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Let $X$ be a noetherian scheme and $(F_n)_{\in \mathbb N}$ an inverse system (indexed by the natural numbers) of locally sheaves of finite rank on $X$ (the ranks may vary). The transition maps $F_{n+1} \rightarrow F_n$ shall be surjective for all $n$.

Then is there a canonical isomorphism

$\mathcal Hom_{\mathcal O_X} (\varprojlim F_n, \mathcal O_X) \simeq \varinjlim \mathcal Hom_{\mathcal O_X}(F_n,\mathcal O_X)$?

Here always Sheaf-Homs are meant, and the limits are at first considered in the category of quasi-coherent modules (where they exist in any case).

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up vote 1 down vote accepted

Let $A$ be a commutative noetherian ring. Let $I\subseteq A$ be an ideal. Assume $A$ is $I$-adically complete.

Let $A_n = A/I^n$. Clearly, $A_n$ is an inverse system, and $A = \varprojlim A_n$.

Hence,

$\operatorname{Hom}_A(\varprojlim A_n ,A) = A $.

However,

$\varinjlim \operatorname{Hom}_A(A_n,A) = \Gamma_I A = \{a\in A : \exists n, I^n a = 0\}$.

In particular, if $A$ is not $I$-torsion, these two are different. For a concrete example, take $A = k[[x]]$, and $I=(x)$.

Edit: this is false, I ignored your assumption about the $F_n$ being projective.

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