DEFECT OF EXACTNESS OF TORUS NERON MODELS OVER A LOCAL-FIELD

Let R be a complete discrete valuation ring with mixed characteristic. Denote by K its field of fractions and by k its residue field. Let 0 --> AK --> BK --> CK --> 0 be an exact sequence of abelian varieties o ver K and consider the corresponding complex of Nerons models 0 --> A --> B --> C --> 0, over R. We assume that the identity component B-k(0 ) Of the special fibre Bk of B is a torus and we study the defect of e xactmess at B in this last sequence.