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.

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.

Equations

$$H^{\nu }(\partial ,H_I^i(R))\cong {\mathrm{Tor}}_{n-\nu }^{A_n(K)}(R^r,H_I^i(R)),$$

Full text

1. Introduction

1.1**.**

Setup: Let $K$ be a field of characteristic zero and $R=K[X_{1},\ldots,X_{n}]$ with standard grading. Let $A_{n}(K)=K\langle X_{1},\ldots,X_{n},\partial_{1},\ldots,\partial_{n}\rangle$ be the $n^{th}$ Weyl algebra over $K$. Consider $A_{n}(K)$ as graded with $\deg X_{i}=1$ and $\deg\partial_{i}=-1$ for $i=1,\ldots,n$. Let $I,J$ be homogeneous ideals in $R$. Fix $i,j\geq 0$ and set $M=H^{i}_{I}(R)$ and $N=H^{j}_{J}(R)$. In [Lyu-1, 2.9], Lyubeznik showed that the local cohomology module $H^{i}_{I}(R)$ is a graded left holonomic $A_{n}(K)$-module for $i\geq 0$.

In [TP6], the first author proved that the de Rham cohomology $H^{\nu}(\partial,H_{I}^{i}(R))$ is concentrated in degree $-n$ for each $\nu\geq 0$. We have a graded isomorphism

[TABLE]

where $R^{r}$ is $R$ considered as a right $A_{n}(K)$-module by the isomorphism $R^{r}=A_{n}(K)/(\partial)A_{n}(K)$. By ${}^{l}R$ we denote $R$ considered as a left $A_{n}(K)$-module via the isomorphism ${}^{l}R\cong A_{n}(K)/A_{n}(K)(\partial)$. Let $L^{\sharp}$ denote the standard right $A_{n}(K)$-module associated to the left $A_{n}(K)$-module $L$ (see LABEL:trans). Then it is easy to check that $(^{l}R)^{\sharp}\cong R^{r}$. Note that $H^{0}_{(0)}(R)=\ ^{l}R$. In [TP5], the first author along with the third author considerably generalized (1.1.1) and proved the following:

Theorem 1.2** (with hypothesis as in 1.1).**

[TP5, Theorem 1.1]* Fix $\nu\geq 0$. Then the graded $K$ vector space $\operatorname{Tor}^{A_{n}(K)}_{\nu}(M^{\sharp},N)$ is concentrated in degree $-n$, i.e., $\operatorname{Tor}^{A_{n}(K)}_{\nu}(M^{\sharp},N)_{j}=0$ for all $j\neq-n$.*

In view of Theorem 1.2, it is natural to investigate $\operatorname{Ext}^{\nu}_{A_{n}(K)}(M,N)$ which is a finite dimensional $K$-vector space for any $\nu\geq 0$, see [BJ, 2.7.15, 1.6.6].

In [TP5, Section 8], the first and third author conjectured that $\operatorname{Ext}_{A_{n}(K)}^{\nu}(M,N)$ is concentrated in degree zero for all $\nu\geq 0$. They gave several examples in support of this conjecture. The main result of this paper is an affirmative answer to this conjecture, i.e., we have

Theorem 1.3** (with hypotheses as in 1.1.).**

[TP5, Conjecture 1.5]* Fix $\nu\geq 0$. The graded $K$-vector space $\operatorname{Ext}^{\nu}_{A_{n}(K)}(M,N)$ is concentrated in degree zero, i.e., $\operatorname{Ext}^{\nu}_{A_{n}(K)}(M,N)_{l}=0$ for $l\neq 0$.*