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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).

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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.

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    $\begingroup$ Atiyah-MacDonald. $\endgroup$
    – Fernando Muro
    Commented Aug 27, 2019 at 22:10
  • $\begingroup$ @WalterNeff Yes, you can call it the Hilbert-Samuel polynomial. $\endgroup$
    – display llvll
    Commented Aug 27, 2019 at 22:36
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    $\begingroup$ +1 for "I would like a proof slicker than Serre's". $\endgroup$
    – roy smith
    Commented Aug 28, 2019 at 22:55
  • $\begingroup$ @roysmith +1 for making me laugh :) $\endgroup$
    – display llvll
    Commented Aug 30, 2019 at 0:39
  • $\begingroup$ @FernandoMuro AM is indeed a great book / intro to CA. $\endgroup$
    – Daniel Donnelly
    Commented Nov 21 at 22:08

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