Duality and the equations of Rees rings and tangent algebras

Abstract

Let $E$ be a module of projective dimension 1 over a Noetherian ring $R$ and consider its Rees algebra $R\left(E\right)$. We study this ring as a quotient of the symmetric algebra $S\left(E\right)$ and consider the ideal $A$ defining this quotient. In the case that $S\left(E\right)$ is a complete intersection ring, we employ a duality between $A$ and $S\left(E\right)$ in order to study the Rees ring $R\left(E\right)$ in multiple settings. In particular, when $R$ is a complete intersection ring defined by quadrics, we consider its module of Kähler differentials ${\Omega }_{R/k}$ and its associated tangent algebras.