# On derived functors of graded local cohomology modules-II

## TL;DR

This paper proves that certain Ext modules of graded local cohomology modules over Weyl algebras are concentrated in degree zero, confirming a previously stated conjecture.

## Contribution

It establishes the degree concentration of Ext modules of graded local cohomology modules over Weyl algebras, confirming a conjecture from earlier work.

## Key findings

- Ext modules are concentrated in degree zero for all ν ≥ 0
- Confirms the conjecture from part I of the series
- Provides structural insight into graded local cohomology over Weyl algebras

## Abstract

Let $R=K[X_1,\ldots, X_n]$ where $K$ is a field of characteristic zero, and let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. We give standard grading on $R$ and $A_n(K)$. Let $I$, $J$ be homogeneous ideals of $R$. Let $M = H^i_I(R)$ and $N = H^j_J(R)$ for some $i, j$. We show that $\Ext_{A_n(K)}^{\nu}(M,N)$ is concentrated in degree zero for all $\nu \geq 0$, i.e., $\Ext_{A_n(K)}^{\nu}(M,N)_l=0$ for $l \neq0$. This proves a conjecture stated in part I of this paper.