  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.
