Let $M$ be a module over an $E_\infty$ ring, $A$. Let $I$ be an $A$-non unital commutative algebra together with an associative map $I \wedge_A M \to M$.

Define ${_A}(M/I^n)$ as the cofiber of $I^{\wedge_A n} \wedge_A M \to M$, and the completion ${_A}M^\wedge_I$ by $\text{lim}_n {_A}M/I^n$.

I want to understand the dependence of the underlying space of $\text{lim}_n {_A}M/I^n$ on $A$. Note that the underlying space of ${_A}M/I^n$ depends on $A$. But sometimes the underlying space of ${_A}M^\wedge_I$ doesn't depend on $A$ since sometimes ${_A}M^\wedge_I=M$.