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Isn't the proof idea at the end misleading without additional comment?
The point there is a unique reduction is missing;
The fact that there is a reduction at all is not at all clear by applying the structure equations. One needs something like induction on the degree of the canonical monomial; thus
where the suummands
have lower degree and thus are reducible by the induction hypothesis.
Including this on the other hand seems overkill.
CSTAR 23:14, 6 Sep 2004 (UTC)
History
I am told that neither
Henri PoincarĂŠ nor
Birkhoff nor
Witt gave a correct proof.
I say the article should say a few words about the history of the theorem and who eventually proved it. At the very least the article should have links to the mathematicians above!
Mhym04:54, 22 March 2006 (UTC)reply
"The first four chains are of the form , where each chain is a closed chain of degree p - 1; therefore by induction, each is the head of an identically zero regular sum. It follows that are identically zero, and therefore each of them can be considered as the head of an identically zero regular sum of degree p."
I don't understand the "It follows that are identically zero" part. This seems to be equivalent to , which I don't believe (the head of an identically zero regular sum isn't necessarily zero). Then again, the authors are only using the weaker assertion that each of is the head of an identically zero regular sum of degree p, and maybe they HAVE some good argument for that which I just don't see. --
Darij (
talk)
09:02, 26 May 2011 (UTC)reply
Birkhoff: father/son
I have a clear impression that actually the wiki-ref. goes to the wrong Birkhoff: there are two of them, father (George David) and son (Garret), and the work cited in the references is the one of G.D. Birkhoff. I'll correct the link. --
Burivykh (
talk)
08:29, 16 July 2010 (UTC)reply
quote: More generally, the PBW theorem as formulated above extends to cases such as where (1) L is a flat K-module, (2) L is torsion-free as an abelian group, (3) L is a direct sum of cyclic modules (or all its localizations at prime ideals of K have this property), or (4) K is a Dedekind domain. See, for example, the 1969 paper by Higgins for these statements.
I don't think condition (1) appears ever in Higgins. It does with "projective" instead of "flat", which is weaker (projectives are always flat, but not conversely). Flatness emerges only at the end of the paper, with respect not to modules but to ring extensions. Am I missing something? --
Darij (
talk)
22:01, 19 April 2011 (UTC)reply
Yes, I was missing something. By Lazard's theorem, a flat module is always the direct limit of some free modules. And by Theorem 8 in Higgins's paper, this yields that the PBW theorem holds whenever L is a free K-module. --
Darij (
talk)
09:02, 26 May 2011 (UTC)reply