Duality and the equations of Rees rings and tangent algebras

TL;DR

This paper explores the structure of Rees rings and tangent algebras of modules over Noetherian rings, employing duality techniques especially in cases where the symmetric algebra is a complete intersection.

Contribution

It introduces a duality approach to analyze Rees rings and tangent algebras, particularly when the symmetric algebra is a complete intersection, extending understanding of their algebraic properties.

Findings

Duality between the defining ideal and the symmetric algebra used to study Rees rings.

Application to modules over complete intersection rings defined by quadrics.

Insights into the structure of tangent algebras via K"ahler differentials.

Abstract

Let $E$ be a module of projective dimension one over a Noetherian ring $R$ and consider its Rees algebra $\mathcal{R}(E)$. We study this ring as a quotient of the symmetric algebra $\mathcal{S}(E)$ and consider the ideal $\mathcal{A}$ defining this quotient. In the case that $\mathcal{S}(E)$ is a complete intersection ring, we employ a duality between $\mathcal{A}$ and $\mathcal{S}(E)$ in order to study the Rees ring $\mathcal{R}(E)$ in multiple settings. In particular, when $R$ is a complete intersection ring defined by quadrics, we consider its module of K\"ahler differentials $\Omega_{R/k}$ and its associated tangent algebras.