Reading on the proof in Atiyah-MacDonald (A-M) of the main theorem of dimension theory, I realized that the inductive step in all three of the inequalities $$ \delta(A) \ge d(A) \ge \dim A \ge \delta(A) $$ uses a non-zero divisor of $A$ to drop one dimension lower. I know that the definitions in A-M generalize to finitely generated modules over semi-local noetherian rings, in which case this non-zero divisor becomes a non-zero divisor of $M$, i.e. an element $a \in A$ which does not annihilate any $m \in M \setminus \{0\}$.
My question is : I am not comfortable yet with the homological techniques of Koszul, but does knowledge of that theory allow some over-kill proof of the main theorem of dimension theory, or is the work in any case still necessary? (And of course, a second part to this question is : has this been done somewhere?)
