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 R (E). We study this ring as a quotient of the symmetric algebra S (E) and consider the ideal A defining this quotient. In the case that S (E) is a complete intersection ring, we employ a duality between A and S (E) in order to study the Rees ring 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 ₑ/₊ and its associated tangent algebras.