Talk:Hilbert's syzygy theorem

Complete rewriting of the article

Why is my version "too technical"? Even if it is, it is clearly superior to the previous "stub" one. Arguably, the "classical" formulation I give (which is essentially Hilbert's own) is less technical than the homological formulation. Of course the modern version is technical, it is impossible to circumvent this. I will add more references soon. Σαμουήλ δυνατός (talk) 19:33, 30 October 2016 (UTC)

To editor Σαμουήλ δυνατός: Before reading this post, I have explained my revert and the reason for which you must wait for a consensus on User talk:Σαμουήλ δυνατός. Please, read WP:BRD and wait for a consensus. D.Lazard (talk) 20:53, 30 October 2016 (UTC)

Could you please clarify in what sense it is too technical? Can you make (for instance) suggestions on how to improve my version? Where do I mix historical details with the mathematical description? I mention where it appears in Hilbert's paper, and then, in a new paragraph, I give a purely mathematical statement. How is this mixing up?

The statement I give is actually the one you can find (in older language) in Hilbert's paper. The previous version contains several mistakes and imperfections, for instance it says that a regular ring must have finite homological dimension (wrong, there is a counterexample by Nagata). Also the usage of "homological dimension" of an R-module is not at all standard usage. The previous version seems like a copy paste from the article on resolution and regular ring. My version, on the other hand, is much more informative for someone that is familiar with homological algebra. I point out nontrivial results and sketch their proof. The previous version didn't contain any proofs, perhaps this makes my version "too technical"?Σαμουήλ δυνατός (talk) 21:05, 31 October 2016 (UTC)

In your version of the article there are many things that deserve to be inserted in the article. However, I recall you that Wikipedia is a collaborative project. Therefore, it is important that other editors may review and improve other's edits. This is the reason for which the usage is to avoid, as far as possible, to dramatically change a whole article in a single edit. It is much better to change (or add) a single section at a time. Also, the articles must have a clear structure in order to allows readers to retrieve easily the information they need (see WP:LAYOUT and MOS:MATH). In your version the detailed description of the content of Hilbert's paper belongs to a section "History", which is presently lacking. You may add such a section without changing the remainder of the article, and I recommend you to do that.

IMO, the section "Statement" must be kept as it is: It states the theorem in a concise way, which requires only, for being understood, knowing what are modules and exact sequences. The statement(s) in your version require either reading a rather technical definition of the sygyzies modules, or knowing already what it the homological dimension. It is what I meant when saying that your version is too technical. As many readers are not "familiar with homological algebra", useful information for people familiar with homological algebra must not appear in section "statement", but must be inserted later in the article, for example in a section "Relation with homological algebra" or "Interpretation in homological algebra."

About proofs, please read WP:WPM/Proofs.

What is your source for saying that "homological dimension" of an R-module is not at all standard usage? As far as I remember, this term has been introduced by Jean-Pierre Serre and is of standard usage since then. D.Lazard (talk) 11:15, 1 November 2016 (UTC)

I chose to rewrite the whole article mainly because it is considered "stub" and because it is very short. I agree that my version could be given a better structure, what I submitted was preliminary anyway (references lacking, etc.). I agree that it requires to read something, but I clarify the relation between the modern statement (using resolutions) and the notion of a "syzygy", which the current version does not even address. The definition of a Syzgy module requires to understand what generators of module are, nothing more, nothing less. So I politely disagree that it is more technical than the current version, I think they are equally technical for the layman. I also include an example, which is lacking in the current version.

As for "homological dimension", I think nowadays almost all modern books (bourbaki, matsumara, weibel, eisenbud, etc.) use "Projective dimension" (which is also better, because there are also the notions "Injective dimension", "flat dimension", etc. - each of these is also a form of homological dimension). The only reference I know that uses "homological dimension" as meaning "projective dimension" is Serre's "algebre locale", which also uses the outdated "codimension homologique" for depth. This is only a minor point, sorry for nitpicking, but I think when an article is meant (as you say) to be understandable by laymen, one should try to keep terminology as standard as possible ("algebre locale" is a standard book, but only experts read it, I think). Overall I think I now understand your criticism better (thanks!), and I will try to prepare a better (structured) version. Σαμουήλ δυνατός (talk) 18:05, 1 November 2016 (UTC)

I have fixed the errors mentioned above. D.Lazard (talk) 15:03, 2 November 2016 (UTC)

To do

Beside some errors, which are presently fixed (at least those of which I am aware), this article is really a stub that has many issues. The version of Σαμουήλ δυνατός is more complete and, technically, much more accurate, but it does not solve the main issues and may be confusing for non-expert readers. Here, I'll try to list what has to be done for solving the issues.

• Lead: It must be more informative on the context and provide a non-technical description of the theorem (what I mean here, will be clear when I'll rewritten the lead, probably today)
• Section Module of relations: this has to be defined, and the link with the first step of a free resolution has to be explained
• Section History: this should be a separate section. I think that Σαμουήλ δυνατός is better than me for writing it
• Remainder of the article: it must certainly be rewritten, but, I have not yet a clear vision of what should appear or not. Probably the relationship with Hilbert series and Castelnuovo–Mumford regularity deserve to be mentioned.

D.Lazard (talk) 15:03, 2 November 2016 (UTC)

I have now rewritten the main part of the article. The first subsection of section "Generalization" may be removed, as its content has been included in the preceding sections. The second subsection, as well as section "The Syzygy theorem and Serre's theorem on regular local rings" of Σαμουήλ δυνατός's version are about the link between global dimension and regularity of Noetherian ringss. As they are, it is not clear that they belong to this article. On the other hand, a section is needed for saying that given an ideal in a polynomial ring, generated by a regular sequence of homogeneous polynomials, then the Koszul complex is an explicit free resolution (note that this article is horribly written and a more explicit description of the complex is needed). Such a section is needed as this is the basis of the proof that the global dimension is not less than the number of indeterminates. Also a sketch of a simple proof would be useful, as, otherwise, the theorem seems "magic". Some thoughts on what should be there? some editing help? D.Lazard (talk) 16:09, 4 November 2016 (UTC)

I think your edits were very good, D.Lazard. I added some history (not much more than I had in my previous version, I could add more, but I think here "less is more"), and rewrote the part on regularity. I agree that a proof (or at least a sketch) would be good. There are some technical points that can be made about the syzgy modules, for instance over a Cohen-Macaulay ring the modules that are k-th syzygies are exactly those having Serre's property (S_k). Do you think it is worth mentioning? It is probably a bit too technical, right? Σαμουήλ δυνατός (talk) 20:06, 6 November 2016 (UTC)

Thanks for your evaluation of my edits. In fact, I had something like this in mind, when I qualified your rewriting as too technical.

IMO, the history section deserve to be expanded by adding more context, namely that Hilbert's article was intended for solving a longstanding problem of invariant theory, and that his article is remarkable as containing the first non-constructive results in algebra, cf. the comments attributed to Gordan that "it is not mathematics but theology", and later, that "theology may sometimes be useful" (see David Hilbert). In other words, I suggest to replace "seminal", which should be avoided per WP:PEACOCK by the reasons for which it is seminal.

On my side, I'll add a section "Computation", which will contain the facts that sygyzies may be computed (Grete Hermann, 1926) in doubly exponential time, that first syzygies may involve polynomials of doubly exponential degree, and that Gröbner bases are presently the most efficient method for computing sygyzies. D.Lazard (talk) 11:10, 7 November 2016 (UTC)