TY - JOUR

T1 - Local cohomology of bivariate splines

AU - Schenck, Hal

AU - Stillman, Mike

N1 - Funding Information:
* Corresponding author. ’ Partially supported by the US Army Research Office through the Mathematical Sciences Institute of Cornell University, contract DAAL 03-92-G-0126. 2 Partially supported by the National Science Foundation through grant DMS 9210805.

PY - 1997/5

Y1 - 1997/5

N2 - We consider the problem of determining the dimension of the space of bivariate splines Ckr(Δ), for all k. This problem is closely related to the question of whether Cr(Δ̂) is a free R-module. The main result is that Cr(Δ̂) is free if and only if |Δ| has genus zero and Ckr(Δ) has the expected dimension for k = r + 1 (and hence for all k). We also obtain several interesting corollaries, including the following simple non-freeness criterion: given a fixed Δ having an edge with both vertices interior, and which does not extend to the boundary, there exists an r0, which can be determined by inspection, such that Cr(Δ̂) is not free for any r ≥ r0.

AB - We consider the problem of determining the dimension of the space of bivariate splines Ckr(Δ), for all k. This problem is closely related to the question of whether Cr(Δ̂) is a free R-module. The main result is that Cr(Δ̂) is free if and only if |Δ| has genus zero and Ckr(Δ) has the expected dimension for k = r + 1 (and hence for all k). We also obtain several interesting corollaries, including the following simple non-freeness criterion: given a fixed Δ having an edge with both vertices interior, and which does not extend to the boundary, there exists an r0, which can be determined by inspection, such that Cr(Δ̂) is not free for any r ≥ r0.

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U2 - 10.1016/S0022-4049(97)00026-1

DO - 10.1016/S0022-4049(97)00026-1

M3 - Article

AN - SCOPUS:0031142011

VL - 117-118

SP - 535

EP - 548

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

ER -