Defect of exactness of toric Néron models on a local field (Q1387914)

\(R\) is a complete discrete valuation ring with mixed characteristic and \(K\) is the field of fractions of \(R\). One considers an exact sequence of abelian varieties \(0\rightarrow A_K\rightarrow B_K\rightarrow C_K\rightarrow 0\) over \(K\) and the corresponding complex of Néron models \(0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0\). In general this sequence is not exact, and in particular the map \(\text{Lie}(B)\rightarrow \text{Lie}(C)\) need not be surjective. The main result of the paper is the following: Assume that \(B\) has multiplicative reduction over \(R\). Then the cokernel of the map \(\text{Lie}(B)\rightarrow \text{Lie}(C)\) is annihilated by \(p^n\) where \(n\) is maximal such that \(K\) contains the \(p^n\)-th roots of unity.