Does anyone know an English reference for the original proof of Hilbert's syzygy theorem? The three proofs that I know use either:

  • the theory of projective dimension and change of rings (plus a step to go from projective to free resolutions)
  • the symmetry of the Tor functors
  • Groebner bases

None of these tools would have been available to Hilbert, and I guess his original proof was much more direct. But, unfortunately, the original reference is in German. Is there an English proof somewhere?

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    $\begingroup$ I am unsure if this was Hilbert's original proof, but Arrondo's Introduction to Projective Varieties contains an elementary proof by induction on the number of variables. $\endgroup$ – Joshua Mundinger Aug 26 '20 at 16:27
  • $\begingroup$ Thank you, I had a quick look at it seems like the kind of argument that would have been available to Hilbert $\endgroup$ – Andrea Ferretti Aug 26 '20 at 16:44

See Theory of Algebraic Invariants

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    $\begingroup$ Thank you, I did not know there was an English translation $\endgroup$ – Andrea Ferretti Aug 26 '20 at 16:48
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    $\begingroup$ this is not a translation of Hilbert's publication on the syzygy theorem in the Mathematische Annalen (which I don't think exists), but a translation of lecture notes by Hilbert's student Sophus Marxsen $\endgroup$ – Carlo Beenakker Aug 26 '20 at 16:51

Actually, there is an English translation of Hilbert's "Über die Theorie der algebraischen Formen" (Mathematische Annalen 36, 473--530, 1890), where the theorem is in Part III of that five-part paper. The translation is in pages 143--224 of "Hilbert's Invariant Theory Papers", Volume VIII of R. Hermann's "Lie Groups: History, Frontiers and Applications", Math. Sci. Press, Brookline, MA, 1978. I don't know if the book is available on-line.


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