ac.commutative algebra - The inverse limit of locally free module

The answer is yes.

1) $A$ is $I$-adic complete implies that $I subset rad(A)$, the intersection of all maximal primes. Indeed, pick any $a in I$. Look at $1-a + a^2 -a^3 ... in A $ (this is where we use completeness). The inverse of this is $1+a$, so $1+a$ is an unit. Since this is true for any $ain I$, we have $Isubset rad(A)$ (see Section 1 Matsumura).

2) It suffices to prove that $M$ is free at any maximal ideal $m$ of $A$. Since $I$ is inside $m$, we may as well replace $A$ by $A_m$ and assume $A$ is local. By assumption then $M/I cong (A/I)^l$, Nakayama Lemma shows that $l$ is the mimimum number of generators of $M$.

Look at the beginning of a minimal resolution of M: $ N to F=A^l to M $.The last map become isomorphism when tensoring with $A/I^n$, so the first map has to become $0$. This means that $N subset I^nF$ for all $n>0$. This forces $N=0$ (use Artin-Rees lemma), thus $M cong F$.