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Let $(A_n)<$ be a projective and inductive system of $\mathbb{Z}/p^n$-modules.

Is then $\operatorname{Hom}{(\projlim A_n, \mathbb{Z}_p)}$ isomorphic to $\operatorname{Hom}(\operatorname{Div}(\injlim A_n), \mathbb{Q}_p/\mathbb{Z}_p)$, resp. under which additional (finiteness?) assumptions does this hold? $\operatorname{Div}$ denotes the maximal divisible subgroup.

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Are you saying to consider both the limit and colimit of the same diagram? So for example for the diagram of finite quotients of $\mathbb{Z}_p$ you would have $\mathbb{Z}_p$ as the limit and $0$ as the colimit? Your claim does not hold in this case. –  jspecter Sep 8 '11 at 6:39
    
$(A_n)$ comes from a $p$-adic sheaf. –  user5262 Sep 8 '11 at 6:41

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