Families of Gorenstein and almost Gorenstein rings

Abstract

Starting with a commutative ring $R$ and an ideal $I$, it is possible to define a family of rings $R(I)_{a,b}$, with $a,b \in R$, as quotients of the Rees algebra $\oplus_{n \geq 0} I^nt^n$; among the rings appearing in this family we find Nagata's idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a,b$; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a,b$. More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.Comment: 15 pages. Roma01.Math.A

New algebraic properties of quadratic quotients of the Rees algebra

“…In a subsequent paper (cf. [5]), the same authors deepen the study of this family of rings in the local case, characterizing when its members are Gorenstein, complete intersection, and almost Gorenstein, proving that these properties do not depend on the particular member chosen in the family, but only on R and I. We also notice that other properties of these quotients have been studied in [10].…”

When is $\mathfrak{m}:\mathfrak{m}$ an almost Gorenstein ring?

“…5. In [4] almost canonical ideals naturally appear characterizing the almost Gorenstein property of some quadratic quotients of the Rees algebra R[It] of R with respect to a proper ideal I. More precisely, if a, b ∈ R, let J denote the contraction of the ideal (t [3,4,19] for the importance of this family of rings.…”

“…In [4] almost canonical ideals naturally appear characterizing the almost Gorenstein property of some quadratic quotients of the Rees algebra R[It] of R with respect to a proper ideal I. More precisely, if a, b ∈ R, let J denote the contraction of the ideal (t [3,4,19] for the importance of this family of rings. If R is a one-dimensional Cohen-Macaulay local ring, [4, Corollary 2.4] says that R(I) a,b is an almost Gorenstein ring if and only if I is an almost canonical ideal of R and z −1 I ∨ is a ring.…”

The Hilbert-Kunz function of some quadratic quotients of the Rees algebra

“…It can be considered as a unified approach to some well-studied constructions: R(I) 0,0 is isomorphic to the Nagata's idealization R ⋉ I [1,21], R(I) −1,0 is isomorphic to the amalgamated duplication R ✶ I of R along I [7,9], R(I) 0,b is isomorphic to the pseudocanonical double cover of R [12,13] provided that I is a canonical ideal of R. Recently the properties of this family have been investigated by many researchers, e.g. in [3,4,10,11,14,23,24].…”