Curently, I'm learning Auslander-Buchsbaum theorem, that is
Any local regular ring is UFD.
Initially, I followed the path which is set in Chapter 4 of the book Commutative Rings by Kaplansky. But things didn't work well. The reason for my difficulties is that in this chapter the author uses so much homological things. I therefore jumped into the Chapter 1 of the book Cohen-Macaulay Ring by Winfried Bruns, but the problem still be the same for the author assumes readers have some experiences with Tor, Ext as well as other homological things.
For clarity, here is my background in Commutative Algebra:
(1) I know basic algebras cover in "Steps in commutative algebra" by R.Y.Sharp or in the first three chapters of Commutative Rings by Kaplansky.
(2) I know some basic properties of tensor product, flat module cover in chapter 2 of Introduction to commutative algebra by Atiyah.
I know all the homological things I need is covered in Weibel's book or any standard homological textbook. But right now, I dont want to learn homological algebra seriously and systematically, rather I want to pick things as small as possible just to understand Auslander-Buchsbaum theorem. I also think it will serve as good motivation to sparks me learn Homological algebra in future.
So, if you already learn this theorem, Please draw me a roadmap to achieve my goal. My perfect roadmap would be as follow:
(i) some Ext, Tor,... just taken from this note A, or from page.... to page of book B,etc.
(ii) some stuffs about projective module,....
.........................................................................
Thank in advance !
Here is a possible path with your definition. For simplicity, let me assume that $R$ is an $R/M=k$ algebra.
Then, $R$ is a UFD if the completion is a UFD.
The completion of $R$ is $k[[x_1,\ldots, x_n]]$ where $n=\dim R$. This is where $\dim R=\dim M/M^2$ is used.
Use Weierstrass preparation/ division theorem to show that the power series ring is a UFD.