# Hilbert Syzygy Theorem - Induction step

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!

• I added the "commutative algebra" tag. – Vladimir Dotsenko Jun 11 '10 at 20:03

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....