Local cohomology modules of a regular affine domain

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 $R$.