Graded components of Local cohomology modules II

Abstract

Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R=A[X_1,\dots ,X_m]$ be a polynomial ring and $A_m(A)=A⟨X_1,\dots ,X_m,{\partial }_1,\dots ,{\partial }_m⟩$ be the $m^{th}$ Weyl algebra over $A$, where ${\partial }_i=\partial /\partial X_i$. Consider both $R$ and $A_m(A)$ as standard graded with $\mathrm{deg}z=0$ for all $z\in A$, $\mathrm{deg}{X}_i=1$, and $\mathrm{deg}{\partial }_i=-1$ for $i=1,\dots ,m$. 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.

Section 10 has been added