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

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.

Theorem 3.15 - From Liu's book.

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

Corollary 3.14 - From Liu's book.

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

The main theorem is Theorem 8.8:

Theorem 8.8 - From Matsumura's book.

The theorems that Matsumura is referring to are:

Theorem 7.7:

Theorem 7.7 - From Matsumura's book.

Theorem 8.1:

Theorem 8.1 - From Matsumura's book.

Theorem 8.6:

Theorem 8.6 - From Matsumura's book.

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!

Tags:

Commutative Algebra

Flatness

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