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 Kähler differentials Ω_R/k and its associated tangent algebras.