Applications of powerful theorems in Bruns -Herzog's book "Cohen-Macaulay Rings" 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. 


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. 
