On projective dimension of spline modules (Q1907510)

A finite collection \(\Delta\) of convex polyhedra in \(\mathbb{R}^d\) is called a polyhedral complex if any face of a member of \(\Delta\) is a member of \(\Delta\), and if the intersection of any two members of \(\Delta\) is a face of both. A simplicial complex is a special case of a polyhedral complex when all faces of \(\Delta\) are simplexes. Consider the set \(S^r (\Delta)\) of all piecewise polynomials on \(\Delta\) which are continuously differentiable of order \(r\geq 0\). With respect to pointwise operations of addition and multiplication, \(S^r (\Delta)\) is an \(R\)-module and it is called the spline module of \(C^r\)-splines on \(\Delta\). The paper deals with the following question: If \(\Delta'\) is a \(d\)-subcomplex of \(\Delta\), when can we say that the projective dimension of \(S^r (\Delta')\) is less than or equal to the projective dimension of \(S^r (\Delta)\)? A wide class of \(d\)-subcomplexes satisfying the desired monotonicity of projective dimension can be characterized. For simplicial complexes, the projective dimension of \(S^r (\Delta\) is proved to be a local concept. This result generalizes earlier observations by Billera and Rose. Further, an example is given, showing that the projective dimension of \(S^r (\Delta)\) for general polyhedral complexes need not to be a local concept.