# Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?

Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely generated $\hat A$-modules. $${\sf mod}(A) \to {\sf mod}(\hat A)$$ $$M\mapsto \hat M$$ Question: Is it true that the functor is injective on isomorphism classes?

Remark: The homomorphism $A\to \hat A$ is faithfully flat. Then $\hat AI\cap A=I$ for any ideal $I$ of $A$. Using this one can check that the functor is injective on isomorphism classes of cyclic modules.

• Yes. This is true more generally if you replace $\widehat{A}$ by any noetherian (commutative) ring faithfully flat over $A$: see EGA IV.2, Proposition 2.5.8 (taking $B=A$ in that Proposition). – abx Aug 15 '17 at 13:12