Graded Components of Local Cohomology Modules II

Let A be a commutative Noetherian ring containing a field of characteristic zero. Let R = A[X-1, ... , X-m] be a polynomial ring and A(m)(A) = A < X-1, ... , X-m, partial derivative(1), ... , partial derivative(m)> be the mth Weyl algebra over A, where partial derivative(i) = partial derivative/partial derivative X-i. Consider standard gradings on R and A(m)(A) by setting deg z = 0 for all z is an element of A, deg X-i = 1, and deg partial derivative(i) = -1 for i = 1, ... , m. We present a few results about the behavior of the graded components of local cohomology modules H-I(i)(R), where I is an arbitrary homogeneous ideal in R. We mostly restrict our attention to the vanishing, tameness, and rigidity properties. To obtain this, we use the theory of D-modules and show that generalized Eulerian A(m)(A)-modules exhibit these properties. As a corollary, we further get that components of graded local cohomology modules with respect to a pair of ideals display similar behavior.