$\mathfrak a$-adic Completion of a Ring is Flat

Question

Let $A$ be a Noetherian ring and $\mathfrak a \subset A$ an ideal of $A$. Denote by $\hat{A_{\mathfrak a}}:= \varprojlim_n A/\mathfrak a^n$ the $\mathfrak a$-adic completion of $A$ wrt $\mathfrak a$.

Why and how to see that $\hat{A_{\mathfrak a}}$ is a flat $A$-module?

Could anybody give sketch for the proof?

My considerations: up to now I know that firstly $\mathfrak a$ is finitely generated.

Remark: As @Max mentioned below one cannot expect that the canonical map $A \to \hat{A_{\mathfrak a}}$ is injective (Krull's intersection thm only works for local Noetherians).

Answer 1

• You can find a solution to this problem in the following document of Conrad Artin–Rees and completions. In particular it is the Corollary 3.2.

• Also, in the book of Qing Liu - Algebraic Geometry and Arithmetic Curves you can find a proof in the Theorem 3.15. I love how Liu wrote the review of commutative algebra in his book and I recommend you to read it.

Here a snap of the proof taken from Liu's book.

The Corollary 3.14 that Liu is referring to is the well-known result:

• Here the approach to this theorem in Matsumura: Commutative Ring Theory, Chapter 3.

The main theorem is Theorem 8.8:

The theorems that Matsumura is referring to are:

Theorem 7.7:

Theorem 8.1:

Theorem 8.6:

Note The notation of the above theorem is: $A$ a noetherian ring, $M$ a finite $A$-module, $N \subset M$ a submodule, and $I$ an ideal of $A$.

I hope this helps you. If you don't understand the references I gave you, don't hesitate to ask!

Answer 2

The Stacks project has a proof, https://stacks.math.columbia.edu/tag/06LE, considering the direct sum over a one element index set.