Let $R$ be a DVR with algebraically closed residue field $k$ and fraction field $K$. Let $f:G\longrightarrow H$ be a morphism of commutative $R$-group schemes. Suppose that $G$ is an abelian scheme over $R$ and that $f_K$ is an embedding. Then, is $f_k$ injective? Is it an embedding?
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Do you really mean commutative $R$-group schemes, rather than abelian schemes over $R$? And if so, do we have any assumptions on $H$? – Quinn Greicius Jan 12 '21 at 01:21
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@QuinnGreicius sorry, as I stated the hypothesis it is a bit confusing, I'll edit the question. – pozio Jan 17 '21 at 06:29