Local cohomology modules of a regular affine domain

TL;DR

This paper investigates the properties of local cohomology modules over regular affine domains, establishing bounds on their injective dimension and support, and proving finiteness of associated primes in polynomial rings over certain algebras.

Contribution

It proves that regular affine domains over characteristic zero fields satisfy specific injective dimension conditions and that polynomial rings over differentiably admissible algebras have finite associated primes for local cohomology modules.

Findings

Regular affine domains over characteristic zero satisfy the injective dimension condition.

Injective dimension of local cohomology modules is at least the support dimension minus one.

Polynomial rings over differentiably admissible algebras have finite associated primes for all local cohomology modules.

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 $R$.