Talk:Koszul complex

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Thanks for that. Aside from your homological program I was wondering if you knew stuff about Fitting ideals, which I've wanted to understand myself. Also, from the point of view of the geometry, we really need an article about Kähler differentials.

Charles Matthews 12:23, 12 Mar 2004 (UTC)

Hi Charles, I do know something about Fitting ideals and years ago I wrote up a longish thing on Kaehler differentials which I never finished and is now long gone. It would be good to do something on it but it will take some time.

Bernard Johnston 12 Mar 2004

The matricies in the example Koszul complex were denoted by psi and phi. I changed this to the more standard d_1 and d_2. Note that the complex has its differentials slightly in the wrong position, wiki doesn't seem to accept the \overset command.

If anyone feels qualified to add this material, it would be nice to see a bit about (a) Koszul relations and (b) the Koszul complex and self-duality. Courtney Gibbons 24 Mar 2011 —Preceding undated comment added 18:33, 24 March 2011 (UTC).[reply]

Too technical

I have tagged this article as too technical, as one needs to know what is a chain complex, and a tensor product of complexes for understanding the given definition. Even if one knows these definitions, but one is not an expert in homological algebra, it remains very difficult to realize what Koszul complex really is.

For a less technical definition, see Hilbert's syzygy theorem#Koszul complex, that I needed to write because this article is not convenient. D.Lazard (talk) 00:16, 11 November 2016 (UTC)[reply]

Maybe it's just me, but I think the definition in terms of exterior powers is more standard (and is at least more familiar to me). So, I've added one. I note Serre's "Algèbre locale, multiplicités" (excuse my French) uses the definition similar to the one given previously. I suppose that one gives a better motivation (but is somehow more cumbersome.) -- Taku (talk) 09:53, 11 November 2016 (UTC)[reply]

It should be noted that Serre's "Algèbre locale, multiplicités" is not a textbook, but a research book; therefore, his objective was not to be pedagogical, but to introduce as shortly as possible the technical tools he needed. Moreover, this book was about local behavior, where tensor product plays an important role. Nevertheless, as far as I remember, the term "complex of exterior algebra" were often used in place of "Koszul complex", in France and at that time.

About the context, the relation with Hilbert's syzygy theorem is worth to be mentioned. This is far to be the only occurrence of Koszul complex in the theory of homogeneous polynomial ideals: The homology of Koszul complex is a key tool in my proof that a zero-dimensional systems of homogeneous polynomials of degree d in n variables may be solved in time

d^{O(n)}.

The theory of the dimension and the degree of projective varieties, as presented in Hilbert series and Hilbert polynomial is also strongly related. D.Lazard (talk) 11:02, 11 November 2016 (UTC)[reply]

It's wikipedia, you can always link back to other articles explaining what you don't understand. Also, why in the world would you want to read about the Koszul complex if you don't know what a chain complex or the tensor product are? It's a more advanced subject. --345Kai (talk) 22:01, 8 October 2017 (UTC)[reply]

The lede is overcomplex. I find the definition to be reasonably understandable (in fact, far more understandable than D.Lazard's claimed "less technical" description in Hilbert's syzygy theorem#Koszul complex.) This article is linked from the end of exterior algebra, It is reasonable for a college sophomore who has read all of the exterior algebra article, and have some reasonable understanding of it, to click through to here, and go "wtf is a ring? wtf is a module?" cause all they've had is reals and complex numbers and vector spaces. (I got exterior algebra in multivariate calculus as a college sophomore) In some later years, one might learn about Lie derivatives (say in physics class, in classical mechanics), and maybe even develop some grasp of Lie algebra cohomology ... which article, BTW, has a really pretty decent "motivation" section that really gets the point across. This article needs a comparable "motivation" or "informal description" section. Using reals, not rings, maybe even mostly repeating what the Lie algebra cohomology "motivation" says. The goal is to informally explain "how to think about it, what to think about it." Something that might say "whenever you have a derivation you can define this exact sequence", and "its great because this means you can have an BRST thing going as a result."

This should include statements about what one can do with it, beyond just saying "its a great tool for homological algebra". Pointers to a few fundamental results that followed from it. (This motivates the reader: "if I want to understand X I first have to understand the Koszul complex"). 67.198.37.16 (talk) 20:40, 18 November 2020 (UTC)[reply]

Imho this complaint is basically that the previous introduction was too long, and hopefully that's been resolved enough to remove the complaint at the top of the page now. 26 Sep 2023 - MM. — Preceding unsigned comment added by MeowMathematics (talk • contribs) 18:11, 26 September 2023 (UTC)[reply]