Let $A$ denote a unital commutative ring. Given a system of ideals $(I_p)_{p\in P}$ indexed by a partially ordered set $P$ such that if $p \leq q$, then $I_p$ is contained in $I_q$, when is
$$GL_n \left(\lim_{\leftarrow} (A/{I_p}) \right) \cong \lim_{\leftarrow} \ GL_n(A/{I_p}) ?$$
Of course this is true when $A$ is $I$-adically complete.
What happens when $A$ is a topological ring and the $GL_n$ 's are endowed with "suitable" topologies making them toplogical groups?
$GL_n$ is a representable functor, $GL_n(A)=Hom_{Ring}(\mathbb{Z}[x_{ij},y| 1 \leq i,j \leq n]/(y \cdot det(x_{ij})-1),A)$, and thus commutes with limits.
The natural map $$GL_n \left(\lim_{\leftarrow} (A/{I_p}) \right) \to \lim_{\leftarrow} \ GL_n(A/{I_p})$$ is always an isomorphism. Just note that $$\lim_{\leftarrow} \ GL_n(A/{I_p}) \subset \prod_{p \in P} GL_n(A/I_p) = GL_n\left( \prod_{p \in P} A/I_p\right)\supset GL_n(\lim_{\leftarrow} (A/{I_p})).$$ It is really easy to see that this gives a natural identification. You do not need to assume that $A$ is commutative or complete.
