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Where can I find a comprehensive treatment of this important result at the level of a very advanced undergraduate/beginning graduate student? What works develop the relevant material in a cohesive and readable way? Assume only a familiarity with basic homological algebra and ring theory on the part of the reader in assessing this question, please. Thank you!

flag	asked Jun 2 2010 at 4:07 lambdafunctor 756●1●7●15

Steven Sam · Accepted Answer · 2010-06-02 07:55:42Z UTC

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The place I learned it from is Chapter 19 of Eisenbud's Commutative Algebra. Most of the proofs in that section do not use material from previous chapters if I recall correctly. The route (which I think is what you are looking for) is to construct the Koszul complex of the residue field of a regular (graded) local ring and also prove the symmetry of the Tor functor, and then use these two facts to get finite global dimension which implies Hilbert's syzygy theorem.

answered Jun 2 2010 at 7:55

Steven Sam
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You need to construct the Koszul resolution $K\to k[X]$ of the polynomial ring $k[X]$ over itself as a bimodule over itself and show that it is in fact a resolution. Using it, you can construct a projective resolution of the same length for every $k[X]$-module $M$ simply applying the functor $(\mathord-)\otimes M$, to get $K\otimes M\to k[X]\otimes M=M$. The "theory" you need is that the complex $K\otimes M$ computes $\mathord{Tor}^{k[X]}(k[X],M)$ (and this is just the definition of $\mathord{Tor}$, and enough theory to prove that $K$ is exact (for example, the Künneth formula over a field) – Mariano Suárez-Alvarez Jun 2 2010 at 13:35

Mariano Suárez-Alvarez · Answer 2 · 2010-06-02 04:26:32Z UTC

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Hilton-Stammbach's book on homological algebra is nice and includes that result, if I recall correctly

I am partial to Cartan-Eilenberg's book, though!

answered Jun 2 2010 at 4:26

Mariano Suárez-Alvarez
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mariano|syzygy ftw. – Harry Gindi Jun 2 2010 at 9:50

Best exposition of the Proof of the Hilbert Syzygy Theorem by Eilenberg-Cartan

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