Cohomology Vanishing theorems over some rings containing nilpotents

(1) Let $(A,\mathfrak{m})$ be complete Noetherian local ring of dimension $d$ and let $P$ be a prime ideal with $G_P(A)={⨁}_{n\ge 0}{P}^n\mathrm{/}{P}^{n+1}$ a domain. Fix $r\ge 1$. If $J$ is a homogeneous ideal of $G_{P^r}(A)$ with $\text{dim }{G}_{P^r}(A)\mathrm{/}J>0$ then the local cohomology module $H_J^d(G_{P^r}(A))=0$. (2) Let $A=K[[X_1,\dots ,X_d]]$ and let $\mathfrak{m}=(X_1,\dots ,X_d)$. Assume $K$ is separably closed. Fix $r\ge 1$. Let $J$ be a homogeneous ideal of $G_{{\mathfrak{m}}^r}(A)$. We show that local cohomology modules $H_J^j(G_{{\mathfrak{m}}^r}(A))=0$ for $j\ge d-1$ if and only if $\text{dim }{G}_{{\mathfrak{m}}^r}(A)\mathrm{/}J\ge 2$ and $\text{Proj }{G}_{{\mathfrak{m}}^r}(A)\mathrm{/}J$ is connected.