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 )