Local cohomology of bivariate splines Original Research Article

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 image is a free R-module. The main result is that image 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 image is not free for any r ≥ r0.