. Thesis. Cornell University (1988) |
| [10] | [Y1] Yuzvinsky, S.: Cohen-Macaulay rings of sections. Adv. Math.63, 172–195 (1987) · Zbl 0613.13011 · doi:10.1016/0001-8708(87)90052-1 |
| [11] | [Y2] Yuzvinsky, S.: Flasque sheaves on posets and Cohen-Macaulay unions of regular varieties. Adv. Math.73, 24–42 (1989) · Zbl 0681.13012 · doi:10.1016/0001-8708(89)90058-3 |
| [12] | [Y3] Yuzvinsky, S.: Cohen-Macaulay seminormalizations of unions of linear subspaces. J. Algebra132, 431–445 (1990) · Zbl 0705.13011 · doi:10.1016/0021-8693(90)90139-F |
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