Deligne's formula

Question

Let $M$ be some $A$-module and $f \in A$. Why do we have an isomorphism $$\varinjlim_n \hom_A(f^n A,M) \cong M_f \text{ ?}$$ Background. Let $X$ be a scheme, $U$ an open subscheme, and $F,G$ quasi-coherent sheaves on $X$. There is a canonical homomorphism $$\varinjlim_{J \subseteq \mathcal{O}_X , J|_{U} = \mathcal{O}_U} \hom_{\mathcal{O}_X}(JF,G) \to \hom_{\mathcal{O}_U}(F|_U,G|_U).$$ Deligne has shows that this is an isomorphism when $X$ is noetherian and $F$ is of finite type (see EGA I (1970), Prop. 6.9.17). In particular, we have the following explicit description of the quasi-coherent sheaf $\tilde{M}$ on $\mathrm{Spec}(A)$ for a noetherian ring $A$ and some $A$-module $M$: $$\tilde{M}(U) = \varinjlim_{J \subseteq A, \tilde{J}|_U = \mathcal{O}_U} \hom_A(J,M)$$ Note that, when $U$ is the complement of $V(I)$, the condition $\tilde{J}|_U = \mathcal{O}_U$ means $I \subseteq \sqrt{J}$, or equivalently $I^n \subseteq J$ for some $n$. This gives an even more explicit formula $$\tilde{M}(\mathrm{Spec}(A) \setminus V(I)) = \varinjlim_n \hom_A(I^n,M).$$ In the paper "Sections of quasi-coherent sheaves" it is claimed that this formula also holds when $U$ is a basic-open subset i.e. $I$ is principal and $A$ is not assumed to be noetherian. But I cannot prove this. The colimit becomes $$\cong \varinjlim_n ~\hom_A(f^n A,M) \cong \varinjlim_n ~\{m \in M : \mathrm{Ann}(f^n) \subseteq \mathrm{Ann}(m)\}.$$ The transition maps multiply with $f$. This colimit has a canonical injection to $\varinjlim_n ~ M \cong M_f = \tilde{M}(D(f))$. But I don't see why it should be surjective. Even when $A$ is noetherian, I don't see a direct argument, without repeating Deligne's proof (which essentially uses the Artin-Rees Lemma).