# Is there a slick proof of the fundamental theorem of dimension theory?

The fundamental theorem of dimension theory in commutative algebra states that given a module $$M$$ over a noetherian local ring $$A$$, we have $$s(M)=\text{dim}(M)=d(M)$$ (where $$s(M)$$ is the infimum of integers $$k$$ such that there exists $$x_1...x_k$$ such that $$M/(x_1...x_k)M$$ is of finite length, and $$d(M)$$ is the degree of the Samuel polynomial of $$M$$)

I know about the standard proof as is written down in Serre's Local Algebra, or in the book Homological Methods for Commutative Algebra [S. Raghavan, R. Balwant Singh, R. Sridharan] (side question: what are other good references for the dim. theorem? Please add details).

Question

I was wondering if there exists some slick / more conceptual proof by applying some more technology, maybe using homological algebra or algebraic geometric methods, or whatever.

• Samuel polynomial = Hilbert polynomial? – Walter Neff Aug 27 at 21:36
• Atiyah-MacDonald. – Fernando Muro Aug 27 at 22:10
• @WalterNeff Yes, you can call it the Hilbert-Samuel polynomial. – GLe Aug 27 at 22:36
• +1 for "I would like a proof slicker than Serre's". – roy smith Aug 28 at 22:55
• @roysmith +1 for making me laugh :) – GLe Aug 30 at 0:39