Let $A$ be an adic ring and $\mathfrak{a}$ be its ideal of definition. Let $\textrm{Spf }A$ denote the set of all open prime ideals $\mathfrak{p}\subset A$, then $\textrm{Spf }A\cong \textrm{Spec }A/\mathfrak{a}\subset \textrm{Spec }A$. So the Zariski topology can be induced to $\textrm{Spf }A$. We denote by $D(f)$ the open set in $\textrm{Spf }A$ where $f\in A$.
Let $(D(f_{i}))_{i}$ be an open covering of $D(f)$. Then how to construct the following exact sequence
$A/\mathfrak{a}^{n}[f^{-1}]\rightarrow\prod_{i} A/\mathfrak{a}^{n}[f_{i}^{-1}]\rightrightarrows\prod_{i,j}A/\mathfrak{a}^{n}[(f_{i}f_{j})^{-1}]$
Here is my attempt: $A/\mathfrak{a}^{n}[f^{-1}]\rightarrow\prod_{i} A/\mathfrak{a}^{n}[f_{i}^{-1}]:\sum_{i}x_{i}(f^{-1})^{i_{1}}\mapsto\prod_{i}x_{i}$ and $\prod_{i} A/\mathfrak{a}^{n}[f_{i}^{-1}]\rightrightarrows\prod_{i,j}A/\mathfrak{a}^{n}[(f_{i}f_{j})^{-1}]:\prod_{i}\sum_{i}x_{i}(f_{i}^{-1})^{k_{i}}\mapsto\prod_{i,j}\sum_{i}x_{i}(f_{i}^{-1})^{k_{i}}-\sum_{j}x_{j}(f_{j}^{-1})^{k_{j}}$
However, it's a mess, I still failed to find the very maps to make the sequence exact.