LOCAL COHOMOLOGY OF BIVARIATE SPLINES

We consider the problem of determining the dimension of the space of b ivariate splines C-k(r)(Delta), for all k. This problem is closely rel ated to the question of whether C-r(<(Delta)over cap>) is a free R-mod ule. The main result is that C-r(<(Delta)over cap>) is free if and onl y if \Delta\ has genus zero and C-k(r)(Delta) has the expected dimensi on for k=r+1 (and hence for all k). We also obtain several interesting corollaries, including the following simple non-freeness criterion: g iven a fixed Delta having an edge with both vertices interior, and whi ch does not extend to the boundary, there exists an r(0), which can be determined by inspection, such that C-r(<(Delta)over cap>) is not fre e for any r greater than or equal to r(0). (C) 1997 Elsevier Science B .V.