Cohomology Vanishing theorems over some rings containing nilpotents
Tony J. Puthenpurakal
TL;DR
The paper addresses vanishing of local cohomology over rings that contain nilpotents, focusing on the associated graded rings $G_{P^r}(A)$ and $G_{ rak m^r}(A)$. It develops a Rees-algebra–based construction and a non-finitely generated module $W^I(A)$ to transfer questions about $G_{I^r}(A)$ to $G_I(A)$ and to control cohomology via Veronese functors, together with D-module or $F$-finite techniques to obtain tame vanishing. The main results show that $H^d_J(G_{P^r}(A))=0$ when $\dim G_{P^r}(A)/J>0$, and that for regular local $A$ with separably closed residue field, $H^j_J(G_{ rak m^r}(A))=0$ for $j\ge d-1$ iff $\dim G_{ rak m^r}(A)/J\ge 2$ and $\mathrm{Proj}\,G_{ rak m^r}(A)/J$ is connected (with a partial converse under equi-characteristic). These results extend Hartshorne–Lichtenbaum-type vanishing to rings with nilpotents by uniting Rees-algebra methods, Veronese transfer, and D-/F-module tameness, offering new structural insights and potential geometric applications to nilpotent-settings.
Abstract
(1) Let $(A,\mathfrak{m})$ be complete Noetherian local ring of dimension $d$ and let $P$ be a prime ideal with $G_P(A) = \bigoplus_{n \geq 0}P^n/P^{n+1}$ a domain. Fix $r \geq 1$. If $J$ is a homogeneous ideal of $G_{P^r}(A)$ with $\text{dim} \ G_{P^r}(A)/J > 0$ then the local cohomology module $H^d_J(G_{P^r}(A)) = 0$. (2) Let $A = K[[X_1, \ldots,X_d]]$ and let $\mathfrak{m} = (X_1, \ldots, X_d)$. Assume $K$ is separably closed. Fix $r \geq 1$. 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 $j \geq d -1$ if and only if $\text{dim} \ G_{\mathfrak{m}^r}(A)/J \geq 2$ and $\text{Proj}\ G_{\mathfrak{m}^r}(A)/J $ is connected.