Graded Components of Local Cohomology Modules II

Abstract

Let A be a commutative Noetherian ring containing a field of characteristic zero. Let R = A[X1,…,Xm] be a polynomial ring and Am(A) = A〈X1,…,Xm, ∂1,…,∂m〉 be the m th Weyl algebra over A, where ∂i = ∂/∂Xi. Consider standard gradings on R and Am(A) by setting $\deg z=0$ for all z ∈ A, $\deg X_{i}=1$ , and $\deg \partial _{i} =-1$ 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 Am(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.