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Does someone know in which books, lecture notes or ... I can find the induction step of the proof of Hilbert Syzygy Theorem? I'd only found the proof for R[x] (e.g. Weibel) and I haven't really an idea how the induction step works. Thank you!

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  • $\begingroup$ I added the "commutative algebra" tag. $\endgroup$
    – Vladimir Dotsenko
    Commented Jun 11, 2010 at 20:03

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I'm not sure exactly which kind of proof you're looking for, but a proof by induction using Gröbner bases is presented in Chapter 15 of Eisenbud's Commutative Algebra, see Corollary 15.11 specifically.

There's also a more abstract homological proof in Chapter 19, see this related question: Best exposition of the Proof of the Hilbert Syzygy Theorem by Eilenberg-Cartan

The first proof is closer in spirit with Hilbert's original proof.

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A proof using Gröbner bases is in Using algebraic geometry by David A. Cox, John B. Little, Donal O'Shea, Theorem 2.1.

However, I was always sure that there should be (at least in the graded case) an inductive proof along the lines of Atiyah-Macdonald's proof of Hilbert--Serre theorem, namely by induction considering the 4-term exact sequence $$0\to K_i\to M_i\to M_{i+1}\to L_i\to0$$ where $K_n$ and $L_n$ are the kernel and the cokernel for the operator of multiplication by $x_n$ (these are modules over the polynomial ring in $n-1$ variables), but something escapes me at the moment, so I just leave it here as a wish....

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