Gorensteinness in Rees algebras of powers of parameter ideals

TL;DR

This paper characterizes when Rees algebras of powers of parameter ideals are Gorenstein in certain Noetherian local rings, providing new examples over non-Cohen-Macaulay rings.

Contribution

It establishes a necessary and sufficient condition for Gorensteinness in these Rees algebras, expanding understanding beyond Cohen-Macaulay cases.

Findings

Rees algebra $ ees(q^d)$ is Gorenstein for parameter ideals $q$ in specific Buchsbaum rings.

Many Gorenstein Rees algebras exist over non-Cohen-Macaulay rings.

Provides explicit conditions linking Gorensteinness to ring properties.

Abstract

This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the $d$-th power of parameter ideals in certain Noetherian local rings of dimension $d\ge 2$. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra $\mathcal{R}(\mathfrak{q}^d)=\oplus_{i\ge 0}\mathfrak{q}^{di}$ is Gorenstein for every parameter ideal $\mathfrak{q}$ that is a reduction of the maximal ideal in a $d$-dimensional Buchsbaum local ring of depth 1 and multiplicity 2.