On derived functors of graded local cohomology modules—II

Abstract

Let $R=K[X_1,\dots ,X_n]$, where $K$ is a field of characteristic zero, and let $A_n\left(K\right)$ be the $n$th Weyl algebra over $K$. We give standard grading on $R$ and $A_n\left(K\right)$. Let $I$, $J$ be homogeneous ideals of $R$. Let $M=H_I^i\left(R\right)$ and $N=H_J^j\left(R\right)$ for some $i,j$. We show that ${Ext}_{A_n\left(K\right)}^{\nu }(M,N)$ is concentrated in degree zero for all $\nu \ge 0$ (i.e., ${Ext}_{A_n\left(K\right)}^{\nu }(M,N{)}_l=0$ for $l\ne 0$). This proves a conjecture stated in Part I of this paper (T. J. Puthenpurakal and J. Singh, On derived functors of graded local cohomology modules, Math. Proc. Cambridge Philos. Soc. 167 (2018), no. 3, 549–565).