Graded components of Local cohomology modules II

TL;DR

This paper investigates the graded components of local cohomology modules over polynomial rings and Weyl algebras, focusing on vanishing, tameness, and rigidity properties using $D$-module theory.

Contribution

It introduces new results on the behavior of graded local cohomology modules, especially generalized Eulerian modules, in the context of Weyl algebras and polynomial rings.

Findings

Graded components exhibit vanishing, tameness, and rigidity properties.

Generalized Eulerian modules over Weyl algebras show these properties.

Components of local cohomology modules with respect to pairs of ideals behave similarly.

Abstract

Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R= A[X_1, \ldots, X_m]$ be a polynomial ring and $A_m(A) = A \langle X_1, \ldots, X_m, \partial_1, \ldots, \partial_m \rangle$ 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 $i=1, \ldots, m$. 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.