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It seems that theorem 1.4.13 and it's corollary of Bruns and Herzog's book Cohen-Macaulay Rings, are powerful tools but I don't see any example that shows the power of it. My original question was an example that shows the power of it in use, but I change it as you see below to be more useful for me and others:

What is your favorite powerful theorem in commutative algebra, especially in the book Cohen-Macaulay Rings by Bruns and Herzog?

Please give an example that shows the power of it in use, with a hint that shows the application of that theorem in that example.

  • I haven't read Bruns-Herzog, but a lot of the interesting results in Eisenbud's The Geometry of Syzygies rely on the fact that $k[x_0, \dots, x_n]$ is C-M. One reason is that Green's work in the 80s connects syzygies to the cohomology of the Koszul complex, which is very connected to depth. – Hoot May 11 '14 at 16:30

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Auslander--Buchsbaum formula.
example: as "11156" says "everybody who read commutative algebra knows examples", but for example: Matt E's solution for my question: (R is a regular local ring of dimension $d$, and $I$ an ideal. If $R/I$ has depth $d − 1$, then $I$ is principal.)

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Theorem $1.2.5$ (Rees). Let $R$ be a Noetherian ring, $M$ a finite $R$-module, and $I$ an ideal such that $IM\neq M$. Then all maximal $M$-sequences in $I$ have the same length $n$ given by $$n=\min \{i: \operatorname{Ext}^i_R(R/I,M)\neq 0\}.$$

It's extremely useful. It has many applications in commutative algebra, local cohomology,... and even dont need an example of usefulness. Everyone who studies commutative algebra knows examples.

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