Duality and the equations of Rees rings and tangent algebras

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 Kähler differentials ${\mathrm{\Omega }}_{R\mathrm{/}k}$ and its associated tangent algebras.