On derived functors of Graded local cohomology modules

TL;DR

This paper investigates derived functors of graded local cohomology modules over polynomial rings in characteristic zero, revealing new structural properties of these modules and their relations with Weyl algebra modules.

Contribution

It establishes novel isomorphisms and concentration results for local cohomology and Tor functors of graded modules over polynomial rings, specifically in characteristic zero, with conjectures extending these findings.

Findings

Local cohomology of Tor modules is isomorphic to injective hulls with specific multiplicities.

Tor functors over Weyl algebra are concentrated in degree -n.

Conjecture that Ext groups are concentrated in degree zero for all i.

Abstract

Let $K$ be a field of characteristic zero and let $R=K[X_1, \ldots,X_n ]$, with standard grading. Let $\mathfrak{m}= (X_1, \ldots, X_n)$ and let $E$ be the $^*$injective hull of $R/\mathfrak{m}.$ Let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. Let $I, J$ be homogeneous ideals in $R$. Fix $i,j \geq 0$ and set $M = H^i_I(R)$ and $N = H^j_J(R)$ considered as left $A_n(K)$-modules. We show the following two results for which no analogous result is known in charactersitc $p > 0$. \begin{enumerate} $H^l_\mathfrak{m}(\Tor^R_\nu(M, N)) \cong E(n)^{a_{l,\nu}}$ for some $a_{l,\nu} \geq 0$. For all $\nu \geq 0$; the finite dimensional vector space $\Tor^{A_n(K)}_\nu( M^\sharp, N)$ is concentrated in degree $-n$ (here $M^\sharp$ is the standard right $A_n(K)$-module associated to $M$). \end{enumerate} We also conjecture that for all $i \geq 0$ the finite dimensional vector space $\Ext^i_{A_n(K)}(M, N)$ is concentrated in degree zero. We give a few examples which support this conjecture.