# Existence of isomorphism mod every power of the maximal ideal

This problem is a continuation of Hensel lemma and rational points in complete noetherian local ring.

Let $$A$$ be a complete noetherian local ring and $$\mathfrak{m}$$ be its maximal ideal.

Assume $$S_1$$ and $$S_2$$ are two finitely generated $$A$$-algebras (no flatness assumption), and $$S_1/\mathfrak{m}^n \cong S_2/\mathfrak{m}^n$$ as $$A$$-algebras for every $$n$$, then do we have $$S_1 \cong S_2$$ as $$A$$-algebras?

Again the problem is about compatibility. If one try to use the Hom functor, there is a subtlety of representability. The problem can also be stated for finite type schemes.

Moreover, I have one similar question: if $$R_1$$, $$R_2$$ are two complete noetherian local rings and $$R_1/\mathfrak{m_1}^n \cong R_1/\mathfrak{m_2}^n$$, then do we have $$R_1 \cong R_2$$ as rings?

The similar question for finitely generated modules is true as $$Isom$$ satisfies Mittag-Leffler condition (an endomorphism of finitely generated module is an isomorphism iff it’s surjective, so this can be checked mod $$\mathfrak{m}$$, and Hom mod $${\mathfrak{m}}^n$$ are modules of finite lengths )