Duality and the equations of Rees rings and tangent algebras

Let be a module of projective dimension 1 over a Noetherian ring and consider its Rees algebra . We study this ring as a quotient of the symmetric algebra and consider the ideal defining this quotient. In the case that is a complete intersection ring, we employ a duality between and in order to study the Rees ring in multiple settings. In particular, when is a complete intersection ring defined by quadrics, we consider its module of Kähler differentials and its associated tangent algebras.