New algebraic properties of quadratic quotients of the Rees algebra

TL;DR

This paper investigates algebraic properties of a family of quotient rings derived from the Rees algebra, providing comprehensive descriptions of their spectra, localizations, and conditions for properties like Cohen-Macaulayness and Gorensteinness.

Contribution

It offers a complete characterization of the spectra and localizations of the rings R(I)_{a,b} and generalizes known results about their Cohen-Macaulay and Gorenstein properties.

Findings

Characterization of the spectrum of R(I)_{a,b}

Conditions for R(I)_{a,b} to be Cohen-Macaulay or Gorenstein

Criteria for R(I)_{a,b} to be an integral domain or reduced

Abstract

We study some properties of a family of rings $R(I)_{a,b}$ that are obtained as quotients of the Rees algebra associated with a ring $R$ and an ideal $I$. In particular, we give a complete description of the spectrum of every member of the family and describe the localizations at a prime ideal. Consequently, we are able to characterize the Cohen-Macaulay and Gorenstein properties, generalizing known results stated in the local case. Moreover, we study when $R(I)_{a,b}$ is an integral domain, reduced, quasi-Gorenstein, or satisfies Serre's conditions.