Un théorème de Rao pour les familles de courbes gauches

The aim of this paper is to prove a generalization of a theorem of Rao for families of space curves, which characterizes the biliaison classes of curves. First we introduce the concept of pseudo-isomorphism: let A be a noetherian ring, a morphism image, where image and image are coherent sheaves on PA3, flat over A, is a pseudo-isomorphism if the induced morphism of functors image (resp. image, resp. image is an isomorphism for all nmuch less-than0 (resp. an isomorphism for all n, resp. a monomorphism for all n). Two sheaves are pseudo-isomorphic, if there exists a chain of pseudo-isomorphisms between them. An N-type resolution for a family of curves image defined by an ideal image is an exact sequence image where image is a locally free sheaf on PA3 and image is (in the case when A is a local ring) a direct sum of invertible sheaves image. We prove the two following results, when the residual field of A is infinite: