A spectral sequence for splines (Q1366269)

Let \(\Delta\) be a connected finite \(d\)-dimensional simplicial complex contained in \(\mathbb{R}^d\). Let \(r\geq 0\) and \(R= \mathbb{R}[x_1, \dots, x_{d+1}]\). The author defines a complex \({\mathcal R}/{\mathcal I}\) of graded modules on \(\Delta\) as follows: \({\mathcal R} (\sigma) =R\) for every simplex \(\sigma\) and \({\mathcal I} (\sigma)= I_\sigma^{r+1}\), \(I_\sigma\) being the homogeneous ideal of the cone on \(\sigma\). \[ 0\to \bigoplus_{\sigma\in \Delta_d} R(\sigma)/ {\mathcal I} (\sigma) @>\partial_d>> \bigoplus_{\sigma \in\Delta_{d-1}} {\mathcal R} (\sigma)/{\mathcal I} (\sigma) @>\partial_{d-1}>> \cdots \bigoplus_{\sigma \in\Delta_0} {\mathcal R} (\sigma)/{\mathcal I} (\sigma) \to 0 \] where \(\partial_i\) is the usual simplicial boundary map. He obtains bounds on the dimension of \(H_i ({\mathcal R}/ {\mathcal I})\) for \(i<n\) and finds a spectral sequence which relates these modules to the spline module. This gives information on the freeness and the projective dimension of the spline module.