Local cohomology of bivariate splines

We consider the problem of determining the dimension of the space of bivariate splines C_k^r(Δ), for all k. This problem is closely related to the question of whether C^r(Δ̂) is a free R-module. The main result is that C^r(Δ̂) is free if and only if |Δ| has genus zero and C_k^r(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 C^r(Δ̂) is not free for any r ≥ r₀.