Local cohomology modules of a regular affine domain
Sayed Sadiqul Islam
TL;DR
The paper investigates when the injective dimension of local cohomology $H^i_I(R)$ matches the dimension of its support, focusing on Lyubeznik functors over regular affine domains in characteristic $0$. Using differentiably admissible $K$-algebras and $\mathcal{D}$-module techniques, it proves a key structure result: if the Bass number $\mu_c(P,\mathcal{T}(R))>0$ for $c=\operatorname{injdim}_R\mathcal{T}(R)$, then $P$ is maximal, which yields $\operatorname{injdim}_R\mathcal{T}(R)=\dim_R(\operatorname{Supp}_R\mathcal{T}(R))$ for regular affine domains (Corollary B). It also establishes a general lower bound $\operatorname{injdim}_R\mathcal{T}(R)\ge \dim_R(\operatorname{Supp}_R\mathcal{T}(R))-1$ and proves finiteness of associated primes for polynomial extensions $S=R[x_1,\dots,x_m]$ (Theorem D). Collectively, these results extend Lyubeznik-type finiteness and injective-dimension phenomena to a broad class of algebras in characteristic $0$, linking local cohomology, holonomic $\mathcal{D}$-modules, and structural properties of Lyubeznik functors.
Abstract
For a Noetherian commutative ring $R$, let $H^i_I(R)$ be the $ i$-th local cohomology module of $R$ with respect to $I$. In \cite{Hel-08}, Hellus posed the question of identifying rings $R$ such that $\operatorname{injdim}_R H^i_I(R)=\operatorname{dim}_R(\operatorname{Supp}_R H^i_I(R))$. In this paper, we show that a regular affine domain over a field of characteristic $0$ satisfies this condition. In fact, we prove that $\operatorname{injdim}_R H^i_I(R)\geq \operatorname{dim}_R(\operatorname{Supp}_R H^i_I(R))-1$ when $R$ is a differentiably admissible $K$-algebra. Indeed, we establish both of these conclusions for a substantially broad class of functors known as Lyubeznik functors. We also prove that if $R$ is a polynomial ring over a differentiably admissible $K$-algebra, then $\operatorname{Ass}_R H^i_I(R)$ is finite for all $i\geq 0$ and for every ideal $I$ of $R$.