When are the Rees algebras of parameter ideals almost Gorenstein graded rings?

TL;DR

This paper investigates conditions under which the Rees algebra of parameter ideals in Cohen-Macaulay local rings is almost Gorenstein, concluding that such cases imply the ring is regular and the parameters form part of a regular system.

Contribution

It establishes a characterization linking almost Gorenstein Rees algebras of parameter ideals to the regularity of the base ring and the nature of the parameters.

Findings

Rees algebra being almost Gorenstein implies the ring is regular.

Parameters form part of a regular system in this case.

The result applies to Cohen-Macaulay local rings of dimension at least 3.

Abstract

Let $A$ be a Cohen-Macaulay local ring with $\operatorname{dim} A = d\ge 3$, possessing the canonical module ${\mathrm K}_A$. Let $a_1, a_2, \ldots, a_r$ $(3 \le r \le d)$ be a subsystem of parameters of $A$ and set $Q= (a_1, a_2, \ldots, a_r)$. It is shown that if the Rees algebra ${\mathcal R}(Q)$ of $Q$ is an almost Gorenstein graded ring, then $A$ is a regular local ring and $a_1, a_2, \ldots, a_r$ is a part of a regular system of parameters of $A$.